Pseudo-periodic Maps and Degeneration of Riemann Surfaces

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan B. Teissier, Paris

For further volumes: http://www.springer.com/series/304

2030

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Yukio Matsumoto Jos´e Mar´ıa Montesinos-Amilibia

Pseudo-periodic Maps and Degeneration of Riemann Surfaces

123

Yukio Matsumoto Gakushuin University Department of Mathematics Mejiro 1-5-1 171-8588 Toshima-ku Tokyo Japan [email protected]

Jos´e Mar´ıa Montesinos-Amilibia Universidad Complutense Facultad de Matem´aticas Departamento de Geometr´ıa y Topolog´ıa Plaza de las Ciencias 3 28040 Madrid Spain [email protected]

ISBN 978-3-642-22533-8 e-ISBN 978-3-642-22534-5 DOI 10.1007/978-3-642-22534-5 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2011934808 Mathematics Subject Classification (2010): 14-XX, 57-XX c Springer-Verlag Berlin Heidelberg 2011  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Dedicated with respect and affection to the memory of Professor Itiro Tamura (1926–1991)

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Preface

In 1944, Nielsen introduced a certain type of mapping classes of a surface which were called by him surface transformation classes of algebraically finite type, [53]. He introduced this type of mapping classes as a generalization of the mapping classes of finite order. By the celebrated Nielsen Theorem [52], the latter classes contain surface homeomorphisms of finite order (For a generalization, see Kerckhoff [30]). A mapping class of algebraically finite type does not necessarily contain a homeomorphism of finite order, but using Nielsen’s theorem [52], one can show that it contains a homeomorphism f satisfying the following conditions (in what follows f will be an orientation-preserving homeomorphism of a closed, connected, oriented surface of genus g, ˙g ): 1. There exists a disjoint union of simple closed curves (which will be called cut curves) C D C 1 [ C2 [    [ C r on ˙g such that f .C / D C , and 2. the restriction of f to the complement of C , f j.˙g  C / W ˙g  C ! ˙g  C is isotopic to a periodic map, namely a homeomorphism of finite order. (Cf. [53, Sect. 14], [22]). In the present memoir, such a homeomorphism (and also a homeomorphism which is isotopic to such a homeomorphism) will be called a pseudo-periodic map. A periodic map is a special case of a pseudo-periodic map. In recent terminology, a homeomorphism f is pseudo-periodic if and only if either it is of finite order or its mapping class [f ] is reducible and all the component mapping classes are of finite order. (See [12,16,22,24,63]). A surface transformation class of algebraically finite type is nothing but a mapping class of a pseudo-periodic map. Nielsen [53] studied these classes extensively and defined several important invariants, for instance, the screw number of f about a cut curve Ci which measures

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the amount of the (fractional) Dehn-twist performed by a certain power f ˛ of f sending Ci to itself; or the character of Ci : whether it is “amphidrome” or not. Here Ci is amphidrome if there is an integer  such that !  !  f  .Ci / D Ci : He asserted in [53] that his invariants were a complete set of conjugacy invariants, meaning that if two pseudo-periodic maps f1 W ˙g.1/ ! ˙g.1/ and f2 W ˙g.2/ ! ˙g.2/ have these same invariants, then their mapping classes Œf1  and Œf2  are equivalent under a certain homeomorphism h W ˙g.1/ ! ˙g.2/ ; i.e. Œf1  D Œh1 f2 h. (For an exact formulation, see [22, Theorem 13.4]). However, his proof of this assertion was rather vague, and we need an invariant (the action of monodromy on the partition graph) which he did not mention explicitly. See Examples 6.3 and 6.4 in Chap. 6. A pseudo-periodic map f is said to be of negative twist if the screw numbers about a certain system of cut curves are all negative (Chap. 3). The purpose of Part I of the present memoir is to construct a complete set of conjugacy invariants for a pseudo-periodic map f of negative twist. We have added to Nielsen’s invariants one more: the action of f on the “partition graph”, which is the action, induced by f , on the configuration of the partition of ˙g obtained by cutting ˙g along a certain system of cut curves fCi griD1. The main result of Part I is roughly stated as follows (see Theorem 6.1 and 6.3 for precise statements): .1/

.1/

.2/

.2/

Theorem 0.1. Let f1 W ˙g !˙g and f2 W ˙g ! ˙g be pseudo-periodic maps of negative twist. Suppose that they have the same values in Nielsen’s invariants and that their actions on the respective partition graphs are equivariantly isomorphic. .1/ .2/ Then there exists an orientation preserving homeomorphism h W ˙g ! ˙g such 1 that Œf1  D Œh f2 h. In the course of the proof, we develop (in Chaps. 3–5) the theory of generalized quotients, which are naturally associated with pseudo-periodic maps, just as ordinary quotient spaces are associated with periodic maps. This makes our proof of Theorem 0.1 unexpectedly long, but the generalized quotients will play an essential role also in the study of the degeneration of Riemann surfaces (in Part II). This was the main reason of our investigation, which therefore concentrated in the study

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of generalized quotients. As a matter of fact, Theorem 0.1 above is just a (non immediate) corollary of our research. The organization of Part I is as follows: In Chap. 1, we review some basic results of Nielsen from [51, 53]. In Chap. 2, we define the “standard form” of a pseudo-periodic map f . Nielsen [53, Sect. 14] constructed a special homeomorphism which served as a standard form, but our standard form is slightly different from his. We prove the existence and the essential uniqueness of the homeomorphism in standard form which is isotopic to a given pseudo-periodic map (Theorem 2.1). In Chap. 3, we introduce the notion of generalized quotients, and in particular, of minimal quotients which are the special case of generalized quotients that satisfy a certain “minimality condition”. According to the definition given in Chap. 3, in order to have a generalized quotient, a pseudo-periodic map f must be in a very special form which we would like to call “superstandard form”. It will be proved that any pseudo-periodic map f of negative twist is isotopic to a pseudo-periodic map in superstandard form having a minimal quotient (Theorem 3.1). In Chap. 4, the following essential uniqueness will be proved (Theorem 4.1): suppose f1 and f2 W ˙g ! ˙g are pseudo-periodic maps of negative twist, both in superstandard form. If they are homotopic, then their respective minimal quotients are isomorphic. By the above existence and uniqueness theorems, we can generalize the definition of minimal quotients to cover any pseudo-periodic map of negative twist not necessarily in superstandard form, i.e., the minimal quotient of a pseudo-periodic map f of negative twist is constructed by first isotoping f to the superstandard form f 0 and then taking the minimal quotient of f 0 , which is declared to be the minimal quotient of f . The minimal quotient captures all of the Nielsen invariants constructed in [53]. Moreover, it will be proved in Part II that the “base space” of the minimal quotient of a pseudo-periodic map f of negative twist is homeomorphic to a (normally minimal) singular fiber of a one-parameter family of Riemann surfaces of genus g around which the topological monodromy is equivalent to [f ]. In Chap. 5, we prove a theorem in elementary number theory, which is basic to the arguments in Chaps. 3 and 4. In Chap. 6, we consider the partition graph and the action of f on it. This action, together with the minimal quotient, determines the conjugacy class of [f ] in Mg (Theorem 6.1). This result is further reformulated in terms of certain cohomology of “weighted graphs” (Theorem 6.3). In Appendix A, we will give a proof of the following theorem: let f and f 0 be (orientation- preserving) periodic maps of a compact surface ˙ each component of which has negative Euler characteristic. Suppose f and f 0 W .˙; @˙/ ! .˙; @˙/ are homotopic as maps of pairs. Then there exists a homeomorphism h W ˙ ! ˙ isotopic to the identity, such that f D h1 f 0 h. This theorem is used in the proof of Theorem 2.1. Among specialists, this theorem seems folklore. A. Edmonds informed, in a letter to the second named

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author, that C. Frohman had proved a stronger result which implied the above theorem. Unfortunately, the authors could not find any reference giving an explicit proof, so we decided to write this appendix. Pseudo-periodic maps of negative twist are closely related to the degeneration of Riemann surfaces. In fact, the topological monodromy around a singular fiber in a one-parameter family of Riemann surfaces is a pseudo-periodic map of negative twist (see [19], also [26, 58]). In Part II of this memoir, we will apply the results in Part I to the topology of degeneration of Riemann surfaces. The main result of Part II is roughly summarized as follows: Theorem 0.2. The topological types of minimal degenerating families of Riemann surfaces of genus g  2, over a disk, which are nonsingular outside the origin, are in a bijective correspondence with the conjugacy classes in the mapping class group Mg represented by pseudo-periodic maps of negative twist. The correspondence is given by the topological monodromy. In the case of g D 1, the validity of Theorem 0.2 is reduced by half: By Kodaira’s classification [32] of singular fibers for genus 1, we see that every pseudo-periodic mapping class (of negative twist) of a torus can be realized as the topological monodromy of a singular fiber. Thus the correspondence is “surjective”, but it is not “injective”. For example, all the multiple fibers of type m I0 (in Kodaira’s notation) have the identity mapping class as their topological monodromy. The assumption g  2 is used almost everywhere in our proof: The existence of an admissible system of cut curves subordinate to a pseudo-periodic map (Lemma 1.1) is essential to the definition of various invariants of the map, and the proof of the existence requires g  2. Also “homotopy implies conjugacy” theorem for periodic maps assumes g  2, because in the proof we apply the hyperbolic geometry (see Appendix A). This theorem is indispensable in the proof of the uniqueness of the standard form (see Theorem 2.1 (ii)). Our arguments in later chapters depend on this uniqueness theorem. We have tried to make the memoir as self-contained as possible, except for the two quotations from [51,53]. (Theorems 1.1 and 1.2 of the present memoir). All the other arguments are elementary. The authors are grateful to Allan Edmonds, Takayuki Oda and Hiroshige Shiga for their useful information and comments. This work started during the first named author’s first visit to Spain (1988) and was completed during his second visit (1991). The first named author would like to express his warmest thanks to the members and staffs of Facultad de Ciencias Matem´aticas, Universidad Complutense de Madrid, for their kind hospitality. Finally but not least at all, the authors deeply thank Srta. Mar´ıa Angeles Bringas for her benevolence, patience, and excellent skill shown in typing this memoir, without which it could have never been published.

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ADDED:SEPTEMBER,20091 The main body of the manuscript of the present memoir was completed in December 1991, and some remaining additional parts in January, 1992. Since then we have not found any occasion to publish this work, for several reasons; the length certainly was one. Another reason, but probably the main one, was the authors’ inability to use Tex. After a long delay of almost two decades, the authors find some unsatisfactory points in the manuscript, for example, it contains too many details, which might be a hindrance for readers who want to get a quick view, but on the other hand, it might help them to understand the details of the argument. Anyway the authors needed to compile these long (elementary and sometimes seemingly trivial) arguments to complete the proof of our theorems. Therefore, we have decided to keep the manuscript in its original form, except for the numbering of the chapters, theorems, propositions, figures, etc. A change we have made is the unification of the two different bibliographies, which were separately attached to each part, into one bibliography at the end. Also we added some references that were published after the completion of our manuscript and some more that we had missed involuntarily or were unknown to us due to our limitations. Unfortunately, the authors cannot be sure even now of the completeness of the augmented bibliography. A pseudo-periodic map would well be called chiral if either it is periodic or all of its screw numbers are of the same sign. A chiral pseudo-periodic map is a pseudo-periodic map of negative twist or of positive twist. If a pseudo-periodic map has both positive and negative screw numbers, it will be called achiral. In Part I of this memoir, we confined ourselves to chiral pseudo-periodic maps (of negative twist). From the viewpoint of surface topology, it would be more natural to treat not only chiral pseudo-periodic maps but also achiral ones, of course. We tried such a general treatment for some time. However, to construct a generalized quotient for an achiral pseudo-periodic map, we are forced to adopt an artificial convention on signs of intersections between the components consisting of a tail-part of the generalized quotient, and we lose the natural uniqueness of the generalized quotient of a pseudoperiodic map. Moreover, our construction of generalized quotients is intended to be applied to topology of degeneration of Riemann surfaces in Part II, and for that purpose, we only need chiral pseudo-periodic maps. For these reasons, we gave up our trial to generalize the construction of generalized quotients to achiral maps. As is immediately seen from the title, the main objects of this memoir are (chiral) pseudo-periodic homeomorphisms and degeneration of Riemann surfaces. Our main point is that these two objects are topologically classified by the same objects, i.e. certain types of “numerical chorizo spaces” together with a cohomology class in the weighted cohomology of their decomposition graphs. This type of chorizo spaces appear as “minimal quotients”of pseudo-periodic homeomorphisms of negative twist, and exactly the same type of chorizo spaces appear also as “normaly minimal

1

Revised in February 2011.

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singular fibers” in one-parameter families of Riemann surfaces. The former objects come from surface topology, while the latter objects come from complex analysis. The authors think interesting that numerical chorizo spaces lie at the common basis of the objects from two different disciplines. The appearance of pseudo-periodic homeomorphisms of negative twist in degenerating families of Riemann surfaces was clarified through the work of Imayoshi [26], Shiga–H. Tanigawa [58], and finally by Earle and Sipe [19]. We should mention here, however, that the pseudo-periodic nature of the monodromy had been observed for the Milnor fibering [44] at an isolated singular point of a complex hypersurface. Brieskorn [14] showed, in general dimensions, that the eigenvalues of the (co)homological monodromy are roots of unity. Lˆe [34] showed, in the case of curves, that the homological monodromy is periodic if the curve is irreducible at the singular point. A’Campo [1] proved that it is not the case if the curve is not irreducible. Also he showed that the geometric (i.e. topological) monodromy is not necessarily periodic, even if the curve is irreducible. A’Campo [2] and Eisenbud – Neumann [20] gave a description of geometric pseudo-periodic monodromies. Finally Lˆe – Michel – Weber [35] proved that the geometric monodromy is pseudo-periodic (“quasi-finie” in their terminology). Michel – Weber [43] gave a detailed description of the negative twist and showed that the geometric monodromy associated to a complex polynomial map from C2 to C (affine case) is also pseudoperiodic of negative twist. During the two decades, after the completion of our manuscript, several related papers have appeared. The most related one is, of course, the anouncement of this memoir, which was published in Bull. A.M.S. in 1994 [42]. This might serve as an introduction to this memoir (see also [40]). Pichon [55] used the pseudo-periodicity of the geometric monodromy to characterize the 3-manifolds that appear as the boundary manifolds of degenerating families of Riemann surfaces over a disk. In both of the papers of Pichon [55] and Lˆe – Michel – Weber [35], Waldhausen’s graph manifolds [66, 67] play an important role. The first authors that put the present memoir to good use were Ashikaga and Ishizaka [7] who gave a complete list of singular fibers in degenerating families of genus 3 (they were more than sixteen hundred!). They very explicitely exploited the algorithm, implicitly contained in the present memoir. It should be noted that the numerical classification of genus 3 singular fibers had been accomplished by Uematsu [64] in 1993 independently of our work. Xiao and Reid [56] proposed the problem of determining all the “atomic” singular fibers, which are defined as such singular fibers that cannot be “split” by any perturbation of the degenerating families. This problem is very interesting from the viewpoint of the present memoir. By our main result, the topological types of singular fibers are classified by the corresponding topological monodromies around them. Then a natural question to be settled would be if all atomic singular fibers (except for “multiple fibers”) correspond to the full .1/-Dehn twist about a certain simple closed curve. Examples of this geometrical situation are contained

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in [3, 29, 38, 39]. (For recent related results, see [5, 6, 8].) Following these trend of ideas, S.Takamura [59, 60] is undertaking a project in solving this problem. The authors would like to thank Professors D. T. Lˆe, J.-P. Brasselet, and M. Oka who showed interest in our work, and especially Professor Lˆe for his explanation on the related results of his own and others. We are also grateful to Professor T. Ashikaga for taking our work seriously and for actively developing related subjects in algebraic geometry and topology, which encouraged us very much. Thanks are also due to Professor Y. Imayoshi for his interest in our results, and for his very benevolent review of our work [28]. In November of 2000 we met in Oberwolfach Professors A’Campo, Weber and Pichon, who encouraged us very strongly to publish our results. We owe to them the final impulse that we needed to conclude the typing of this memoir that has eventually lead to its publication. Tokyo and Madrid, September 2009

Yukio Matsumoto Jos´e Mar´ıa Montesinos-Amilibia

Acknowledgement The authors are very grateful to the referees for their careful reading, valuable comments and suggestions. The first named author has been supported by Grantin-Aid for Scientific Research (No. 20340014), J.S.P.S. The second named author has been supported by MEC, MTM2009-07030.

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Contents

Part I

Conjugacy Classification of Pseudo-periodic Mapping Classes

1

Pseudo-periodic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Nielsen’s Results on Pseudo-periodic Maps . . . . . .. . . . . . . . . . . . . . . . . . . .

3 3 6

2

Standard Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1 Definitions and Main Theorem of Chap. 2 . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Periodic Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Non-amphidrome Annuli . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4 Amphidrome Annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5 Proof of Theorem 2.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

17 17 20 20 35 47

3

Generalized Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Definitions and Main Theorem of Chap. 3 . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Proof of Theorem 3.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Quotient of .B 0 ; f jB 0 / . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4 Re-normalization of a Rotation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5 Re-normalization of a Linear Twist . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6 Re-normalization of a Special Twist. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7 Completion of Theorem 3.1. (Existence) .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .

53 53 60 61 61 71 83 91

4

Uniqueness of Minimal Quotient . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Main Theorem of Chap. 4 .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Structure of  1 .arch/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Structure of  1 .tail/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Structure of  1 .body/ . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Completion of the Proof of Theorem 4.1. (Uniqueness) . . . . . . . . . . . . . 4.6 General Definition of Minimal Quotient . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 Conjugacy Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

93 93 97 103 107 113 118 119

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A Theorem in Elementary Number Theory . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 131 5.1 Proof of Theorem 5.1. (Uniqueness) . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 132 5.2 Proof of Theorem 5.1. (Existence).. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 135

6

Conjugacy Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Partition Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.2 Weighted Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Completion of the Proof of Theorem 6.1 .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Weighted Cohomology .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

Part II

145 149 151 162 164

The Topology of Degeneration of Riemann Surfaces

7

Topological Monodromy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Proof of Theorem 7.1.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Construction of  and fNi gsiD1 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 The Decomposition F D A [ B . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.4 Construction of a Monodromy Homeomorphism .. . . . . . . . . . . . . . . . . . . . 7.5 Negativity of Screw Numbers.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.6 Completion of the Proof of Theorem 7.1 .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .

8

Blowing Down Is a Topological Operation . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 189

9

Singular Open-Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 Completion of the Proof of Theorem 7.2 .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 Characterization of the Triples .S; Y; c/ That Come from Pseudo-periodic Maps. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 Completion of the Proof of Theorem 9.2 .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.4 Concluding Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

A

173 175 176 177 177 181 185

199 211 213 218 220

Periodic Maps Which Are Homotopic . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 221

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 233 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 237

Part I

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Chapter 1

Pseudo-periodic Maps

1.1 Basic Definitions In this Chapter, we will review some basic results from Nielsen [51, 53]. We will begin with the definition of a pseudo-periodic map. Hereafter, all surfaces will be oriented, and all homeomorphisms between them will be orientationpreserving. Let ˙g be a closed connected surface of genus g  1. Definition 1.1 (Compare [53, Sect. 8]). A homeomorphism f W ˙g ! ˙g is a pseudo-periodic map if f is isotopic to a homeomorphism f 0 W ˙g ! ˙g which satisfies the following conditions: (i) there exists a disjoint union of simple closed curves C D C 1 [ C2 [    [ C r on ˙g such that f 0 .C / D C , and (ii) the restriction   f 0 j ˙g  C W ˙g  C ! ˙g  C of f 0 to the complement of C , is isotopic to a periodic map, i.e. a mapping of finite order. We call fCi griD1 a system of cut curves subordinate to f . It may be empty if f is isotopic to a periodic map. It is easy to see that a homeomorphism of a torus f W T2 ! T2

Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5 1, © Springer-Verlag Berlin Heidelberg 2011

3

4

1 Pseudo-periodic Maps

is pseudo-periodic if and only if jTrace .f /j  2, where f W H1 .T 2 I Z/ ! H1 .T 2 I Z/ is the induced homomorphism. In what follows, we will assume g  2 unless otherwise stated. Lemma 1.1. Let f W ˙g ! ˙g be a pseudo-periodic map. Then there exists a system of cut curves fCi griD1 subordinate to f such that (i) Ci does not bound a disk on ˙g ; i D 1; 2; : : : ; r; and (ii) Ci is not parallel to Cj if i ¤ j . Proof. Let fCj 0 gsj D1 be any system of cut curves subordinate to f . Isotoping f , we may assume 1 0 [ [ f @ Cj 0 A D Cj 0 : j

j

0

If a cut curve Cj bounds a disk, we simply remove Cj 0 . Suppose that Ci 0 and Cj 0 bounds an annulus A. Since g  2, A is uniquely determined by Ci 0 and Cj 0 . Let m be the smallest positive integer such that f m .A/ D A. We equivariantly isotop f on A [ f .A/ [    [ f m1 .A/ so that f m maps the “center line” CA of A onto itself. Then we omit Ci 0 and Cj 0 and also their images under the iteration of f , and add instead, CA together with its images. Two curves Ci 0 and Cj 0 are replaced by a curve CA . Proceeding in this way, we arrive at a system of cut curves fCi griD1 satisfying (i) and (ii) of the lemma. t u We call a system of cut curves which satisfies conditions (i) and (ii) of Lemma 1.1 an admissible system ([12, 22]). Let fCi griD1 be an admissible system of cut curves subordinate to a pseudoperiodic map f W ˙g ! ˙g : Then Nielsen [53, Sect. 12] introduced the “screw number” s.Ci / for each curve Ci in the system. The definition is as follows: We may assume ! [ [ Ci D Ci : f i

Let Cj be a fixed cut curve in the the system.

i

1.1 Basic Definitions

5

Let ˛ be the smallest positive integer ˛ such that !  !  f ˛. C j / D C j !  where C j denotes Cj with an orientation assigned, so f ˛ preserves Let b and b 0 be the connected components of ˙g  C such that Cj belongs to their adherence in ˙g , where r [ C D Ci : i D1 0

0

It is possible that b D b . Let ˇ (resp. ˇ ) be the smallest positive integer such that 0 f ˇ .b/ D b (resp. f ˇ .b 0 / D b 0 ). Clearly ˛ is a common multiple of ˇ and ˇ 0 .  Since f j ˙g  C is isotopic to a periodic map, there exists a positive integer nb such that .f ˇ jb/nb Š idb . We choose the smallest number among such integers and denote it by nb again. Likewise we choose nb0 .> 0/ for b 0 . Let L be the least common multiple of nb ˇ and nb 0 ˇ0 . Then f L is isotopic to the identity on b and b 0 . Thus, on the union b [ Cj 0 [ b 0 ; f L is isotopic to the result of a number of full Dehn twists performed about Cj . Let e.2 Z/ be this number of full twists. We adopt the following convention for the sign of e: Convention (). The sign convention of e is depicted in Fig. 1.1. (Compare [13, p. 166], where the orientation of the surface is opposite to ours; and compare [46, Fig. 3 and p. 158]). Since the system fCi griD1 is admissible, neither b nor b 0 is an annulus or a disk. Nielsen [53] proved that the number e is well-defined. Definition 1.2 ([53, Sect. 12]). The rational number e˛=L is called the screw number of f about Cj and is denoted by s.Cj /. It measures the “amount of Dehntwist” performed by f ˛ about Cj . If s.Cj / D 0, we can equivariantly isotop f on b [ Cj [ b 0 and on its images under the iteration of f so that the result of the isotopy is a periodic map of b [ Cj [ b 0 : Thus we can omit Cj , together with its images under the iteration of f , keeping the remaining system admissible.

6

1 Pseudo-periodic Maps

Fig. 1.1 Positive and negative Dehn twists

Iterating this omission, we arrive at a cut system in which all cut curves have non-zero screw number. Definition 1.3 (Cf. [12,22]). An admissible system of cut curves fCi griD1 is precise if s.Ci / ¤ 0 for each Ci . Nielsen did not use the terminology “precise system”; he talked, instead, about “division of the surface into complete kernels” ([53, Sect. 13]) meaning the decomposition of the surface by a precise system. Definition 1.4 ([53, Sect. 10]). An essential simple closed curve C on ˙g is said to be amphidrome with respect to a homeomorphism f W ˙g ! ˙g !  !  !  if there is an integer  such that f  . C / is freely homotopic to  C , where C and !   C denote the same C with the opposite directions assigned.

1.2 Nielsen’s Results on Pseudo-periodic Maps The main result of Nielsen’s paper [53] is the following: Theorem 1.1 ([53, Sect. 15], cf. [22, p. 90]). Let f W ˙g ! ˙g

1.2 Nielsen’s Results on Pseudo-periodic Maps

and

7

f 0 W ˙g ! ˙g

be pseudo-periodic maps, and let fCi griD1 and fCj 0 gsj D1 be precise systems of cut curves subordinate to f and f 0 respectively. Suppose f is homotopic to f 0 . Then (i) there is a homeomorphism h W ˙g ! ˙g which is isotopic to the identity and such that h.C / D C 0 , where C D

r [

Ci

i D1

and C0 D

s [

Cj 0 :

i D1

In particular, r D s. (ii) For each cut curve Ci in fCi griD10 the number ˛ and the screw number s.Ci / are equal to the corresponding numbers for h.Ci /. (iii) For each cut curve Ci in fCi griD10 its character of being amphidrome or not (with respect to f ) coincides with the same character of h.Ci / (with respect to f 0 ). (iv) For each connected component b of ˙g  C , the integers ˇ and nb are equal to the corresponding integers for h.b/. (v) For each connected component b of ˙g  C , the periodic map of b which is isotopic to f ˇ jb is conjugate to the periodic map of h.b/ which is isotopic to .f 0 /ˇ jh.b/: (Here we are assuming f .C / D C and f 0 .C 0 / D C , which can be achieved by isotoping f and f 0 if necessary). Nielsen [53, Sect. 15] also asserted that conversely these invariants determine the conjugacy class of the mapping class to which f belongs. This, however, is not the case. In fact, the converse as formulated as Theorem 13.4 of [22] has counter examples. See Sect. 7, Examples, 3 and 4, of the present paper. If in the statement of Theorem 1.1 we add the following condition: (vi) the actions of f and f 0 on the respective partition graphs are conjugate (Here the partition graph of .˙g ; f / is a graph whose vertices and edges correspond to connected components of ˙g  C and cut curves fCi griD10 , respectively. Similarly for .˙g ; f 0 /),

8

1 Pseudo-periodic Maps

then Nielsen’s assertion is correct at least for pseudo-periodic maps of negative twists (see Sect. 4 for the definition). The conditions (i)–(vi) become sufficient. See Sect. 7 for details. The part of Nielsen’s result stated above is certainly correct, because he developed his theory so that it depends only on the homotopy class of f . For the construction of h in assertion (i), consult, for example, Casson’s lecture notes ([16, Lemma 2.4]). Note that by the above theorem we can speak of the precise system of cut curves subordinate to f , provided that we identify two isotopic systems up to indexing of the curves. As is obvious from assertion (v), we must study periodic maps as a prerequisite to Theorem 1.1. Nielsen [51] studied periodic maps of surfaces and established a complete set of conjugacy invariants, which we will now describe. Let ˙ be a compact surface with or without boundary, f W˙ !˙ a periodic map of order n > 1. Let p be a point of ˙. Then there is a positive integer m D m.p/ such that the points p; f .p/; : : : f m1 .p/ are distinct and f m .p/ D p. If m D n, the point p is called a simple point, while if 0 < m < n, p is called a multiple points. In particular, if m D 1, p is a fixed point. Since we are assuming that f is orientation-preserving, a multiple point is an isolated, interior point of ˙. For later use, we consider a more general situation than is needed here. Let !  !  !  !  C D C1 [ C2 [[ C s be a set of oriented and disjoint simple closed curves in a surface ˙, and let g be !  !  !  a map g W ˙ ! ˙ such that g. C / D C and gj C is periodic. We will define the  !  ! notion of the valency of a curve C 2 C with respect to g. !  !  Let m be the smallest positive integer such that gm . C / D C . The restriction !  !  gm j C is a periodic map of C of order, say,  > 0. Then m D n (D the order of !  !  gj C ). Let q be any point on C , and suppose that the images of q under the iteration !  of g m are situated on C in this order. fq; g m .q/; g 2 m .q/; : : : ; g .1/ m .q/g !  when viewed in the direction of C . Here, by Convention (), the integer  satisfies 0   <  and gcd.; / D 1, so  D 0 iff  D 1. Define an integer ı as follows:

1.2 Nielsen’s Results on Pseudo-periodic Maps

ı  1 .mod /;

9

0ı<

!  (NB. ı D 0 iff  D 1). Then the action of g m on C is topologically equivalent to the rotation of angle 2ı= in a circle. If  is known,  and ı give the same information. !  Definition 1.5 ([51, Sect. 2]). The triple .m; ; / is called the valency of C with respect to g. If the orientation of C is reversed, then m and  do not change, while  .mod / changes sign. Returning to our original periodic map f of order n acting on a surface ˙, we define the valency of a boundary curve as its valency (with respect to f ) assuming it has the orientation induced by the surface ˙. The valency of a multiple point p is defined to be the valency of the boundary curve @Dp0 orientated from the inside of an invariant disk neighborhood Dp . The valency .m; ; / of a curve C (with respect to f ) is common to its images f .C /; f 2 .C /; : : : ; f m1 .C /; so this notion descends to the quotient space ˙=f . The quotient space ˙=f is an orbifold ([62], cf. [57]). Its underlying space is a compact surface. A multiple point p 2 ˙ corresponds to a cone point 2 ˙=f . Thus we can speak of the valency of a cone point of ˙=f . Also we can speak of the valency of a boundary curve of ˙=f . Theorem 1.2 ([51, Sect. 11]). Let f W˙ !˙ and

f 0 W ˙0 ! ˙0

be periodic maps of the same order of compact, connected, mutually homeomorphic surfaces ˙ and ˙ 0 . Then f and f 0 are conjugate if and only if one can establish a bijective correspondence between the set of the boundary curves and the cone points of ˙=f and the set of the boundary curves and the cone points of ˙ 0 =f 0 , respectively, so that corresponding boundary curves (and cone points) have the same valency. We remark that f W ˙ ! ˙ and f 0 W ˙ 0 ! ˙ 0 are conjugate if, by definition, there is an orientation-preserving homeomorphism h W ˙ ! ˙ 0 such that f D h1 f 0 h. (Beware that Nielsen’s original statement [51, Sect. 11] is a little confusing with regard to orientation, but correct). Corollary 1.1 ([51, Sect. 11]). Let f W˙ !˙

10

and

1 Pseudo-periodic Maps

f 0 W ˙0 ! ˙0

be periodic maps of the same order, of compact connected mutually homeomorphic surfaces ˙ and ˙ 0 . If ˙ and ˙ 0 are closed surfaces, and f and f 0 have no multiple points, then f and f 0 are conjugate. Theorem 1.2, Corollary 1.1, and Theorem 1.3 below hold regardless of the genus of the surfaces. Example 1.1 ([51, Sect. 6]). Parametrize a torus T 2 by R2 =Z2 . Define periodic maps of order 5 f1 ; f2 W T 2 ! T 2 as follows: f1 .x; y/ D .x; y C 1=5/; f2 .x; y/ D .x; y C 2=5/: Then f1 and f2 are conjugate. In fact, a homeomorphism h W T 2 ! T 2 defined by h.x; y/ D .3x C 5y; 4x C 7y/ gives the conjugation: f1 D h1 f2 h: For later use (especially in Sect. 7), we need a somewhat more detailed version of Theorem 1.2. Theorem 1.3. Let f W˙ !˙ and

f 0 W ˙0 ! ˙0

be periodic maps of the same order, of compact connected, mutually homeomorphic surfaces ˙ and ˙ 0 . Suppose they have no multiple points. Let M D ˙=f and M 0 D ˙ 0 =f 0 be the respective quotient spaces, and suppose there is a homeomorphism H W M ! M0 which preserves the valencies of the boundary curves. Then there exists another homeomorphism H0 W M ! M0 and a homeomorphism

h W ˙ ! ˙0

1.2 Nielsen’s Results on Pseudo-periodic Maps

11

such that (i) the diagram h

˙0 ? ?f 0 y

h

˙0 ? ? 0 y

˙ ! ? ? fy ˙ ! ? ? y H0

M ! M 0 commutes: (,  0 being the natural projections) (ii) H 0 j@M D H j@M . This theorem is essentially proved by Nielsen [51], but our version is slightly stronger. We now give an outline of the proof. Let n be the order of the periodic maps f , f 0 . First we recall the definition of the monodromy exponent ! from [51] (which was denoted by  in [51, Sect. 2]. Let l W Œ0; 1 ! M be a loop, l.0/ D l.1/. Let

lQ W Œ0; 1 ! ˙

be a lift. Then there is an integer k .mod n/ such that Q Q f k .l.0// D l.1/; and !.l/ is defined to be k .mod n/. Nielsen calls the mod n integer !.l/ the monodromy exponent of the (directed) loop l, which depends only on the homology class of l. We will call the homomorphism ! W H1 .M / ! Z=n the monodromy exponent, too. Suppose M has genus q and s boundary components. Choosing a base point O on M , Nielsen draws a canonical system of 2q C s curves ˛1 ; ˇ1 ; : : : ; ˛q ; ˇq ; 1 ; : : : ; s on M . (Fig. 1.2). Then he defines eight special homeomorphism (called by him “generating mappings”)

12

1 Pseudo-periodic Maps b2

Fig. 1.2 Figure 1 of [51] aq

a2

b1

bq a1 g1 g2 gs

S1 ; : : : ; S8 W M ! M: He properly chooses a combination of generating mappings, so that when it is applied to the original system of curves, a new canonical system (again denoted by ˛1 ; ˇ1 ; : : : ; ˛q ; ˇq ; 1 ; : : : ; s ) satisfying the following conditions !.˛1 / D 1 !.˛i / D 0; 2  i  q; !.ˇi / D 0; 1  i  q; is obtained. (See [51, Sect. 5]). Suppose we have drawn on M 0 a canonical system of 2q C s curves ˛1 0 ; ˇ1 0 ; : : : ; ˛q 0 ; ˇq 0 ; 1 0 ; : : : ; s 0 with the same properties as above. Recall (in our version) that we are given a homeomorphism H W M ! M0 which preserves the valencies of the boundary curves. This H does not in general preserve the above canonical systems, of course. We will construct a new homeomorphism H0 W M ! M0

1.2 Nielsen’s Results on Pseudo-periodic Maps

13

satisfying (A) H 0 maps the canonical system on M to the one on M 0 , and (B) H 0 j@M D H j@M . We must be careful here because it is impossible in general to construct such a homeomorphism unless we change properly the choice of the loops 1 0 ; : : : ; s 0 . Let s . M / be a disk with s holes which contains 1 ; 2 ; : : : ; s , and such that M  s is a connected surface (with one boundary component) containing ˛1 ; ˇ1 ; : : : ; ˛q ; ˇq : Since .˛1 ; ˇ1 ; : : : ; ˛q ; ˇq ; 1 ; : : : ; s / is a cannonical system, such a s exists. (See Fig. 1.3). We take a similar disk with s holes s 0 in M 0 , which contains 1 0 ; 2 0 ; : : : ; s 0 and does not contain

˛1 0 ; ˇ1 0 ; : : : ; ˛q 0 ; ˇq 0 :

Suppose the loops 1 ; 2 ; : : : ; s are attached to the base point O on M in this order (as in Fig. 1.3). Let i be the boundary curve of M around which the loop i goes, i D 1; 2; : : : ; s. We denote the image H. i / by i 0 .

bq aq b1 Δs

g1

a1

g2 Γ1 gs

Γ2

Fig. 1.3 A disk with s holes s

Γs

14

1 Pseudo-periodic Maps

Fig. 1.4 Rechosen loops 10 ; 20 ; : : : ; s0 g ′s

g 1′ Δs′

Γ2′

Γ1′

Γs

g 2′

Beware that i 0 does not necessarily go around the boundary curve i 0 D .H i /. We must re-choose i 0 as follows: look at the disk with s holes s 0 . We re-choose 1 0 ; 2 0 ; : : : ; s 0 inside s 0 so that (1) i 0 goes around i 0 , and (2)

1 0 ; 2 0 ; : : : ; s 0

are attached to O 0 in this order. (See Fig. 1.4) Now we can define a homeomorphism H0 W M ! M0 such that

H 0 .˛i / D ˛i 0 ; H 0 .ˇi / D ˇi 0 ; H 0 .j / D j 0

and H 0 j@M D H j@M: .i:e: H 0 . j / D j 0 /;

i D 1; : : : ; qI j D 1; : : : ; s:

Since !.˛1 / D !.˛1 0 / D 1; !.˛2 / D !.˛2 0 / D 0; : : : ; !.ˇq / D !.ˇq 0 / D 0 and H preserves the valencies of the boundary curves, we infer that H0 W M ! M0

1.2 Nielsen’s Results on Pseudo-periodic Maps

15

preserves the monodromy exponents of curves. Now follow Nielsen’s argument in [51, Sect. 11] and construct a homeomorphism h W ˙ ! ˙0 such that f D h1 f 0 h and  0 h D H 0 . This completes the outline of the proof of Theorem 1.3. t u

•

Chapter 2

Standard Form

Given a pseudo-periodic map f W ˙g ! ˙g Nielsen constructed a special homeomorphism which is homotopic to f and plays the role of a “standard form” in the mapping class of f , [53, Sect. 14]. In this Chapter, we will construct a similar standard form, slightly different from Nielsen’s, and will show its essential uniqueness. He wanted to avoid fixed points which might appear in annular neighborhoods of cut curves, while we do not care about such fixed points (Compare [22, Theorem 13.3]).

2.1 Definitions and Main Theorem of Chap. 2 Definition 2.1. Let A be an annulus and  W Œ0; 1  S 1 ! A a parametrization (i.e. homeomorphism), where S 1 D R=Z. A homeomorphism f WA!A which does not interchange the boundary components of A is called a linear twist with respect to , if f .t; x/ D .t; x C at C b/; .t; x/ 2 Œ0; 1  S 1 ; for some a; b 2 Q. We say simply that f WA!A

Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5 2, © Springer-Verlag Berlin Heidelberg 2011

17

18

2 Standard Form

is a linear twist if f is a linear twist with respect to a certain parametrization  W Œ0; 1  S 1 ! A: Definition 2.2. Let A and  W Œ0; 1  S 1 ! A be as above. A homeomorphism f WA!A which interchanges the boundary components of A is called a special (piecewiselinear) twist with respect to , if

f .t; x/ D

8    1 ˆ ˆ < 1  t; x  3a t  3 ; 0  t  .1  t; x/; ˆ ˆ : 1  t; x  3a t  2  ; 3

for some a 2 Q. If f WA!A is a special twist with respect to a certain parametrization  W Œ0; 1  S 1 ! A; we simply say that f is a special twist. Remark 2.1. Let  W Œ0; 1  S 1 ! Œ0; 1  S 1 be defined by .t; x/ D .1  t; x/: If f WA!A is a special twist with respect to  W Œ0; 1  S 1 ! A; then this same f is also a special twist with respect to  W Œ0; 1  S 1 ! A: Now we define our standard form.

1 3 2 3

t 

1 3 2 3

t 1

2.1 Definitions and Main Theorem of Chap. 2

19

Definition 2.3. A pseudo-periodic map f W ˙g ! ˙g is said to be in standard form if the following conditions are satisfied: (i) There exists a system of disjoint annular neighborhoods fAi griD1 of the precise system of cut curves subordinate to f , such that f .A / D A ; where A D

r [

Ai :

i D1

(ii) The map f j ˙g  A W ˙g  A ! ˙g  A is periodic. (iii) Let ki be the smallest positive integer such that f ki .Ai / D Ai ; i D 1; 2; : : : ; r: (iii)-a If f ki j Ai W Ai ! Ai does not interchange the boundary components of Ai , then f ki j Ai is a linear twist. (iii)-b If f ki j Ai W Ai ! Ai interchanges the boundary components of Ai , then f ki j Ai is a special twist. This whole chapter is devoted to the proof of the following theorem. Theorem 2.1 (cf. [53, Sect. 15] [22, Theorem 13.3]). (i) Any pseudo-periodic map f W ˙g ! ˙g is isotopic to a pseudo-periodic map in standard form. (ii) Suppose two pseudo-periodic maps in standard form f; f 0 W ˙g ! ˙g are homotopic, then there is a homeomorphism h W ˙g ! ˙g isotopic to the identity, such that f D h1 f 0 h.

20

2 Standard Form

The uniqueness statement (ii) above is stronger than the original one ([53, Sect. 15]). The proof requires considerations on periodic parts, non-amphidrome annuli, and amphidrome annuli. These preliminary considerations occupy the main body of this chapter.

2.2 Periodic Part We need the following theorem. Theorem 2.2. Let f and f 0 be periodic maps of a compact surface ˙ each component of which has negative Euler characteristic. Suppose f; f 0 W .˙; @˙/ ! .˙; @˙/ are homotopic as maps of pairs. Then there exists a homeomorphism h W ˙ ! ˙ isotopic to the identity, such that f D h1 f 0 h. This theorem seems folklore among specialists. A. Edmonds informed, in a letter to the second named author, that C. Frohman had proved a stronger result in his thesis which implied Theorem 2.2. Unfortunately the authors could not find any reference giving an explicit proof. A proof will be given in Appendix A.

2.3 Non-amphidrome Annuli Lemma 2.1 (L INEARIZATION). Let A be an annulus, f WA!A a homeomorphism which does not interchange the boundary components. Suppose f j @A W @A ! @A is periodic. Then there exists an isotopy f W A ! A; such that

0    1;

f0 D f; f j @A D f j @A; and

f1 W A ! A is a linear twist. (An isotopy as this will be referred to hereafter as “rel:@”). Proof. Let @0 A and @1 A denote the two boundary components. Since f j @A W @A ! @A

2.3 Non-amphidrome Annuli

21 (1,b + n)

b+n

x

~ Γ ~ I

t

(0,a)

a

Fig. 2.1 Linearization

is periodic, there are homeomorphisms ' W S 1 ! @0 A and W S 1 ! @1 A such that f '.x/ D '.x C a/; f for a; b 2 Q. Note that ' and homeomorphism

.x/ D

.x C b/;

are homotopic as maps from S 1 into A. Thus there is a  W Œ0; 1  S 1 ! A

such that .0; x/ D '.x/; .1; x/ D

.x/

([21, Lemma 2.4]). Now we identify A with Œ0; 1  S 1 through . We lift the curve f .Œ0; 1  f0g/ to the universal covering Œ0; 1  R. Let IQ be the lift starting at .0; a/. Then it ends at .1; b C n/, for some n 2 Z. (See Fig. 2.1) Let Q be a line segment in Œ0; 1R joining .0; a/ and .1; bCn/. Then Q projects to a “linear” arc  in Œ0; 1  S 1 connecting .0; a/ to .1; b/. By an innermost arc argument, f is rel.@ isotopic to a homeomorphism f 0 W Œ0; 1  S 1 ! Œ0; 1  S 1 satisfying

f 0 .Œ0; 1  f0g/ D :

Also by composing f 0 with an isotopy which keeps @A pointwise fixed and moves along  , f 0 becomes isotopic to a homeomorphism f .2/ such that f .2/ .t; 0/ D .t; / 2 ;

22

2 Standard Form

i.e., f .2/ preserves the t-level when restricted to a map from Œ0; 1  f0g onto  . Define f .3/ W Œ0; 1  S 1 ! Œ0; 1  S 1 by f .3/ .t; x/ D .t; x C a.1  t/ C .b C n/t/: f .3/ is a linear twist which coincides with f .2/ on @A [  . Thus by the Alexander trick, f .2/ is isotopic to f .3/ keeping the points of @A [  fixed. t u It will be convenient to extend the notion of screw number of a curve to an annulus. Let r [ Ai A D i D1

be a disjoint union of annuli f W A ! A a homeomorphism. Suppose the restriction of f to the boundary @A D

r [

@Ai

i D1

is periodic. Let Ai be an annulus in A . Let ˛ be the smallest positive integer such that (i) f ˛ .Ai / D Ai ; and (ii) f ˛ does not interchange the boundary components. Let l be a non-zero integer such that f l j @Ai D the identity. Then l is a multiple of ˛, and f l W Ai ! Ai is the result of e full Dehn-twists, e being an integer. Definition 2.4. The rational number e˛= l is called the screw number of f in Ai and is denoted by s.Ai /. It measures the amount of Dehn twist performed by f ˛ in Ai . The number s.Ai / is independent of the choice of l. Of course if fAi griD1 is an invariant system of annular neighborhoods of a precise cut system fCi griD1 subordinate to a pseudo-periodic map f W ˙g ! ˙g0 then s.Ai / D s.Ci /;

i D 1; 2; : : : ; r:

An annulus Ai is said to be amphidrome if there is an integer  such that f  .Ai / D Ai and f  interchanges the boundary components. !  The valency .m; ; / of a boundary curve C of A oriented by the orientation !  induced from A is defined as the valency of  C with respect to the periodic map @A ! @A :

2.3 Non-amphidrome Annuli

23

As a corollary to (the proof of) Lemma 2.1, we have the following: S Corollary 2.1. Let A D riD1 Ai be a disjoint union of annuli, f WA !A a homeomorphism such f j @A ! @A is periodic. Let Ai be a non-amphidrome annulus of A with @Ai D @0 Ai [ @1 A1 : Let .m0i ; 0i ; i0 / and .m1i ; 1i ; i1 / be the valencies of @0 Ai , and @1 Ai , respectively, s.Ai / the screw number. Then (i) the equality m0i D m1i holds; and (ii) the number s.Ai / C ıi0 =0i C ıi1 =1i is an integer, where the integer ıi is determined by i ıi  1 .mod  i / and 0  ıi <  i ;

D 0; 1:  ! Proof. The oriented Ai will be denoted by Ai ; the induced orientation in @Ai by !  !  @ A i . Remember that the valency of @Ai is defined as the valency of @ A i . (i) m i was defined to be the smallest positive integer such that

!  !  f mi .@ A i / D @ A i ;

D 0; 1:

Since Ai is not amphidrome, !  !  f m .@ A i / D @ A i if and only if f m .Ai / D Ai ; This proves m0i D m1i .

D 0; 1:

24

2 Standard Form

Fig. 2.2 The orientation of the annulus used in Corollary 2.1

t=1

x t

t=0 x

(ii) Let us give an orientation to Œ0; 1  S 1 as Fig. 2.2 indicates. Then the orientation of f1g  S 1 given by the x-direction is ! @1 .Œ0; 1  S 1 / and the orientation of f0g  S 1 given by the x-direction is ! @0 .Œ0; 1  S 1 /: Let m denote the common number m0i D m1i , and consider f m W Ai ! Ai as the homeomorphism (not yet normalized) f W A ! A of Lemma 2.1. Take a parametrization  W Œ0; 1  S 1 ! Ai for which f m .0; x/ D .0; x C a/ and f m .1; x/ D .1; x C b/ for some a; b 2 Q, and identify Ai with Œ0; 1  S 1 through . Then by the geometric meaning of ıi = i . D 0; 1/, we have ıi1  b .mod 1/; 1i ıi0  a .mod 1/: 0i Recall that if, as in Lemma 2.1, f m .Œ0; 1  f0g/

2.3 Non-amphidrome Annuli

25

is lifted to a curve in Œ0; 1  R joining .0; a/ to .1; b C n/, then s.Ai / D .b C n  a/: (Remember the Convention () on the sign of a Dehn twist.) Therefore, s.Ai / C

ıi0 ı1 C i1  .b C n  a/  a C b  0 .mod 1/ 0 i i t u

Corollary 2.2. Let f WA!A be a linear twist with respect to a parametrization  W Œ0; 1  S 1 ! A: Then the equation giving the linear twist is determined, up to the ambiguity of an integer l, by the screw number s.A/ and the valency .m0 ; 0 ; 0 / of @0 A.f0g  S 1 / as follows:

  ı0 f .t; x/ D  t; x  s.A/t  0 C l ; 

l 2 Z;

where ı 0 is determined by ı 0 0  1 .mod 0 /;

0  ı 0 < 0 :

The proof is immediate from the argument of Corollary 2.1. Lemma 2.2 (U NIQUENESS

OF LINEARIZATION).

Let

f; f 0 W A ! A be linear twists of an annulus A. Suppose that f j @A D f 0 j @A; and that the screw number of f in A is equal to the screw number of f 0 in A. Then there is an isotopy

26

2 Standard Form

h W A ! A;

0    1;

such that (i) h0 D idA , 0 0 (ii) h1  .f j @A/ h D f j @A.D f j @A/ on @A, and 1 0 (iii) f D h1 f h1 . Remark 2.2. This shows that the quality of being linear is independent of the parametrization  up to a sort of isotopy given by conditions (i), (ii) and (iii) above. Essentially this isotopy is boundary equivariant. Proof (of Lemma 2.2). Let  and  0 W Œ0; 1  S 1 ! A be the parametrizations for which f and f 0 are linear twists respectively. By considering  0  instead of  0 , if necessary, we may assume .f0g  S 1 / D  0 .f0g  S 1 / and

.f1g  S 1 / D  0 .f1g  S 1 /;

where  W Œ0; 1  S 1 ! Œ0; 1  S 1 is defined by .t; x/ D .1  t; x/: For a while we confine ourselves to the boundary component @0 A D .f0g  S 1 /; and denote the restrictions  j f0g  S 1 W f0g  S 1 ! @0 A and

 0 j f0g  S 1 W f0g  S 1 ! @0 A

simply by  W S 1 ! @0 A and

 0 W S 1 ! @0 A;

respectively. By the definition of a (Q-) linear twist, the action of f j @0 A D f 0 j @0 A on @0 A is topologically equivalent to a rotation of finite order, say,  > 0.

2.3 Non-amphidrome Annuli

27

Since  W S 1 ! @0 A and

 0 W S 1 ! @0 A

are “linear” parametrizations for this same action we have   1 1 . 0 /1  x C D . 0 /1 .x/ C   (Recall S 1 D R=Z here). However, . 0 /1  W S 1 ! S 1 need not be a “linear” rotation. This requires additional technicality in the first half of the proof below. Working in the universal covering R, we define an isotopy l W R ! R; by

0    1;

l .x/ D .1  /. 0 /1 .x/ C x;

Obviously we have

x2R

l0 D . 0 /1 ; l1 D idR ;

and l .x C 1=/ D l .x/ C 1=: The last property assures that l projects to an isotopy of S 1 D R=Z which we denote by the same notation l W S 1 ! S 1 . It satisfies l0 D . 0 /1 ; l1 D idS 1 ; and l .x C 1=/ D l .x/ C 1=: Define g W @0 A ! @0 A; .0    1/ by

g D  0 l  1 :

Then g0 D id and

g1 D  0  1 :

28

2 Standard Form

Also g satisfies the condition g1 .f 0 j @0 A/g D f 0 j @0 A on @0 A: To see this, recall that . 0 /1 .f 0 j @0 A/ 0 W S 1 ! S 1 and

 1 .f 0 j @0 A/ D  1 .f j @0 A/ W S 1 ! S 1

are the same rotation of order ;  1 .f 0 j @0 A/.x/ D . 0 /1 .f 0 j @0 A/ 0 .x/ D x C

ı ; 

where ı is an integer such that gcd.ı; / D 1. On the other hand,  1 g1 .f 0 j @o A/g .x/ D l1 . 0 /1 .f 0 j @0 A/ 0 l .x/   ı D l1 l .x/ C by (2.1 )  DxC Thus

ı : 

 1 g1 .f 0 j @0 A/g  D  1 .f 0 j @0 A/;

so

g1 .f 0 j @0 A/g D f 0 j @0 A

as asserted. We extend g W @0 A ! @0 A to an isotopy g  W A ! A;

0    1;

in such a way that g0 D idA ;

g  j @0 A D g ;

and g  j @1 A D identity:

Then the isotopy g  satisfies conditions (i) and (ii) of Lemma 2.2, and .g 1 /1 f 0 g 1 W A ! A is a linear twist with respect to the parametrization .g 1 /1  0 W Œ0; 1  S 1 ! A:

(2.1)

2.3 Non-amphidrome Annuli

29

Moreover, this parametrization .g 1 /1  0 satisfies .g1 /1  0 j f0g  S 1 D  j f0g  S 1 Therefore, taking

.g 1 /1 f 0 g 1

instead of f 0 if necessary, we may assume the parametrization  0 W Œ0; 1  S 1 ! A for f 0 satisfies

 0 j f0g  S 1 D  j f0g  S 1 :

Applying the same argument to f1g  S 1 , we may assume  0 j f0; 1g  S 1 D  j f0; 1g  S 1 : This completes the preparatory argument. Now by the technique of the proof of Lemma 2.1, there exists an isotopy ˚ W Œ0; 1  S 1 ! Œ0; 1  S 1 ;

0    1;

such that 1. ˚0 D . 0 /1 , 2. ˚1 is linear, that is

˚1 .t; x/ D .t; x C at C b/

for some a; b 2 Q, and 3. the restriction ˚ j f0; 1g  S 1 equals . 0 /1  j f0; 1g  S 1 (D the identity of f0; 1g  S 1 ). Define h W A ! A; by

0    1;

h D  0 ˚  1 :

Then h satisfies (i) h0 D idA , (ii) h j @A D id@A , in particular 0 0 h1  .f j @A/h D f j @A:

Moreover, one can verify 0 .h1 1 f h1 /.t; x/ D .t; x C ct C d / for some c; d 2 Q:

30

2 Standard Form

0 Thus the homeomorphisms f and h1 1 f h1 W A ! A are both linear with respect to the same ; they coincide on the boundary @A, and have the same screw number 0 by the assumption. Therefore, (iii) f D h1 t u 1 f h1 .

Putting together Lemmmas 2.1 and 2.2, and generalizing them to a disjoint union of annuli, we obtain the following theorem. Theorem 2.3. (i) (L INEARIZATION ) Let A D

r [

Ai

i D1

be a disjoint union of annuli, f WA !A a homeomorphism such that f .Ai / D Ai C1 ;

i D 1; 2; : : : ; r  1;

and f .Ar / D A1 . Suppose that f j @A W @A ! @A is periodic and that no annulus in A is amphidrome with respect to f . Then f is rel.@ isotopic to a homeomorphism f 0 W A ! A such that .f 0 /r j Ai W Ai ! Ai is a linear twist for each i D 1; 2; : : : ; r. (ii) (U NIQUENESS OF L INEARIZATION) Let f; f 0 W A ! A be homeomorphisms such that f .Ai / D f 0 .Ai / D Ai C1 ; and

i D 1; 2; : : : ; r  1;

f .Ar / D f 0 .Ar / D A1 :

Suppose that .f /r j Ai and .f 0 /r j Ai W Ai ! Ai are linear twists for each i D 1; 2; : : : ; r, and that f j @A D f 0 j @A , and that f; f 0 W A ! A

2.3 Non-amphidrome Annuli

31

are mutually isotopic by a rel:@ isotopy. Then there is an isotopy h W A ! A ;

0    1;

such that 1. h0 D idA 0 0 2. h1  .f j @A / h D f j @A .D f j @A / on @A ; and 1 0 3. f D h1 f h1 : Proof (of (i) L INEARIZATION). Let  W Œ0; 1  S 1 ! A1 be a parametrization for which f r W A1 ! A1 is “linear” on @A1 . We adopt f i 1  W Œ0; 1  S 1 ! Ai as a parametrization of Ai , i D 1; 2; : : : ; r. By essentially the same argument as in Lemma 2.1, f j Ar W Ar ! A1 is rel:@ isotopic to a map

f 0 j Ar W Ar ! A1

which is “linear” with respect to the parametrizations f r1  W Œ0; 1  S 1 ! Ar and  W Œ0; 1  S 1 ! A1 : We define by setting

f0 WA !A f 0 j Ai D f j Ai W Ai ! Ai C1;

for i D 1; 2; : : : ; r  1, and by taking the above f 0 j Ar W Ar ! A1 , for i D r. Then .f 0 /r j A1 W A1 ! A1

32

2 Standard Form

satisfies .f 0 /r .t; x/ D .f 0 j Ar /.f 0 j Ar1 /    .f 0 j A1 /.t; x/ D .f 0 j Ar /f r1 .t; x/ D .t; x C at C b/; for some a; b 2 Q Thus

.f 0 /r j A1 W A1 ! A1

is a linear twist with respect to  W Œ0; 1  S 1 ! A1 . Similarly, if r  2, .f 0 /r j A2 W A2 ! A2 satisfies .f 0 /r f .t; x/ D .f 0 j A1 /.f 0 j Ar /    .f 0 j A2 /.f 0 j A1 /.t; x/ D .f j A1 /.f 0 j Ar /f r1 .t; x/ D .f j A1 /.t; x C at C b/ D f .t; x C at C b/: Thus

.f 0 /r j A2 W A2 ! A2

is a linear twist with respect to f  W Œ0; 1  S 1 ! A2 ; t u

and so on. This proves (i). Proof (of (ii) U NIQUENESS

OF LINEARIZATION ).

Applying Lemma 2.2 to

f r j A1 W A1 ! A1 and

.f 0 /r j A1 W A1 ! A1 ;

we find an isotopy g.1/ W A1 ! A1 ; such that

0    1;

g0 D idA1 , .g.1/ /1 ..f 0 /r j @A1 /g.1/ D .f 0 /r j @A1 ;

and

.1/

f r j A1 D .g1 /1 ..f 0 /r j A1 /g.1/ : .1/

Define isotopies g.i / W Ai ! Ai ;

0    1;

i D 2; : : : ; r

2.3 Non-amphidrome Annuli

33

by the formula g.i / D .f j Ai 1 /    .f j A1 /g.1/ .f j A1 /1    .f j Ai 1 /1 (or equivalently, by an inductive formula g.i / D .f j Ai 1 /g.i 1/ .f j Ai 1 /1 ; i D 2; : : : ; r) and define an isotopy g W A ! A ;

0    1;

by setting g j Ai D g.i / ;

i D 1; 2; : : : ; r

Then g satisfies go D idA , g1 .f 0 j @A /g D f 0 j @A ; and

.g11 f 0 g1 /r j A1 D f r j A1 :

Therefore, replacing f 0 by g11 f 0 g1 if necessary, we may assume .f 0 /r j A1 D f r j A1 : By the assumption, f W .A ; @A / ! .A ; @A / is rel.@ isotopic to

f 0 W .A ; @A / ! .A ; @A /:

In particular, there is an isotopy h.2/  W A2 ! A2 ;

0    1;

such that h0 D idA2 , h j @A2 D id@A2 , and h1 .f j A1 / D f 0 j A1 . Then .2/ .f j A2 /.h1 /1 W A2 ! A3 .2/

.2/

is rel.@ isotopic to

.2/

f 0 j A2 W A2 ! A3 ;

so there is an isotopy h.3/  W A3 ! A3 ;

0    1;

such that h0 D idA3 , h j @A3 D id@A3 , and h1 .f j A2 /.h1 /1 D f 0 j A2 . .3/

.3/

.3/

.2/

34

2 Standard Form

Proceeding in this way, we can construct isotopies h.i / W Ai ! Ai ; .i /

0    1; i D 2; : : : ; r;

.i /

such that h0 D idAi , h j @Ai D id@Ai , and .i 1/ 1

.i /

h1 .f j Ai 1 /.h1 We will examine

/

D f 0 j Ai 1 :

.f j Ar /.h1 /1 W Ar ! A1 : .r/

For this purpose, set .r/

P D h1 .f j Ar1 /.f j Ar2 /    .f j A1 / W A1 ! Ar : Then .r1/ 1 .r1/ / h1 .f

.r/

P D h1 .f j Ar1 /.h1 .3/ h1 .f

.2/ .2/ j A2 /.h1 /1 h1 .f

0 r1

D .f /

.r2/ 1

j Ar2 /.h1

/



j A1 /

j A1 :

We have .f j Ar /.h1 /1 P D f r j A1 D .f 0 /r j A1 D .f 0 j Ar /P .r/

implying .f j Ar /.h1 /1 D f 0 j Ar W Ar ! A1 .r/

Finally define an isotopy h W A ! A ;

0    1;

by setting h j A1 D id; h j Ai D h.i / ;

i D 2; : : : ; r:

Then, 1 0 h0 D idA ; h j @A D id@A ; and f 0 D h1 f h1 1 (i.e., f D h1 f h1 )

as asserted.

t u

2.4 Amphidrome Annuli

35

2.4 Amphidrome Annuli Lemma 2.3 (S PECIALIZATION). Let A be an annulus, f WA!A an (orientation-preserving) homeomorphism which interchanges the boundary components. Suppose f j @A W @A ! @A is periodic. Then there exists a rel.@ isotopy f W A ! A; such that f0 D f and

0    1;

f1 W A ! A

is a special twist. We need a sublemma. Sublemma 1 Let A and f WA!A be as in Lemma 2.3. Then there is a parametrization  W Œ0; 1  S 1 ! A such that f .0; x/ D .1; x C a/ f .1; x/ D .0; x  a/ for some a 2 Q. Proof. Let @A D @0 A [ @1 A. Since f 2 j @0 A W @0 A ! @0 A is a rotation of finite order, there is a parametrization ' W S 1 ! @0 A such that f 2 '.x/ D '.x  2b/

36

2 Standard Form

for some b 2 Q. Take a parametrization W S 1 ! @1 A for which f '.x/ D

.x C a/

holds, where a 2 Q is arbitrary at this point. Now, f

.x/ D .f 2 '/.x C a/ D '.x C a  2b/

We have already chosen the number a satisfying f '.x/ D

.x C a/

and we want a to satisfy also f

.x/ D '.x  a/:

To attain this a 2 Q must be chosen so that a  2b  a .mod 1/; that is a  b or b C

1 .mod 1/: 2

This ambiguity cannot be settled now. Since ' W S 1 ! @0 A and W S 1 ! @1 A are homotopic as maps from S 1 into A, there is a homeomorphism  W Œ0; 1  S 1 ! A such that .0; x/ D '.x/ and .1; x/ D Then

.x/.

f .0; x/ D .1; x C a/; f .1; x/ D .0; x  a/;

2.4 Amphidrome Annuli

37

as required. But remember that there are two possible values for a .mod 1/.

t u

This sublemma has the following corollary. Corollary 2.3. Let A D

r [

Ai

i D1

be a disjoint union of annuli, f WA !A a homeomorphism such that f j @A W @A ! @A is periodic. Suppose Ai be an amphidrome annulus in A with respect to f . Let .m0i ; 0i ; i0 / and .m1i ; 1i ; i1 / be the valencies of @0 A and @1 A, respectively, and s.Ai / the screw number of f in Ai . Then (i) m0i D m1i D an even number, (ii) .0i ; i0 / D .1i ; i1 / (iii) .1=2/s.Ai / C ıi =i is an integer, where ıi is an integer determined by i ıi  1 .mod i / and 0  ıi < i . (Here i denotes 0i D 1i , and i denotes i0 D i1 .) (iv) .i ; i /is uniquely determined by s.Ai /. Proof. Let k be the smallest positive integer such that f k .Ai / D Ai . Since Ai is amphidrome, f k interchanges the boundary components of Ai . Thus 2k is the smallest positive integer such that f 2k .Ai / D Ai does not interchange the boundary components. This implies (i) m0i D m1i D 2k. Obviously, f k j @0 A W @0 A ! @1 A is equivariant with respect to the actions of f 2k j @0 A on @0 A and f 2k j @1 A on @1 A. This proves (ii) .0i ; i0 / D .1i ; i1 /. To prove (iii), consider f k W A i ! Ai as f in sublemma 1 and take a parametrization  W Œ0; 1  S 1 ! Ai

38

2 Standard Form

there. We identify Ai with Œ0; 1  S 1 through . We give to Œ0; 1  S 1 the same orientation as in the proof of Corollary 2.1 (Fig. 2.2). Note that f k .0; x/ D .1; x C a/; f k .1; x/ D .0; x  a/: Let us pass to the universal covering Œ0; 1  R. Let IQ be the lift of the arc f k .Œ0; 1  f0g/ which starts at .0; a/ 2 Œ0; 1  R: Then it ends at .1; a C n/ for some n 2 Z. Thus there is a lift fQk W Œ0; 1  R ! Œ0; 1  R of f k W Œ0; 1  S 1 ! Œ0; 1  S 1 satisfying fQk .0; x/ D .1; x C a C n/; fQk .1; x/ D .0; x  a/: We have .fQk /2 .0; x/ D .0; x  2a  n/; .fQk /2 .1; x/ D .1; x C 2a C n/: The curve f 2k .Œ0; 1f0g/ is lifted to a curve joining .0; 2an/ to .1; 2aCn/. This implies s.Ai / D 4a  2n: (2.2) (Recall the convention of the sign of a Dehn twist in Convention ().) On the other hand, by the geometric meaning of ıi =i , we have ıi  2a .mod 1/: i Thus

ıi 1  2a  n C 2a  0 .mod 1/; s.Ai / C 2 i

which proves (iii). Since 0  i < 1 and i ıi  1 .mod i /, assertion (iv) follows from (iii).

t u

2.4 Amphidrome Annuli

39

Now we can fix the ambiguity of the number a unsettled in Sublemma 1. We observed in the above proof that 1 s.Ai / D 2a  n: 2 Hence by shifting a by 1=2 if necessary, we can make n an even number. In other words, we can (and will) take a so that 1 a   s.Ai / .mod 1/: 4 Let us restate Sublemma 1 as Corollary 2.4, taking this choice into account. Corollary 2.4. Let A and f WA!A be as in Lemma 2.3. Let s.A/ be the screw number of f in A. Then there is a parametrization  W Œ0; 1  S 1 ! A such that   1 f .0; x/ D  1; x  s.A/ ; 4   1 f .1; x/ D  0; x C s.A/ : 4 Remark 2.3. In this way f 2 rotates both sides of A by half the screw number in opposite directions. Proof (of Lemma 2.3). Let  W Œ0; 1  S 1 ! A be the parametrization of Corollary 2.4, and identify A with Œ0; 1  S 1 through this . Let fQ W Œ0; 1  R ! Œ0; 1  R be the lift of f W Œ0; 1  S 1 ! Œ0; 1  S 1 such that

Then

  1 Q f .1; x/ D 0; x C s.A/ : 4 1 fQ.0; x/ D .1; x  s.A/ C n/ 4

40

2 Standard Form

for some n 2 Z, but this n must be 0 because, as we observed in the proof of Corollary 2.3, if the curve fQ.Œ0; 1  f0g/ connects .0; 1=4s.A// and .1; 1=4s.A/ C n/, then 1 s.A/ D 4. s.A//  2n: 4 (See (2.2) in the proof.) This implies n D 0, and we get   1 Q f .0; x/ D 1; x  s.A/ : 4 Define a piecewise-linear homeomorphism fQ0 W Œ0; 1  R ! Œ0; 1  R by setting 8   3 1 1 ˆ ˆ < 1  t; x C 4 s.A/ t  3 ; 0  t  3 ; 0 1 fQ .t; x/ D .1  t; x/;  t  23 ; 3 ˆ    ˆ : 1  t; x C 3 s.A/ t  2 ; 2  t  1: 4 3 3 Note that

fQ0 j f0; 1g  R D fQ j f0; 1g  R:

This homeomorphism fQ0 projects to a special twist f 0 W Œ0; 1  S 1 ! Œ0; 1  S 1 : By essentially the same argument in the proof of Lemma 2.1, f W Œ0; 1  S 1 ! Œ0; 1  S 1 is rel:@ isotopic to f 0 . This completes the proof of Lemma 2.3.

t u

The following corollary will be obvious from the above argument. Corollary 2.5. Let f W A ! A be a special twist with respect to a parametrization  W Œ0; 1  S 1 ! A: Then, in contradistinction with the case of linear twists (Corollary 2.2), the equation defining a special twist is uniquely determined by the screw number s.A/ as follows:

2.4 Amphidrome Annuli

f .t; x/ D

41

8    3 1 ˆ ˆ < 1  t; x C 4 s.A/ t  3 ; .1  t; x/; ˆ ˆ : 1  t; x C 3 s.A/ t  2 ; 4 3

Lemma 2.4 (U NIQUENESS

OF SPECIALIZATION).

0  t  13 ; 1 3 2 3

 t  23 ;  t  1:

Let

f; f 0 W A ! A be special twists of an annulus A. Suppose that f j @A D f 0 j @A and that the screw number of f in A is equal to the screw number of f 0 in A. Then there is an isotopy h W A ! A; 0    1; such that (i) h0 D idA , 0 0 (ii) h1  .f j @A/h D f j @A .D f j @A/on@A, and 0 (iii) f D h1 f h : 1 1 Proof (of Lemma 2.4). The idea is the same as in Lemma 2.2. Let  and  0 W Œ0:1  S 1 ! A be the parametrizations with respect to which f and f 0 are special twists. After a preliminary isotopy, we may assume  j f0; 1g  S 1 D  0 j f0; 1g  S 1 : (Cf. the proof of Lemma 2.2.). Then, the next Claim shows that  and  0 differ by an even number of full twists. Claim (A). There exist lifts Q Q 0 W Œ0; 1  R ! AQ ; of

;  0 W Œ0; 1  S 1 ! A

such that Q .Q 0 /1 .0; x/ D .0; x C m/; Q x/ D .1; x  m/; .Q 0 /1 .1; for some m 2 Z.

42

2 Standard Form

Proof (of Claim (A)). For simplicity, we identify A with Œ0; 1  S 1 through  0 . By Corollary 2.5, f 0 W Œ0; 1  S 1 ! Œ0; 1  S 1 is lifted to

fQ0 W Œ0; 1  R ! Œ0; 1  R

e0 is a piecewise-linear arc such that the image of the interval Œ0; 1  f0g under f connecting .0; 1=4s/ and .1; 1=4s/, where s D s.A/. Since f j @A D f 0 j @A, and the screw number s.A/ is common to f and f 0 , f W Œ0; 1  S 1 ! Œ0; 1  S 1 can also be lifted to

fQ W Œ0; 1  R ! Œ0; 1  R

such that f .Œ0; 1  f0g/ is a (not necessarily piecewise-linear) arc connecting .0; 1=4s/ and .1; 1=4s/. In particular,   1 fQ.0; x/ D 1; x  s ; 4

  1 fQ.1; x/ D 0; x C s : 4

By Corollary 2.5, there is a lift Q Œ0; 1  R via Q 0 / Q W Œ0; 1  R ! A.D such that

Q Q 1 fQ.Œ0; 1  f0g/ ./

is a piecewise-linear arc connecting .0; 1=4s/ and .1; 1=4s/. Since  j f0; 1g  S 1 D  0 j f0; 1g  S 1 ; Q satisfies Q .0; x/ D .0; x C m/; Q .1; x/ D .1; x C n/; for some m, n 2 Z. Then   1 Q Q 1 fQ.0; x/ D 1; x  m  n  s ; 4   1 Q x/ D 0; x  m  n C s : Q 1 fQ.1; 4

2.4 Amphidrome Annuli

43

Q m C n D 0. Thus By our choice of , .Q 0 /1 Q W Œ0; 1  R ! Œ0; 1  R t u

satisfies the formula stated in Claim (A).

Proof (of Lemma 2.4: continued). Just as in the proof of Lemma 2.2, there is an isotopy ˚ W Œ0; 1  S 1 ! Œ0; 1  S 1 such that 1. ˚0 D . 0 /1 , 2. ˚1 is linear;

˚1 .t; x/ D .t; x  2mt C m/;

m being the integer of Claim (A). 3. The restriction ˚ j f0; 1g  S 1 equals

. 0 /1  j f0; 1g  S 1

which is the identity on f0; 1g  S 1 . Claim (B). If f 0 W A ! A is a special twist with respect to a parametrization  0 W Œ0; 1  S 1 ! A; then f 0 is also a special twist with respect to  0 ˚1 W Œ0; 1  S 1 ! A; where ˚1 W Œ0; 1  S 1 ! Œ0; 1  S 1 is given by ˚1 .t; x/ D .t; x  2mt C m/ for some m 2 Z. Proof (of Claim (B)). Compute ˚11 . 0 /1 f 0  0 ˚1 .t; x/: But beware that it is here where the full force of Claim (A) is used. Define an isotopy h W A ! A;

0    1;

t u

44

2 Standard Form

by

h D  0 ˚  1 :

Then h satisfies (i) h0 D idA , (ii) h j @A D id@A . Moreover, using Claim (B), one can verify that 0 1 0 1 0 0 1 h1 1 f h1 .D ˚1 . / f  ˚1  / 0 is a special twist with respect to  W Œ0; 1  S 1 ! A. Both h1 1 f h1 and f are special twists with respect to the same parametrization

 W Œ0; 1  S 1 ! A: Since they coincide on @A and have the same screw number in A then (iii) f D 0 h1 u t 1 f h1 . Let us generalize Lemmas 2.3 and 2.4 to the case of a disjoint union of annuli. Theorem 2.4. (i) (S PECIALIZATION) Let A D

r [

Ai

i D1

be a disjoint union of annuli, f WA !A a homeomorphism such that f .Ai / D Ai C1 ;

i D 1; 2; : : : ; r  1;

and f .Ar / D A1 . Suppose that f j @A W @A ! @A is periodic and that each Ai is amphidrome with respect to f . Then f is rel.@ isotopic to a homeomorphism f0 WA !A such that

.f 0 /r j Ai W Ai ! Ai

is a special twist for each i D 1; 2; : : : ; r. (ii) (U NIQUENESS OF S PECIALIZATION) Let f; f 0 W A ! A be homeomorphisms such that

2.4 Amphidrome Annuli

45

f .Ai / D f 0 .Ai / D Ai C1 ; and

i D 1; 2; : : : ; r  1;

f .Ar / D f 0 .Ar / D A1 :

Suppose that f r j Ai and

.f 0 /r j Ai W Ai ! Ai

are special twists for each i D 1; 2; : : : ; r; and that f j @A D f 0 j @A ; and that

f; f 0 W A ! A

are mutually isotopic by a rel.@ isotopy. Then there is an isotopy h W A ! A;

0    1;

such that (i) h0 D idA , 0 0 (ii) h1  .f j @A/h D f j @A.D f j @A/ on @A, and 1 0 (iii) f D h1 f h1 . Proof (of (i) S PECIALIZATION). Since A1 is amphidrome, f r W A1 ! A1 interchanges the boundary components. Take a parametrization  W Œ0; 1  S 1 ! A1 for which f r W A1 ! A1 satisfies 

 1 f .0; x/ D  1; x  s.A1 / ; 4   1 f r .1; x/ D  0; x C s.A1 / : 4 r

(See Corollary 2.4). As in the proof of Theorem 2.3 (i), we adopt f i 1  W Œ0; 1  S 1 ! Ai

46

2 Standard Form

as a parametrization of Ai , i D 1; 2; : : : ; r. By our choice of  W Œ0; 1  S 1 ! A1 ; the map f j Ar W Ar ! A1 trivially satisfies   1 .0; x/ D  1; x  s.A1 / ; .f j Ar /f 4   1 .f j Ar /f r1 .1; x/ D  0; x C s.A1 / : 4 r1

Then f j Ar W Ar ! A1 is rel:@ isotopic to a homeomorphism f 0 j Ar W Ar ! A1 defined by

0

.f j Ar /f

r1

.t; x/ D

8    3 1 ˆ ˆ < 1  t; x C 4 s.A1 / t  3 ; .1  t; x/; ˆ ˆ : 1  t; x C 3 s.A / t  2  ; 1 4 3

by setting

1 3 2 3

 t  23 ;  t  1: t u

(See the proof of Lemma 2.3). We define

0  t  13 ;

f0 WA !A f 0 j Ai D f j Ai W Ai ! Ai C1;

for i D 1; 2; : : : ; r  1, and by taking the above f 0 j Ar W Ar ! A1 for i D r. Then by the same argument as in the proof of Theorem 2.3(i), we can show that .f 0 /r j Ai W Ai ! Ai is a special twist with respect to the parametrization f i 1  W Œ0; 1  S 1 ! Ai ;

2.5 Proof of Theorem 2.1

47

for i D 1; 2; : : : ; r. Clearly f 0 is rel.@ isotopic to f . We are done. Proof (of (ii) U NIQUENESS

OF

S PECIALIZATION ). Applying Lemma 2.4 to f r j A1 W A1 ! A1 ;

and

.f 0 /r j A1 W A1 ! A1

we find an isotopy g.1/ W A1 ! A1 ;

0    1;

.1/

such that g0 D idA1 , .g.1/ /1 ..f 0 /r j @A1 /g.1/ D .f 0 /r j @A1 and

f r j A1 D .g1 /1 ..f 0 /r j A1 /g1 : .1/

.1/

Then the rest of the proof is exactly the same as the proof of Theorem 2.3 (ii). This completes the proof. t u Now we are in a position to prove the main theorem of this Chap. 2.

2.5 Proof of Theorem 2.1 Proof (of (i)). We must show that a given pseudo-periodic map f W ˙g ! ˙g is isotopic to a pseudo-periodic map in standard form. Let fCi griD1 be the precise system of cut curves subordinate to f (Chap. 1). We may assume f .C / D C ; S C being riD1 Ci . We choose a system of annular neighborhoods fAi griD1 of fCi griD1 : We may assume f .A / D A ; where A D

r [

Ai :

i D1

By the definition of a pseudo-periodic map, f j .˙g  C / W ˙g  C ! ˙g  C

48

2 Standard Form

is isotopic to a periodic map. Then f jB W B ! B is also isotopic to a periodic map (Cf. [52]), where B D ˙g  I nt.A /; so we may assume that f j B is already periodic. Decompose the finite set fAi griD1 into cyclic orbits under the permutation caused by f , and decompose A into A .1/ [ A .2/ [    [ A .s/ accordingly. Since f cyclically permutes the connected components of A . / , all the annuli contained in A . / are simultaneously non-amphidrome or amphidrome for each D 1; 2; : : : ; s. Apply Theorem 2.3 (i) or Theorem 2.4 (i) as the case may be, then f j A . / W A . / ! A . / is rel.@ isotopic to a homeomorphism f 0 j A . / W A . / ! A . / . /

such that, for each annulus Aj in A. / , .f 0 j A . / /r W Aj ! Aj . /

. /

is a linear twist or a special twist, where r denotes the number of the annuli in A . / . Applying this isotopy for each D 1; 2; : : : ; s, we get a pseudo-periodic map f 0 W ˙g ! ˙g in standard form, which is isotopic to the original f W ˙g ! ˙g : t u

This completes the proof of (i). Proof (of (ii)). Suppose we are given two pseudo-periodic maps f; f 0 W ˙g ! ˙g

in standard form. Suppose they are homotopic. We will show the existence of a homeomorphism h W ˙g ! ˙g isotopic to the identity and such that f D h1 f 0 h.

2.5 Proof of Theorem 2.1

49

By Theorem 2.2 (i), there exists a homeomorphism g W ˙g ! ˙g which is isotopic to the identity and sends the precise system of cut curves for f to that for f 0 . Replacing f 0 by g1 f 0 g, we may assume that the precise system of cut curves is common to f and f 0 . Let us denote it by fCi griD1 . Let fAi griD1 and fA0i griD1 be the systems of annular neighborhoods of fCi griD1 which are invariant under the action of f and of f 0 , respectively. Again there exists a homeomorphism g W ˙g ! ˙g which is isotopic to the identity and sends Ai to A0i . Replacing S f 0 by g 1 f 0 g, we r 0 may assume that fAi gi D1 is common to f and f . Let us denote riD1 Ai by A and ˙g  I nt.A / by B as before. t u Claim (C). f j B and f 0 j B are homotopic as maps of pairs: .B; @B/ ! .B; @B/: Proof. We may assume that fCi griD1 are closed geodesic with respect to a certain metric ˙g . Since f; f 0 W ˙g ! ˙g are homotopic, they are isotopic, [10, 21]. Let f W ˙g ! ˙g ;

0    1;

be the isotopy with f0 D f and f1 D f 0 . This isotopy gives a homeomorphism F W ˙g  Œ0; 1 ! ˙g  Œ0; 1 defined by F .p; / D .f .p/; / for .p; / 2 ˙p  Œ0; 1: Since ˙g  Œ0; 1 is an irreducible 3-manifold (cf. [68]), we can apply an innermost disk argument to achieve F .C  Œ0; 1/ D C  Œ0; 1: Also by moving F by a rel.@ isotopy, we may assume F .A  Œ0; 1/ D A  Œ0; 1:

50

2 Standard Form

Then F j B  Œ0; 1 W B  Œ0; 1 ! B  Œ0; 1 gives a homotopy between f j B and f 0 j B.

t u

Now we can apply Theorem 2.2, and obtain a homeomorphism hjB W B ! B isotopic to the identity idB and such that f j B D .h j B/1 .f 0 j B/.hB/: h j B extends to a homeomorphism h W ˙g ! ˙g which is isotopic to the identity. Replacing f 0 by h1 f 0 h, we may assume f j B D f 0 j B: Claim (D). f j A and f 0 j A W .A ; @A / ! .A ; @A / are isotopic by a rel.@ isotopy. Proof. Let Ai be an annulus of A . f and f 0 cause the same permutation on the set of annuli fAi griD1 because f j B D f 0 j B: We have

f .A1 / D f 0 .A1 /;

which we denote A2 , for simplicity. Let L be a “straight” line in A1 connecting a point p0 2 @0 A1 and another p1 2 @1 A1 . Then the images f .L/ and f 0 .L/ are arcs in A2 connecting q0 WD f .p0 / D f 0 .p0 / and

q1 WD f .p1 / D f 0 .p1 /:

The arcs f .L/ and f 0 .L/ are homotopic in A2 fixing the end points fq0 ; q1 g. (Proof. Note that .f 0 /1 f W ˙g ! ˙g is a pseudo-periodic map, because .f 0 /1 f j B D idB :

2.5 Proof of Theorem 2.1

51

If the assertion above is not correct then .f 0 /1 f would have non-zero screw number about C1 , the center line of A1 . But this is impossible by Theorem 2.2, because .f 0 /1 f W ˙g ! ˙g is homotopic to id˙g by the assumption of Theorem 2.1(ii)). Then by the ordinary innermost arc argument, together with the Alexander trick, f j A1 W A1 ! A2 is rel.@ isotopic to

f 0 j A1 W A1 ! A2 :

Doing the same argument for each Ai , i D 1; 2; : : : ; r, we get Claim (B). As we did in the proof of Theorem 2.1 (i), decompose A into cyclic orbits A .1/ [ A .2/ [    [ A .s/ under the action of f . Consider an orbit A . / . By the assumption of Theorem 2.1 . / (ii), f r j Aj and .f 0 /r j Aj W Aj ! Aj . /

. /

. / . /

are simultaneously linear twists or special twists, where Aj is an annulus of A . / and r D #.A . / /: Also by Claim (B), f j A . / and f 0 j A . / W .A . / ; @A . / / ! .A . / ; @A . / / are isotopic by a rel.@ isotopy. Then we can apply Theorem 2.3 (ii) or Theorem 2.4 (ii) as the case may be, and obtain an isotopy . / h. / ! A . / ;  WA

0    1;

such that . /

1. h0 D id: . / . / 2. .h /1 .f 0 j @A. / /h D f 0 j @A . / .D f j @A . / / on @A . / ; and . / 1 . / . / 3. f j A D .h1 / .f 0 j A . / /h1 : The condition (2) above says that . / h. / ! A . /  WA

is equivariant on @A . / with respect to the action of f 0 j @A . / .D f j @A . / /:

52

2 Standard Form

Therefore, we can extend the isotopy h. /  ;

0    1;

equivariantly into collar neighborhoods .@B/  Œ0; "/ of B so that, beyond the collar, the extension gives the identity on B  .@B/  Œ0; "/: Doing the same construction and the extension for each D 1; 2; : : : ; s, we get an isotopy h W ˙g ! ˙g ; 0    1; such that (a) h .A / D A and h .B/ D B; (b) h0 D id; (c) h j B ! B is equivariant with respect to f 0 j B D f j B, i.e. .h j B/1 .f 0 j B/.h j B/ D f 0 j B.D f j B/; and . /

(d) h j A . / D h ; D 1; 2; : : : ; s: Now, since and

f j B D .h1 j B/1 .f 0 j B/.h1 j B/

f j A . / D .h1 j A . / /1 .f 0 j A . / /.h1 j A . / /; D 1; 2; : : : ; s;

it is evident that

0 f D h1 1 f h1 :

This completes the proof of Theorem 2.1 (ii).

t u

Chapter 3

Generalized Quotient

3.1 Definitions and Main Theorem of Chap. 3 A periodic map f on a surface ˙ defines a quotient space ˙=f . In the case of a pseudo-periodic map f W ˙ ! ˙; however, the quotient space ˙=f would not be any reasonable space, if the term “quotient space” is taken in the usual sense, i.e., the orbit space under the action of f . To adjust this, we introduce the following definition. Definition 3.1. A pseudo-periodic map f W ˙g ! ˙g is said to be of negative twist if each cut curve Ci has negative screw number in the precise system of cut curves subordinate to f . Thus, the purpose of this and the next chapters is to generalize the notion of quotient spaces so that a pseudo-periodic map of negative twist always has its “generalized quotient space”. Let us begin by recalling Bers’ definition of Riemann surfaces with nodes [11]: A Riemann surface with nodes, S , is a connected complex space such that every p 2 S has arbitrarily small neighborhoods isomorphic either to the set jzj < 1 in C or to the set jzj < 1, jwj < 1, zw D 0 in C2 . In the second case, p is called a node. When we disregard the complex structure, we will call S a chorizo space1 . A chorizo space with boundary is defined in the obvious way. Following Bers [11], we call every component of S -fnodesg a part of S . The closure of a part is an irreducible component. We call a neighborhood Np of a node p (isomorphic to

1

chorizo D Spanish sausage, good to be eaten fried with some wine.

Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5 3, © Springer-Verlag Berlin Heidelberg 2011

53

54

3 Generalized Quotient

the set jzj < 1, jwj < 1, z w D 0 in C2 ) a nodal neighborhood. We sometimes consider a closed nodal neighborhood N p corresponding to jzj  1, jwj  1, w z D 0. A nodal neighborhood Np has two banks (cf. [11]) corresponding to fjzj < 1; w D 0g and fz D 0; jwj < 1g. A connected component of S

[

Np

p D node

is called a closed part. (Of course, a closed part may have a non-empty boundary!). The following definition generalizes (at least in the case when S1 is non-singular) the notion of a deformation S1 ! S which was introduced by Bers ([11]). Definition 3.2. Let ˙ be a surface, S a chorizo space. A continuous map W˙ !S is called a pinched covering if the following conditions are satisfied: (i)  is surjective, (ii) for each nodal neighborhood Np , each connected component A of  1 .Np / is an (open) annulus, jA W A ! Np maps the central-line C of A to the node p, and j.A  C / W A  C ! Np  fpg is the disjoint union of two cyclic coverings over the two punctured banks, and (iii) for each part P 0 , j 1 .P 0 / W  1 .P 0 / ! P 0 is an orientation-preserving covering (in the usual sense, though  1 .P 0 / may not be connected). In what follows, surfaces and chorizo spaces will be compact. In that case, given a pinched covering  W ˙ ! S; each covering j 1 .P 0 / W  1 .P 0 / ! P 0 in it is finite-sheeted, so we can attach this number of sheets to the part P 0 as its multiplicity. We may consider this multiplicity to be attached to the closed part P , which P 0 contains, or to the irreducible component  which contains P 0 , as well. Definition 3.3. A chorizo space S is numerical if to each irreducible component is attached a positive integer (called the multiciply).

3.1 Definitions and Main Theorem of Chap. 3

55

In a numerical chorizo space, each bank D of a nodal neighborhood Np gets its multiplicity from the irreducible component to which D belongs. Henceforth all chorizo spaces will be numerical. Now we give a definition of a generalized quotient of a pseudo-periodic map f W ˙g ! ˙g : This definition may be considered to be a preliminary one, because it provides a generalized quotient only for a very restricted type of pseudo-periodic maps with negative twist. The final definition will appear almost at the end of the next chapter, after a uniqueness theorem is proved. Definition 3.4. A pinched covering  W ˙g ! S is called a generalized quotient of a pseudo-periodic map f W ˙g ! ˙g if the following conditions are satisfied: (i) A system of closed nodal neighborhoods fN p gp D node is fixed in S . For every node p,  1 .N p / is preserved by f as a set, i.e. f . 1 .N p // D  1 .N p /; and for every connected component Pi of [ S Np ; .Np D Int.N p //; p D node

 1 .Pi / is also preserved by f as a set i.e. f . 1 .Pi // D  1 .Pi /: S (ii) For each connected component Pi of S  p D node Np ; f j 1 .Pi / W  1 .Pi / !  1 .Pi / is a periodic map of order mi , mi being the multiplicity of Pi , and j 1 .Pi / W  1 .Pi / ! Pi is the quotient map by f j 1 .Pi /. (NB. By the definition of a pinched covering, f j 1 .Pi / has no multiple points).

56

3 Generalized Quotient

(iii) For each node p,  1 .N p / consists of m .D gcd.m1 ; m2 // annuli, where m1 ; m2 are the multiplicities of the banks of N p , and f j 1 .N p / W  1 .N p / !  1 .N p / permutes these annuli cyclically. (iv) For each annulus A in  1 .N p /, f m jA W A ! A is a linear twist (Chap. 2), and its screw number is equal to 1=n1 n2 , where n1 D m1 =m, n2 D m2 =m. (v) For each node p, there exist parametrizations of the banks fzj jzj  1g ! D1 ; fzj jzj  1g ! D2 sending O to p, such that if we identify Di with fzj jzj  1g through these parametrizations (i D 1; 2) and define Ti W Di ! Di .i D 1; 2/ by ! 1 .1  jzj/ ; T1 .z/ D z exp m2 ! p 1 T2 .z/ D z exp .1  jzj/ ; m1 p

then the following identities hold: Ti  D f;

on  1 .Di / .i D 1; 2/:

The definition of a generalized quotient is motivated by the results of Chap. 7. Remark 3.1. 1. When we talk about a generalized quotient, a system of closed nodal neighborhoods fN p gp D node will always be fixed in S , so that it satisfies the above conditions, and a closed part P will always be a connected component of [ S Np ; p D node

Np being the interior of N p . 2. If  W ˙g ! S

3.1 Definitions and Main Theorem of Chap. 3

57

is a generalized quotient of a pseudo-periodic map f W ˙g ! ˙g ; then f is almost in standard form. The difference is that the system of cut curves f 1 .p/gp D node may not be a precise system (Chap. 1), but in compensation, the screw numbers are restricted to be as in (iv) above, i.e. they depend on the multiplicities of the banks of p. We will say that such an f is in superstandard form. 3. In a generalized quotient  W ˙g ! S ; every annulus in  1 .N p / is non-amphidrome. Amphidrome annuli are hidden in the global configuration of S , in a sense that will be clarified later (see Chap. 4). Lemma 3.1. Let  W ˙g ! S be a generalized quotient of a pseudo-periodic map f W ˙g ! ˙g of negative twist. Let p 2 S be a node. Let N p be the fixed closed nodal neighborhood of p. Let D1 ; D2 be the two banks of N p . Thus, N p is the one point union D1 _D2 . Let m1 ; m2 be the multiplicities of D1 ; D2 , respectively. Suppose the boundary curves of D1 and D2 , @D1 and @D2 , are oriented as the boundaries of the closed parts P1 ; P2 which are adjacent to @D1 ; @D2 , respectively, and let .i ; i ; i / be the valency of the curves in  1 .@Di / with respect to the periodic action of f W  1 .Pi / !  1 .Pi /;

i D 1; 2:

Then (i) 1 D 2 D gcd.m1 ; m2 /, (ii) 1 D m1 = gcd.m1 ; m2 /, 2 D m2 = gcd.m1 ; m2 /, (i.e. 1 D n1 ; 2 D n2 , in the notation of the definition of a generalized quotient, (iv)), and (iii) 1  n2 .mod n1 /, 2  n1 .mod n2 /. Proof. (i) Since f W  1 .Np / !  1 .Np / causes a cyclic permutation of the components, the numbers 1 ; 2 are equal to the number of the annuli contained in  1 .Np /, which equals gcd.m1 ; m2 / by the definition of a generalized quotient. (See the proof of Corollary 2.1, (i)). (ii) Remember that the multiplicity mi is the order of the periodic map f j 1 .Pi / W  1 .Pi / !  1 .Pi /:

58

3 Generalized Quotient

Then f j 1 .@Di / has order mi , while by (i)  1 .@Di / contains m components, m being gcd.m1 ; m2 /. Thus for each component @DQ i of  1 .@Di /, f m W @DQ i ! @DQ i has order mi =m. This is i by definition. (iii) Let ı1 and ı2 be defined by 1 ı1  1 .mod 1 /;

0  ı1 < 1 ;

2 ı2  1 .mod 2 /;

0  ı2 < 2 :

Applying Corollary 2.1 (ii) to the disjoint union of annuli  1 .N p / whose members have screw number 1=n1 n2 , we have 

1 ı1 ı2 C C D ˛ an integer n1 n2 n1 n2

(Remember 1 D n1 ; 2 D n2 .) Then n2 ı1 C n1 ı2 D 1 C ˛n1 n2 ; from which we have n2 ı1  1 .mod n1 /; and n1 ı2  1 .mod n2 /. Since gcd.n1 ; n2 / D 1, we obtain 1  n2 .mod n1 /; and 2  n1 .mod n2 /.

t u

Proposition 3.1. Let  W ˙g ! S be a generalized quotient of a pseudo-periodic map f W ˙g ! ˙g : Let 0 be an irreducible component of S with multiplicity m0 . Let fp1 ; p2 ; : : : ; pk g be the set of the intersection points of 0 with the other irreducible components. Let mi be the multiplicity of the irreducible component which intersects 0 at pi .i D 1; 2; : : : ; k/:

3.1 Definitions and Main Theorem of Chap. 3

59

Then m 1 C m2 C    C m k is divisible by m0 . Proof. We add the self-intersection points of 0 , fpkC1 ; : : : ; pl g; to fp1 ; p2 ; : : : ; pk g; and set mi D m0 for i D k C 1; : : : ; l. Let P be the closed part contained in 0 : P D 0 

l [

Int.Di /;

i D1

where Di D 0 \ N pi . We denote @Di by @i P . Let ! W H1 .P / ! Z=m0 be the monodromy exponent associated with  1 .P / !  1 .P /=f D P: (See the outlined proof of Theorem 1.3 for the definition. Also see [51]). Let .i ; i ; i / be the valency of @i P (with respect to the periodic map f j 1 .P / W  1 .P / !  1 .P /I all curves in the preimage of @i P have the same valency, and, as we remarked in Chap. 1, this is, by definition, the valency of @i P with respect to f j 1 .P //. Then, by definition of valency [51, Sect. 2], !.Œ@i P /  i i .mod m0 /, By Lemma 3.1 (i), (iii), we have i D gcd.m0 ; mi / and i  mi = gcd.m0 ; mi / .mod .m0 /= gcd.m0 ; mi //. Thus i i  mi .mod m0 /: Since

l X i D1

Œ@Pi  D Œ@P  D 0

60

3 Generalized Quotient

in H1 .P /, we have l X

mi D

i D1

l X

!.Œ@i P /  0 .mod m0 /:

i D1

Remember that mi D m0 for i D k C 1; : : : ; l. Then k X

mi  0 .mod m0 /:

i D1

t u

(Cf. [51, (4.6)]).

Definition 3.5. A generalized quotient  W ˙ ! S is called a minimal quotient if the multiplicities of S satisfy the following minimality conditions: (i) if an irreducible component 0 is a sphere (without self-intersection) and intersects only one other component i , in one point exactly, then mi =m0  2 (NB. mi =m0 is an integer by Proposition 3.1), and (ii) if an irreducible component 0 is a sphere (without self-intersection) and intersects the union of the other components exactly in two points fp1 ; p2 g, then .m1 C m2 /=m0  2. (NB. .m1 C m2 /=m0 is also an integer). Our main result in Chap. 3 is the following: Theorem 3.1 (Existence). Let f W ˙g ! ˙g be a pseudo-periodic map of negative twist. Then f is isotopic to a pseudo-periodic map f 0 W ˙g ! ˙g whose minimal quotient

 W ˙g ! S

exists. Uniqueness of the minimal quotient will be proved in Chap. 4.

3.2 Proof of Theorem 3.1 By Theorem 2.1, a pseudo-periodic map f W ˙g ! ˙g is isotopic to a pseudo-periodic map in standard form. Thus we may assume that f is already in standard form.

3.4 Re-normalization of a Rotation

61

Let fAi griD1 be the annular neighborhood system of precise cut curves fCi griD1 (required in the definition of standard form in Chap. 2) such that S (a) f .A / D A and f .B/ D B, where A D riD1 Ai and B D ˙g  Int.A /, (b) f jB W B ! B is a periodic map, and (c) f causes linear twists or special twists, or both, in A . Let D be a disjoint union of invariant disk neighborhoods of all the multiple points in B, and let B0 denote B  Int.D/. Minimal quotients will be constructed for .A ; f jA /, .B0 ; f jB0 /, and .D; f jD/, separately.

3.3 Quotient of .B0 ; f jB 0 / This is the easiest. We have only to take the quotient jB0 W B 0 ! B 0 =.f jB0 / by the periodic action f jB0 W B 0 ! B 0 : Since we have deleted all the multiple points, B 0 =.f jB0 / is a surface free from cone points. Each connected component P of B 0 =.f jB 0 / acquires its multiplicity as the number of the sheets over it. By formal reasons, we denote the numerical surface B 0 =.f jB 0 / by Ch.B0 / and the quotient by  W B 0 ! Ch.B0 /:

3.4 Re-normalization of a Rotation We will construct a generalized quotient for .D; f jD/. First, let us consider the case in which f permutes the disks of D cyclically. Assume m [ DD ˛ ˛D1

and f .˛ / D ˛C1 ;

for ˛ D 1; 2; : : : ; m  1;

Also assume that f m W 1 ! 1

and f .m / D 1 :

62

3 Generalized Quotient

is a rotation of angle 2 ı=, where ı;  are integers satisfying gcd.ı; / D 1;

0 < ı < :

Then the same rotation appears in the other disks ˛ ;

˛ D 2; : : : ; m:

Let  be defined by ı  1 .mod /; and

0 <  < :

Note that .m; ; / is the valency of the centers of the ˛ ’s. Now let .n0 ; n1 ; : : : ; nl /;

l  1;

be a sequence of positive integers satisfying (˛) n0 D ; n1 D   , (ˇ) n0 > n1 >    > nl D 1, and ( ) ni 1 C ni C1  0 .mod ni /, i D 1; 2; : : : ; l  1. Remark 3.2. 1. Condition . / is empty, if l D 1, i.e., if    D 1. 2. Conditions .ˇ/; . / imply gcd.n0 ; n1 / D gcd.n1 ; n2 / D    D gcd.nl1 ; nl / D 1 3. Since ni 1 > ni , we have .ni 1 C ni C1 /=ni  2; for i D 1; 2; : : : ; l  1. 4. Integers n0 ; n1 ; : : : ; nl satisfying .˛/, .ˇ/, . / exist uniquely by the Euclidean algorithm. Claim (E). Let .n0 ; n1 ; : : : ; nl / be the above integers. Then for each k with 0 < k < l we have l1 X i Dk

where ık satisfies

nk  ık 1 D ni ni C1 nk

gcd.ık ; nk / D 1;

I for some integer ık ;

0 < ı k < nk ;

3.4 Re-normalization of a Rotation

63

and for k D 0 we have l1 X i D0

1 ı D : ni ni C1 

This claim is nothing but Lemma 5.2. (3), which will be proved in Chap. 5. Remark 3.3. Ashikaga and Ishizaka [9] noticed that the formula of Claim (E) can be used to prove the reciprocity law of the Dedekind sum. See also [25, 47]. Consider 1 to be a unit disk in C and divide it by 2l concentric circles

2l1 ; 2l2 ; : : : ; 1 ; 0 of radii 1=2l; 2=2l; : : : ; .2l  1/=2l; 1; into 2l  1 annuli and a disk (NB. 0 D @1 ). Let 01 denote the central disk bounded by 2l1 , and let Ai denote the annulus between i and i 1 . For convenience, Ai is called a “black” annulus if i is odd and a “white” annulus if i is even. We identify the disk ˛ with 1 through f ˛1 W 1 ! ˛ ;

˛ D 2; : : : ; m;

and we will deform the last step f W m ! 1 by a rel:@ isotopy so that, if

f 0 W m ! 1

denotes the resulting homeomorphism, then the composition f 0 f m1 W 1 ! 1 satisfies the following conditions: (i) f 0 f m1 preserves

2l1 ; 2l2 ; : : : ; 1 ; 0 ; (ii) on a white annulus A2k , f 0 f m1 W A2k ! A2k

64

3 Generalized Quotient

is a rotation of angle k D 1; 2; : : : ; l  1;

2 ık =nk ; ık ; nk being as in Claim (E), (iii) on the central disk 01 ,

f 0 f m1 W 01 ! 01 is the identity, and (iv) on a black annulus A2k1 , f 0 f m1 W A2k1 ! A2k1 is a linear twist with screw number k D 1; 2; : : : ; l:

1=nk1 nk ; (Twists occur only in black annuli.)

Example 3.1 (Fig. 3.1). Set  D 10; ı D 3. Then  D 7. n0 D 10; n1 D 10  7 D 3; n2 D 2; n3 D 1; where l D 3. P 0′ A1 A2

P ′1

A3

P ′2

A4 A5

P ′3

P ′4

Δ 1′ P6

P5

P4

P3

P2

P1

P0

Fig. 3.1 f 0 f m1 W 1 ! 1 maps the arc P0 P1 P2 P3 P4 P5 P6 onto the arc P00 P10 P20 P30 P40 P5 P6

3.4 Re-normalization of a Rotation

65

1 1 D ; n2 n3 2

ı2 D 1:

1 1 1 1 2 C D C D ; n1 n2 n2 n3 32 2 3

ı1 D 1

1 1 1 1 1 1 7 C C D C C D ; n0 n1 n1 n2 n2 n3 10  3 32 2 10

ı0 D ı D 3:

In the general case, the picture is essentially the same as in this example. We first define f 0 f m1 on the white annuli (or on 01 ) to be the rotations prescribed by Condition (ii) (or as the identity). Then the screw numbers in the black annuli are automatically adjusted as prescribed by Condition (iv), because of Claim (E). (Of course, we must avoid “artificial” full Dehn twists.) On each black annulus, this twist is rel:@ isotopic to a linear twist by Theorem 2.3. (See also Lemma 2.1.) Thus we get the required f 0 W m ! 1 : Define f0 WD !D by setting f 0 j˛ D f j˛ ;

˛ D 1; 2; : : : ; m  1;

0

0

f jm D the above f W m ! 1 : Then f WD !D is rel:@ isotopic to f 0 W D ! D; and in each ˛ , .f 0 /m W ˛ ! ˛ behaves just as described by conditions (i), (ii), (iii) and (iv). This completes the re-normalization of f W D ! D; and we say that the re-normalization f0 WD !D is in superstandard form.

and

66

3 Generalized Quotient

Fig. 3.2 Chorizo space Ch.D /

Next we will construct a chorizo space Ch.D/ and a pinched covering  W D ! Ch.D/ so that  W D ! Ch.D/ is a generalized quotient of the re-normalized f 0 W D ! D: Take a disk D0 and l spheres S1 ; S 2 ; : : : ; Sl : Let q0 be the center of D0 , and qiC (resp. qi ) the north (resp. the south) pole of Si , i D 1; 2; : : : ; l. Identify q0 with q1C , and qi with qiCC1 for i D 1; 2; : : : ; l  1. Then we have our (not yet numerical) chorizo space Ch.D/. See Fig. 3.2. The pinched covering  W D ! Ch.D/ is constructed as follows. First we will define  W 1 ! Ch.D/: Let Ck . 1 / be the center-line of the black annulus A2k1 .k D 1; 2; : : : ; l/. The complement, A2k1  Ck consists of two half-open annuli, A02k1 and A002k1 , where A02k1 denotes the outer one. Then the pinched covering  W 1 ! Ch.D/ is defined by setting 1. .Ck / D qkC ; 2. the map

k D 1; 2; : : : ; l;

j.A002k1 [ A2k [ A02kC1 / W A002k1 [ A2k [ A02kC1 ! Sk  fqkC ; qk g is an nk -fold cyclic covering whose covering translations are generated by the rotation of angle 2=nk .

3.4 Re-normalization of a Rotation

67

3. the map jA01 W A01 ! D0  fq0 g is an n0 .D /-fold cyclic covering whose covering translations are generated by the rotation of angle 2=n0 . 4. the map j01 [ A002l1 W 01 [ A002l1 ! Sl  fqlC g is a homeomorphism. These conditions determine a pinched covering  W 1 ! Ch.D/: The image .A2k1 / of a black annulus A2k1 is a closed nodal neighborhood N k of qkC consisting of two banks  / .A02k1 [ Ck / .DW Dk1

and .A002k1 [ Ck / .DW DkC /: Before constructing the projection  W ˛ ! Ch.D/ for the other disks ˛ ;

˛ D 2; : : : ; m;

 , for each k D 1; : : : ; l. The we must parameterize the two banks, DkC , Dk1 parametrizations  fzj jzj  1g ! DkC ; Dk1  are chosen so that, if we identify DkC and Dk1 with fzj jzj  1g through these parametrizations,  j1 W A02k1 [ Ck ! Dk1

is described as   p p r b  r exp. 1/ D exp. 1nk1 / ab

68

3 Generalized Quotient

and j1 W A002k1 [ Ck ! DkC is described as   p p br exp. 1nk /;  r exp. 1/ D bc where the annuli A02k1 [ Ck and A002k1 [ Ck are regarded, for simplicity, as A02k1 [ Ck D

n

o p r exp. 1/ j b  r  a; 0    2 ;

n

o p r exp. 1/ j c  r  b; 0    2 :

and A002k1 [ Ck D

 It is easy to verify that with these parametrizations of DkC , Dk1 the following identities hold:  m / .j1 / D .j1 /.f 0 /m .Tk1

on

A02k1 [ Ck

on

A002k1 [ Ck

and .TkC /m .j1 / D .j1 /.f 0 /m where    W Dk1 ! Dk1 Tk1

and TkC W DkC ! DkC are defined by  Tk1 .z/

and TkC .z/ respectively.

! p 1 D z exp .1  jzj/ ; mnk ! p 1 D z exp .1  jzj/ ; mnk1

3.4 Re-normalization of a Rotation

69

Now we are in a position to construct  W ˛ ! Ch.D/; The definition is as follows: 8  ˛1 ˛1 1 ˆ <.Tk1 / .j1 /Œ.f /  j˛ D .TkC /˛1 .j1 /Œ.f /˛1 1 ˆ : ˛1 1 ˛1 .j1 /Œ.f /



on f

for ˛ D 2; : : : ; m:

on f ˛1 .A02k1 [ Ck / .k D 1; 2; :::; l/; on f ˛1 .A002k1 [ Ck / .k D 1; 2; :::; l/; .A2k /

and f ˛1 .01 / .k D 1; 2; :::; l  1/:

Taking the disjoint union, we obtain the pinched covering D

m [

.j˛ / W D ! Ch.D/:

˛D1

Let Pk denote the closed part Sk  .Int.DkC / [ Int.Dk //: The re-normalization f 0 W D ! D preserves  1 .N k / and  1 .Pk / (condition (i) of generalized quotients). For each Pk , f 0 W  1 .Pk / !  1 .Pk / is a periodic map of order mnk , and  W  1 .Pk / ! Pk is its projection (condition (ii)). For each node qkC , the number of annuli in  1 .N k / is m, and f 0 W  1 .N k / !  1 .N k / cyclically permutes the components (condition (iii)). f 0 W  1 .N k / !  1 .N k / is a linear twist in each annulus, and the screw number is 1=nk1 nk (condition (iv)). Finally condition (v) is easily verified by the construction above. Thus attaching the multiplicities mn0 ; mn1 ; : : : ; mnl .D m/ to the components D0 ; S1 ; : : : ; Sl ; we can make Ch.D/ a numerical chorizo, and  W D ! Ch.D/ a generalized quotient of f 0 .

70

3 Generalized Quotient

The above argument can be summarized as follows: Lemma 3.2. Let D be a disjoint union of m disks,  1 ;  2 ; : : : ; m : Suppose that a homeomorphism f W D ! D permutes these disks cyclically, and that for each ˛ .D 1; 2; : : : ; m/, f m W  ˛ ! ˛ is a rotation of angle 2ı=, where ı;  are integers satisfying gcd.ı; / D 1;

0 < ı < :

Let  be an integer such that ı  1 .mod / and 0< < (meaning that the centers of the ˛ ’s have the valency .m; ; /). Let l  1;

.n0 ; n1 ; : : : ; nl /;

be the sequence of positive integers determined by n0 D ; n1 D   ; n0 > n1 >    > nl D 1 and ni 1 C ni C1  0 .mod ni /;

i D 1; 2; : : : ; l  1:

Let Ch.D/ be the chorizo space as shown in Fig. 3.2, and attach the multiplicities mn0 ; mn1 ; : : : ; mnl .D m/; to the components D0 ; S 1 ; : : : ; S l : Then there exist a pinched covering  W D ! Ch.D/ and a rel:@ isotopy from f WD !D

3.5 Re-normalization of a Linear Twist

71

to f0 WD !D such that  W D ! Ch.D/ is a generalized quotient of f 0 . Remark 3.4. 1. By Remark 3.2, 3) before Claim (E), the sphere components S 1 ; S2 ; : : : ; Sl of Ch.D/ satisfy the minimality condition. 2. The passage from the cone D=f to the chorizo space Ch.D/ corresponds to blowing up the cone point (see [33] for a description of the blowing up process). In the general case when f W D ! D does not necessarily permute the disks cyclically, decompose the disks into cyclic orbits : D D D .1/ [ D .2/ [ : : : [ D .s/ ; and define  W D ! Ch.D/ to be the disjoint union

s [

D . / !

D1

s [

Ch.D . / /:

D1

3.5 Re-normalization of a Linear Twist Let A D

m [

A˛

˛D1

be a disjoint union of annuli. Suppose that a homeomorphism f WA !A cyclically permutes the components, and that f m W A˛ ! A˛ is a linear twist (of negative screw number) for each ˛ D 1; 2; : : : ; m.

72

3 Generalized Quotient

We will construct a chorizo space Ch.A / and a pinched covering  W A ! CH.A /; and will show that f WA !A is rel:@ isotopic to a homeomorphism f0 WA !A such that  W A ! Ch.A / is a generalized quotient of f 0 . We may assume f .A˛ / D A˛C1 ;

˛ D 1 ; : : : ; m  1;

and f .Am / D A1 . Let us identify A1 with Œ0; 1  S 1 (as we did in Chap. 2), S 1 being parametrized as R=Z. Let .m; 0 ; 0 / and .m; 1 ; 1 / be the valencies of the boundary curves @0 A1 D f0g  S 1 and @1 A1 D f1g  S 1 with respect to f WA !A: We introduce integers ı0 ; ı1 as usual: 0 ı0  1 .mod 0 /;

0  ı0 < 0 ;

1 ı1  1 .mod 1 /;

0  ı1 < 1 I

(ı0 D 0 iff 0 D 1. Similarly for ı1 ). Finally let s be the screw number of

3.5 Re-normalization of a Linear Twist

73

f WA !A in A˛ ;

˛ D 1; : : : ; m;

which is independent of ˛. Note that s < 0 by our assumption. By Corollary 2.1, jsj C ı0 =0 C ı1 =1 is an integer. Claim (F). 1. There exists uniquely a sequence of positive integers .n0 ; n1 ; : : : ; nl /;

l  1;

satisfying the following conditions .˛/ ."/: (˛) (ˇ) ( ) (ı) (")

n0 D 0 , nl D 1 , n1  0 .mod 0 /, nl1  1 .mod 1 /, ni 1 C ni C1  0 .mod ni /, i D 1; 2; : : : ; l  1, .ni 1 C ni C1 /=ni  2, i D 1; 2; : : : ; l  1, and P l1 i D0 1=.ni ni C1 / D jsj.

2. The above sequence .n0 ; n1 ; : : : ; nl / has additional properties: ( ) for each k with 0 < k < l, we have X 1 ı0 dk C D 0 n n nk i D0 i i C1 k1



for some integer dk coprime with nk , and () X ı1 ı0 1 C  .mod 1/: 0 n n  1 i D0 i i C1 l1



Claim (F), (1) is nothing but Theorem 5.1, which will be proved in Chap. 5. Claim (F), (2) ( ) is Lemma 5.2 (1). The last property, (), follows from (") and the fact that jsj C ı0 =0 C ı1 =1 is an integer. Note that gcd.ni ; ni C1 / D 1 for i D 0; : : : ; l  1 because of (˛), (ˇ) and ( ). Let us proceed taking the above claim for granted.

74

3 Generalized Quotient

Divide A1 D Œ0; 1  S 1 by 2.l  1/ circles

i D fi=.2l  1/g  S 1 ;

i D 1; 2; : : : ; 2l  2

into 2l  1 annuli. We set

0 D @0 A1 D f0g  S 1 and

2l1 D @1 A1 D f1g  S 1 : Let Zi denote the annulus between i 1 and i . As in the case of a rotation, we call Zi a “black” annulus if i is odd and a “white” annulus if i is even. We identify the ˛-th annulus A˛ with A1 through f ˛1 W A1 ! A˛ ;

˛ D 2; : : : ; m;

and will deform the last step f W A m ! A1 by a rel.@ isotopy so that the resulting f 0 W Am ! A1 satisfies the following conditions: (i) The map f 0 f m1 W A1 ! A1 preserves

0 ; 1 ; : : : ; 2l1 ; (ii) on a white annulus Z2k , f 0 f m1 W Z2k ! Z2k is a “rotation” sending .t; x/ to .t; x C dk =nk /; dk ; nk being as in Claim (F), . /,

k D 1; 2; : : : ; l  1;

3.5 Re-normalization of a Linear Twist

75

(iii) on a black annulus Z2k1 , f 0 f m1 W Z2k1 ! Z2k1 is a linear twist with screw number 1=.nk1 nk /; and (iv) on 0 D @0 A1 ,

k D 1; : : : ; l;

f 0 f m1 D f m

is a rotation sending .0; x/ to

.0; x  ı0 =0 /;

and on 1 D @1 A1 , f 0 f m1 D f m is a rotation sending .1; x/ to .1; x C ı1 =1 /: The last condition (iv) is nothing but the definition of the valencies .m; 0 ; 0 / and .m; 1 ; 1 /: (Remember the orientation conventions in Chap. 2. See also Corollary 2.2.) We add (iv) just as a remark. Example 3.2 (Fig. 3.3). Set 0 D 5;

0 D 2;

1 D 4;

ı0 D 3;

ı1 D 3:

1 D 3:

Then We assume s D 7=20: (NB. jsj C ı0 =0 C ı1 =1 D 7=20 C 3=5 C 3=4 D 1/: Then .n0 ; n1 ; n2 ; n3 / D .5; 2; 3; 4/

76

3 Generalized Quotient P5

P ′5 1 –– 4

Z5

P ′4

Z4

P 3′

P3 1 –– 3

Z3

P ′2

Z2 t

P4

P2

P ′1

P1 1 –– 2

Z1

P0

P ′0 0

3 –– 5

x

Fig. 3.3 f 0 f m1 W A1 ! A1 maps the arc P0 P1 P2 P3 P4 P5 onto the arc P00 P10 P20 P30 P40 P50

and l D 3: (See the proof of Theorem 5.1 in which there is an algorithm to construct these numbers.)   

ı0 1 3 1 1 C D C D ; 0 n0 n1 5 52 2

d1 D 1

1 1 3 1 1 1 ı0 C C D C C D ; 0 n0 n1 n1 n2 5 52 23 3

d2 D 1

1 1 1 1 ı0 1 1 1 3 C C D ; C C C D C 0 n0 n1 n1 n2 n2 n3 5 52 23 34 4   ı1  .mod 1/ d3 D 1: 1

In the general case, we have essentially the same picture. To get f 0 W Am ! A1 ; we first define

f 0 f m1

on the white annuli to be the prescribed rotations (Condition (ii)). Then the screw numbers in the black annuli are automatically adjusted as prescribed by Condition

3.5 Re-normalization of a Linear Twist

77

(iii), thanks to Claim (F). (Again we must avoid artificial full twists.) On each black annulus this twist is rel.@ isotopic to a linear twist (Theorem 2.3 and Lemma 2.1). Thus we have obtained the desired f 0 W Am ! A1 : Define f0 WA !A by setting f 0 jA˛ D f jA˛ ;

˛ D 1; 2; : : : ; m  1;

0

f jAm D the above f 0 : Then f WA !A is rel:@ isotopic to f0 WA !A and in each A˛ , .f 0 /m W A˛ ! A˛ behaves just as described by Conditions (i), (ii), (iii) and (iv). This completes the re-normalization of f WA !A; and we say that the re-normalized f0 WA !A is in superstandard form. We will next construct a chorizo space Ch.A / and a pinched covering  W A ! Ch.A / so that  W A ! Ch.A / is a generalized quotient of the renormalized f 0 . Take two disks D0 ; Dl and l  1 spheres S1 ; S2 ; : : : ; Sl1 : Let q0 and q1 be the centers of D0 and Dl , respectively, and let qiC and qi be the north and the south poles of Si as before. The chorizo space Ch.A / is constructed as shown in Fig. 3.4 (though not yet numerical).

78

3 Generalized Quotient D0

S1

q0

Sl–1

q +1

q –1

q +2

Dl

q –l–1

ql

Fig. 3.4 Chorizo space Ch.A /

The construction of  W A ! Ch.A / is almost the same as that of  W D ! Ch.D/: First we define  W A1 ! Ch.A /: For this, take the center-line Ck of a black annulus Z2k1 ;

k D 1; 2; : : : ; l

and define .Ck / to be the node qkC 2 Ch.A /: The complement Z2k1  Ck consists of two half-open annuli 0 ; Z2k1

00 Z2k1

0 (Z2k1 denoting the lower one with respect to the t-level). Define 00 0 00 0 jZ2k1 [ Z2k [ Z2kC1 W Z2k1 [ Z2k [ Z2kC1 ! Sk  fqk ; qkC g

to be the nk -fold cyclic covering whose covering translations are generated by the rotation .t; x/ ! .t; x C 1=nk / (in the .t; x/-picture), k D 1; : : : ; l  1. Finally, define jZ10 W Z10 ! D0  fq0 g and 00 00 W Z2l1 ! Dl  fql g jZ2l1

to be the obvious n0 .D 0 /-fold and nl .D 1 /-fold cyclic coverings, respectively. The construction of jA˛ W A˛ ! Ch.A /

3.5 Re-normalization of a Linear Twist

79

for the other annuli A˛

˛ D 2; : : : ; m

proceeds similarly to that of j˛ W ˛ ! Ch.D/ for ˛ D 2; : : : ; m: We repeat it for completeness. First we parametrize the two banks of N k WD .Z2k1 /; 0 00  / and .Z2k1 / be denoted by Dk1 for each k D 1; : : : ; l. Let the banks .Z2k1 C  and Dk , respectively. We parametrize them so that if we identify Dk1 and DkC with

fz j jzj  1g through the parametrizations, 0  jA1 W Z2k1 [ Ck ! Dk1

is described as .t; x/ D

p bt exp. 1 2 nk1 x/ ba

and 00 jA1 W Z2k1 [ Ck ! DkC

is described as .t; x/ D

p t b exp. 1 2 nk x/; cb

where the annuli 0 [ Ck Z2k1

and 00 [ Ck Z2k1

are regarded, for simplicity, as 0 Z2k1 [ Ck D f.t; x/j a  t  b; x 2 R=Zg ; 00 [ Ck D f.t; x/j b  t  c; x 2 R=Zg : Z2k1  Note that with these parametrizations of Dk1 ; DkC , the following identities hold:  m / .jA1 / D .jA1 /.f 0 /m .Tk1

0 on Z2k1 [ Ck

80

3 Generalized Quotient

and .TkC /m .jA1 / D .jA1 /.f 0 /m

00 on Z2k1 [ Ck ;

where    Tk1 W Dk1 ! Dk1

and TkC W DkC ! DkC are defined by ! p 1  .z/ D z exp .1  jzj/ Tk1 m nk and ! 1 D z exp .1  jzj/ ; m nk1 p

TkC .z/

respectively, just as in the case of Ch.D/. The construction of jA˛ W A˛ ! Ch.A / for ˛ D 2; : : : ; m is as follows: 8  ˛1 ˛1 1 ˆ / ˆ <.Tk1 / .jA1 /.f C ˛1 ˛1 1 jA˛ D .Tk / .jA1 /.f / ˆ ˆ :.jA /.f ˛1 /1 1

0 on f ˛1 .Z2k1 [ Ck /; k D 1;    ; l; 00 on f ˛1 .Z2k1 [ Ck /; k D 1;    ; l;

on f ˛1 .Z2k /;

k D 1;    ; l:

Taking the disjoint union, we get the pinched covering D

m [

.jA˛ / W A ! Ch.A /:

˛D1

The images N k WD .Z2k1 /; k D 1; : : : ; l make the system of closed nodal neighborhoods of q1C ; : : : ; qlC which should be fixed in the definition of a generalized quotient. To make Ch.A / numerical, attach

3.5 Re-normalization of a Linear Twist

81

the multiplicities m n0 ; m n1 ; : : : ; m nl to the components D0 ; S1 ; : : : ; Sl1 ; Dl : With the system fNk glkD1 and the multiplicities, one can easily check Conditions (i) (v) (of Def. 3.4) to be satisfied by a generalized quotient of f0 WA !A: We summarize the above argument. Lemma 3.3. Let A be a disjoint union of m annuli, A1 ; A2 ; : : : ; Am : Suppose that a homeomorphism f WA !A permutes these annuli cyclically, and that for each ˛ .D 1; 2; : : : ; m/, f m W A˛ ! A˛ is a linear twist of negative screw number s. Let .m; 0 ; 0 / and .m; 1 ; 1 / be the valencies of the two boundary curves of A˛ (which are independent of ˛). Let .n0 ; n1 ; : : : ; nl /; l  1; be the sequence of positive integers which are determined by n0 D 0 ; nl D 1 ; n1  0 .mod 0 /; nl1  1 .mod 1 /; ni 1 C ni C1  0 .mod ni /; i D 1; 2; : : : ; l  1; .ni 1 C ni C1 /  2; i D 1; 2; : : : l  1; ni and

l1 X i D0

1 D jsj: ni ni C1

(For the existence and the uniqueness of such a sequence, see Theorem 5.1.)

82

3 Generalized Quotient

Let Ch.A / be the chorizo space as shown in Fig. 3.4, and attach the multiplicities m n0 ; m n1 ; : : : ; m nl ; to the components D0 ; S1 ; : : : ; Sl1 ; Dl : Then there exist a pinched covering  W A ! Ch.A / and a rel:@ isotopy from f WA !A to f0 WA !A such that  W A ! Ch.A / is a generalized quotient of f 0 . Remark 3.5. By the property of .n0 ; n1 ; : : : ; nl /; the sphere components S1 ; S2 ; : : : ; Sl1 in Ch.A / satisfy the minimality condition. In the general case when f WA !A does not necessarily permute the annuli cyclically, decompose the annuli into cyclic orbits: A ! A .1/ [ A .2/ [ : : : [ A .s/ ; and define  W A ! Ch.A / to be the disjoint union s [ D1

A . / !

s [ D1

Ch.A . / /:

3.6 Re-normalization of a Special Twist

83

3.6 Re-normalization of a Special Twist Let A D

m [

A˛

˛D1

be a disjoint union of annuli. Suppose that a homeomorphism f WA !A cyclically permutes the components, and that for each ˛ D 1; 2; : : : ; m, f m W A˛ ! A˛ is a special twist (in the sense of Chap. 2) of negative screw number. By the definition of a special twist, every annulus in A is amphidrome. To distinguish the present case from the previous (linear) case, we will denote the disjoint union of annuli by Asp in the case of specials twist, and by Aln in the case of linear twists. We may assume f .A˛ / D A˛C1 ;

˛ D 1; : : : ; m  1;

f .Am / D A1

as before. We will construct a chorizo space Ch.Asp / and a pinched covering  W Asp ! Ch.Asp /; and will show that f W Asp ! Asp is rel:@ isotopic to a homeomorphism f 0 W Asp ! Asp such that  W Asp ! Ch.Asp / is a generalized quotient of f 0 . This, however, has been essentially done by the compound nature of a special twist. Let us identify A1 with Œ0; 1  S 1 ; where S 1 is R=Z. Let s.<0/ be the screw number of f in A1 . Then by Corollary 2.5, we have that

84

3 Generalized Quotient

8   3 1 ˆ ˆ < 1  t; x C 4 s t  3 f m .t; x/ D .1  t; x/ ˆ ˆ :1  t; x C 3 s t  2  4

3

for 0  t  13 ; for for

1 3 2 3

 t  23 ;  t  1:

We decompose A1 D Œ0; 1  S 1 into three parts:  1 0;  S 1; 3 Let A01 denote



1 2  S 1; ; 3 3



2 ; 1  S 1: 3

 1  S 1  A1 ; 0; 3

and for each ˛ D 1; 2; : : : ; 2m; set A0˛ D f ˛1 .A01 /: Clearly A0mC1 D



2 ; 1  S 1  A1 ; 3

and more generally A0˛ [ A0˛Cm  A˛ ;

˛ D 1; 2; : : : ; m:

Let Aln0 denote the disjoint union 2m [

A0˛ :

˛D1

If f W Asp ! Asp is restricted to f W Aln0 ! Aln0 ; it cyclically permutes the 2m annuli. Claim (G). The map f 2m W A01 ! A01 is a linear twist with screw number 1=2s.

3.6 Re-normalization of a Special Twist

85

Proof. Using the above expression of f m , we compute f 2m to get 

f

2m

 3 1 .t; x/ D t; x  st C s ; 2 2

Since A01 is parametrized as

0t 

1 : 3

 1  S 1; 0; 3

the screw number is the coefficient of 3t, which is 1=2s. (Cf. Corollary 2.2).

t u

Let .2m; 0 ; 0 / be the valency of the “outer” boundary @0 A01 D f0g  S 1 with respect to f W Aln0 ! Aln0 : Let ı0 be the integer determined by 0 ı0  1 .mod 0 /;

0  ı0  0 :

(NB. ı0 D 0 iff 0 D 1). Then 1 ı0 =0   s .mod 1/; 2 see Corollary 2.3. The valency of the “inner” boundary @1=3 A01 D f1=3g  S 1 is equal to .2m; 1; 0/: (This is geometrically obvious, but also can be seen from the expression of f 2m .t; x/ in the proof of Claim (G).) Applying Claim (F) (1), we obtain a unique .n0 ; n1 ; : : : ; nl / of positive integers such that (˛ 0 ) (ˇ 0 ) ( 0 ) (ı 0 ) ("0 )

n0 D 0 , nl D 1, n1  0 .mod 0 /, ni 1 C ni C1  0 .mod ni /; i D 1; 2; : : : ; l  1, .n i D 1; 2; : : : ; l  1; and 1 C ni C1 /=ni  2; Pil1 i D0 1=ni ni C1 D 1=2jsj.

In fact, in the present case, the integers .n0 ; n1 ; : : : ; nl / can be determined more easily.

86

3 Generalized Quotient

Case 1. ı0 D 0; then 0 D 1, and 1 jsj 2 is an integer k.> 0/ because 1 ı0   s .mod 1/: 0 2 The sequence is nothing but a sequence of k C 1 copies of 1’s: .1; 1; : : : ; 1/: „ ƒ‚ … kC1

Case 2. ı0 > 0; determine .n0 ; n1 ; : : : ; nk / using the Euclidean algorithm, so that n0 D ; n1 D ;

n0 > n1 >    > nk D 1;

ni 1 C ni C1  0 .mod ni /;

i D 1; : : : ; k  1:

Then by Lemma 5.2(3) (in Chap. 5), we have k1 X i D0

1 ı0 D : ni ni C1 0

Since ı0 1 D an integer (say h)  0; jsj  2 0 we can add h copies of 1’s to obtain .n0 ; n1 ; : : : ; nl / D .n0 ; : : : ; nk ; 1; 1; : : : ; 1/: „ ƒ‚ … h

In this case l D k C h. Remark 3.6. In both cases, .n0 ; n1 ; : : : ; nl / is in the form n0 > n1 >    > nk D nkC1 D    D nl D 1; for some k with 0  k  l.

3.6 Re-normalization of a Special Twist

87

D0

S1

Sl –1

Dl

2mn0

2mn1

2mnl –1

2m

Fig. 3.5 Chorizo space Ch.Aln0 /

By Lemma 3.3, we have a generalized quotient  W Aln0 ! Ch.Aln0 / of the re-normalized f 0 W Aln0 ! Aln0 : The shape of Ch.Aln0 / is shown in Fig. 3.5. We have next to consider the middle annulus  1 2  S 1  A1 : ; B10 D 3 3 For ˛ D 1; : : : ; m; we set B˛0 D f ˛1 .B10 /; and let B0 denote the disjoint union m [

B˛0 :

˛D1

The m-th iteration of f restricted to B˛0 gives a “180ı-rotation” f m W B˛0 ! B˛0 around an axis. See Fig. 3.6. Let p1˛ ; p2˛ be the fixed points of f m W B˛0 ! B˛0 and let D ˛ ;

D 1; 2

88

3 Generalized Quotient

Fig. 3.6 f m W B˛0 ! B˛0 is a “180ı -rotation”

180° P a1 B ′a P a2

ch(

″) =

2m

valency (m,2,1)

valency (2m,1,0)

Fig. 3.7 Chorizo space Ch.B 00 /

be a small invariant disk neighborhood of p ˛ . We denote the disjoint union m [

D ˛

˛D1

by D . The complement B00 D B0  .Int.D1 / [ Int.D2 // has no multiple point, and f acts on B 00 as a periodic map of order 2m. We regard the quotient B 00 =f as a chorizo space (without nodes) and denote it by Ch.B00 /, whose shape is shown in Fig. 3.7. As for the union of m disks D , Lemma 3.2 can be applied to produce a chorizo space Ch.D /. See Fig. 3.8. To obtain a generalized quotient corresponding to f W Asp ! Asp ; we have only to paste together the chorizo spaces Ch.Aln0 /; Ch.B 00 /; Ch.D /;

D 1; 2;

along their boundaries. The multiplicities match up naturally. The shape of Ch.Asp / is shown in Fig. 3.9. It consists of a disks and l C 2 spheres.

3.6 Re-normalization of a Special Twist

89

Fig. 3.8 Chorizo space Ch.D /; D 1; 2

D ′n

S n′

2m

m

Fig. 3.9 Chorizo space Ch.Asp /

The pinched covering  W Asp ! Ch.Asp / is the union of  W Aln0 ! Ch.Aln0 /;  W B 00 ! Ch.B 00 / and  W D ! Ch.D /;

D 1; 2:

The homeomorphism f W Asp ! Asp can be re-normalized over Ch.Aln0 /, and Ch.D /; D 1; 2; without being changed over Ch.B00 /. We summarize the above argument. Lemma 3.4. Let Asp be a disjoint union of m annuli, A1 ; A2 ; : : : ; Am : Suppose that a homeomorphism f W Asp ! Asp

90

3 Generalized Quotient

permutes these annuli cyclically, and that for each ˛ D 1; 2; : : : ; m; the map f m W A˛ ! A˛ is a special twist of negative screw number s. Let .2m; ; / be the valency common to the two boundary curves of A˛ (which is independent of ˛). Let .n0 ; n1 ; : : : ; nl /; l  1; be the sequence of positive integers which are uniquely determined by n0 D ;

n1   .mod /;

n0 > n1 >    > nk D nkC1 D    D nl D 1; ni 1 C ni C1  0 .mod ni /; k1 X i D0

for some k with 0  k  l;

i D 1; 2; : : : ; l  1;

and

1 1 D jsj: ni ni C1 2

Let Ch.Asp / be the chorizo space as shown in Fig. 3.9, and attach the multiplicities 2mn0 ; 2mn1 ; : : : ; 2mnl . D 2m/; m; m; to the components D0 ; S1 ; : : : ; Sl ; S10 ; S20 : Then there exist a pinched covering  W Asp ! Ch.Asp / and a rel.@ isotopy from f W Asp ! Asp to f 0 W Asp ! Asp such that  W Asp ! Ch.Asp / is a generalized quotient of f 0 . Note that the sphere components of Ch.Asp / satisfy the minimality condition.

3.7 Completion of Theorem 3.1. (Existence)

91

In the general case when f W Asp ! Asp does not necessarily permute the annuli cyclically, decompose the annuli into cyclic orbits as before: .s/ Asp D A.1/ sp [    [ Asp ;

and define  W Asp ! Ch.Asp / to be the disjoint union

s [ D1

Asp. / !

s [

Ch.Asp. / /:

D1

3.7 Completion of Theorem 3.1. (Existence) Remember that we had a pseudo-periodic map f W ˙g ! ˙g (of negative twist) in standard form and we tried to isotop f in order that the resulting f 0 would have its minimal quotient. At the beginning of the proof, we took a system of annular neighborhoods A of a precise cut system, on which linear twists or special twists, or both, take place. Outside A , f is a periodic map. Let B denote the outside: B D ˙g  Int.A /: We also took a disjoint union D of invariant disk neighborhoods of all the multiple points of f jB W B ! B; and set B 0 D B  Int.D/: We further classified the annuli in A into Asp and Aln according to their character of being amphidrome or non-amphidrome. Now, on all these spaces, B 0 ; D; Aln ; and Asp ; f is rel:@ isotopic to f 0 in superstandard form (f 0 jB 0 is the same as f jB 0 ), and f 0 , when restricted to one of these spaces, has a generalized quotient  W B 0 ! Ch.B 0 /;

92

3 Generalized Quotient

 W D ! Ch.D/;  W Aln ! Ch.Aln /; or  W Asp ! Ch.Asp /: We paste together these chorizo spaces along their boundaries. The multiplicities match up, and we obtain a chorizo space S D Ch.B 0 / [ Ch.D/ [ Ch.Aln / [ Ch.Asp / and a pinched covering  W ˙g ! S , which is a generalized quotient of f 0 . It remains to show that the sphere components in S satisfy the minimality condition. We have already checked it for spheres in Ch.D/; Ch.Aln /;

and Ch.Asp /;

so let  be a sphere component (without self-intersection) which is not contained in Ch.D/ [ Ch.Aln / [ Ch.Asp /: Then the closed part P in  is a connected component of Ch.B0 /. This means that each connected component of  1 .P / is a component of B 0 which has negative Euler characteristic because B0 D B  Int.D/ and B is the complement of the annular neighborhoods A of a precise cut system. Therefore .P / < 0; which implies that the sphere component  intersects the other irreducible components in more than two points. Thus the minimality condition is trivially satisfied by . This completes the proof of Theorem 3.1. t u

Chapter 4

Uniqueness of Minimal Quotient

4.1 Main Theorem of Chap. 4 Definition 4.1. A homeomorphism H W S ! S0 between chorizo spaces is numerical if it preserves the orientation and the multiplicity on each part. In this chapter, we will prove the following: Theorem 4.1 (Uniqueness). Let f and f 0 W ˙g ! ˙g be pseudo-periodic maps of negative twist. Suppose there are minimal quotients  W ˙g ! S and

 0 W ˙g ! S 0

of f and f 0 respectively (thus f and f 0 are in superstandard form). If f and f 0 are mutually homotopic, then there exist a homeomorphism h W ˙g ! ˙g and a numerical homeomorphism H W S ! S0 such that

Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5 4, © Springer-Verlag Berlin Heidelberg 2011

93

94

4 Uniqueness of Minimal Quotient

(i) h is isotopic to the identity, (ii) f D h1 f 0 h, and (iii)  0 h D H  W h

˙g ! ? ? y

˙g ? ? 0 y

S ! S 0 H

We may assume that one of these quotients, say,  0 W ˙g ! S 0 is constructed by the method of Chap. 3 as the union of C h.B0 /; C h.D/; C h.Aln / and C h.Asp /: We will study the structure of the other one  W ˙g ! S . Let us start with a set of general definitions. Definition 4.2. An irreducible component  of a chorizo space S is said to be trivial if  is a sphere without self-intersection and intersects the union of the other irreducible components in one or two points. Otherwise  is said to be essential. Definition 4.3. Let  be a trivial component. Assume there exist a sequence  D 0 ; 1 ; : : : ; m of trivial components such that i intersects i C1 ; i D 0; : : : ; m  1, and the last m intersects only m1 in one point exactly. Then  is called a trivial component of type I (of course, all members in the sequence are trivial components of type I, see Fig. 4.1). A trivial component which does not touch any trivial component of type I is said to be of type II. Definition 4.4. Let T1 (resp. T2 ) be the subspace of S which is the union of all trivial components of type I (resp. type II). A connected component of T1 (resp. T2 ) will be called a trivial array of type I (resp. type II). Hereafter, a system of closed nodal neighborhoods fN p gp D node will be fixed in S (as in the case of the “base chorizo” of a generalized quotient. See Remark 1) before Lemma 3.1).

4.1 Main Theorem of Chap. 4

95

Fig. 4.1 Decomposition of a chorizo space

As before, from now on, when we talk about a closed nodal neighborhood N p and a closed part P , we always will be talking about a member of the fixed system fN p gp D node and a connected component of S

[

Np ;

p D node

respectively, where Np D Int.N p /:

96

4 Uniqueness of Minimal Quotient

Definition 4.5. Any union of closed nodal neighborhoods and closed parts is called a subchorizo space . Definition 4.6. A tail of S is a minimal subchorizo space containing a given trivial array of type I (see Fig. 4.1). The number of the trivial components contained in the array is the length of the tail, which is always  1. Definition 4.7. An arch of S is a minimal subchorizo space containing a given trivial array of type II. The number of the trivial components contained in the array is the length of the arch, which is  1. In general, there still remain some nodes which are not in tails nor in arches. The closed nodal neighborhood of such a node is called an arch of length 0. (Such a node is either a self- intersection point of an essential component or an intersection point between two essential components.) Let ARCH D

s [

ARCH

D1

denote the union of all arches (of length  0) in S . Definition 4.8. A connected component of S  Int.ARCH/ is called a body. Here is an elementary lemma. Lemma 4.1. A body contains one and only one closed part with negative Euler characteristic. Proof. Note that there is a bijective correspondence between closed parts and irreducible components: P $  (such that P   ), and that a closed part P has negative Euler characteristic if and only if the corresponding irreducible component  is an essential component. (A “pinched torus” is the only example of an essential component that contains a closed part of Euler characteristic 0. But this does not appear in the “base chorizo space” of a surface ˙g with genus g  2.) Suppose a body BDY ( being a suffix) contains a closed part P whose Euler characteristic is  0. Then P belongs to a trivial component of type I (because trivial components of type II have been deleted), thus it belongs to a tail. But the tail is attached to an essential component  ( being a suffix), and the closed part P of  belongs to the BDY . This proves that BDY  contains a closed part P with .P / < 0. Since two essential components are joined only by arches, two closed parts in different essential components cannot belong to the same connected component of S  int.ARCH /. Thus BDY  contains only one closed part with negative Euler characteristic. t u

4.2 Structure of  1 .arch/

97

Definition 4.9. The closed part with negative Euler characteristic is called the core part of the body. Remark 4.1. A body is a subchorizo space which is a union of its core part P and the tails that are attached to P . Figure 4.1 shows an example of the decomposition of a chorizo space. Now we return to the generalized quotient  W ˙g ! S of Theorem 4.1. Remember that it is a minimal generalized quotient of a pseudoperiodic map f W ˙g ! ˙g of negative twist. Let us examine the structure of  1 .arch/;  1 .tail/; and  1 .body/:

4.2 Structure of  1 .arch/ Let ARCH be an arch. To make the notation consistent with that of Lemma 3.3, let l 1 denote the length of the arch. ARCH contains two attaching banks D0 , Dl . If the length is  1 (i.e. if l  2) it also contains l  1 spheres S1 ; : : : ; Sl1 which are trivial components of type II. Suppose D0 \ S1 D fq1 g; Si 1 \ Si D fqi g;

i D 2; : : : ; l  1

and Sl1 \ Dl D fql g. Let m0 ; ml .or mi / denote the multiplicities of D0 , Dl (or Si ), i D 1; : : : ; l  1. By Proposition 3.1, mi 1 C mi C1  0 .mod mi /;

i D 1; : : : ; l  1:

Thus gcd.mi 1 ; mi / is independent of i.D 1; : : : ; l/, which will be denoted by m. Let N i be the closed nodal neighborhood of qi , i D 1; : : : ; l. Then by the definition of a generalized quotient (Chap. 3),  1 .N i /

98

4 Uniqueness of Minimal Quotient

consists of m annuli which are permuted by f W  1 .N i / !  1 .N i / cyclically. The closed part Pi contained in the sphere i D 1; : : : ; l  1;

Si ; is an annulus. Thus

 1 .Pi /

is also a disjoint union of annuli, and the number of the annuli in  1 .Pi / is equal to m. These “narrow” annuli in  1 .N i /; and those in

 1 .Pi /;

i D 1; : : : ; l; i D 1; : : : ; l  1;

are pasted together along their boundaries to make a disjoint union of m, “long” annuli, A1 ; A2 ; : : : ; Am : Thus  1 .ARCH / D

m [

A˛ :

˛D1

The map

f W  1 .ARCH / !  1 .ARCH /

permutes A1 ; A2 ; : : : ; Am cyclically. Divide the unit interval Œ0; 1 into 2l  1 small intervals I1 ; I2 ; : : : ; I2l1 ; 

where Ij D

 j j 1 : ; 2l  1 2l  1

Lemma 4.2. There are parametrizations ˛ W Œ0; 1  S 1 ! A˛ .S 1 D R=Z/;

˛ D 1; : : : ; m;

4.2 Structure of  1 .arch/

99

such that (i) (ii)

 1 .N i / D  1 .Pi / D

m [

˛ .I2i 1  S 1 /;

˛D1 m [

˛ .I2i  S 1 /;

i D 1; : : : ; l; i D 1 ; : : : ; l  1;

˛D1

(iii) the map ˛1 f m ˛ W I2i 1  S 1 ! I2i 1  S 1 ; . for each ˛ D 1; : : : ; m/ is a linear twist with screw number  where ni 1 D

mi 1 ; m

1 ; ni 1 ni

ni D

mi ; m

i D 1; : : : ; l;

and (iv) the map ˛1 f m ˛ W I2i  S 1 ! I2i  S 1 . for each ˛ D 1; : : : ; m/ is a rotation sending .t; x/ to

  di ; t; x C ni

where di is an integer coprime with ni , i D 1; : : : ; l  1. (v) Let .m; 0 ; 0 / and .m; 1 ; 1 / be the valencies of @0 A˛ D ˛ .f0g  S 1 / and @1 A˛ D ˛ .f1g  S 1 /; respectively. Define the integers ı0 , ı1 as usual: 0  ıi < i ; i ıi  1 .mod i /; Then

i D 0; 1:

  ı0 f ˛ .0; x/ D ˛ 0; x  0 m

100

4 Uniqueness of Minimal Quotient

  ı1 : f ˛ .1; x/ D ˛ 1; x C 1

and

m

Proof. First we fix our attention at the “first annulus” A1 . Set Y2i 1 D  1 .N i / \ A1 (a “black” annulus) and

Y2i D  1 .Pi / \ A1 (a “white” annulus):

Of course, A1 D

2l1 [

Yj :

j D1

By the definition of a generalized quotient, there is a parametrization of a black annulus 1 2i 1 W I2i 1  S ! Y2i 1 ; for each i D 1; : : : ; l, such that 1 m 2i 1 f

2i 1

W I2i 1  S 1 ! I2i 1  S 1

is a linear twist with screw number 

1 : ni 1 ni

We can assume  2i 1

  2i  2  S 1 D Y2i 2 \ Y2i 1 ; 2l  1

 2i 1

  2i  1 1  S D Y2i 1 \ Y2i : 2l  1

For a white annulus Y2i , we take a preliminary parametrization 0 2i

satisfying and so that

0 2i

W I2i  S 1 ! Y2i ;

0 2i jf.2i 1/=.2l1/gS 1

D

2i 1 jf.2i 1/=.2l1/gS 1

0 2i jf2i=.2l1/gS 1

D

2i C1 jf2i=.2l1/gS 1 ;

is “linear” on the boundary of Y2i . Since  W Y2i ! Pi

4.2 Structure of  1 .arch/

101

is an ni -fold cyclic covering over an annulus Pi , there is an integer di , coprime with ni , such that on @Y2i the following hold: fm

0 2i

fm

0 2i

 

2i  1 ; x 2l  1 2i ; x 2l  1

 D

0 2i

D

0 2i



 

di 2i  1 ; xC 2l  1 ni di 2i ; xC 2l  1 ni

 ;  :

Then the parametrization 0 2i

W I2i  S 1 ! Y2i

can be rectified (by an isotopy rel.@) to a parametrization W I2i  S 1 ! Y2i

2i

satisfying condition (iv) of the lemma. The desired parametrization for A1 ; 1 W Œ0; 1  S 1 ! A1 ; is defined to be the union of the above parametrizations 1

[

[[

2

2l1 :

For the other annuli A˛ , ˛ D 2; : : : ; m, define ˛ to be the f ˛1 1 W Œ0; 1  S 1 ! A˛ : (We are assuming A˛ D f ˛1 .A1 /.) It is easy to check conditions (i) (v) for these parametrizations ˛ ;

˛ D 1; : : : ; m: t u

Definition 4.10. The parametrizations ˛ W Œ0; 1  S 1 ! A˛ ;

˛ D 1; : : : ; m;

are called superstandard parametrizations for f . The homeomorphism f W

m [ ˛D1

A˛ !

m [ ˛D1

A˛

102

4 Uniqueness of Minimal Quotient

is in superstandard form with respect to these parametrizations. Corollary 4.1 (to Lemma 4.2). (i) The equations describing f m W A˛ ! A˛ with respect to the superstandard parametrizations ˛ W Œ0; 1  S 1 ! A˛ ;

˛ D 1; : : : ; m;

are uniquely determined by the series of numbers .n0 ; n1 ; : : : ; nl /: (ii) In particular, the valencies of @0 A˛ and @1 A˛ with respect to f are equal to .m; n0 ; 0 / and .m; nl ; 1 /, respectively, where 0 and 1 are determined by 0  0 < n0 ; 0  n1 .mod n0 /;

0  1 < nl ; 1  nl1 .mod nl /:

The screw number of f in A˛ is equal to 

l1 X

1 : ni ni C1

i D0

(iii) Conversely, the length of the ARCH and the multiplicities m0 ; m1 ; : : : ; m l of D0 ; S1 ; : : : ; Sl1 ; Dl are determined by the valencies .0 ; 0 ; 0 / and .1 ; 1 ; 1 / of @0 A˛ and @1 A˛ , together with the screw number of f in A˛ , ˛ being any of 1; : : : ; m. Proof. (i) The numbers ı0 = 0 and ı1 = 1 which describe the action of f m on @A˛ are determined by . 0 ; 0 / and . 1 ; 1 / but these two pairs are determined by .n0 ; n1 ; : : : ; nl1 ; nl / as 0 D n0 ;

1 D nl ;

0  0 < n0 ;

0  n1 .mod n0 /;

0  1 < nl ;

1  nl1 .mod nl /:

4.3 Structure of  1 .tail/

103

See Lemma 3.1. The numbers di ; i D 1; : : : ; l  1, which describe the action of f m on the white annuli Y2i .i D 1; : : : ; l  1/ might look uncontrolable. But one can find them by accumulating the twists 1=.ni 1 ni / starting from level t D 0: k X 1 ı0 dk  C D ; k D 1; : : : ; l  1: 0 ni 1 ni nk i D1 (Remember our sign convention for twists. Cf. Corollary 2.2. See also Fig. 3.3.) (ii) follows immediately from (i). (iii) The gcd.mi 1 ; mi / D m is determined by the valency of @0 A˛ (or @1 A˛ /I m D 0 D 1 . The numbers .n0 ; n1 ; : : : ; nl / satisfy n0 D 0 ; nl D 1 ;

n1  0 .mod 0 /;

nl1  1 .mod 1 /

(by (ii)), ni 1 C ni C1  0 .mod ni /;

i D 1; : : : ; l  1

(by Proposition 3.1), .ni 1 C ni C1 /=ni  2;

i D 1; : : : ; l  1

(by the minimality assumption on S ), and l1 X i D0

1 D js.A˛ /j ni ni C1

by (ii). Therefore, the sequence .n0 ; n1 ; : : : ; nl / is uniquely determined by . 0 ; 0 /, . 1 ; 1 / and s.A˛ /. See Theorem 5.1 This proves (iii). t u

4.3 Structure of  1 .tail/ Let TL , be a tail (of length l  1). TL contains an attaching bank D0 and l spheres S1 ; : : : ; Sl , which are trivial components of type I. Suppose fq1 g D D0 \ S1 ; fqi g D Si 1 \ Si ;

i D 2; : : : ; l:

104

4 Uniqueness of Minimal Quotient

Let N i be the closed nodal neighborhood of qi .i D 1; : : : ; l/, Pi the closed part in Si .i D 1; : : : ; l/. The multiplicities of D0 ; S1 ; : : : ; Sl are denoted as usual: m0 ; m 1 ; : : : ; m l ; m D gcd.mi 1 ; mi / (being independent of i ), and ni D mi =m. The following lemma is proved by the same argument as Lemma 4.2. Here we use the polar coordinates .r; /, 0  r  1, 0    2, for the unit disk D and set   1 0 D .r; / j 0  r  ; 2l   2l  j 2l  j C 1 ; r  Ij  S 1 D .r; / j 2l 2l

j D 1; 2; : : : ; 2l  1:

(N.B. Ij  S 1 is on the outside of Ij C1  S 1 ). Lemma 4.3.  1 .TL / consists of m disks 1 ; 2 ; : : : ; m , and f W  1 .TL / !  1 .TL / cyclically permutes them: f . ˛ / D ˛C1 ;

˛ D 1; : : : ; m  1;

f . m / D 1 :

There are parametrizations ˛ W D ! ˛ ;

˛ D 1; : : : ; m;

such that (i)  1 .N i / D

m [

˛ .I2i 1  S 1 /; i D 1; : : : ; l:

˛D1

(ii)  1 .Pi / D (iii) 

1

.Pl / D

m [

˛D1 m [

˛ .I2i  S 1 /; i D 1; : : : ; l  1; ˛ . 0 /;

˛D1

(iv) ˛1 f m ˛ W I2i 1  S 1 ! I2i 1  S 1 is a linear twist with screw number 

1 ; ni 1 ni

˛ D 1; : : : ; m; i D 1; : : : ; l:

4.3 Structure of  1 .tail/

105

(v) ˛1 f m ˛ W I2i  S 1 ! I2i  S 1 is a rotation sending   2ıi .r; / to r;  C ; ni where ıi is an integer coprime with ni ; ˛ D 1; : : : ; m; i D 1; : : : ; l  1: (vi) ˛1 f m ˛ W 0 ! 0 is the identity. (vii) Let .m; ; / be the valency of @ ˛ , and let ı be determined by 0  ı < ; ı  1 .mod /: Then

  2ı f ˛ .1; / D ˛ 1;  C : m

Definition 4.11. The parametrizations ˛ W D ! ˛ ; ˛ D 1; : : : ; m; are called superstandard parametrizations for f . The homeomorphism f W

m [

˛ !

˛D1

m [

˛

˛D1

is in superstandard form with respect to these parametrizations. Corollary 4.2 (to Lemma 4.3). (i) The equations describing f m W ˛ ! ˛ with respect to the superstandard parametrizations ˛ W D ! ˛ ;

˛ D 1; : : : ; m;

are uniquely determined by the series of numbers .n0 ; n1 ; : : : ; nl /: (ii) In particular, the valency of @ ˛ with respect to f is equal to .m; n0 ; n0  n1 /: (iii) Conversely, the length of the TL , and the multiplicities m0 ; m1 ; : : : ; ml

106

4 Uniqueness of Minimal Quotient

of D0 ; S1 ; : : : ; Sl are determined by the valency .; ; / of @ ˛ , ˛ being any of 1; : : : ; m. Proof (of Corollary 4.2). First we prove (ii). This is nothing but an application of Lemma 3.1, but we must be careful about the orientation: In the present case, the !  valency of @ ˛ is the valency of @ ˛ (Cf. Chap. 1) unlike Lemma 3.1 where the boundary curves are oriented from the outside (see the explanation immediately before Corollary 2.1). This discrepancy results in the appearance of n0  n1 . This proves (ii). To prove (i), we have only to note that the integers ık , k D 1; : : : ; l  1, which describe the action of f m on ˛ .I2k  S 1 /, are determined by the accumulation of the twists: l1 X 1 nk  ık D ; k D 1; : : : ; l  1: ni ni C1 nk i Dk

To prove (iii), we need a claim. Claim (H). n0 > n1 >    > nl D 1;

l  1:

Proof (of Claim). By Proposition 3.1, ml divides ml1 , and so ml D gcd.ml1 ; ml / D m; implying nl D 1. By the minimality assumption, we have ml1 =ml  2: Thus ml1 > ml . Also by the minimality assumption mi 1 C mi C1  2mi ;

i D 1; : : : ; l  1;

we have mi 1  mi  mi  mi C1 : Starting with ml1  ml > 0, we have inductively mi 1  mi > 0, i D 1; : : : ; l: t u Now assertion (iii) can be proved as follows: m D ; n0 D ; n1 D  ;

4.4 Structure of  1 .body/

107

(by (ii)), and ni 1 C ni C1  0 .mod ni /;

i D 1; : : : ; l  1;

(by Proposition 3.1). Then the sequence .n0 ; n1 ; : : : ; nl / is uniquely determined by . ; / using the Euclidean algorithm. t u

4.4 Structure of  1 .body/ Let BDY  be a body of S . Lemma 4.4. The Euler characteristic . 1 .BDY  // is negative, except for the body BDY.m/ depicted by Fig. 4.2. For this body the Euler characteristic of  1 .BDY.m// is zero. Proof. As was remarked after Lemma 4.1, BDY  is a union of its core part P0 and the tails TL1 ; TL2 ; : : : ; TLk attached to P0 . Let m0 be the multiplicity of P0 . Then all the attaching disks D.1/ ; D.2/ ; : : : ; D.k/ of TL1 ; TL2 ; : : : ; TLk have the same multiplicity m0 . Let mi be the multiplicity of the sphere S.i / in TLi next to D.i / (i.e. such that S.i / \ D.i / ¤ ;/, i D 1; : : : ; k. We denote the valency of a curve in  1 [email protected] / / by .i ; i ; i /. Then, by Corollary 4.2. (ii), i D gcd.m0 ; mi /; i D m0 =i > 1: (NB. i > 1 because of Claim (H) in the proof of Corollary 4.2). By Lemma 4.3,  1 .TLi / consists of i disks. Therefore, . 1 .BDY  // D m0 .P0 / C

k X

i

i D1

D m0 " D m0

k X 1 .P0 / C i D1 i

!

# k  X 1 .P 0 /  ; 1 i i D1

108

4 Uniqueness of Minimal Quotient

Fig. 4.2 BDY.m/

where P 0 denotes the surface obtained from BDY  by replacing each tail TLi .i D 1; : : : ; k/ by a disk. Now assuming . 1 .BDY  //  0; let us see what happens. Since 1  1= i > 0, .P 0 / must be  0. If k D 0, then .P0 / D .P 0 /  0: This is impossible because P0 is a core part. So k > 0, and .P 0 / > 0. If P 0 is a sphere,  1 .BDY  / must coincide with ˙g . This is again impossible because .˙g / < 0. Thus P 0 is a disk. Since 1  1= i  1=2, k is 1 or 2. If k D 1, then P0 is an annulus, contradicting .P0 / < 0. Thus k D 2, and 1 D 2 D 2. From i D m0 = gcd.m0 ; mi /; m0 must be an even number, say 2m, and gcd.m0 ; mi / D m: Then mi D m, and the length of TLi must be 1; i D 1; 2. (See Claim (H) in the proof of Corollary 4.2.) Thus we are led to the BDY.m/ of Fig. 4.2, for which . 1 .BDY.m/// D 0 by the above formula.

t u

Corollary 4.3 (to Lemma 4.4). Let BDY.m/ be the body shown in Fig. 4.2. Then  1 .BDY.m// consists of m annuli, B10 ; B20 ; : : : ; Bm0 (this notation for the annuli is to be consistent with the case of C h.Asp / of Chap. 3).

4.4 Structure of  1 .body/

109

Fig. 4.3 .f 0 /m W B˛0 ! B˛0 is a “180ı -rotation”

180°

B ′α

f W  1 .BDY.m// !  1 .BDY.m// permutes these annuli cyclically, and is rel.@ isotopic to f 0 such that, for each ˛ D 1; : : : ; m, .f 0 /m W B˛0 ! B˛0 is a 180ı rotation of B˛0 around the axis in Fig. 4.3. The proof is left to the reader. (Consider the monodromy exponent in the proof of Proposition 3.1.) Lemma 4.5. Let ARCH0 D D0 _ S1 _    _ Sl1 _ Dl be an arch attached (by the bank Dl ) to the body BDY.m/ D P0 [ TL1 [ TL2 of Fig. 4.2. Let m0 ; m1 ; : : : ; ml1 ; ml be the multiplicities of D0 ; S1 ; : : : ; Sl1 ; Dl : Then (i) m0  m1  : : :  ml D 2m, and the gcd.mi 1 ; mi / is equal to 2m (independently of i D 1; 2; : : : ; l) (ii)  1 .ARCH0 [ BDY.m// consists of m annuli A1 ; A2 ; : : : ; Am ; and f W  1 .ARCH0 [ BDY.m// !  1 .ARCH0 [ BDY.m//

110

4 Uniqueness of Minimal Quotient

cyclically permutes them. The m-th power f m W A˛ ! A˛ interchanges the two boundary curves of the annulus, ˛ D 1; : : : ; m. (iii) The screw number s.A˛ / of f in A˛ is equal to 2

l1 X i D0

1 ; ni ni C1

and the valency of a boundary curve of A˛ with respect to f is equal to .2m; n0 ; 0 /; where ni D mi =2m, and 0 is determined by

0  n1 .mod n0 /:

0  0 < n0 ;

Proof. (i) The multiplicity ml of the bank Dl must be the same as the multiplicity of the core part P0 of BDY.m/ to which Dl is attached. Thus ml D 2m. We apply Proposition 3.1 to the irreducible component 0 containing P0 . Then ml1 C m C m  0 .mod 2m/: This means ml1  0 .mod 2m/: Thus gcd.ml1 ; ml / D 2m;

and ml1  2m D ml :

By the minimality assumption on S , we have mi 1  mi  mi  mi C1;

i D 1; : : : ; l  1:

Hence (i) follows. (ii) By Lemma 4.2 and (i),  1 .ARCH0 / consists of 2m annuli A01 ; A02 ; : : : ; A02m permuted by f cyclically. By Corollary 4.3,  1 .BDY.m// consists of m annuli B10 ; B20 ; : : : ; Bm0 : We may assume, for ˛ D 1; : : : ; m, A0˛ \ B˛0 ¤ ;;

and

A0˛Cm \ B˛0 ¤ ;:

4.4 Structure of  1 .body/

111

Let A˛ be the annulus A0˛ [ B˛0 [ A0˛Cm ;

˛ D 1; 2; : : : ; m:

Then  1 .ARCH0 [ BDY.m// is a disjoint union of m annuli, A1 ; A2 ; : : : ; Am ; again permuted cyclically by f . Since f m interchanges A0˛ and A0˛Cm , it interchanges the boundary components of A˛ . (iii) The screw number of f in A˛ is the sum of those in A0˛ and A0˛Cm . These two are equal to l1 X 1 ; (Corollary 4.1.(ii)).  ni ni C1 i D0 Thus s.A˛ / D s.A0˛ / C s.A00˛ / D 2

l1 X i D0

1 : ni ni C1

The assertion on the valency follows from Corollary 4.1(ii).

t u

With the same notation as in Lemma 4.5, we have the following Lemma 4.6. There are parametrizations ˛ W Œ0; 1  S 1 ! A˛ ;

.S 1 D R=Z/;

˛ D 1; : : : ; m;

such that (i) ˛ jŒ0; 1=3  S 1 W Œ0; 1=3  S 1 ! A0˛ and ˛ jŒ2=3; 1  S 1 W Œ2=3; 1  S 1 ! A0˛Cm are superstandard parametrizations for fj

2m [

A0˛ ;

˛D1

in the sense of Lemma 4.2. (ii) Let D1˛ ; D2˛ be the disks . B˛0 / which are over the small tails TL1 ; TL2 of BDY.m/. Then ˛ , restricted to ˛1 .D1˛ /; ˛1 .D2˛ /, provides Sm superstan˛ dard parametrizations in the sense of Lemma 4.3 for f j ˛D1 D1 , and Sm ˛ f j ˛D1 D2 , respectively. (iii) On the boundary @A˛ , the following holds   1 m f ˛ .0; x/ D ˛ 1; x  s.A˛ / ; 4   1 f m ˛ .1; x/ D ˛ 0; x C s.A˛ / : 4

112

4 Uniqueness of Minimal Quotient

The parametrizations ˛ are constructed separately for the three parts: • the annular part 2m [

A0˛ ;

˛D1

• the periodic part

m [

.B˛0  Int.D1˛ / [ Int.D2˛ //;

˛D1

and • the disk parts

m [

S2m

D1˛ ;

˛D1

m [

D2˛ :

˛D1

In the construction for ˛D1 A0˛ , we can choose any “linear” coordinates for black annuli (see the proof of Lemma 4.2). Thus we can choose them so that the boundary behaviour of f m satisfies (iii). (See Sublemma 1, and Corollary 2.4.) The details are left to the reader. Definition 4.12. The parametrizations ˛ W Œ0; 1  S 1 ! A˛ ; ˛ D 1; : : : ; m, are called superstandard parametrizations for f . The homeomorphism f W

m [ ˛D1

A˛ !

m [

A˛

˛D1

is in superstandard form with respect to these parametrizations. Remark 4.2. The equation of a special twist given in Chap. 2 was symmetric under the interchange of the parametrization .t; x/ ! .1  t; x/ But the equations describing f in superstandard form (in the sense of Lemma 4.6) have no such symmetry in general; the symmetry remains only on the boundary action of f (Lemma 4.6(iii)). This asymmetry will cause no problem because all treatments on “superstandard level” will be done separately for the annular, periodic, and disk parts. S 0 Corollary 4.4 (to Lemma 4.6). (i) The equations describing f 2m j 2m ˛D1 A˛ with respect to the superstandard parametrizations     1 2  S1 [ ; 1  S 1; ˛ j 0; 3 3

˛ D 1; : : : ; m;

4.5 Completion of the Proof of Theorem 4.1. (Uniqueness)

113

are uniquely determined by the series of numbers .n0 ; n1 ; : : : ; nl /: (ii) Conversely, the length of ARCH0 and the multiplicities m0 ; m 1 ; : : : ; m l ; of D0 ; S1 ; : : : ; Dl are determined by the valency .; ; / of @0 A˛ and the screw number of f in A˛ , ˛ being any of 1; : : : ; m. This corollary is an interpretation of Corollary 4.1. Definition 4.13. The body BDY.m/ depicted by Fig. 4.2 is called a special body (with multiplicity 2m). The other bodies are called ordinary bodies. Definition 4.14. The union of the special body BDY.m/ and an arch ARCH0 attached to BDY.m/ is called a special tail. A special tail is denoted by SPL ( being a suffix). Definition 4.15. An arch which is not attached to special bodies is called an ordinary arch.

4.5 Completion of the Proof of Theorem 4.1. (Uniqueness) Remember that we had two pseudo-periodic maps f; f 0 W ˙g ! ˙g (of negative twist) in superstandard form, and that we were given their minimal quotients  W ˙g ! S and  0 W ˙g ! S 0 : Assuming that f and f 0 are homotopic, we want to show that these quotients are essentially the same (in the sense of (i), (ii), (iii) of Theorem 4.1). Let us assume that  0 W ˙g ! S 0 is constructed by the method of Chap. 3, as the union of C h.B 0 /; C h.D/; C h.Aln / and C h.Asp /: Then its structure is well understood. We examine the structure of  W ˙g ! S:

114

4 Uniqueness of Minimal Quotient

Let ARCH ı (resp. BDY ı , SPL) denote the collection of ordinary arches (resp. ordinary bodies, special tails) in S . S is decomposed as follows: S D ARCH ı [ BDY ı [ SPL: By Corollary 4.1,  1 .ARCH ı / is a disjoint union of non-amphidrome annuli, in which the screw number of f is non-zero. By Lemma 4.4,  1 .BDY ı / is a disjoint union of compact surfaces, which have negative Euler characteristics. Finally by Lemma 4.5,  1 .SPL/ is a disjoint union of amphidrome annuli, in which the screw number of f is non-zero. Therefore, the decomposition of ˙g ˙g D  1 .ARCH ı / [  1 .BDY ı / [  1 .SPL/ corresponds to the decomposition by the precise system of cut curves subordinate to f W ˙g ! ˙g (Chap. 1). Let us deform f and f 0 (which are in superstandard form) back into standard 0 forms f and f . N.B. Linear and special twists for f will be considered with respect to the superstandard parametrizations provided by Lemmmas 4.2 and 4.6. In the annuli where f are to be special twists, the property (iii) of the parametrizations (symmetry on the boundary) imposed by Lemma 4.6 assures that f can be deformed into special twists in the annuli without changing the parametrizations. See the proof of 0 Theorem 2.4(i). “Linear” and “special” parametrizations for f already exist in 0 Aln and Asp , because f 0 was constructed starting from f in standard form. See Chap. 3. 0

The standard forms f and f preserve the decompositions ˙g D  1 .ARCH ı / [  1 .BDY ı / [  1 .SPL/; and ˙g D Aln [ B [ Asp ; where B D ˙g Int.Aln /[Int.Asp /, respectively. See the proof of Theorem 2.1(i), at the end of Chap. 2. 0 By our assumption of Theorem 4.1, f and f are mutually homotopic. Then by Theorem 2.1 (ii) (Uniqueness of standard form), there is a homeomorphism h W ˙g ! ˙g

4.5 Completion of the Proof of Theorem 4.1. (Uniqueness)

115

isotopic to the identity, such that 0

f D h1 f h: The homeomorphism h preserves the decomposition i.e. h. 1 .ARCH ı // D Aln ; h. 1 .BDY ı // D B;

h. 1 .SPL// D Asp

(See the proof of Theorem 2.1 (ii).) Let us look at the linear part h1 . 1 .ARCH ı // more closely. Let ARCHı be an ordinary arch. By Lemma 4.2,  1 .ARCHı / is a disjoint union of annuli A1 ; A2 ; : : : ; Am which are permuted by f cyclically: f .A˛ / D A˛C1 ;

˛ D 1; : : : ; m  1;

f .Am / D A1 :

Thus  1 .ARCHı / is a cyclic orbit of the permutation on the annuli in  1 .ARCH ı / caused by f . Denote h.A˛ / by A0˛ . The disjoint union Aln D A01 [ A02 [    [ A0m ./

is a cyclic orbit of the permutation on the annuli in Aln caused by f 0 . Take an annulus A1 in  1 .ARCHı /. By Corollary 4.1 (i), the equations describing f m W A1 ! A1 in superstandard form are uniquely determined by the series of numbers .n0 ; n1 ; : : : ; nl /: The same thing can be said on A01 and .f 0 /m W A01 ! A01 ; i.e. the equations describing .f 0 /m W A01 ! A01 in superstandard form are uniquely determined by the series of numbers .n00 ; n01 ; : : : ; n0l 0 /

116

4 Uniqueness of Minimal Quotient

carrying the same meaning as .n0 ; n1 ; : : : ; nl /: By Corollary 4.1 (iii), the numbers .n0 ; n1 ; : : : ; nl / are determined by the valencies of @0 A1 and @1 A1 , together with the screw number of f in A1 . The numbers .n00 ; n01 ; : : : ; n0l 0 / are determined in the same fashion (again by Corollary 4.1 (iii)). But we have 0

f D h1 f h; which implies that the corresponding valencies and screw numbers are equal. Therefore, .n0 ; n1 ; : : : ; nl / D .n00 ; n01 ; : : : ; n0l 0 /;

l D l 0;

and the equations for f m W A1 ! A1 and

.f 0 /m W A01 ! A01

coincide. When we constructed (in Chap. 2) the homeomorphism h W ˙g ! ˙g ; we adjusted so that

hjA1 W A1 ! A01

is linear with respect to the parametrizations 1 W Œ0; 1  S 1 ! A1 and

10 W Œ0; 1  S 1 ! A01 0

with which the linearity of f and f are constructed, respectively. Also hjA1 preserves the t-levels of these parametrizations. (See the proof of Lemma 2.2.) Remember that these parametrizations 1 ; 10 are superstandard parametrizations for f and f 0 . Since the equations for f m W A1 ! A1

4.5 Completion of the Proof of Theorem 4.1. (Uniqueness)

and

117

.f 0 /m W A01 ! A01

(with respect to 1 and 10 ) coincide, and hjA1 W A1 ! A1 is linear with respect to 1 and 10 , we have f m D h1 .f 0 /m h on A1 : Moreover, the two homeomorphisms f;

h1 f 0 h W . 1 .ARCHı /; @ 1 .ARCHı // ! . 1 .ARCHı /; @ 1 .ARCHı // 0

are isotopic rel: @ 1 .ARCHı /, because f (resp.f 0 ) is isotopic to f (resp.f ) ./ rel: @ 1 .ARCHı / (resp. rel: @Aln /, and 0 f D h1 f h: Then using this isotopy and the fact that f m D h1 .f 0 /m h, we can proceed as in the second half of the proof of Theorem 2.3 (ii), and can find an isotopy h0 W  1 .ARCHı / !  1 .ARCHı /;

0 1

such that h00 D id h0 j@ 1 .ARCHı / D id f D .h01 /1 h1 f 0 h .h01 /: Replacing h with h h01 , we may assume f D h1 f 0 h on  1 .ARCHı /. Now it is easy to see that h projects to a numerical homeomorphism H W ./ ARCHı ! C h.Aln / which makes the diagram commute: h

 1 .ARCHı / ! ? ? y ARCHı

H

./

Aln ? ? 0 y ./

! C h.Aln /

S S ./ Taking the disjoint unions   1 .ARCHı /,  Aln , we have a homeomorphism h W  1 .ARCH ı / ! Aln

118

4 Uniqueness of Minimal Quotient

and a numerical homeomorphism H W ARCHı ! C h.Aln / which make the diagram commute: h

 1 .ARCHı / ! ? ? y

Aln ? ? 0 y

H

ARCHı

! C h.Aln /

We are done in the part  1 .ARCHı /. To carry out the construction in the part  1 .BDY ı /, divide  1 .BDYı / into periodic parts and rotational parts, the latter being over the tails, adjust h so that it preserves this decomposition, and apply Lemma 4.3 and Corollary 4.2 to the rotational parts. Finally in  1 .SPL/, the construction is done separately: in the linear part over the “special” arches, in the periodic parts over the closed parts, and in the rotational parts over the small tails. In these constructions, use the parametrizations provided by Lemma 4.6. Putting together these results, we obtain a homeomorphism h W ˙g ! ˙g and a numerical homeomorphism H W S ! S 0 such that (i) h is isotopic to the identity, (ii) f D h1 f 0 h, and (iii)  0 h D H. This completes the proof of Theorem 4.1.

t u

4.6 General Definition of Minimal Quotient Let f W ˙g ! ˙g be a pseudo-periodic map of negative twist. By Theorem 3.1 (Existence), f is isotopic to f 0 in superstandard form which has its minimal quotient  W ˙g ! S: By Theorem 4.1 (Uniqueness), this quotient  W ˙g ! S is independent of the choice of the superstandard form f 0 of f . Thus the following definition makes sense: Definition 4.16. The above quotient  W ˙g ! S

4.7 Conjugacy Invariance

119

is called the minimal quotient of f W ˙g ! ˙g : It is also called the minimal quotient of the mapping class Œf  (2 Mg ) to which f belongs. Notation To make f explicit in the notation of its minimal quotient, we will denote it by  W ˙g ! S Œf : Proposition 4.1. Let f W ˙g ! ˙g be a pseudo-periodic map of negative twist, then the following diagram homotopically commutes: f

˙g ! ˙g ? ? ? ? y y S Œf  ! S Œf  id: .D/

The proof is immediate.

4.7 Conjugacy Invariance Theorem 4.2. Let f1 W ˙g.1/ ! ˙g.1/ and f2 W ˙g.2/ ! ˙g.2/ be pseudo-periodic maps of negative twist. Suppose there exists a homeomorphism h W ˙g.1/ ! ˙g.2/ such that f1 ' h1 f2 h (homotopic). Then there exist a homeomorphism h0 W ˙g.1/ ! ˙g.2/ and a numerical homeomorphism H W S Œf1  ! S Œf2 

120

4 Uniqueness of Minimal Quotient

such that (i) h0 is isotopic to h, and (ii) 2 h0 D H1 , i.e. the following diagram commutes: .1/

˙g ? ? 1 y

h0

.2/

! ˙g ? ? y 2 H

S Œf1  ! S Œf2  Here 1 W ˙g.1/ ! S Œf1  and 2 W ˙g.2/ ! S Œf2  denote the respective minimal quotients. (iii) Suppose that fi0 W ˙g.i / ! ˙g.i / is a pseudo-periodic map in superstandard form which is isotopic to fi W ˙g.i / ! ˙g.i / and that fi0 gives the minimal quotient i W ˙g.i / ! S Œfi ;

i D 1; 2:

Then f10 D .h0 /1 f20 h0 . Proof. Let f10 and f20 be the superstandard forms of f1 and f2 , respectively. There is a numerical homeomorphism H2 W S Œh1 f20 h ! S Œf20  such that the following diagram commutes: .1/

˙g ? ? 0y

h

!

.2/

˙g ? ? y 2

S Œh1 f20 h ! S Œf20  D S Œf2  : H2

H2 is simply the projection of h under the identifications  0 and 2 . Since f10 ' h1 f20 h (homotopic), there exist a homeomorphism h1 W ˙g.1/ ! ˙g.1/

4.7 Conjugacy Invariance

121

and a numerical homeomorphism H1 W S Œf10  ! S Œh1 f20 h such that h1 is isotopic to the identity and the following diagram commutes (Theorem 4.1): h1

.1/

!

˙g ? ? 1 y

.2/

˙g ? ? 0 y

S Œf10  ! S Œh1 f20 h : H1

Also we have f10 D .h1 /1 .h1 f20 h/ h1 . Define a homeomorphism h0 W ˙g.1/ ! ˙g.2/ to be hh1 and a numerical homeomorphism H W S Œf1  ! S Œf2  to be H2 H1 , then h0 is isotopic to h and the following diagram commutes: .1/

˙g ? ? 1 y

h0

.2/

! ˙g ? ? y 2

S Œf1  ! S Œf2  H

t u 0

0

0

Definition 4.17. The pinched coverings  W ˙ ! S and  W ˙ ! S are isomorphic if there exist a homeomorphism h W ˙ ! ˙0 and a numerical homeomorphism H W S ! S0 such that  0 h D H. Corollary 4.5 (to Theorem 4.2). The isomorphism class of the minimal quotient  W ˙g ! S Œf 

122

4 Uniqueness of Minimal Quotient

Fig. 4.4 1=5-turn f W ˙5 ! ˙5

1 f :– turn : Σ5 → Σ5 5

P1 P2

Fig. 4.5 The minimal quotient of the 1=5-turn f W ˙5 ! ˙5

is a conjugacy invariant of Œf  2 Mg , i.e. the isomorphism class is independent of the choice of the representative of the conjugacy class of Œf  in the group Mg . Example 4.1. 15 -turn of ˙5 f W ˙5 ! ˙5 ; (See Fig. 4.4) , valency of P1 D .1; 5; 1/ (NB: D 1; ı D 1): n0 D 5; n1 D 5  1 D 4; n2 D 3; n3 D 2; n4 D 1; Lemma 3.2, valency of P2 D .1; 5; 4/ (NB: D 4; ı D 4): n0 D 5; n1 D 5  4 D 1; The minimal quotient S Œf : see Fig. 4.5.

Lemma 3.2.

4.7 Conjugacy Invariance

123

Fig. 4.6 2=5-turn f W ˙5 ! ˙5

Fig. 4.7 The minimal quotient of the 2=5-turn f W ˙5 ! ˙5

Example 4.2. 25 -turn of ˙5 f W ˙5 ! ˙5 ; (See Fig. 4.6) , valency of P1 D .1; 5; 3/ (NB: D 3; ı D 2): n0 D 5; n1 D 5  3 D 2; n2 D 1; valency of P2 D .1; 5; 2/, (NB: D 2; ı D 3): n0 D 5; n1 D 5  2 D 3; n2 D 1: The minimal quotient S Œf : see Fig. 4.7. Example 4.3. (See Figs. 4.8 and 4.9). f jB1 ' id; f j.B2  C2 / ' 180ırotation; C2 is amphidrome: S.C2 / D 2;

124

4 Uniqueness of Minimal Quotient

Fig. 4.8 Example 4.3

Fig. 4.9 f j.B2  C2 / is a “180ı ”-rotation

180°

(1,2,1)

C1 is non-amphidrome: S.C1 / D  3=2 with .n0 ; n1 ; n2 / D .1; 1; 2/ (see Theorem 5.1). The minimal quotient S Œf  is depicted in Fig. 4.10. Proposition 4.2. Let  W ˙g ! S be a generalized quotient of a pseudo-periodic map f W ˙g ! ˙g : Let BDY0 be a body in S . Let P0 be the core part of BDY0 ; 0 the irreducible component (with the multiplicity m0 ) containing P0 . Suppose that 0 intersects the other irreducible components in k points, p1 ; p2 ; : : : ; pk ; and that the multiplicity of the irreducible component which intersects 0 at pi is mi .i D 1; 2; : : : ; k/. (i) If P0 is of genus 0 (namely, a sphere with a number of disks deleted), then the number of the connected components of  1 .BDY0 / is equal to gcd.m0 ; m1 ; : : : ; mk /. (ii) If P0 is of genus  1, then the number of the connected components of  1 .BDY0 / is a common divisor of m0 ; m1 ; : : : ; mk . (iii) In both cases, f W 1 .BDY0 / !  1 .BDY0 / permutes the connected components cyclically.

4.7 Conjugacy Invariance

125

Fig. 4.10 The minimal quotient SŒf  of Example 4.3

The numbers of the connected components of  1 .BDY0 / and  1 .P0 / are equal, so we have only to consider  1 .P0 /. The proof is a simple application of the monodromy exponent ! W H1 .P0 / ! Z=m0 considered in the proof of Proposition 3.1. See also [[51], Sect. 2]. If P0 has positive genus, the behaviour of ! “around the genus” cannot be controlled by the multiplicities downstairs. The ambiguity thus arises in (ii). Assertion (iii) is an immediate consequence of the definition of a generalized quotient. The details are left to the reader. We conclude Chap. 4 with the following proposition, which shows that the number of the connected components of the preimage of a body is a strong local invariant. Let f1 W ˙g.1/ ! ˙g.1/ and f2 W ˙g.2/ ! ˙g.2/ be pseudo-periodic maps of negative twist, each being in superstandard form. Let 1 W ˙g.1/ ! S .1/ and 2 W ˙g.2/ ! S .2/ be the respective minimal quotients. We assume that there is a numerical homeomorphism H W S .1/ ! S .2/ : Up to isotopy, H may be assumed to preserve the systems of closed nodal .1/ .2/ .1/ neighborhoods. Let BDY0 be a body in S .1/ , BDY0 D H.BDY0 / the image of .1/ BDY0 .

126

4 Uniqueness of Minimal Quotient

Proposition 4.3. If the numbers of the connected components of 11 .BDY0 / and .2/ 21 .BDY0 / are equal, then there is a homeomorphism .1/

h0 W 11 .BDY0 / ! 21 .BDY0 / .1/

.2/

and a numerical homeomorphism H 0 W BDY0 ! BDY0 .1/

.2/

such that (i) the following diagram commutes: h0

.2/

h0

.2/

11 .BDY0 / ! 21 .BDY0 / ? ? ? ?f f1 y y2 .1/

11 .BDY0 / ! 21 .BDY0 / ? ? ? ? 1 y y 2 .1/

.1/

BDY0

H0

.2/

!

BDY0

and (ii) H 0 [email protected]/ D H [email protected]/ . 0

0

.1/ BDY0

is a special body (Fig. 4.2), the proposition follows from CorolProof. If .1/ .1/ .2/ lary 4.3. Thus we assume that BDY0 is an ordinary body. Let P0 (resp. P0 ) be .1/ .2/ the core part of BDY0 (resp. BDY0 ). H W S .1/ ! S .2/ preserves the multiplicities, so it also preserves the valencies of the boundary curves .1/ .2/ of P0 and P0 . (See Lemma 3.1. Also see the proof of Proposition 3.1 for the .i / explanation of the valencies of the boundary curves of P0 .) Thus there are the .1/ same number of circles over a boundary curve i of P0 and over the boundary .2/ curve H.i / of P0 . Let .1/

.1/

.2/

.2/

.1/ .2/ .resp. Q1 ; Q2 ; : : : ; Qm / Q 1 ; Q 2 ; : : : ; Qm

be the connected components of 11 .P0 / .resp. 21 .P0 //; .1/

.2/

4.7 Conjugacy Invariance

127 .1/

.2/

which are cyclically permuted by f1 (resp. f2 ). By the above remark, Q˛ and Qˇ have the same number of boundary components, ˛; ˇ D 1; : : : ; m. Moreover, since .1/ .2/ P0 and P0 are homeomorphic and have the same multiplicity, say m0 , we have .11 .P0 // D .21 .P0 // .< 0/; .1/

.2/

implying .2/

.Q˛.1/ / D .Qˇ /; .1/

˛; ˇ D 1; : : : ; m:

.2/

Therefore, Q˛ and Qˇ are homeomorphic, ˛; ˇ D 1; : : : ; m. We may assume .1/

f1 .Q˛.1/ / D Q˛C1 ;

˛ D 1; : : : ; m  1; .1/

.1/ f1 .Qm / D Q1 ; .2/

and similarly for Qˇ . Apply Theorem 1.3 to .1/

.1/

.2/

.2/

f1m W Q1 ! Q1 and

f2m W Q1 ! Q1 ; both of which are periodic maps of order m0 =m. Then there exist a homeomorphism H 0 W P0

.1/

.2/

! P0

and a homeomorphism .1/

.2/

h1 W Q1 ! Q1 such that 1. the following diagram commutes:

0

h1

.2/

h0

.2/

.1/

! Q1 ? ? y 2

.1/

! P0

P0

0

.2/

! Q1 ? ?f m y2

Q1 ? ? 1 y

and 2. H 0 j@P .1/ D H j@P .1/ .

h1

.1/

Q1 ? ? f1m y

128

4 Uniqueness of Minimal Quotient

For ˛ D 2; : : : ; m, define h˛ W Q˛.1/ ! Q˛.2/ by setting h˛ D .f2 /˛1 h1 .f1 /1˛ ; and define

h W 11 .P0 / ! 21 .P0 / .1/

.2/

to be the union h1 [ h2 [    [ hm : Then it is easy to check on

.1/ 11 .P0 /

that

hf1 D f2 h and H 0 1 D 2 h: .1/

Suppose P0 has k tails .1/

.1/

.1/

TL1 ; TL2 ; : : : ; TLk ; .1/

TL.1/  being attached to @ P0 . Set D H.TL.1/ TL.2/   /;

 D 1; : : : ; k:

Let . ;  ;  / be the valencies of

@ P0.1/ ;

 D 1; : : : ; k:

Since H W S .1/ ! S .2/ preserves these valencies, .2/

@ P0

.1/

. D H.@ P0 //

has the same valency . ;  ;  /;

 D 1; : : : ; k:

By Lemma 4.3 and Corollary 4.2, 1 .2/ 11 .TL.1/  / (resp. 2 .TL /)

4.7 Conjugacy Invariance

129

consists of  disks .2/ .1/ ;˛ (resp. ;˛ ),

˛ D 1; : : : ;  ;

permuted cyclically by f1 (resp. f2 ), and .1/ .f1 / W .1/ ;˛ ! ;˛

and .2/ .f2 / W .2/ ;˛ ! ;˛

have the identical superstandard expressions, ˛ D 1; : : : ;  . Thus, defining .2/ h;˛ W .1/ ;˛ ! ;˛

to be the “identity” with respect to the superstandard parametrizations (˛ D 1; : : : ;  ), we can extend h W 11 .P0 / ! 21 .P0 / .1/

to

h0 D h [ .

[

.2/

h;˛ / W 11 .BDY 0 / ! 11 .BDY 0 /; .1/

.2/

;˛

which projects to a homeomorphism (again denoted by) H 0 W BDY 0 ! BDY 0 : .1/

.2/

Obviously they satisfy h0 f1 D f2 h0

and H 0 j@BDY .1/ D H j@BDY .1/ : 0

0

t u

•

Chapter 5

A Theorem in Elementary Number Theory

In this chapter we will prove the following theorem which was essential to the arguments in Chaps. 3 and 4. Theorem 5.1. Let .0 ; 0 ; ı0 / and .1 ; 1 ; ı1 / be two triples of integers, each satisfying the following two conditions (i), (ii): (i)  >   0;  > ı  0, and (ii)  ı  1 .mod  /,  D 0; 1. Let s be a positive rational number such that (iii) s C .ı0 =0 C ı1 =1 / is an integer. Then there exists uniquely a sequence of positive integers .n0 ; n1 ; : : : ; nl /,

(1) (2) (3) (4) (5)

l  1;

with the following properties: n0 D 0 ; nl D 1 , n1  0 .mod 0 /, nl1  1 .mod 1 /, ni 1 C ni C1  0 .mod ni /, i D 1; : : : ; l  1, .ni 1 C ni C1 /=ni  2, i D 1; : : : ; l  1, and l1 X i D0

1 D s. ni ni C1

Remark 5.1. 1.  D ı D 0 if and only if  D 1. If  > 1, then  ; ı  1. 2. The sequence .n0 ; n1 ; : : : ; nl / satisfies gcd.ni 1 ; ni / D 1;

i D 1; 2; : : : ; l;

Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5 5, © Springer-Verlag Berlin Heidelberg 2011

131

132

5 A Theorem in Elementary Number Theory

because of properties (1),(2) and (3). (NB. gcd. ;  / D 1.) 3. For a sequence with “length” l D 1, properties (3) and (4) are empty.

5.1 Proof of Theorem 5.1. (Uniqueness) The uniqueness of .n0 ; n1 ; : : : ; nl / is contained in the following lemma: Lemma 5.1. Let .m0 ; m1 ; : : : ; mk / and .n0 ; n1 ; : : : ; nl /; k  1; l  1; be sequences of positive integers satisfying (i) m0 D n0 ; mk D nl , (ii) m1  n1 .mod m0 /; mk1  nl1 .mod mk /, (iii) mi 1 C mi C1  0 .mod mi /; i D 1; 2; : : : ; k  1, nj 1 C nj C1  0 .mod nj /; j D 1; 2; : : : ; l  1, (iv) .mi 1 C mi C1 /=mi  2; i D 1; 2; : : : ; k  1, .nj 1 C nj C1 /=nj  2; j D 1; 2; : : : ; l  1; and (v)

k1 X i D0

X 1 1 D : mi mi C1 n n i D0 j j C1 l1

Then we have k D l; and mi D ni ; i D 0; 1; : : : ; l: Claim (I). Let .m0 ; m1 ; : : : ; mk /;

k  1;

be a sequence of positive integers such that .mi 1 C mi C1/=mi  2;

i D 1; 2; : : : ; k  1:

If m0  mk , then m0  m1 . Proof. If k D 1, the lemma is trivial. Let us assume k  2. Suppose, on the contrary, that m0 < m1 . The condition .mi 1 C mi C1 /=mi  2 is rewritten as mi  mi 1  mi C1  mi ;

i D 1; : : : ; k  1:

Hence if m0 < m1 , then m1 < m2 and inductively m0 < m1 < m2 <    < mk : This contradicts the assumption m0  mk . Thus m0  m1 .

t u

5.1 Proof of Theorem 5.1. (Uniqueness)

133

Proof (of Lemma 5.1.). By the symmetry of the statement, we may assume m0  mk . By Claim (I), m0  m1 . By assumption (i), n0 D m0 and nl D mk , so n0  nl . Then by Claim (I) again, n0  n1 . Now we have m0  m1 > 0, n0  n1 > 0: Then assumptions (i), (ii) imply m1 D n1 , which together with assumption (iii) implies m2  n2 .mod m1 /. By assumption (v) and m0 D n0 ; m1 D n1 , we have k1 X i D1

X 1 1 D : mi mi C1 nj nj C1 j D1 l1

Thus the subsequences .m1 ; : : : ; mk / and .n1 ; : : : ; nl / satisfy the five conditions corresponding to (i)(v). By induction on l, we have k D l and .m1 ; : : : ; mk / D .n1 ; : : : ; nl /: We know already m0 D n0 . Thus the lemma follows.

t u

The uniqueness of .n0 ; : : : ; nl / is now established. Before proving the existence, we will show a basic lemma. Lemma 5.2. Let .a0 ; a1 ; : : : ; au /,

u  1;

be a sequence of positive integers satisfying (i) gcd.ai 1 ; ai / D 1; i D 1; : : : ; u. (ii) ai 1 C ai C1  0 .mod ai /; a D 1; 2; : : : ; u  1. Then 1.

u1 X i D0

1 d0 eu  C .mod 1/, where d0 ; eu are positive integers determined ai ai C1 a0 au

by

0 < d0  a0 , 0 < eu  au ,

d0 a1  1 .mod a0 /; eu au1  1 .mod au /:

(NB. Here, for convenience, the inequalities are 0 < d0  a0 , 0 < eu  au , not as usual 0  d0 < a0 ; 0  eu < au ; d0 D a0 if and only if a0 D 1. Similarly for eu ). 2. Moreover, if ai  2, i D 1; : : : ; u  1, then u1 X i D0

d0 eu 1 D C  1: ai ai C1 a0 au

134

5 A Theorem in Elementary Number Theory

3. In particular, if ai  2; i D 1; : : : ; u  1, and au D 1, then u1 X i D0

1 d0 D : ai ai C1 a0

Claim (J). Let a; a0 be positive integers which are mutually coprime. Let d; e 0 be defined as follows 0 < d  a; da0  1 .mod a/ 0 < e 0  a0 ; e 0 a  1 .mod a0 /: Then

1 d e0 D  1: C aa0 a a0 Proof. Obviously da0 C e 0 a  1 .mod a/ and  1 .mod a0 /. Thus da0 C e 0 a D 1 C aa0 z

for a certain z 2 Z, because gcd.a; a0 / D 1. Hence 1 d e0 C z: C 0 D a a aa0 By the assumption 0 < d  a; 0 < e 0  a0 , we have 1=aa0 < .d=a C e 0 =a0 /  2. Therefore, z D 1. u t Proof (of Lemma 5.2). Define two sequences of positive integers .d0 ; d1 ; : : : ; du1 /,

.e1 ; e2 ; : : : ; eu /

as follows: 0 < di  ai ; 0 < ei  ai ;

di ai C1  1 .mod ai /; i D 0; 1; : : : ; u  1; ei ai 1  1 .mod ai /; i D 1; 2; : : : ; u:

By the assumption (ii), ai 1 C ai C1  0 .mod ai /, we have di C ei  0 .mod ai /, i.e. di ei C D an integer: ai ai By Claim (J), 1 di ei C1 D C  1; ai ai C1 ai ai C1

i D 0; : : : ; u  1:

5.2 Proof of Theorem 5.1. (Existence)

135

Thus, adding up the equalities 1 d0 e1 D C 1 a0 a1 a0 a1 d1 e2 1 D C 1 a1 a2 a1 a2 ::::::::::::::: du1 eu 1 D C 1 au1 au au1 au and taking into account that d1 e2 d2 e1 C ; C ; ::: a1 a1 a2 a2 are integers, we get u1 X i D0

d0 eu 1  C .mod 1/: ai ai C1 a0 au

This proves (1). To prove (2), note that if ai  2, then di =ai C ei =ai D 1 because in this case, 0 < di < ai ; 0 < ei < ai and 0 < .di =ai C ei =ai / < 2. Then the above addition gives u1 X 1 d0 eu D C  1: a a a0 au i D0 i i C1 To prove (3), note that, if au D 1, then eu D 1. Putting this in (2), we get u1 X i D0

d0 1 D : ai ai C1 a0 t u

5.2 Proof of Theorem 5.1. (Existence) First we consider the general case where 0 ; 1 > 1 and hence 0 ; ı0 ; 1 ; ı1  1. Let .a0 ; a1 ; : : : ; au / be a sequence of positive integers satisfying 1. a0 D 0 ; a1 D 0 ; 2. a0 > a1 >    > au D 1; 3. ai 1 C ai C1  0 .mod ai /; i D 1; 2; : : : ; u  1: Such a sequence can be obtained by the Euclidean algorithm.

136

5 A Theorem in Elementary Number Theory

Likewise let .b0 ; b1 ; : : : ; bv / be a sequence of positive integers satisfying 1. b0 D 1 ; b1 D 1 ; 2. b0 > b1 >    > bv D 1; 3. bj 1 C bj C1  0 .mod bj /; j D 1; 2; : : : ; v  1: By Lemma 5.2. (3), u1 X i D0

1 ı0 D ; ai ai C1 0

v1 X

1 ı1 D : b b 1 j D0 j j C1 Let K denote s  .ı0 =0 C ı1 =1 /, which is an integer by assumption (iii) of Theorem 5.1. We have K  1; because s > 0 and 0 < .ı0 =0 C ı1 =1 / < 2: Three cases are distinguished: K > 0; K D 0; K D 1. Case I. K > 0. We define the required sequence .n0 ; n1 ; : : : ; nl / to be .a0 ; a1 ; : : : ; au ; 1; : : : ; 1; bv ; : : : ; b1 ; b0 /: „ ƒ‚ … K1

This sequence has property (5) of Theorem 5.1: l1 X i D0

1 ı0 ı1 D CK C D s: ni ni C1 0 1

The other properties (1)(4) are obvious by the construction. Case II. K D 0. We define the sequence .n0 ; n1 ; : : : ; nl / to be .a0 ; a1 ; : : : ; au1 ; 1; bv1 ; : : : ; b1 ; b0 /: Recall that au D bv D 1. Case III. K D 1. This is the most complicated case. We give several definitions. These definitions are motivated by the chain diagrams in [37] (Fig. 5.1): Definition 5.1. An -sequence  of length l  1 is a finite sequence  D .n0 ; "1 ; n1 ; "2 ; : : : ; "l ; nl / n0

Fig. 5.1 Chain diagram

n1 e1

n2 e2

n l –1 el –1

el

nl

5.2 Proof of Theorem 5.1. (Existence)

137

where the ni ’s denote positive integers (called the multiplicities of  ) and "i is a sign "i 2 fC; g. Moreover, multiplicities and signs are required to satisfy the following ‘-condition’: "i ni 1 C "i C1 ni C1  0 .mod ni /;

i D 1; 2; : : : ; l  1:

()

Of course, if l D 1, this condition is empty. Remark 5.2. The -condition implies that gcd.ni 1 ; ni / is independent of i . Definition 5.2. The s-number s. / of an -sequence  D .n0 ; "1 ; n1 ; "2 ; : : : ; "l ; nl / is a rational number defined by s. / D

l1 X "i C1 n n i D0 i i C1

Definition 5.3. Let  D .n0 ; "1 ; n1 ; "2 ; : : : ; "l ; nl / be an -sequence. (i) Suppose there exists an i 2 f1; 2; : : : ; l  1g such that "i ni 1 C "i C1 ni C1 D ˙ni : Then the -sequence  0 D .n0 ; "1 ; : : : ; "i 1 ; ni 1 ; "0 ; ni C1 ; "i C2 ; : : : ; nl / is said to be obtained from  by blowing down the vertex of multiplicity ni , where "0 D sign Œ"i "i C1 ."i ni 1 C "i C1 ni C1 / D sign ."i C1 ni 1 C "i ni C1 /: (ii) Suppose there exists an i 2 f1; 2; : : : ; l  1g such that "i ni 1 C "i C1 ni C1 D 0: Then ni 1 D ni C1 and "i D "i C1 . The -sequence  00 D .n0 ; "1 ; : : : ; ni 2 ; "i 1 ; n0 ; "i C2; ni C2 ; : : : ; nl / is said to be obtained by contracting  at the vertex of multiplicity ni , where n0 D ni 1 D ni C1 :

138

5 A Theorem in Elementary Number Theory

It is not difficult to see that the operations of blowing down and contraction actually give an -sequence. The operations do not change the s-numbers. For simplicity, from now on, a positive sign "i will be omitted from the notation of an -sequence. For example, an -sequence .5; ; 3; C; 2; C; 1/ will be denoted as .5; ; 3; 2; 1/. An -sequence in which every sign is positive is called a positive -sequence. Such a sequence will be denoted as .n0 ; n1 ; : : : ; nl / by the above convention. Lemma 5.3. If a positive -sequence  D .n0 ; n1 ; : : : ; nl /; l  1, satisfies n0 > n1 >    > nl , then 1 s. / < : nl .nl1  nl / Proof. For each i D 1; 2; : : : ; l  1; there is a positive integer zi such that ni 1 C ni C1 D zi ni : The assumption n0 > n1 >    > nl implies zi  2: Thus ni 1 C ni C1  2 ni , or equivalently, ni 1  ni  ni  ni C1 ;

i D 1; 2; : : : ; l  1:

Setting d D nl1  nl .> 0/, we inductively have nlj  nl C jd;

j D 0; 1; : : : ; l:

Then, s. / D

l1 X i D0



D

1 ni ni C1

l1 X

1 .nl C jd / Œnl C .j C 1/ d  j D0 l1  X j D0

<

1 1  nl C jd nl C .j C 1/ d



1 d

1 : nl d t u

5.2 Proof of Theorem 5.1. (Existence)

139

Lemma 5.4. Let  be an -sequence .a0 ; a1 ; : : : ; au ; ; bv ; bv1 ; : : : ; b0 / with a negative sign between the vertices of multiplicities au and bv . Suppose that  satisfies the following conditions, (i)(iv): (i) (ii) (iii) (iv)

u  1; v  1, a0 > a1 >    > au ; b0 > b1 >    > bv , au1  bv , bv1  au , and s. / > 0.

Then (1) or (2) occurs: 1.  can be contracted to a positive -sequence  0 which cannot be blown down anymore, or 2.  can be blown down to an -sequence  0 with one negative sign which satisfies the conditions corresponding to (i)(iv). Moreover, in both cases, the terminal multiplicities a0 , b0 , of  remain unchanged in  0 . Remark 5.3. An -sequence .23; 7; 5; ; 2; 5; 23/ shows that both (1) and (2) can occur simultaneously. In this case, by convention, we will always choose contraction. Proof (of Lemma 5.4). Suppose au D bv1 , then  can be contracted at the vertex of multiplicity bv . The resulting  0 is a positive -sequence .a0 ; a1 ; : : : ; au .D bv1 /; bv2 ; : : : ; b0 /: Since a0 > a1 >    > au .D bv1 / < bv2 <    < b0 ; this -sequence cannot be further blown down. Similarly, if au1 D bv ;  can be contracted to a positive -sequence, which cannot be further blown down. Henceforth we will assume au1 > bv and bv1 > au . By the -condition, we have  au1  bv D ˛au .9˛ W a positive integer/; () bv1  au D ˇbv .9ˇ W a positive integer/: Case A. ˛ D 1 and ˇ D 1 W In this case, au1 D au C bv D bv1 : We will prove that u D v D 1 is impossible. Suppose on the contrary that u D v D 1. Then  must be .a0 ; a1 ; ; b1 ; b0 /

140

5 A Theorem in Elementary Number Theory

with a0 D a1 C b1 D b0 . We would have s. / D

1 1 1  C D 0; a0 a1 a1 b1 b1 b0

which contradicts condition (iv) on  . Thus u D v D 1 is impossible.If u  2, then, using au1  bv D au , we can blow down  at the vertex of multiplicity au to obtain an -sequence  0 D .a0 ; : : : ; au1 ; ; bv ; bv1 ; : : : ; b0 /; in which u  1  1. This  0 satisfies the conditions corresponding to (i)(iv). If u  2, the argument is the same. Case B. ˛ D 1 and ˇ  2 W Since au1  bv D au ;  can be blown down to an -sequence  0 D .a0 ; : : : ; au1 ; ; bv ; bv1 ; : : : ; b0 /: We will show that .i/0 u  1  1, and .iii/0 bv1 > au1 . (The -sequence trivially satisfies the conditions corresponding to (ii) and (iv)). Proof (of .i/0 ). Let us suppose u D 1 and show that this leads to a contradiction. Equations () with ˛ D 1, ˇ  2, produce 

a0  bv D a1 bv1  a1  2bv :

Hence it follows bv  bv1 C a0  0: 0

 would have the form .a0 ; ; bv ; bv1 ; : : : ; b0 /: Then X 1 1 1 1 C < C a0 bv b b a0 bv bv .bv1  bv / i D0 i i C1 v1

s. / D s. 0 / D  (by Lemma 5.3) D

bv1 C bv C a0  0 by (  ): a0 bv .bv1  bv /

This contradicts condition (iv) on  . Thus u  1  1.

(  )

5.2 Proof of Theorem 5.1. (Existence)

141

Proof (of .iii/0 ). From () with ˛ D 1; ˇ  2, we have  au1 D au C bv ; bv1  au C 2bv : Thus bv1 > au1 . Case C. ˛  2 and ˇ D 1: The argument is the same as in Case B. Case D. ˛  2 and ˇ  2: From ./ with ˛  2; ˇ  2; we have  au1  bv  2au ; bv1  au  2bv : Rewriting these inequalities, we obtain  au1  au  au C bv ; bv1  bv  au C bv :

(?)

Then s. / D

u1 X i D0

X 1 1 1  C ai ai C1 au bv j D0 bj bj C1 v1

<

1 1 1 C  au .au1  au / au bv bv .bv1  bv /



1 1 1 C  au .au C bv / au bv bv .au C bv /

(Lemma 5.3) by (?)

D 0: This contradicts condition (iv) on  . Thus Case D is impossible. Finally we can check that in every case discussed above, the terminal multiplicities a0 ; b0 remain unchanged. This completes the proof of Lemma 5.4. t u Let us continue the proof of Theorem 5.1 (existence) in Case (III); K D 1. Remember that .a0 ; a1 ; : : : ; au / is a positive -sequence satisfying a0 D 0 ;

a1 D 0 ;

a0 > a1 >    > au D 1;

and .b0 ; b1 ; : : : ; bv / is a similar sequence satisfying b0 D  1 ;

b1 D 1 ;

b0 > b1 >    > bv D 1:

142

5 A Theorem in Elementary Number Theory

We define an -sequence  with one negative sign by  D .a0 ; a1 ; : : : ; au ; ; bv ; bv1 ; : : : ; b0 /: Since we are considering the general case where 0 > 0  1 and 1 > 1  1, we have u  1 and v  1 (assumption (i) of Lemma 5.4). By the choice of .a0 ; a1 ; : : : ; au / and .b0 ; b1 ; : : : ; bv /; we have a0 > a1 >    > au and b0 > b1 >    > bv (assumption (ii) of Lemma 5.4). Since au D bv D 1, we have au1 > bv and bv1 > au (assumption (iii)). By Lemma 5.2 (3), we have u1 X i D0

ı0 1 D ; ai ai C1 0

and

v1 X

ı1 1 D : b b 1 j D0 j j C1

Thus s. / D

u1 X i D0

D

X 1 1 1  C ai ai C1 au bv j D0 bj bj C1 v1

ı0 ı1 1C D s > 0: 0 1

This is assumption (iv) of Lemma 5.4. Therefore, applying Lemma 5.4 to  recursively, we will obtain a positive -sequence 0 D .n0 ; n1 ; : : : ; nl /; l  1; which can no longer be blown down. This is the desired sequence; let us check the properties, (1)(5). Since the terminal multiplicities a0 ; b0 of  remain unchanged in 0 , we have n0 D a0 D 0 ;

and nl D b0 D 1

( property (1) ):

Property (2) will be checked later. Property (3) for 0 is obvious because 0 is an -sequence.

5.2 Proof of Theorem 5.1. (Existence)

143

Property (4) is true because 0 cannot be blown down. Property (5) is true because neither contraction nor blowing down change the s-numbers, and  has been chosen so that s. / D s. Let us check property (2) for 0 . Let 0 00 D .a00 ; a10 ; : : : ; ah0 ; ; bk0 ; bk1 ; : : : ; b00 /;

h  1; k  1;

be the -sequence one step before 0 ; 0 is obtained from 00 by contraction. 00 has property (2) because h  1; k  1; a10 D a1 D 0 and b10 D b1 D 1 . Suppose 0 is obtained by contracting 00 at the vertex of the multiplicity ah0 . If h  2, then a10 remains in 0 , and 0 has property (2). If h D 1, we have 0 ; : : : ; b00 /: 0 D .a00 .D bk0 /; bk1

By the -condition on 0 00 D .a00 ; a10 ; ; bk0 ; bk1 ; : : : ; b00 /; 0 0  0 .mod bk0 /. Thus bk1  a10  0 .mod a00 / and we have a10 C bk1 0 ; : : : ; b00 / 0 D .a00 ; bk1

has property (2). If 0 is obtained by contracting 00 at the vertex of multiplicity bk0 , the argument is the same. This completes the proof of Theorem 5.1 in the general case where 0 ; 1 > 1. Three special cases remain to be examined. Special case (1). 0 D 1, 1 > 1. In this case 0 D ı0 D 0 and the integer k D sı1 =1 must be non-negative. Let .b0 ; b1 ;    ; bv / be an -sequence satisfying b0 D 1 ;

b1 D 1 ;

b0 > b1 >    > bv D 1:

Then the sequence .1; : : : ; 1; bv ; bv1 ; : : : ; b0 / „ ƒ‚ … k

is the desired one. Special case (2). 0 > 1, 1 D 1. The argument is the same as in Special case (1).

144

5 A Theorem in Elementary Number Theory

Special case (3). 0 D 1 D 1. In this case, s D k D a positive integer. The desired sequence is .1; 1; : : : ; 1/ „ ƒ‚ … kC1

t u

The proof of Theorem 5.1 is now completed. Remark 5.4. The sequence .n0 ; n1 ; : : : ; nl / of Theorem 5.1 satisfies one of the following inequalities:

1. n0 > n1 >    > ni D ni C1 D    D nj .D 1/ < nj C1 <    < nl , for some i; j with 0 < i < j < l. 2. n0 > n1 >    > ni < ni C1 <    < nl , for some i with 0 < i < l. 3. n0 > n1 >    > nj D nj C1 D    D nl .D 1/, for some 0 < j  l. 4. n0 > n1 >    > nl .> 1/. 5. .1 D/ n0 D n1 D    D ni < ni C1 <    < nl , for some 0  i < l. 6. .1 0: Then .a0 ; a1 ; a2 ; : : : ; au / D .27; 17; 7; 4; 1/;

uD4

.b0 ; b1 ; b2 ; : : : ; bv / D .17; 10; 3; 2; 1/;

v D 4:

 D .27; 17; 7; 4; 1; ; 1; 2; 3; 10; 17/ # 1 (meaning “blowing down at 1”) .27; 17; 7; 4; 1; ; 2; 3; 10; 17/ #2 .27; 17; 7; 4; 1; ; 3; 10; 17/ #1 .27; 17; 7; 4; ; 3; 10; 17/ #4 .27; 17; 7; ; 3; 10; 17/ #3 .27; 17; 7; ; 10; 17/ #7 .27; 17; ; 10; 17/ # contracting 10 0 D .27; 17/:

Chapter 6

Conjugacy Invariants

Let f W ˙g ! ˙g be a pseudo-periodic map of negative twist. According to Corollary 4.5, the isomorphism class of the minimal quotient  W ˙g ! S Œf  and, in particular, the numerical homeomorphism type of S Œf  are conjugacy invariants of Œf  2 Mg . However, the converse is not true: the minimal quotient does not necessarily determine the conjugacy class of Œf . (Nielsen [50, Sect. 15] incorrectly claims that this converse is true (compare Theorem 13.4 of [22] where this same claim is repeated)). Example 6.1. In Fig. 6.1, f1 j˙2  C1 ' 180ırotation around the (vertical) axis; s.C1 / D 2. The minimal quotient space S Œf1  is depicted in Fig. 6.2. Example 6.2. In Fig. 6.3, f2 j˙2  C1 ' 180ırotation around the (horizontal) axis; s.C1 / D 2. The minimal quotient space S Œf2  is depicted in Fig. 6.4. The minimal quotient spaces S Œf1  and S Œf2  are numerically homeomorphic, but Œf1  and Œf2  are not conjugate, because the amphidrome curve in Example 6.1 is null-homologous while it is not the case in Example 6.2. In fact, the minimal quotients 1 W ˙g ! S Œf1  and 2 W ˙g ! S Œf2  are not isomorphic. (Proof. Let P0 be the closed part (Chap. 4) in the irreducible component of genus 1, which is contained in both minimal quotient spaces. Then 11 .P0 / has two connected components, while 21 .P0 / is connected). Remark 6.1. The pseudo-periodic maps f1 and f2 are the topological monodromies of the singular fibers 2-2I00 and 3-II10 in Namikawa-Ueno’s classification [49], respectively. They pointed out that these singular fibers have the same configuration but different monodromies.

Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5 6, © Springer-Verlag Berlin Heidelberg 2011

145

146

6 Conjugacy Invariants

Fig. 6.1 Example 6.1: f1 j˙2  C1 is a “180ı -rotation” and s.C1 / D 2

Fig. 6.2 The minimal quotient of f1 of Example 6.1

Fig. 6.3 Example 6.2: f2 j˙2  C1 is a “180ı -rotation” and s.C1 / D 2

a

C1 Σ2 180°

Fig. 6.4 The minimal quotient of f2 of Example 6.2

S Example 6.3. In Fig. 6.5, f3 j˙6  5iD1 Ci ' 1=5 turn rotation; s.Ci / D 1, i D 1; 2; :::; 5. The minimal quotient space S Œf3  is depicted in Fig. 6.6. S Example 6.4. In Fig. 6.7, f4 j˙6  5iD1 Ci ' 2=5 turn rotation; s.Ci / D 1, i D 1; 2; :::; 5. The minimal quotient space S Œf4  is depicted in Fig. 6.8 The minimal quotients 3 W ˙6 ! S Œf3 

6 Conjugacy Invariants

147

a ¦3

C2

C1

C3

C5 C4

Fig. 6.5 Example 6.3: f3 j˙6 

S5 iD1

Ci is a 1=5-turn and s.Ci / D 1

Fig. 6.6 The minimal quotient of f3 of Example 6.3

and 4 W ˙6 ! S Œf4  are isomorphic as pinched coverings, but Œf3  and Œf4  are not conjugate, because the actions of f3 and f4 on the “partition graphs” are different. (See Corollary 6.1). Remark 6.2. f4 is not conjugate to .f3 /2 ; the minimal quotient space of .f3 /2 , S Œ.f3 /2 , is depicted in Fig. 6.9. The purpose of this chapter is to show that the action of a pseudo-periodic map f on its partition graph gives the information lacking in the minimal quotient and that that action, together with the minimal quotient, determines the conjugacy class of Œf  in Mg . (Theorems 6.1 and 6.3).

148

6 Conjugacy Invariants

a

¦4

C2

C1

C3

C5 C4

¦4

Fig. 6.7 Example 6.4: f4 j˙6 

Fig. 6.8 The minimal quotient of f4 of Example 6.4

S5 iD1

Ci is a 2=5-turn and s.Ci / D 1

b

5

Fig. 6.9 The minimal quotient of .f3 /2

6.1 Partition Graphs

149

6.1 Partition Graphs Definition 6.1. A graph is a 1-dimensional finite cell complex. A 1-cell is called an edge, a 0-cell a vertex. There may be loops. There may be loops sharing the same vertex. Also there may be a component consisting of one vertex. A homeomorphism ' W X ! X 0 between graphs is called an isomorphism if it preserves the graph structure. Definition 6.2. Let f W ˙g ! ˙g be a pseudo-periodic map of negative twist, fCi griD1 the precise system of cut curves subordinate to f (Chap. 1). The partition graph X Œf  associated with f is the graph whoseSvertices are in one-to-one correspondence to the connected components of ˙g  riD1 Ci and whose edges are in one-to-one correspondence to the cut curves fCi griD1 . An edge e.Ci / joins vertices v.b/ and v.b 0 / if and only if Ci is in the adherence of the connected component b and at the same time in that of the connected component b 0 . It may happen that b D b 0 , in which case e.Ci / is a loop. By Nielsen’s theorem (Theorem 2.2), the isomorphism class of the partition graph X Œf  is a conjugacy invariant of Œf . Definition 6.3. The collapsing map  W ˙g ! X Œf  is defined as follows: Let fAi griD1 be the annular neighborhood system of fCi griD1. Each Ai can be identified with Œ0; 1  S 1 through a parametrization i W Œ0; 1  S 1 ! Ai : The map  W ˙g ! X Œf  S shrinks each connected component of ˙g  riD1 Int.Ai / to the corresponding vertex and projects each annulus Ai D Œ0; 1  S 1 onto its first factor Œ0; 1 which is identified with the corresponding edge. We assume that  maps a cut curve Ci to the middle point of the edge e.Ci /. It is not difficult to prove that the collapsing map  W ˙g ! X Œf  is well-defined up to isotopy of ˙g . (Cf. Nielsen’s theorem, Theorem 2.2.) The standard form f (“fitted” to fAi griD1) projects to an isomorphism 'Œf  W X Œf  ! X Œf  such that 'Œf  je is linear on each edge e.

150

6 Conjugacy Invariants

Of course, we have the commutative diagram: f

!

˙g ? ? y

˙g ? ? y

X Œf  ! X Œf : 'Œf 

Since X Œf  is a finite graph, the map 'Œf  is a periodic map. Definition 6.4. 'Œf  W X Œf  ! X Œf  is said to be the periodic map induced by f . f acts on X Œf  through 'Œf  . .1/

.1/

.2/

.2/

Lemma 6.1. Let f1 W ˙g ! ˙g and f2 W ˙g ! ˙g be pseudo-periodic maps of negative twist. Suppose there exists a homeomorphism h W ˙g.1/ ! ˙g.2/ such that f1 ' h1 f2 h (homotopic), then there exist a homeomorphism h0 W ˙g.1/ ! ˙g.2/ and an isomorphism ˚ W X Œf1  ! X Œf2  such that (i) h0 is isotopic to h, and (ii) the following diagram commutes: .1/

˙g ? ? 1 y

h0

.2/

! ˙g ? ? y2 ˚

X Œf1  ! X Œf2  ? ? ?'Œf  'Œf1  ? y y 2 ˚

X Œf1  ! X Œf2 : Proof. The lemma follows from the uniqueness of the standard form (Theorem 2.1). t u Corollary 6.1. The (equivariant) isomorphism class of .X Œf ; 'Œf  / is a conjugacy invariant of Œf  2 Mg .

6.2 Weighted Graphs

¦1

151

¦2

¦3

¦4

Fig. 6.10 Partition graphs of fi ’s in Examples 6:i; i D 1; 2; 3; 4

¦1

¦2

¦3

¦4

Fig. 6.11 Refined partition graphs of fi ’s in Examples 6:i; i D 1; 2; 3; 4

Examples. Partition graphs and the actions of the f ’s corresponding to Examples 6.1 through 6.4. See Fig. 6.10. Definition 6.5. Let f W ˙g ! ˙g be a pseudo-periodic map of negative twist. The refined partition graph XŒf  associated with f is obtained from X Œf  by subdividing each edge e.Ci / corresponding to an amphidrome curve Ci into two edges by the middle point. The middle point becomes a new vertex v.Ci / of X Œf . The collapsing map  W ˙g ! XŒf  is defined to be the same map  W ˙g ! X Œf  as before. (Thus  maps an amphidrome curve Ci to the vertex v.Ci /.) Examples. Refined partition graphs corresponding to Examples 6.1 through 6.4 (Fig. 6.11).

6.2 Weighted Graphs An advantage of the refined partition graph X Œf  is that the periodic action of 'Œf  has no multiple points in the interior of an edge. Thus if we take the quotient space Y Œf  D X Œf ='Œf  , the projection map  W X Œf  ! Y Œf  sends each edge of X Œf  homeomorphically onto an edge of Y Œf .

152

6 Conjugacy Invariants

Fig. 6.12 Weighted graphs

Definition 6.6. A graph Y is weighted if each vertex (and each edge) carries a positive integer called the weight. An isomorphism  W Y ! Y 0 between weighted graphs is called a weighted isomorphism if  preserves the weights. The quotient graph Y Œf  becomes a weighted graph as follows: the weight of a vertex v (resp. an edge e) is defined to be ]1 .v/ (resp. ]1 .e/). Definition 6.7. Y Œf  is called the weighted graph associated with f W ˙g ! ˙g . Examples. Weighted graphs corresponding to Examples 6.1 through 6.4 (Fig. 6.12): The weighted graph Y Œf  has another meaning. It serves as the decomposition diagram of the base chorizo space of the minimal quotient  W ˙g ! S Œf , i.e. the vertices of Y Œf  are in one-to-one correspondence to the bodies, and their edges are in one-to-one correspondence to the arches of S Œf  (Cf. Chap. 4). More precisely, we have the following Lemma 6.2. There exists a map  W S Œf  ! Y Œf  satisfying the following conditions: (i)  shrinks a body of S Œf  to a vertex and maps an arch onto an edge of Y Œf . (ii)  induces a one-to-one correspondence between the set of the bodies (resp. arches) and the set of the vertices (resp. edges). (iii) For a body BDY of S Œf , the weight of the vertex .BDY / is equal to the number of the connected components of  1 .BDY /, and for an arch ARCH

of S Œf , the weight of the edge .ARCH / is equal to the number of the annuli in  1 .ARCH /. (iv) The following diagram “almost” commutes in the sense that  can be deformed on amphidrome annuli so that the diagram commutes with the modified  0 : 0

˙g ! X Œf  ? ? ? ? y y S Œf  ! Y Œf : 

Proof. Let f be the standard form of f . Let BDY be a body of S Œf . If BDY is an ordinary body, then  1 .BDY / consists of a certain number of compact surfaces with negative Euler characteristic (Lemma 4.4). Pick up one component Bj , map

6.2 Weighted Graphs

153

it by  to a vertex of X Œf , and project it down to a vertex of Y Œf . The image .BDY / is defined to be this vertex. Since f permutes the components of  1 .BDY / cyclically (Proposition 4.2), the vertex .BDY / is well-defined, independently of the choice of the connected component of  1 .BDY /. The weight of .BDY / is equal to the number of the vertices of X Œf  which are over the vertex .BDY /, but by the definition of  W ˙g ! X Œf ; this number is equal to the number of the connected components of  1 .BDY /. If BDY is a special body, BDY.m/, with multiplicity 2m (see Fig. 4.2), then  1 .BDY.m// consists of m annuli B10 ; B20 ; : : : ; Bm0 ; (Corollary 4.3). Let ARCH 0 be the (special) arch attached to BDY.m/. Then  1 .ARCH 0 [ BDY.m// consists of m amphidrome annuli, A1 ; A2 ; : : : ; Am ; and Bi0 is the middle one-third of Ai . Pick up one annulus Ai .  W ˙g ! X Œf  maps the center-line Ci of Ai to the vertex v.Ci / of X Œf . The image .BDY.m// is defined to be the vertex .v.Ci // 2 Y Œf , which is well-defined because f permutes fA1 ; A2 ; : : : ; Am g cyclically. The weight of the vertex .BDY.m// is certainly m. Deform  W ˙g ! X Œf  Sm on i D1 Ai so that the resulting  0 sends each middle one-third Bi0 of Ai to the vertex v.Ci /, then we have  0 D  on  1 .BDY.m//. Let ARCH be an (ordinary or special) arch of S Œf . The preimage 1  .ARCH / consists of a certain number of annuli, which are permuted by f cyclically (Lemma 4.2). Pick up an annulus A0i in  1 .ARCH /. If ARCH is ordinary, A0i is mapped by  onto an edge e.Ci / of X Œf . Project down e.Ci / onto an edge of Y Œf  to get  W ARCH ! .e.Ci //: Clearly we have  D  on A0i .

154

6 Conjugacy Invariants

If ARCH is special, then A0i is an outer one-third of an annulus Ai (Lemma 4.5). The annulus A0i is mapped by  0 (which has been deformed so that it sends Bi0 to the vertex v.Ci /) onto an edge e 0 .Ci / of X Œf , where e 0 .Ci / is a half of the edge e.Ci / of X Œf . Project down e 0 .Ci / onto an edge of Y Œf  to obtain  W ARCH ! .e 0 .Ci //: Then we have  0 D  on A0i . Putting together the above partially defined maps, we get the desired “collapsing” map  W S Œf  ! Y Œf : t u

The properties (i)(iv) are now easy to check. Definition 6.8. The map  W S Œf  ! Y Œf  is called the collapsing map of the chorizo space S Œf . Corollary 6.2. Let  W S Œf  ! Y Œf  be the collapsing map.

(i) The weight of the edge .ARCH / is the gcd of the successive multiplicities on ARCH . (ii) Let BDY be a body of S Œf . If the core part P0 of BDY is of genus 0, then the weight of the vertex .BDY / is equal to gcd.m0 ; m1 ; : : : ; mk /, where m0 , m1 ; : : : ; mk have the same meaning as in Proposition 4.2. (iii) If P0 is of genus  1, the weight of the vertex .BDY / is a common divisor of m 0 , m1 ; : : : ; mk . (iv) The weight of a vertex v 2 Y Œf  is a common divisor of the weights of the edges emanating from v. The proof is straightforward by Lemmmas 6.2, 4.2, and Proposition 4.2. Assertion (iv) follows from (i)(iii). Remark 6.3. The weighted graph Y Œf  generalizes the graph considered in [18,48]. Theorem 6.1. Let f1 W ˙g.1/ ! ˙g.1/

and

f2 W ˙g.2/ ! ˙g.2/

be pseudo-periodic maps of negative twist, both in superstandard form. Let 1 W ˙g.1/ ! S Œf1 

and

2 W ˙g.2/ ! S Œf2 

6.2 Weighted Graphs

155

be the respective minimal quotients. Then there exists a homeomorphism h W ˙g.1/ ! ˙g.2/ such that f1 D h1 f2 h if and only if there exist a numerical homeomorphism H W S Œf1  ! S Œf2 ; a weighted isomorphism  W Y Œf1  ! Y Œf2  and an isomorphism ˚ W X Œf1  ! XŒf2  such that the diagram commutes: 1

 X Œf1  ? ? ˚y

 X Œf1  ? ? ˚y

S Œf2  ! Y Œf2 

 X Œf2 

 X Œf2 :

2

1

'Œf1 

S Œf1  ! Y Œf1  ? ? ? ? Hy y

2

.1/

'Œf2 

.2/

Remark 6.4. The homeomorphism h W ˙g ! ˙g obtained above does not necessarily project to the given numerical homeomorphism H . Proof (of Theorem 6.1). The “only if” part is easy to prove. The “if”part is essential. We may assume H W S Œf1  ! S Œf2  preserves the systems of closed nodal neighborhoods. .2/ .1/ Let BDY .1/ be a body in S Œf1 , BDY D H.BDY / its image. Since .1/ .2/  1 .BDY .1/ / D 2 H.BDY / D 2 .BDY /; .2/ 1 the number of connected components of 11 .BDY .1/ / and that of 2 .BDY / are equal (see Lemma 6.2(iii)). Then by Proposition 4.3, there exist a homeomorphism 1 .2/ h W 11 .BDY .1/ / ! 2 .BDY /

and a numerical homeomorphism .2/ H 0 W BDY .1/ ! BDY

such that 1. f1 D .h/1 f2 h on 11 .BDY .1/ /, 2. 2 h D H 0 1 on 11 .BDY .1/ and /, .1/ 3. H 0 j@BDY .1/ D H j@BDY .

156

6 Conjugacy Invariants

.2/ 1 Note that the connected components of 11 .BDY .1/ / and 2 .BDY / are permuted cyclically by f1 and f2 , respectively, and that the equality (1) implies that .2/ 1 h W 11 .BDY .1/ / ! 2 .BDY / preserves these cyclic orders. Thus, replacing h l by h.f1 / with a certain integer l, we may assume the following equality holds:

4. 2 h D ˚1

on  1 .BDY .1/ /.

(Here 1 W ˙g.1/ ! XŒf1 

and

2 W ˙g.2/ ! X Œf2 

denote the respective collapsing maps). Let ARCH .1/

be an arch in S Œf1 , .1/ ARCH .2/

D H.ARCH /

its image. Since H W S Œf1  ! S Œf2  .2/ 1 preserves the multiplicities, 11 .ARCH .1/

/ and 2 .ARCH / consists of the same number of annuli (Lemma 4.2), and by Corollary 4.1, there exists a homeomorphism 1 .2/ h W 11 .ARCH .1/

/ ! 2 .ARCH /

such that 5. f1 D .h/1 f2 h on 11 .ARCH .1/

/, and .1/ 1 6. 2 h D H1 on 1 .ARCH /. By the same reason as in case of BDY , h can be assumed to satisfy 7. 2 h D ˚1

on

11 .ARCH .1/

/.

Now our task it to paste together these homeomorphisms fh j 11 .BDY .1/ /g

and .1/

fh j 11 .ARCH .1/

/g

.2/

to produce a homeomorphism h W ˙g ! ˙g . If it is done, the pasted h will satisfy f1 D .h/1 f2 h by (1) and (5). To see if this pasting is possible, let us fix our attention at an edge e of Y Œf1  and the terminal vertices 0 and 1 . (It may happen that 0 D 1 , when e is a loop). Let .1/ .1/ BDY .1/ the be the body in S Œf1  such that .BDY / D  . D 0; 1/, ARCH .1/ .1/ arch in S Œf1  such that .ARCH / D e. BDY has a certain number of boundary .1/ components, D 0; 1. We assume ARCH.1/ is attached to a boundary curve 0 of .1/ .1/ .1/ BDY 0 and to a boundary curve 1 of BDY 1 .

6.2 Weighted Graphs

157

Take a connected component B0 of 11 .BDY 0 /. Over the boundary curve .1/ .1/ 0 , there are a certain number (say c0 ) of boundary curves of B0 . Let them be .1/

.1/

.1/ .1/ .1/ Q0;1 ; Q0;2 ; : : : ; Q0;c0 :

Consider one of them, say Q0;1 . There is an annulus A.1/ over ARCH .1/ such that .1/

.1/ Q0;1 is one of its two boundary curves. Suppose the other boundary curve of A.1/ is .1/ Q1;1 , where .1/ .1/ .1/ Q1;1 ; Q1;2 ; : : : ; Q1;c1 .1/

are the boundary curves of a certain connected component, say B1 , of .1/ .1/ 11 .BDY 1 /, which lie over Q1 . .2/ .1/ .2/ .1/ .2/ .1/ Set B0 D h.B0 /, B1 D h.B1 / and A.2/ D h.A.1/ /. Also set Q0;˛ D h. Q0;˛ / .2/ .1/ (˛ D 1; : : : ; c0 ) and Q D h. Q / (ˇ D 1; : : : ; c1 ). 1;ˇ

1;ˇ

By (4) and (7), it is clear that one of the two boundary curves of A.2/ is among .2/ .2/ .2/ f Q0;1 ; Q0;2 ; : : : ; Q0;c0 g

and the other is among .2/ .2/ .2/ f Q1;1 ; Q1;2 ; : : : ; Q1;c1 g; .1/

.1/

but to paste together the homeomorphisms hjB0 , hjB1 and hjA.1/ , we need the equality .2/ .2/ @A.2/ D Q0;1 [ Q1;1

to hold. However, we cannot expect this in general. The difficulty will be overcome by the following lemma: Lemma 6.3. Let ˙ be a compact connected surface whose Euler characteristic is negative, f W ˙ ! ˙ a periodic map of order m0 without multiple points. Let M D ˙=f be the quotient space with the projection  W ˙ ! M . Take a boundary component 1 of M , and let Q1;1 ; Q1;2 ; : : : ; Q1;

be the totality of the preimages of 1 under . Suppose they are indexed so that f . Q1;˛ / D Q1;˛C1 , ˛ D 1; : : : ;  1, and f . Q1; / D Q1;1 . Then there exists a homeomorphism j W ˙ ! ˙ such that (A) jf D fj (B) j. Q1;˛ / D Q1;˛C1 , ˛ D 1; : : : ;  1, and j. Q1; / D Q1;1 , S

(C) j j.@˙  ˛D1 Q1;˛ / D id.

158

6 Conjugacy Invariants

Fig. 6.13 A band connected sum N.C /\N. 1 / is identified with 2

Proof. Suppose M has k boundary components, 1 , 2 , : : :, k , each oriented from the inside. Let ! W H1 .M / ! Z=m0 be the monodromy exponent (see Theorem 1.3, Proposition 3.1, and [51, Sect. 2]). Set !. i / D mi , i D 1; 2; : : : ; k. Case 1. M has genus  1. In this case, there exists a simple closed (oriented) curve C in M such that !.C / D 1, ([51, Sect. 5]). Take an annular neighborhood N.C / of C and a collar neighborhood N. 1 / of 1 , then make a band connected sum N.C /\N. 1/ of N.C / and N. 1 / inside M . (Fig. 6.13). Identifying N.C /\N. 1 / with a disk with two holes 2 , we perform a full Dehn-twist inside

2 about a curve parallel to the boundary @0 2 not homologous to 1 nor to C , then rotate the boundary curve @C 2 parallel to C , .m1  1/ times. (See Fig. 6.14 where m1 D 2.) This gives a homeomorphism J W 2 ! 2 with J j@ 2 D id. We extend J W 2 ! 2 to a homeomorphism J WM !M by defining J j.M Int 2 / to be the identity. Let c be an arc in 2 connecting @C 2 (the boundary curve parallel to C ) and the boundary curve @0 2 . Then since !.J.c/  c/ D !. 1 / C !.C /  .m1 C 1/!.C / D 0; Fig. 6.14; we can lift J to a homeomorphism j W˙ !˙

6.2 Weighted Graphs

159

∂0Δ2

c

γ1

Δ2 Γ1 C

J

J(c)

J(γ 1)

Fig. 6.14 Perform a full Dehn twist about a curve parallel to @0 2 , then rotate @C 2 .m1  1/ times

such that j j.˙   1 .Int. 2 // D id: By the definition of the monodromy exponent ! W H1 .M / ! Z=m0 ; there are gcd.m0 ; m1 / boundary curves of ˙ over the boundary curve 1 of M . [51, Sect. 2]. Thus, in particular, the number of the boundary curves over 1 is equal to gcd.m0 ; m1 /. Let 1 be an arc in 2 connecting 1 and @0 2 . Since the arc 1 is mapped to J.1 / satisfying !.J.1 /  1 / D !. 1 / C !.C / D m1 C 1; Fig. 6.14; the lifted homeomorphism j W˙ !˙ sends Q1;˛ to Q1;˛C1 (˛ D 1; : : : ;  1) and Q1; to Q1;1 .The homeomorphism j satisfies jf D fj because j is a lift of a homeomorphism of M and j has fixed points in ˙. This completes the proof in Case 1.

160

6 Conjugacy Invariants

Fig. 6.15 A band connected sum N. 1 /\N. 2 / is identified with 2

Γk

Γ2

Γ1

Case 2. M is of genus 0. Since .M / D ..˙//=m0 < 0, M has more than two boundary curves 1 ; 2 ; : : : ; k ,

k  3:

Take collar neighborhoods N. 1 / and N. 2 / of 1 and 2 , respectively, and make a band connected sum N. 1 /\N. 2 / inside M . (See Fig. 6.15). Identifying N. 1 /\N. 2 / with a disk with two holes 2 , we perform a full Dehn twist about a curve parallel to the boundary curve @0 2 other than 1 or 2 . (See Fig. 6.16). This gives us a homeomorphism J2 W 2 ! 2 which extends by the identity to a homeomorphism J2 W M ! M . We lift J2 to a homeomorphism j2 W ˙ ! ˙ such that j2 j.˙   1 .Int 2 // D id. Let i be an arc in 2 which connects @0 2 and i , i D 1, 2. Then !.J2 .1 /  1 / D m1 C m2 !.J2 .2 /  2 / D m1 C m2 ; Fig. 6.16: Thus j2 maps each boundary curve Q1;˛ (over 1 ) to Q1;˛Cm2 (the indices are considered to be in Z= D Z= gcd.m0 ; m1 /, and each boundary curve Q2;ˇ (over 2 ) to Q2;ˇCm1 (the indices are considered to be in Z= gcd.m0 ; m2 /). Apply this homeomorphism j2 m2 times, and set j20 D .j2 /m2 . Then j20 maps Q1;˛ to Q1;˛C.m2 /2 , and Q2;ˇ to Q2;ˇCm2 m1 D Q2;ˇ . A certain amount of rotation about Q2;ˇ might be caused by j20 , but it can be rotated back to the identity, equivariantly with respect to f . The modified j20 sends the boundary curve (over 1 ) of index ˛ to 2 the boundary curve (over S 1 ) of index ˛ C .m2 / , ˛ D 1; : : : ; , and restricts to Q the identity of @˙  ˛D1 1;˛ . We do the same construction using 1 and i .i D 2; 3; : : : ; k/. Then obtain a homeomorphism ji0 , for each i D 2, 3, : : :, k, which sends the boundary curve (overS 1 ) of index ˛ to the one of index ˛ C .mi /2 ,

and restricts to the identity on @˙  ˛D1 Q1;˛ . The .ji0 /’s commute with f by the same reason as in Case 1. Since ˙ is connected, we have:

6.2 Weighted Graphs

161

γ1

γ2

∂0Δ2 Γ2

Γ1

J2

J 2(γ 2)

J 2(γ 1)

Fig. 6.16 Perform a full Dehn twist about a curve parallel to @0 2

gcd.m0 ; m1 ; m2 ; : : : ; mk / D 1;

[51, Sect. 4(4.4)];

which implies gcd.m0 ; m1 ; .m2 /2 ; : : : ; .mk /2 / D 1: Let l0 , l1 , : : :, lk be the integers such that l0 m0 C l1 m1 C l2 .m2 /2 C    C lk .mk /2 D 1: Then the homeomorphism j W ˙ ! ˙, defined by j D .j20 /l2 .j30 /l3    .jk0 /lk ; satisfies the conditions (A), (B), (C), of Lemma 6.3. This completes the proof of Lemma 6.3 in Case 2. u t

162

6 Conjugacy Invariants

6.3 Completion of the Proof of Theorem 6.1 Proof. Let us return to the situation before Lemma 6.3. Let 0 (resp. 1 ) be the .2/ .2/ number of the connected components of 21 .BDY 0 / (resp. 21 .BDY 1 /), and let .2/ .2/ .2/ P0 (resp. P1 ) be the core part of BDY 0 (resp. BDY 1 ). Set PQ0 D 21 .P0 / \ B0 , .2/ and PQ1 D 21 .P1 / \ B1 . Recall that PQ0 has c0 boundary curves .2/ .2/ .2/ Q0;1 ; Q0;2 ; : : : ; Q0;c0 .2/

over the boundary curve 0 boundary curves

.1/ .2/ D H. 0 / of BDY 0 , and likewise PQ1 has c1

.2/ .2/ .2/ Q1;1 ; Q1;2 ; : : : ; Q1;c1 .2/

.1/

.2/

over the boundary curve 1 D H. 1 / of BDY 1 . Applying Lemma 6.3 to the periodic map .f2 / 0 W PQ0 ! PQ0 ; we obtain a homeomorphism j0 W PQ0 ! PQ0 such that (A) j0 .f2 / 0 D .f2 / 0 j0 , .2/ .2/ (B) j0 . Q0;˛ / D Q0;˛C1 (˛ 2 Z=c0 ), Sc0 .2/ (C) j0 j.@PQ0  ˛D1 Q / D id.

and

0;˛

Extend j0 W PQ0 ! PQ0 by the identity to .2/

.2/

j0 W B0 ! B0

having the same properties (A), (B), (C). Then replacing the homeomorphism .1/

.2/

h W B0 ! B0

by .j0 /l h, with a certain integer l, we can cyclically change the positions of the .2/ Q0;˛ ’s (without moving c0 [ .2/ .2/ @B0  Q0;˛ ˛D1

at all), and adjust so that one of the two boundary components of A.2/ .D h.A.1/ // .2/ .1/ is attached to Q0;1 .D h. Q0;1 //. By the same argument, we can find a homeomorphism j1 W PQ1 ! PQ1

6.3 Completion of the Proof of Theorem 6.1

163

with the similar properties (A), (B), (C) and extend it to .2/

.2/

j1 W B1 ! B1 : .2/ Again changing cyclically the positions of the Q1;˛ ’s using j1 , we can adjust so that .2/ .1/ the other boundary component of A.2/ is attached to Q .D h. Q //. 1;1

.1/

1;1

.1/

Note that the modified hjB0 and hjB1 still satisfy .1/

.1/

8. .f2 / 0 .hjB0 / D .hjB0 /.f1 / 0 .1/ .1/ 9. .f2 / 1 .hjB1 / D .hjB1 /.f1 / 1

and

because of property (A). .1/ .1/ Now we can paste the homeomorphisms hjB0 , hjB1 and hjA.1/ . We must be a .1/ .1/ little careful, because on the intersection Q0;1 D B0 \ A.1/ the homeomorphisms .1/

hjB0 and hjA.1/ might be different by a certain amount of rotation. The same .1/ .1/ thing can be said on the intersection Q1;1 D B1 \ A.1/ . Thus, in general, we must compose hjA.1/ W A.1/ ! A.2/ with a certain linear twist l W A.2/ ! A.2/ .1/

.1/

so that the composed lh coincides with hjB0 (resp. hjB1 ) on the boundary curve .1/ .1/ Q0;1 (resp. Q1;1 ). If this is done, denoting the composition lh W A.1/ ! A.2/ again by h W A.1/ ! A.2/ ; we will have the pasted homeomorphism .1/

.1/

.2/

.2/

h W B0 [ A.1/ [ B1 ! B0 [ A.2/ [ B1 : Let m be the number of the annuli in 11 .ARCH .1/ /. We extend the above homeomorphism (equivariantly) to .1/

.1/

.2/

.2/

h˛ W .f1 /˛1 .B0 [ A.1/ [ B1 / ! .f2 /˛1 .B0 [ A.2/ [ B1 /;

˛ D 1; : : : ; m;

by the formula 10. h˛ D .f2 /˛1 .h/.f1 /1˛ . .1/

.1/

Note that if .f1 /˛1 .B0 / D B0 , then ˛  1 is a multiple of 0 , say k 0 , and we have

164

6 Conjugacy Invariants

.f2 /˛1 .hjB0 /.f1 /1˛ D .f2 /k 0 .hjB0 /.f1 /k 0 D hjB0 .1/

.1/

.1/

.1/

.1/

by (8). Similarly, if .f1 /˛1 .B1 / D B1 , then .1/

.1/

.f2 /˛1 .hjB1 /.f2 /1˛ D hjB1

by (9). Therefore, formula (10) gives well-defined homeomorphisms h˛ , ˛ D 1, : : :, m, and since 11 .BDY 0 [ ARCH .1/ [ BDY 1 / D .1/

.1/

m [

.1/

.1/

.2/

.2/

.f1 /˛1 .B0 [ A.1/ [ B1 /;

˛D1

and 21 .BDY 0 [ ARCH .2/ [ BDY 1 / D .2/

.2/

m [

.f2 /˛1 .B0 [ A.2/ [ B1 /;

˛D1

we have a homeomorphism, hD

m [

h˛ W 11 .BDY 0 [ ARCH .1/ [ BDY 1 / ! 21 .BDY 0 [ ARCH .2/ [ BDY 1 /; .1/

.1/

.2/

.2/

˛D1

such that f2 h D hf1 . (This last equality is verified using Corollary 4.1.) So far we have confined ourselves to the part over an edge e of Y Œf1 . But the same argument can be carried out for the part over every edge e, and we can paste up the component homeomorphisms 1 .1/ fhj11 .BDY .1/ /g and fhj .ARCH /g

(after proper modifications as we did above) to produce a homeomorphism h W ˙g.1/ ! ˙g.2/ such that f1 D .h/1 f2 h. This completes the proof of Theorem 6.1.

t u

6.4 Weighted Cohomology We would like to reformulate Theorem 6.1 in terms of the “weighted cohomology” of a weighted graph which will be introduced below. Let Y be a weighted graph.

6.4 Weighted Cohomology

165

Definition 6.9. A 0-cochain c 0 is an operation which assigns to each vertex v an element of a cyclic group Z=W .v/, where W .v/ denotes the weight attached to v. e an element A 1-cochain c 1 is an operation which assigns to each oriented edge ! !  !  e /. 1 of Z= gcd.W .v/; W .w//, where @ e D v  w, such that c . e / D c 1 .! Let C 0 .Y / and C 1 .Y / denote the groups of 0-cochains and 1-cochains, respectively. Then the coboundary operator ı W C 0 .Y / ! C 1 .Y / is defined as follows: e / D c 0 .v/  c 0 .w/ 2 Z= gcd.W .v/; W .w//; .ı.c 0 //.! e D v  w. where @! Definition 6.10. The weighted cohomology group HW1 .Y / or Y is defined by HW1 .Y / D C 1 .Y /=ıC 0 .Y /: Let X be a graph, ' W X ! X an isomorphism such that 'je is linear for each edge e. Since we are considering only finite graphs, ' is a periodic map. We assume the following condition: ' has no multiple points in the interior of an edge. Then we have the quotient graph Y D X='. Let  W X ! Y be the projection. By condition (), each edge of X is mapped onto an edge of Y homeomorphically. Y naturally becomes a weighted graph: the weight of a vertex  is ] 1 ./, and the weight of an edge e is ]1 .e/. In what follows we will construct a cohomology class in HW1 .Y / which classifies the (equivariant) isomorphism class of .X; '/. Let  be a vertex of Y . The preimage 1 ./ contains W ./ vertices, Q 1 , Q 2 , : : :, Q W ./ . We will always assume that the indices are elements of Z=W ./ and satisfy '.Q ˛ / D Q ˛C1 ;

˛ D 1; 2; : : : ; W ./:

Suppose all the vertices in 1 ./ have been indexed as above, for every vertex  of Y . Let e be an oriented edge of Y with @e D v  w. Let eQ be an edge of X which is a lift of e. Suppose @eQ D Q ˛  wQ ˇ , where ˛ 2 Z=W ./ and ˇ 2 Z=W .w/. Then we define a 1-cochain c 1 2 C 1 .Y / by setting c 1 .e/ D ˛  ˇ 2 Z= gcd.W ./; W .w//: The value ˛ˇ in Z= gcd.W ./; W .w// is independent of the choice of a lift of e. To see this, let eQ 0 be another lift of e. Suppose @eQ 0 D Q   wQ  , where  2 Z=W ./ and  2 Z=W .w/. Since .e/ Q D .eQ 0 / D e, there exists a power of ', say ' l , such l 0 that ' .e/ Q D eQ , which implies

166

6 Conjugacy Invariants

' l .Q ˛ / D Q 

and ' l .wQ ˇ / D wQ  :

By our assumption on the manner of indexing, we have ˛ C l D  2 Z=W ./;

and ˇ C l D  2 Z=W .w/:

Thus ˛  ˇ D    in Z= gcd.W ./; W .w// as asserted. Note that the operation of indexing the vertices in 1 ./ has an ambiguity, i.e. the choice of Q 1 2 1 ./ is arbitrary for each . We will show that this ambiguity does not affect the cohomology class of c 1 . In fact, suppose the vertices Q 1 , Q 2 , : : :, Q W ./ are re-indexed as Q 1Ck , Q 2Ck , : : :, Q W ./Ck . The amount of the shift, k, depends on v, so we denote it by k.v/. Then, on an edge e with @e D   w, the new 1-cochain c 1 takes the following value: c 1 .e/ D .˛Ck.//.ˇCk.w// D .˛ˇ/C.k./k.w// 2 Z= gcd.W ./; W .w//: Therefore, defining a 0-cochain c 0 2 C 0 .Y / by c 0 ./ D k./; we have, c 1 D c 1 C ı.c 0 /: Thus we have proved the following: Lemma 6.4. There is a well-defined cohomology class Œc 1  2 HW1 .Y / associated with the pair .X; '/. Also the above argument has already proved the following: Corollary 6.3. Let c 1 2 C 1 .Y / be any 1-cochain representing the cohomology class Œc 1  2 HW1 .Y / associated with the pair .X; '/. Then we can properly index the vertices in  1 ./ for every  so that the associated 1-cochain is exactly the given cochain c 1 . Suppose we have indexed the vertices of 1 ./ for every . In the second assertion (ii) of the following lemma, c 1 2 C 1 .Y / is the associated 1-cochain with this indexing. Let e be an oriented edge of Y and assume @e D   w. Lemma 6.5. (i) The weight W .e/ is a common multiple of the weights W .v/, W .w/. (ii) A vertex vQ ˛ 2 1 .v/ is joined to a vertex w Q ˇ 2 1 .w/ by an edge over e if 1 and only if ˛  ˇ D c .e/ 2 Z= gcd.W .v/; W .w//. Moreover, if vQ ˛ and w Q ˇ are joined by an edge over e there are exactly W .e/=lcm.W .v/; W .w// edges over e connecting vQ ˛ and wQ ˇ . Proof. (i) is left to the reader. We will prove the first claim of (ii). The “only if” part is already proved. We will prove the converse, i.e. if ˛  ˇ D c 1 .e/ 2 Z= gcd

6.4 Weighted Cohomology

167

.W .v/; W .w//, then vQ ˛ and w Q ˇ are joined by an edge over e. Proof. There is certainly an edge eQ over e which joins vQ ˛ to a vertex over w, say w Q  . By the “only if” part, ˛   D c 1 .e/ 2 Z= gcd.W .v/; W .w//. Thus ˇ

.mod gcd.W .v/; W .w///;

in other words, ˇ D  C nG;

G D gcd.W .v/; W .w//:

There are integers k, l, such that kW .v/ C lW .w/ D G: Then ' nkW .v/ sends vQ ˛ to v˛CnkW .v/ D vQ ˛ (because ˛ 2 Z=W .v/) and sends wQ  to wQ  CnkW .v/ D wQ  CnGnlW .w/ D wQ ˇ because ˇ,  2 Z=W .w/ and ˇ D  C nG. Therefore, ' nkW .v/ .e/ Q connects vQ ˛ and wQ ˇ . The second claim of (ii) is left to the reader. u t Let W0 be the gcd of the weights of all the vertices of Y , and let H1 .Y / be the ordinary homology group of Y . A cohomology class c 2 HW1 .Y / defines a homomorphism c W H1 .Y / ! Z=W0 ; P P which sends a 1-cycle mi ei to mi c.ei / 2 Z=W0 . Lemma 6.6. X is connected if and only if Y is connected and c W H1 .Y / ! Z=W0 is onto, where c 2 HW1 .Y / is the cohomology class associated with .X; '/. The proof is left to the reader as an exercise. The (equivariant) isomorphism class of .X; '/ is classified as follows: Theorem 6.2. Let .X1 ; '1 / and .X2 ; '2 / be pairs, each consisting of a graph and a (periodic) isomorphism on it. Let Y1 D X1 ='1 and Y2 D X2 ='2 be the respective quotient graphs. Then there exists an isomorphism ˚ W X1 ! X2 such that '1 D ˚ 1 '2 ˚ if and only if there exists a weighted isomorphism  W Y1 ! Y 2 such that   .c2 / D c1 , where   W HW1 .Y2 / ! HW1 .Y1 /

168

6 Conjugacy Invariants

is the induced isomorphism and ci 2 HW1 .Yi / is the cohomology class associated with .Xi ; 'i /, i D 1; 2. Proof. The “only if” part is trivial. We will prove the “if” part. We may assume that the vertices of X1 and X2 are properly indexed so that the 1-cochain c21 associated with .X2 ; '2 / is pulled back by  exactly to the 1-cochain c11 associated with .X1 ; '1 /, (Corollary 6.3). .0/ .0/ Let X1 (resp. X2 ) denote the set of the vertices of X1 (resp. X2 ). For each vertex v of Y1 , define a bijective map ˚v W 11 .v/ ! 21 . .v// by sending a vertex vQ ˛ in 11 .v/ to the vertex vQ 0˛ in 21 . .v// having the same index ˛ as vQ ˛ . This is possible because 11 .v/ and 21 . .v// contain the same number of vertices (for  W Y1 ! Y2 preserves the weights). The set f˚v gv gives a bijection .0/

.0/

˚ .0/ W X1 ! X2 : For each edge e of Y1 , there exists an edge eQ of X1 which lies over e. Let us fix eQ for a while. Let @e D v  w. Suppose @eQ D vQ ˛  wQ ˇ , where vQ ˛ 2 11 .v/ and wQ ˇ 2 11 .w/. Then there exists an edge e 0 of X2 , lying over  .e/ and connecting vQ 0˛ and wQ 0ˇ , because ˇ  ˛ D c11 .e/ D c21 . .e//, (Lemma 6.5). Define ˚0 W 11 .e/ ! 21 . .e// Q to '2k .eQ 0 / for each k  1. Since the weights by sending eQ to eQ 0 , and sending '1k .e/ k of e and  .e/ are the same, '1 .e/ Q D eQ if and only if '2k .eQ 0 / D eQ 0 . This means that ˚0 W 11 .e/ ! 21 . .e// is well-defined and that it is equivariant with respect to '1 and '2 . Now the set f˚v gv [ f˚0 g0 gives the desired isomorphism ˚ W X 1 ! X2 : This completes the proof of Theorem 6.2.

t u

Remark 6.5. The isomorphism ˚ W X1 ! X2 obtained above makes the diagram commute: ˚ X1 ! X2 ? ? ? ?2 1 y y Y1 ! Y2 

Combining Theorems 6.1 and 6.2, we obtain the following theorem, which is the main theorem of Part I.

6.4 Weighted Cohomology

169

Theorem 6.3. Let f1 W ˙g.1/ ! ˙g.1/

and

f2 W ˙g.2/ ! ˙g.2/

be pseudo-periodic maps of negative twist, both in superstandard form. Let 1 W ˙g.1/ ! S Œf1 

and

2 W ˙g.2/ ! S Œf2 

be the respective minimal quotients. Then there exists a homeomorphism h W ˙g.1/ ! ˙g.2/ such that f1 D h1 f2 h if and only if there exist a numerical homeomorphism H W S Œf1  ! S Œf2  and a weighted isomorphism  W Y Œf1  ! Y Œf2  such that (i) the following diagram commutes: H

S Œf1  ! S Œf2  ? ? ? ?2 1 y y Y Œf1  ! Y Œf2  

(ii)   .cŒf2 / D cŒf1 , where   W HW1 .Y Œf2 / ! HW1 .Y Œf1 / is the induced isomorphism, and cŒfi  2 HW1 .Y Œfi / is the cohomology class associated with .X Œfi ; 'Œfi  /, i D 1; 2. Remark 6.6. Theorem 6.3 says that the triple .S Œf ; Y Œf ; cŒf / is a complete set of conjugacy invariants of the mapping class Œf  of a pseudo-periodic map f W ˙g ! ˙g with negative twist. Triples .S; Y; c/ coming from a pseudo-periodic map f of negative twist will be characterized in Part II. (See Theorem 9.2).

•

Part II

•

Chapter 7

Topological Monodromy

Definition 7.1 (Cf. [48]). A triple .M; D; / is called a degenerating family of Riemann surfaces of genus g (abbreviated as degenerating family of genus g) if (i) M is a complex surface, (ii) D D f 2 C j jj < 1g, (iii) W M ! D is a surjective proper holomorphic map, (iv) for each  2 D, the fiber F D 1 ./ is connected, and (v) jM  W M  ! D  is a smooth (i.e. C 1 ) fiber bundle whose fiber is a Riemann surface of genus g, where D  D D  f0g and M  D M  1 .0/. If futhermore .M; D; / satisfies the following condition (vi), .M; D; / is said to be minimal. (vi) No fiber F contains a smoothly embedded sphere of self-intersection number 1. A fiber F ,  ¤ 0, will be called a general fiber. The central fiber F0 may contain a singular point of , in which case F0 will be called singular fiber. Let us choose a base point 0 2 D  , and let l W Œ0; 2 ! D  be a loop based at 0 (l.0/ D l.2/ D 0 ) which turns around 0 once in the positive direction. Then there is a continuous family of (orientation-preserving) homeomorphisms h W ˙g ! Fl. /

.0    2/;

where ˙g is a fixed surface of genus g. Definition 7.2. f D h1 0 h2 W ˙g ! ˙g is called a monodromy homeomorphism of the family .M; D; /. Given a degenerating family .M; D; / of genus g, a monodromy homeomorphism f W ˙g ! ˙g

Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5 7, © Springer-Verlag Berlin Heidelberg 2011

173

174

7 Topological Monodromy

b

is well-defined up to isotopy and conjugation. More precisely, let Mg denote the set of conjugacy classes in Mg (= the mapping class group of ˙g ). Then the conjugacy class hf i 2 Mg to which the mapping class Œf  of f belongs is determined by .M; D; / and is independent of the various choices involved. See [45].

b

b

Definition 7.3. hf i 2 Mg is called the topological monodromy of .M; D; /. Henceforth we will assume g  2. The main effort of Chap. 7 will be devoted to the proof of Theorem 7.1 ([19, 26, 27, 58]). A monodromy homeomorphism f W ˙g ! ˙g of a degenerating family .M; D; / of genus g is a pseudo-periodic map of negative twist. (See Chap. 3 for the definition of a pseudo-periodic map of negative twist.) This is proved by Earle and Sipe [19, Sect. 7] using Teichm¨uller space theory. See [26,58] for preceding important results. We give our topological proof here because it will clarify the topological structure of .M; D; / whose understanding is a key step to the main result of Part II. Definition 7.4. Two degenerating families of genus g, .M; D; / and .M 0 ; D 0 ; 0 /, are topologically equivalent (or have the same topological type) if there exist orintation-preserving homeomorphisms  W M ! M 0 and  W D ! D 0 such that (i) .0/ D 0, and (ii) 0  D  . Let Sg denote the set of all topological types of degenerating families of genus g which are minimal. Then since topologically equivalent degenerating families have the same topological monodromy, the map  W Sg ! Mg sending .M; D; / to its topological monodromy hf i is well-defined. By Theorem 7.1, the image of  is contained in the set of Pg of conjugacy calsses represented by pseudo-periodic maps of negative twist. The main result of Part II is the following.

b

Theorem 7.2. The map  W Sg ! Pg is bijective for g  2. The proof will be given in Chap. 9. By Dehn and Nielsen [53, Sects. 22, 23], Mg Š Aut.1 ˙g /=I nn.1 ˙g /. Thus we have Corollary 7.1. If .M; D; / is minimal, the action of the monodromy on the fundamental group 1 ˙g modulo inner automorphisms determines the topological type of .M; D; /. It has been pointed out by Namikawa and Ueno [49] that the action of the monodromy on the homology group H1 .˙g I Z/ does not necessarily determine the topological type of .M; D; /. The algebraic and combinatorial description of

7.1 Proof of Theorem 7.1

175

the action of the monodromy on 1 ˙g has been given in the case of stable curves by Asada, Matsumoto and Oda [4], for number theoretic purposes. As a special case of Corollary 7.1, we have a topological proof of Corollary 7.2 (Cf. [26, 48]). If .M; D; / is minimal and if the action of monodromy on 1 ˙g is an inner automorphism, then F0 is non-singular. Strictly speaking, the implication of Corollary 7.1 is this: under the condition of Corollary 7.2, F0 is topologically equivalent to a non-sigular fiber. However, this actually assures that F0 is analytically non-singular. See Corollary 8.4. Although the authors could not find any references stating the result clearly, Corollary 7.2 might not be a new result because it can be immediately proved by combining Theorem 1.6 of [26] and Lemma 3.2 of [48]. Corollary 7.3. Given a pseudo-periodic map of negative twist f W ˙g ! ˙g , there exists a degenerating family .M; D; / whose monodromy homeomorphism around F0 is equal (up to isotopy and conjugation) to f . Essentially the same result as Corollary 7.3 is announced by Earle and Sipe [19, p. 92].

7.1 Proof of Theorem 7.1 In what follows, F0 D 1 .0/ will have only normal crossings (nodes). This can be attained by blowing up F0 (without affecting the topological monodromy around it), see [33]. The closure of a connected component of F0  fnodesg is called an irreducible component of F0 . Each irreducible component is a smoothly immersed closed Riemann surface, and F0 is the union of these irreducible components: F0 D 1 [ 2 [    [ s : Let p be a generic point of i (namely, a point which is not a node). Then we can find local coordinates .z1 ; z2 / of M such that .z1 .p/; z2 .p// D .0; 0/ and W M ! D is locally written as .z1 ; z2 / D .z2 /mi : The positive integer mi does not depend on the choice of p and is determined by i . The number of mi is called the multiplicity of i , and the coordinates .z1 ; z2 / satisfying the above condition are called admissible coordinates centered at p. To make the multiplicities explicit, it will be convenient to denote F0 as a divisor: F0 D m1 1 C m2 2 C    C ms s :

176

7 Topological Monodromy

Let p be a crossing point of i and j (the case i D j is not excluded), then we can find local coordinates .z1 ; z2 / of M such that .z1 .p/; z2 .p// D .0; 0/ and W M ! D is locally written as .z1 ; z2 / D .z1 /mj .z2 /mi : Again such coordinates .z1 ; z2 / will be called admissible coordinates centered at p. The plan of the proof is as follows. We decompose a general fiber F which is near to F0 into two parts A and B . A is a union of annuli, and B is a union of compact surfaces. A monodromy homeomorphism f W F ! F will be constructed so that it preserves this decomposition. f acts on B as a periodic map, while it gives twisting to A . This will prove that f is a pseudo-periodic map. The proof of the negativity of twisting requires a closer look, which will be taken in the second half of Chap. 7. At each node p 2 F0 , we will construct a polydisk p . Let denote the union [p p . Then A D F \ , and B D F \ .M  Int. //. We will also construct a system of tubular neighborhoods fNi gsiD1 of the irreducible components f i gsiD1. This is to control the behaviour of f outside .

7.2 Construction of  and fNi gsi D1 Let p be a node of F0 . We fix admissible coordinates .z1 ; z2 / centered at p. Define a polydisk p by

p D f.z1 ; z2 / j jz1 j  "; jz2 j  "g; where " is a sufficiently small number (> 0) chosen independently of p. Let be the union [p p , p running over all the nodes of F0 . Let Ni be a tubular neighborhood of i , i D 1; 2; : : : ; s. Ni has the structure of a D 2 -bundle over i . We choose the projections i W Ni ! i so as to be “compatible” with the fixed admissible coordinates in p , in other words, if p is an intersection point of i and j , the projections i and j should satisfy i .z1 ; z2 / D z1 ;

j .z1 ; z2 / D z2 :

Also Ni must be very thin. To fix the idea, we will assume that the radius of the fiber is  "=2. (See Fig. 7.1) Finally we make admissible coordinates outside compatible with the projections, i ’s, as follows. Let p be a generic point of i , .z1 ; z2 / admissible coordinates centered at p. We deform .z1 ; z2 / by a C 1 -isotopy which moves them in the z1 -direction so that with respect to the resulting coordinates the projection i W Ni ! i is locally written as 1 .z1 ; z2 / D z1 ; and the equation

.z1 ; z2 / D .z2 /mi be still valid.

7.4 Construction of a Monodromy Homeomorphism

Δp

177

Δq

Ni

p

q

Θi

Fig. 7.1 Polydisks p , q , and a tubular neighborhood Ni of i

The new coordinates .z1 ; z2 / will no longer give the complex structure of M , but our argument in Chap. 7 will be in the topological category, so this will not cause any essential difficulty.

7.3 The Decomposition F D A [ B If  ¤ 0 is a complex number sufficiently close to 0, we define A D F \ ;

B D F \ .M  Int. //:

7.4 Construction of a Monodromy Homeomorphism p Let e./ denote exp. 1/. Let ı be a small positive real number. We will construct a continuous family of homeomorphisms h W Fı ! Fe. /ı

.0    2/

starting with h0 D id . Then f D h2 W Fı ! Fı will be a monodromy homeomorphism of .M; D; /. The construction will be done separately on the Aı -part and on the Bı -part. We will begin with Bı -part. For a general (small) , let Bi; denote F \ .Ni  Int. //. Note that B D Ss L L i D1 Bi; (disjoint union). Let i D i  . i \ Int. //. i is a compact surface, generally with boundary. Then Bi; is an mi -fold cyclic covering over L i with the projection i jBi; W Bi; ! L i (mi D the multiplicity of i ) because, around each

178

7 Topological Monodromy

point p 2 L i , Bi; and L i are given by .z2 /mi D 

and z2 D 0

respectively (with respect to admissible coordinates) and the projection i is given by i .z1 ; z2 / D z1 . Now consider the special value  D e./ı. For each  2 Œ0; 2, define h jBi;ı W Bi;ı ! Bi;e. /ı locally by      z2 h .z1 ; z2 / D z1 ; e mi using admissible coordinates. If .z2 /mi D ı, then Œe.=mi / z2 mi D e./ı: Thus h sends Bi;ı into Bi;e. /ı . Note that h jBi;ı respects the 2-disk fibers of i W Ni ! L i because .z1 ; z2 / are compatible with i . Also note that the second coordinates z2 , z02 , in two overlapping systems of admissible coordinates .z1 ; z2 / and .z01 ; z02 /, differ by an mi -th root of i unity: z2 D e.2k=mi /z02 (for some k 2 Z), because zm .z1 ; z2 / D .z01 ; z02 / D 2 D 0 mi .z2 / . Thus the local definition of h jBi;ı is independent of the choice of admissible coodinates, and h jBi;ı is globally defined over Bi;ı . h jBi;ı W Bi;ı ! Bi;e. /ı is a homeomorphism which depends continuously on . Clearly h0 jBi;ı D id , and h2 jBi;ı W Bi;ı ! Bi;ı is a periodic map of order mi , which is a covering translation of Bi;ı ! L i . We define a continuous family of homeomorphisms h jBı W Bı ! Be. /ı to be the disjoint union [siD1 .h jBi;ı /. Then h0 jBı D id , and h2 jBı is a periodic map of order lcm.m1 ; m2 ; : : : ; ms /. The construction on the Bı -part is completed. For a general (small) , let Ap; denote F \ p . If p is an intersection point of i and j , then (with admissible coordinates) Ap; D f.z1 ; z2 / j .z1 /mj .z2 /mi D ; jz1 j  "; jz2 j  "g; and A D [p Ap; (disjoint union). Each Ap; is a disjoint union of annuli. To see this, consider an annulus on p \ i , defined by f.z1 ; 0/ j "0  jz1 j  "g; where "0 D .jj="mi /1=mj . See Fig. 7.2.

7.4 Construction of a Monodromy Homeomorphism

179 Θj

Fig. 7.2 Ap; is a disjoint union of annuli

z2 Ap,x

Θi

p e′

e

z1

Ap; is an mi -fold cyclic covering over this annulus, with the projection .z1 ; z2 / 7! .z1 ; 0/. Thus Ap; is a disjoint union of annuli. The boundary of Ap; is divided into two parts: Ap; \ f.z1 ; z2 / j jz1 j D "g and Ap; \ f.z1 ; z2 / j jz2 j D "g; which we call the i -boundary and the j -boundary, respectively. The i -boundary is a torus link of type .mi ; mj / in @ p  S 3 as well as the j -boundary. The number of the connected components of such a link is gcd.mi ; mj /, which is denoted by m. Ap; is homeomorphic to .i -boundary/  Œ"0 ; ". Thus Ap; consists of m annuli. Now putting  D ı (sufficiently small, for instance  1=.2"/), we will define a homeomorphism h jAp;ı W Ap;ı ! Ap;e. /ı : Let t D t.jz1 j; jz2 j/ be an auxiliary, real-valued function on 0p (D p  f.z1 ; z2 / j jz1 j D jz2 j D "g) defined by ( "jz j 2 if jz1 j  jz2 j  "; j/ t D 2."jz1"jz 1j 1  2."jz2 j/ if "  jz1 j  jz2 j: (See Fig. 7.3) Then the homeomorphism h jAp;ı is defined as follows:       .1  t/ t z1 ; e z2 : h .z1 ; z2 / D e mj mi It is easy to see that h jAp;ı is in fact a homeomorphism of Ap;ı onto Ap;e. /ı , and that it preserves the value of the function t.jz1 j; jz2 j/. Let Cp;e. /ı denote the set of

180

7 Topological Monodromy

Fig. 7.3 "0 D .ı="mi /1=mj ; "00 D .ı="mj /1=mi

(e ′,e )

|z2|

t=0

e

(e ,e )

1 t=– 4

1 t=– 2

t=1 t=

(e,e″ )

3 – 4

e

(0, 0)

|z1|

the center-lines of the annuli Ap;e. /ı : \

f.z1 ; z2 / j jz1 j D jz2 jg  \ 1 : D Ap;e. /ı .z1 ; z2 / j t D 2

Cp;e. /ı D Ap;e. /ı

Then h jAp;ı sends Cp;ı onto Cp;e. /ı homeomorphically. Cleary, h jAp;ı depends continuously on , and h0 jAp;ı D id . On the i -boundary of Ap;ı , where t D 1, we have      z2 ; h .z1 ; z2 / D z1 ; e mi which coincides with h jBi;ı on Ap;ı \ Bi;ı . On the j -boundary, where t D 0, we have      h .z1 ; z2 / D e z1 ; z2 ; mj which coincides with h jBj;ı on Ap;ı \ Bj;ı . Since Aı D [p Ap;ı , we define a continuous family of homeomorphisms h jAı W Aı ! Ae. /ı S as the disjoint union p .h jAp;ı /. h jAı and h jBı coincide on their common boundaries, thus they give a continuous family of homeomorphisms h D h jAı

[

h jBı W Fı ! Fe. /ı

.0    2/;

with h0 D id . The monodromy homeomorphism f W Fı ! Fı is the final stage of h W f D h2 .

7.5 Negativity of Screw Numbers

181

S Let Cı D p Cp;ı be the disjoint union of the center-lines of the annuli. Then f preserves Cı , and f jFı Cı W Fı  Cı ! Fı  Cı is isotopic to a periodic map because Fı  Cı is the union of Bı and open collars Aı  Cı on @Bı and the restriction f jBı D f2 jBı is a periodic map as we saw before. We have proved the following: Lemma 7.1. The monodromy homeomorphism f W Fı ! Fı is a pseudo-periodic map.

7.5 Negativity of Screw Numbers Fix an intersection point p of i and j . We look at the action of f on Ap;ı more closely. Define positive reals "0 , "00 by "0 D



ı "mi

1=mj ;

"00 D



1=mi

ı " mj

:

Then ."; "00 / 2 p is a point of the i -boundary of Ap;ı . Let A0p;ı denote the component of Ap;ı which contains the point ."; "00 /. We will construct a parametrization  W Œ0; 1  S 1 ! A0p;ı : For this, note that for each t 2 Œ0; 1, there is a unique pair of reals .r; s/ satisfying 0  r < "; 0  s < "; r mj s mi D ı;

and t.r; s/ D t;

where t.jz1 j; jz2 j/ is the function introduced just before the definition of h jAp;ı . In fact, the point .r; s/ is the intersection of the curve r mj s mi D ı and the line t.r; s/ D t. (See Fig. 7.3.) Since .r; s/ is uniquely determined by t 2 Œ0; 1, we denote it by .r.t/; s.t//. Then .r.0/; s.0// D ."0 ; "/; .r.1/; s.1// D ."; "00 /; and .r.t/; s.t// determines the curve Œ0; 1 ! A0p;ı joining ."0 ; "/ 2 .j  boundary/ to ."; "00 / 2 .i  boundary/. Let ni , nj be the positive integers given by ni D where m D gcd.mi ; mj /.

mi ; m

nj D

mj ; m

182

7 Topological Monodromy

We define the parametrization  W Œ0; 1  S 1 ! A0p;ı as follows: .t; x/ D .e.2 ni x/r.t/; e.2 nj x/s.t//; where t 2 Œ0; 1 and x 2 S 1 D R=Z. If x D 0, .t; 0/ (0  t  1) is the curve .r.t/; s.t//, and if t D 1, .1; x/ (0  x  1) is a torus knot of type .ni ; nj /, which is the i -boundary of A0p;ı . It is readily seen that  W Œ0; 1  S 1 ! A0p;ı is a homeomorphism. For an integer a, let us denote by Aap;ı the component of Ap;ı containing the point ."; e.2=mi /a "00 /. Then, by the property of a torus link of type .mi ; mj /, Aap;ı D Abp;ı

iff

a  b .mod m/:

The monodromy homeomorphism f jAp;ı .D h2 jAp;ı / W Ap;ı ! Ap;ı satisfies

  a   aC1 ! 2 2 00 " D "; e "00 ; .f jApı / "; e mi mi

so f jAp;ı cyclically permutes the m components of Ap;ı : A0p;ı ; A1p;ı ; : : : ; Am1 p;ı : The m-th power .f jAp;ı /m maps each component to itself. Using the parametrization  W Œ0; 1  S 1 ! A0p;ı ; .f jAp;ı /m W A0p;ı ! A0p;ı is described as follows:    2.1  t/ m .f jAp;ı / .t; x/ D e e.2 ni x/r.t/; mj    2 t m e.2 nj x/s.t/ : e mi m

To simplify this expression, we introduce positive integers k, l by nj k  1 .mod ni /;

0 < k  ni ;

ni l  1 .mod nj /;

0 < l  nj :

and

7.5 Negativity of Screw Numbers

183

(Note that k D ni if and only if ni D 1. Similarly for l.) Then  e

2.1  t/ mj



 2.1  t/ C 2 ni x nj    t l  ; D e 2 ni x C nj ni nj

m

e.2 ni x/ D e

and 

2 t e mi

m



 2 t e.2 nj x/ D e  2 nj x ni    t l  : D e 2 nj x C nj ni nj

Thus we obtain the following description:   l t  .f jAp;ı /m .t; x/ D  t; x C : nj ni nj

( )

Claim. k=ni C l=nj D 1 C 1=.ni nj / This claim is nothing else than Claim (J) just after the statement of Lemma 5.2 (Chap. 5). Put t D 0 in . /, then   l .f jAp;ı / .0; x/ D  0; x C : nj m

( 0)

Put t D 1 in . / and apply Claim, then   k : .f jAp;ı /m .1; x/ D  1; x  ni

( 1)

We have proved the following: Theorem 7.3 (Cf. [17, Sect. 3]). Let p be an intersection of i and j whose multiplicities are mi and mj respectively. Let m denote gcd.mi ; mj /. Then (i) Ap;ı D Fı \ p consists of m annuli, A0p;ı ; A1p;ı ; : : : ; Am1 p;ı ; and the monodromy f jAp;ı W Ap;ı ! Ap;ı cyclically permutes these annuli.

184

7 Topological Monodromy

Fig. 7.4 ni D 3; k D 1; screw number D 1=12 x

nj = 4 l=3

t I

f(I )

(ii) The m-th power .f jAp;ı /m maps each annulus onto itself preserving the i - and j -boundaries. (iii) Set ni D mi =m and nj D mj =m. Define the integers k, l by nj k  1 .mod ni /, 0 < k  ni , and ni l  1 .mod nj /, 0 < l  nj . Then by a certain parametrization a W Œ0; 1  S 1 ! Aap;ı , the action of .f jAp;ı /m W Aap;ı ! Aap;ı is described as   t l .f jAp;ı /m a .t; x/ D a t; x C  ; nj ni nj in particular on the i -boundary, as x 7! x 

k ; ni

x 7! x C

l : nj

and on the j -boundary, as

(iv) s.Aap;ı / D 1=.ni nj /, where s.Aap;ı / is the screw number of f jAp;ı W Ap;ı ! Ap;ı in Aap;ı . Clemens [17, Sect. 3] described the monodromy at a normal crossing point in general dimensions. The above theorem is regarded as a detailed version of his result in dimension two. By (iii) of Theorem 7.3, f jAp;ı is a linear twist. Thus the screw number s.Aap;ı / is the coefficient of t, that is, 1=.ni nj /. See Corollary 2.2. (Beware that the direction of x in Chap. 2 is opposite to the one here. See Fig. 7.4, and remark after the example below.) This proves (iv) of Theorem 7.3.

7.6 Completion of the Proof of Theorem 7.1

185

Example 7.1. Figure 7.4 illustrates the action of .f jAp;ı /m in the case mi D 3, mj D 4; here m D 1. Remark 7.1. The orientation of Ap;ı in Fig. 7.4 is the natural one of the complex curve .z1 /mj .z2 /mi D ı, which coincides with the orientation induced on Ap;ı from the natural orientation of the z1 -axis by the projection Ap;ı ! f.z1 ; 0/ j "0  jz1 j  "g;

.z1 ; z2 / 7! .z1 ; 0/:

See Fig. 7.2.

7.6 Completion of the Proof of Theorem 7.1 Let us return to the decomposition Fı D Aı [ Bı . Let Cı be the disjoint union of center-lines of Aı . As we did in the proof of Lemma 2.1, an admissible system of cut curves is obtained from Cı by deleting inessential curves, and inserting (non-amphidrome or amphidrome) curves between parallel curves which are to be deleted. In the latter case, the screw numbers of the deleted curves (which are negative by Theorem 7.3 (iv)) are inherited by the inserted curves. Thus in the resulting admissible system, each curve has negative screw number. This completes the proof of Theorem 7.1. t u Let .M; D; / be a degenerating family of genus g  2. We assume as before that the central fiber F0 has only normal crossings. Let f D h2 W Fı ! Fı be the monodromy homeomorphism constructed earlier in this chapter. Theorem 7.4. f W Fı ! Fı is in superstandard form, and there exists a pinched covering  W Fı ! F0 which is a generalized quotient of f (in the sense of Chap. 3). Proof. We continue to use the same notation as in the proof of Thorem 7.1. Recall the following decomposition of Fı and F0 : Fı D

[

Ap;ı [

p

F0 D

[ p

[

Bi;ı

i

Ap;0 [

[

L i

i

The pinched covering  W Fı ! F0 will be constructed according to this decomposition, firstly on Bi;ı and then on Ap;ı . As a matter of fact, the construction jBi;ı W Bi;ı ! L i has already been done. It is simply the restriction to Bi;ı of the projection map i W Ni ! i of a tubular neighborhood Ni . In terms of admissible coordinates .z1 ; z2 / centered at a generic point of L i which are compatible with i , the projection jBi;ı is given by

186

7 Topological Monodromy

.z1 ; z2 / D z1 : Also Bi;ı is locally written as .z2 /mi D ı and L i as z2 D 0; mi being the multiplicity of i . Thus jBi;ı W Bi;ı ! L i is an mi -fold cyclic covering. Next let us consider Ap;ı . Let p be an intersection point of i and j whose multiplicities are mi and mj , respectively. The polydisk p was defined by

p D f.z1 ; z2 / j jz1 j  "; jz2 j  "g; where .z1 ; z2 / are admissible coordinates centered at p. Also Ap;ı D Fı \ p D f.z1 ; z2 / j jz1 j  "; jz2 j  "; .z1 /mj .z2 /mi D ıg; and Ap;0 is a closed nodal neighborhood of p consisting of two banks D1 D f.z1 ; 0/ j jz1 j  "g and D2 D f.0; z2 / j jz2 j  "g. We will define jAp;ı W Ap;ı ! Ap;0 as follows: 8  < 0; ".jz2 jjz1 j/  z2 2 D2 "jz j jz j .z1 ; z2 / D  ".jz jjz j/1 z 2  1 2 1 :  ; 0 2 D1 "jz2 j jz1 j

if

jz1 j  jz2 j  ";

if

"  jz1 j  jz2 j:

To see the geometric nature of this map, recall the definition of the parametrization  W Œ0; 1  S 1 ! A0p;ı ; introduced before Theorem 7.3:  .t; x/ D e.2 ni x/r.t/; e.2 nj x/s.t/ ; p where e./ D exp. 1/, t 2 Œ0; 1, x 2 S 1 D R=Z, ni D mi =m and nj D mj =m, m being gcd.mi ; mj /. The pair of functions .r.t/; s.t// satifies r.t/mj s.t/mi D ı

and t.r.t/; s.t// D t;

where the function t.jz1 j; jz2 j/ was defined on p  f.z1 ; z2 / j jz1 j D jz2 j D "g by setting ( "jz j  1 2  if jz1 j  jz2 j  "; 2."jz1 j/ t.jz1 j; jz2 j/ D  21 "jz1 j 1  2."jz2 j/  2 if "  jz1 j  jz2 j: (See Fig. 7.3). Then observing that   1 ".jz2 j  jz1 j/ D 2"  t.jz1 j; jz2 j/ "  jz1 j 2

if jz1 j  jz2 j  ";

7.6 Completion of the Proof of Theorem 7.1

187

and that   ".jz1 j  jz2 j/ 1 D 2" t.jz1 j; jz2 j/  "  jz2 j 2

if "  jz1 j  jz2 j;

we have (  0; 2" 12  t e.2 nj x/ ; .t; x/ D   2" t  12 e.2 ni x/; 0 ;

0  t  12 ; 1 2

 t  1:

Thus the map jAp;ı W Ap;ı ! Ap;0 pinches the center-line t D 1=2 of A0p;0 to p D .0; 0/, and when restricted to the half-open annuli A0p;ı  ft D 1=2g, it becames nj -fold and ni -fold cyclic coverings over the punctured banks D2  fpg and D1  fpg, respectively. Identifying D2 and D1 with the unit disk fz j jzj  1g through the parametrizations z 7! .0; "z/ and z 7! ."z; 0/, define T2 W D2 ! D2 and T1 W D1 ! D1 by setting    .1  jzj/ T2 .z/ D ze mi and

 T1 .z/ D ze

  .1  jzj/ : mj

Recall that the monodromy homeomorphism f D h2 W Fı ! Fı was defined, within p , as follows:       2.1  t/ 2 t z1 ; e z2 ; f .z1 ; z2 / D e mj mi where t D t.jz1 j; jz2 j/. Then it is easy to verify T2  D f

  1 on Ap;ı \ t  2

T1  D f

  1 on Ap;ı \ t  : 2

and

188

7 Topological Monodromy

S Thus if we define the pinched covering  W Fı ! F0 to be the union p .jAp;ı / [ S i .jBi;ı /,  satisfies condition (v) in the definition of a generalized quotient given in Chap. 3. (Note that  is well-defined because jAp;ı and jBi;ı coincide on the intersection Ap;ı \ Bi ı whenever it is non-empty.) The other conditions (i)–(iv) in the same definition are readily checked on  W Fı ! F0 and f W Fı ! Fı , using the results of this Chap. 7. This completes the proof of Theorem 7.4. t u

Chapter 8

Blowing Down Is a Topological Operation

The notation will be the same as in the previous chapter. The following fact is wellknown. Suppose F0 has only normal crossings. Proposition 8.1. Let 0 be an irreducible component of F0 with multiplicity m0 . Let fp1 ; p2 ; : : : ; pk g be the set of intersection points between 0 and the other irreducible components of F0 . Let mi be the multiplicity of the irreducible component which intersects 0 at pi .i D 1; 2; : : : ; k/. Then m1 C m2 C    C mk is divisible by m0 , and the quotient .m1 C m2 C    C mk /=m0 is equal to 0  0 , where 0  0 denotes the self-intersection number of 0 as a 2-cycle in M . Proof. By the definition of the multiplicities, it is clear thatPa general fiber is homologous in M to the divisor-expression of F0 , m0 0 C j ¤0 mj j . Since 0 does not intersect a general fiber, we have 0 0 D 0  @m0 0 C

X

1 mj j A

j ¤0

D m0 0  0 C .m1 C m2 C    C mk /: The proposition follows now easily from this.

t u

We will call an irreducible component  a .1/-curve if it is a smoothly embedded 2-sphere with    D 1. Definition 8.1. F0 is said to be normally minimal if F0 has only normal crossings and every irreducible component  which is a .1/-curve intersects the other irreducible components in more than two points. (If F0 is normally minimal, we cannot blow down any irreducible component without producing a singular point which is not a normal crossing). Corollary 8.1. Suppose F0 has only normal crossings. F0 is normally minimal if and only if the following conditions are satisfied Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5 8, © Springer-Verlag Berlin Heidelberg 2011

189

190

8 Blowing Down Is a Topological Operation

(i) If an irreducible component 0 is an embedded sphere and intersects only one other component i in one point exactly, then the multiplicity m0 of 0 divides the multiplicity mi of i and mi =m0  2. (ii) If an irreducible component 0 is an embedded sphere and intersects the other components only in two points fp1 ; p2 g, then the multiplicity m0 of 0 divides m1 Cm2 and .m1 Cm2 /=m0  2, where mi is the multiplicity of the irreducible component i intersecting 0 at pi , i D 1, 2. Remark 8.1. Corollary 8.1 motivated the definition of a minimal quotient given in Chap. 3. Essentially the same argument as in the proof of Proposition 8.1 can be done if F0 is not assumed to have only normal crossings. In this general case, irreducible components and multiplicities are defined as follows (Cf. [32]). A point p of F0 is called a generic point of multiplicy m if there exist coordinates .z1 ; z2 / around p such that z1 .p/ D z2 .p/ D 0 and .z1 ; z2 / D .z2 /m . It is easy to see that the multiplicity is locally constant, thus constant on a connected component of the set of generic points of F0 . The closure of such a connected component is called an irreducible component of F0 . The meaning of the multiplicity of an irreducible component would be evident. Let F0 D m1 1 C m2 2 C    C ms s be the expression of F0 as a divisor (in the general case). Then by the same reason as in the case of normal crossings, we have the following equalities i  .m1 1 C m2 2 C    C ms s / D 0;

i D 1; 2; : : : ; s:

()

Using ./, we can prove Proposition 8.2. Assume the general fiber F has genus  1. Let i , i D 1; 2, be the irreducible components of F0 which are .1/-curves. Then 1 and 2 do not intersect: 1 \ 2 D ;. Proof. Note that if i \ j ¤ ; and i ¤ j , then i  j > 0, where i and j are any two irreducible components of F0 (see [65, Sect. 20], [45]). We will show that the assumption 1 \ 2 ¤ ; leads to contradiction. Case 1. 1  2 > 1. Putting i D 1 in ./, we have 0 D m1 C m2 1  2 C m3 1  3 C    > m1 C m2 :

(A)

Putting i D 2, we have 0 D m1 2  1  m2 C m3 2  3 C    > m1  m2 : Obviously .A/ contradicts .B/. Case 2. 1  2 D 1. Putting i D 1 in ./, we have

(B)

8 Blowing Down Is a Topological Operation

191

0 D m1 C m2 C m3 1  3 C     m1 C m2 :

(A0 )

in which the equality holds iff 1 does not intersect i , i  3. Putting i D 2, we have 0 D m1  m2 C m3 2  3 C     m1  m2 :

(B 0 )

in which the equality holds iff 2 does not intersect i , i  3. Clearly .A0 / contradicts .B 0 / unless the equalities hold in .A0 / and .B 0 /. Therefore m1 D m2 and, since F0 is connected, F0 consists of 1 and 2 only. i.e. the shape of F0 is very restricted: F0 D m1 C m2 ;

1  2 D 1:

Blow down 2 , then F0 D m10 , where 10 is a smoothly embedded 2-sphere with 10 10 D 0. This implies that the general fiber F is a disjoint union of m 2-spheres, which contradicts the assumption of Proposition 8.2. This completes the proof of Proposition 8.2. t u Let SOg denote the set of all topological types of degenerating families .M; D; / of genus g in which the central fibers F0 are normally minimal. Let O W SOg ! Pg denote the map sending .M; D; / to the topological monodromy around F0 D 1 .0/. Recall that we defined in Chap. 7 a similar set Sg consisting of all topological types of minimal degenerating families of genus g and a similar map  W Sg ! Pg . The purpose of the present chapter is to prove Theorem 8.1. There exists an onto map ˇ W SOg ! Sg such that  ı ˇ D b . Remark 8.2. ˇ W SOg ! Sg is given by blowing down .1/-curves. Proof of 8.1. Let .M; D; / be a degenerating family of genus g. If there are .1/-curves in .M; D; /, we blow down one of them. Generally speaking, there still remain .1/-curves, and/or some irreducible components of 1 .0/ may presently change into .1/-curves after blowing down. Then we blow down one of these .1/-curves. Repeating this process finitely many times, we get a minimal degenerating family .M ; D;  / of genus g. Lemma 8.1. .M ; D;  / is independent of the ordered sequence of the .1/curves according to which they are blown down. Proof. We call the .1/-curves in .M; D; / the .1/-curves of the 1-st generation. They are disjoint by Proposition 8.2. If we blow down them simultaneously, some irreducible components may change into .1/-curves. We call them the .1/-curves of the 2-nd generation. They are disjoint. If we blow down them simultaneously,

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8 Blowing Down Is a Topological Operation

some irreducible components may again change into .1/-curves, which we call the .1/-curves of the 3-rd generation, and so on. This process terminates after finitely many steps, and we get a minimal degenerating family .Mst ; D; st /. We call it the standard minimal degenerating family obtained from .M; D; /. It will be shown that .M ; D;  / is nothing but .Mst ; D; st /. Let .1 ; 2 ; : : : ; k / be the ordered sequence of (appropriately re-indexed) .1/-curves such that .M ; D;  / is obtained from .M; D; / by blowing down i after i 1 for i D 2; : : : ; k. The first 1 must belong to the 1-st generation. Suppose j be the .1/-curve in the sequence which belongs to the 1-st generation and is the nearest to 1 in the sense of the sequence. Then by Proposition 8.2, any .1/-curve in the sequence between 1 and j is disjoint from j . Thus we can change the position of j to the one next to 1 without affecting the final result, so that the sequence .1 ; j ; 2 ; : : : ; j 1 ; j C1 ; : : : ; k / gives the same .M ; D;  /. Repeating this argument and re-indexing the irreducible components if necessary, we get a new sequence giving .M ; D;  /, .1 ; 2 ; : : : ; k /; in which there exists an index i.1/ such that 1 ; 2 ; : : : ; i.1/ are .1/-curves of the 1-st generation, while i.1/C1 ; i.1/C2 ; : : : ; k are not. Note that there are left in M no .1/-curves of the 1-st generation other than 1 ; 2 ; : : : ; i.1/ because if there were any, .M ; D;  / would contain the .1/-curves of the 1-st generation. This contradicts the minimality of .M ; D;  /. Thus 1 ; 2 ; : : : ; i.1/ all are .1/-curves of the 1-st generation. After blowing down 1 ; : : : ; i.1/ , we get a new degenerating family .M.1/ ; D; .1/ / in which .1/-curves of the 2-nd generation may appear in general. In that case, the .i.1/ C 1/-th curve i.1/C1 must be of the second generation, and we can repeat the same argument in the manifold M.1/ as in M .

8 Blowing Down Is a Topological Operation

Proceeding in this way, we get a sequence giving .M ; D;

193  /,

.1 ; 2 ; : : : ; i.1/ ; i.1/C1 ; : : : ; i.2/ ; : : : ; i.n/ / in which 1 ; 2 ; : : : ; i.1/ are of the first generation, and i.l1/C1 ; i.l1/C2 ; : : : ; i.l/ all are .1/-curves of the l-th generation, for l D 2; : : : ; n. This implies that .M ; D;  / is nothing but .Mst ; D; st /, as asserted. t u By Lemma 8.1 we may call .M ; D;  / the minimal degenerating family obtained from .M; D; /. Now let .M; D; / and .M 0 ; D 0 ; 0 / be two degenerating families of genus g which are topologically equivalent. The point in proving Theorem 8.1 is to show that the minimal degenerating families .M ; D;  / and .M0 ; D 0 ; 0 / obtained from .M; D; / and .M 0 ; D 0 ; 0 /, respectively, are also topologically equivalent. To prove this we need two more lemmas and some corollaries. To state Lemma 8.2, let us recall the Milnor fibering from [45]. Let p be a point of F0 D 1 .0/. We consider as a holomorphic function defined around p, Let S" be a 3-sphere of radius " > 0 centered at p, where " is sufficiently small. The Milnor fibering of at p is a fibering over S 1 defined by j j

W S"  K " ! S 1 ;

where K" D S" \ 1 .0/. Its typical fiber is called the Milnor fiber. Let  W M ! M 0 and  W D ! D 0 be orientation-preserving homeomorphisms such that .0/ D 0 and 0  D  . Lemma 8.2. The Milnor fiber of at p 0 D .p/.

at p is homeomorphic to the Milnor fiber of

0

Proof. Let D" be the 4-ball centered at p of radius ". If the complex number c is sufficiently close to 0, the interior of F";c D D" \ 1 .c/ is homeomorphic to the Milnor fiber of at p (see [45]). Suppose " and "0 are two small numbers with 0 < "0 < ". If jcj is small enough, .D"  IntD"0 / \ 1 .c/ is homeomorphic to .ıF";c /  Œ"0 ; ", which is a disjoint union of annuli. Cf. [37]. We take likewise a small 4-ball ı centered at p 0 of radius ı > 0. Suppose ı, " and "0 are chosen so that D" b 1 .ı / b D"0 where A b B means A  Int.B/. See Fig. 8.1.

194

8 Blowing Down Is a Topological Operation y –1(c)

y –1(0) p De ′ Λ–1(Δd) De

Fig. 8.1 Topological invariance of Milnor fibers 0 Setting Fı;.c/ D ı \ . 0 /1 ..c//, we see that each connected component (D an annulus) of .D" Int.D"0 //\ 1 .c/ contains exactly one component (D a circle) 0 of 1 .@Fı;.c/ / and that the two boundary components of the annulus are separated 0 /. Then it follows, by the annulus theorem, that by the component of 1 .@Fı;.c/ 1 0 each component of F";c   .IntFı;.c/ / is homeomorphic to an annulus. Thus F";c 0 is homeomorphic to Fı;.c/ , whose interior is homeomorphic to the Milnor fiber of 0 at p 0 if ı is sufficiently small. This completes the proof of Lemma 8.2 t u

Corollary 8.2. If p .2 F0 / is a generic point of multiplicity m, then p 0 D .p/ .2 F00 / is a generic point of the same multiplicity. Proof. By the definition of a generic point (given after Corollary 8.1), there exist local coordinates .z1 ; z2 / centered at p such that .z1 ; z2 / D .z2 /m around p. Now decompose 0 (which is considered as a holomorphic function around p 0 D .p/) into irreducible factors: 0

with fi .p 0 / D 0; i D 1; : : : ; r;

D f1m1 f2m2    frmr ;

where r is equal to the number of the branches of F00 D . 0 /1 .0/ passing through p 0 . This number r is a topological invariant. (This follows from the fact that a 3-sphere of a small radius centered at p0 intersects F00 transversely in a link of r components. See [45].) Note that only one branch of F0 D 1 .0/, namely z2 D 0, passes through p and that  maps .F0 ; p/ homeomorphically onto .F00 ; p 0 /. Thus r D 1 and we may assume 0

D .f1 /m1 ;

where f1 is irreducible with f1 .p 0 / D 0.

8 Blowing Down Is a Topological Operation

195

The Milnor fiber of 0 at p 0 is a disjoint union of m1 copies of the (connected) Milnor fiber of f 0 at p 0 , while the Milnor fiber of at p is clearly a disjoint union of m open 2-disks. Thus by Lemma 8.2, m1 D m and the Milnor fiber of f1 at p 0 is an open 2-disk. It follows that f1 is non-singular at p 0 (see [45]), which proves that p 0 is a generic point of multiplicity m. t u Recall that an irreducible component of F0 D 1 .0/ is the closure of a connected component of the set of generic points of F0 . The following corollary follows from Corollary 8.2. Corollary 8.3. If  is an irreducible component of F0 , then ./ is an irreducible component of F00 . The multiplicitics of  and ./ are equal. F0 is non-singular if and only if every point of F0 is a generic point of multiplicity 1. Thus we have Corollary 8.4. If F0 D 1 .0/ is non-singular and if .M; D; / is topologically equivalent to .M 0 ; D 0 ; 0 /, then F00 D . 0 /1 .0/ is non-singular. The next lemma will give a topological characterization of a .1/-curve. Working for a while in a more general setting, let  be a compact complex irreducible curve (namely, a closed Riemann surface which may be singular) in a complex manifold M of complex dimension 2. Let p1 ; p2 ; : : : ; pk be the singular points of , and let Di be a small 4-ball centered at pi . Let T be a tubular neighborhood S  of Sk Sk k   i D1 . \ IntDi / in M  i D1 IntDi . We will call N D T [ i D1 Di an admissible neighborhood of . (See Fig. 8.2.) It can be proved that N   is homeomorphic to @N  Œ0; 1/. See [45].

Θ

p1 D1

pk p2 D2

Fig. 8.2 Admissible neighborhood of 

Dk

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8 Blowing Down Is a Topological Operation

Lemma 8.3. An irreducible curve  is a smoothly embedded 2 -sphere with    D ˙1 if and only if N   is simple connected. Proof. The “only if” part is trivial. Conversely we will prove that if N  is simply connected, then  is a smoothly embedded 2-sphere with    D ˙1. Since N   Š @N  Œ0; 1/, the assumption is equivalent to the simple connectivity of @N . If  has self-intersection points or has positive genus, neither  nor @N is simply connected. Thus if @N is simply connected,  is a topologically embedded 2-sphere. Let p1 ; p2 ; : : : ; pk be the singular points of , and let Di , i D 1; : : : ; k, be 4-balls, each centered at pi . Let T be a tubular neighborhood of L   small S S k D   i D1  \ IntDi in M  kiD1 IntDi , as before. Let Ei denote the complement @Di  Int.@Di \ T /, for i D 1; : : : ; k. Then we have @N D @T [ S k i D1 Ei . Case 1. k  2. @T is an orientable S 1 -bundle over L (D a sphere with k holes ' a bouquet of k  1 S 1 s). Thus @T Š L  S 1 , whose identification will be fixed throughout. Firstly, we will see that the inclusion @E1 D @T \ E1 ! @T induces an injective homomorphism 1 .@E1 / ! 1 .@T / between fundamental L  S 1 , and groups. In fact, @E1 is identified with .a boundary component of / since k  2, any non-zero multiple of any boundary component of L is not nullL Secondly we note that the knot @D1 \  is non-trivial in @D1 homotopic in . because p1 is a singular point [45]. Thus the inclusion @E1 ! E1 also induces an injective homomorphisms 1 .@E1 / ! 1 .E1 / [54]. By Van Kampen’s theorem and [36, Theorem 4.3], the homomorphisms i .@T / ! 1 .@T [ E1 / and

i .E1 / ! 1 .@T [ E1 /

are injective. We can repeat the above argument inductively to show that 1 .@T [ E1 [    [ Ei 1 / ! 1 .@T [ E1 [    [ Ei / and i .Ei / ! 1 .@T [ E1 [    [ Ei / are injective, for i D 1; 2; : : : ; k. In particular, 1 .@N / is non-trivial. This contradicts the assumption 1 .@N / Š f1g. Case 2. k D 1. The manifold @N D @T [ E1 is obtained from the 3-sphere @D1 by Dehn-surgery along the knot K D @D1 \ . By the Burau and K¨ahler theorem, K is a (non-trivial) iterated torus knot [45], which has property P , [15]. Thus 1 .@N / 6Š f1g, contradicting the assumption.

8 Blowing Down Is a Topological Operation

197

So far we have proved that if @N is simply connected, then k D 0, i.e.  has no singular points and is a smoothly embedded 2-sphere. In this case, it holds that 1 .@N / Š Z= l, where l D j  j. Thus if @N is simply connected, we have    D ˙1. This completes the proof of Lemma 8.3. t u Corollary 8.5. If  is a .1/-curve, then so is ./. The proof is left to the reader. Remark 8.3. If  is a .1/-curve, the boundary @N of the admissible neighborhood is diffeomorphic to 3-sphere. Completion of the proof of Theorem 8.1 Let .M; D; / and .M 0 ; D 0 ; 0 / be degenerating families of genus g  1. Assuming that they are topologically equivalent under orientation-preserving homeomorphisms .; / W .M; D/ ! .M 0 ; D 0 / such that .0/ D 0 and 0  D  , we will prove that the minimal degenerating families .M ; D;  / and .M0 ; D 0 ; 0 / obtained from .M; D; / and .M 0 ; D 0 ; 0 /, respectively, are topologically equivalent. Let .1 ; 2 ; : : : ; k / be an ordered sequence of .1/-curves giving .M ; D;  /. The operation of blowing down 1 is equivalent to the following one: delete 1 from M and compactify the created end by adding one point. By Corollary 8.5, .1 / is a .1/-curve in M 0 . Thus, by this interpretation, blowing down 1 and .1 / in M and M 0 , respectively, gives topologically equivalent degenerating families .M.1/ ; D; .1/ / 0 0 and .M.1/ ; D 0 ; .1/ /. Repeating this argument for i , i D 1; 2; : : : ; k, we can 0 0 conclude that .M ; D;  / is topologically equivalent to .M.k/ ; D 0 ; .k/ / which 0 0 0 is obtained from .M ; D ; / by blowing down .i /, for i D 1; 2; : : : ; k in 0 0 this order. By Corollary 8.5, .M.k/ ; D 0 ; .k/ / is minimal, and by Lemma 8.1, 0 0 0 0 0 0 .M.k/ ; D ; .k/ / is nothing but .M ; D ;  /. Thus .M ; D;  / is topologically equivalent to .M0 ; D 0 ; 0 / as asserted. We have completed the proof that for g  1 the map ˇ W SOg ! Sg which sends ŒM; D;  to ŒM ; D;   is well-defined. Moreover it is onto because any minimal degenerating family .M ; D;  / can be blown up to an .M; D; / whose central fiber has only normal crossings, [33]. Finally, for g  2, the diagram O SOg ! Pg ? ? ? ?D ˇy y

Sg ! Pg : 

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8 Blowing Down Is a Topological Operation

commutes because blowing down does not affect the topological monodromy around the central fiber. This completes the proof of Theorem 8.1. t u Remark 8.4. For g D 1, we have the commutative diagram: O SO1 ! fconjugacy classes in SL.2; Z/g ? ? ? ?D ˇy y

S1 ! fconjugacy classes in SL.2; Z/g: 

Chapter 9

Singular Open-Book

In this chapter, we will complete the proof of Theorem 7.2. We wish to show that the monodromy correspondence  W Sg ! Pg is bijective for g  2. In Chap. 8, we proved the existence of an onto map ˇ W SOg ! Sg such that the diagram commutes (Theorem 8.1): O SOg ! Pg ? ? ? ?D ˇy y

Sg ! Pg : 

Clearly, the desired bijectivity of  will follow from that of . O To prove that O is bijective, we take a pseudo-periodic map of negative twist, f W ˙g ! ˙g ; and will construct a normally minimal degenerating family of genus g, .M; D;

/

whose topological monodromy coincides with f up to isotopy and conjugation. This will give a well-defined map O W Pg ! SOg ; which will be the inverse of . O

Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5 9, © Springer-Verlag Berlin Heidelberg 2011

199

200

9 Singular Open-Book

The construction proceeds in two steps: The 1-st step (mapping cylinder). We may assume that the pseudo-periodic map f W ˙g ! ˙g is in superstandard form and has the minimal quotient  W ˙g ! S , (Theorem 3.1). Take the disjoint union ˙g  Œ0; 1 [ S and identify each .y; 0/ 2 ˙g  Œ0; 1 with .y/ 2 S . The resulting quotient space C is the mapping cylinder of . C is not in general a manifold. The 2-nd step (Singular open-book). The notion of open-book decomposition of a manifold was introduced by Winkelnkemper [69], and independently by Tamura [61] under the terminology of “spinnable structure”. Our construction here is very similar to theirs, except that the “binding” is a chorizo space instead of a manifold. The “page” is the mapping cylinder C , and the characteristic map fQ W C ! C is the trace of a certain deformation of f W ˙g ! ˙g , which we will now describe. Recall that for each node p of S , we fixed a closed nodal neighborhood Np (Chap. 3), and that both banks of Np were parametrized so that they were identified with fz j jzj  1g, (see the definition of generalized quotient in Chap. 3). Now we take an isotopy Ir W S ! S parametrized by r .0 < r  1/ such that (i) I1 D id S , and (ii) for each closed nodal neighborhood Np and for each bank Dp of Np , parametrized as fz j jzj  1g, we have .Ir j Dp /.z/ D rz. Using fIr g0
0 < r  1:

Note S that f1 D f and that fr commutes with  W ˙g ! S over Sr WD S  pDnode Ir .Np /, i.e. induces the identity of Sr . (N.B. By the definition of a generalized quotient given in Chap. 3, f is a collection of covering translations over S S  pDnode Np .) Thus as r ! 0, this induced map converges to the identity of S . S (But the “limit” of fr , as r ! 0, fails to be continuous along pDnode  1 .p/.) The definition of the characteristic map fQ W C ! C is the following: (

.fr .y/; r/ if c D .y; r/ 2 ˙g  Œ0; 1; r > 0 fQ.c/ D c if c 2 S:

9 Singular Open-Book

201

limit of fr as r ! 0 is not continuous along S As we 1remarked, the 1  .p/, but  .p/ collapses to a point in S . Thus this discontinuity pDnode Q causes no problem and f W C ! C is a homeomorphism satisfying fQjS D idS . Our singular open-book M is constructed first by taking the mapping torus of fQ W C ! C and then collapsing the “sub-torus” corresponding to id S D fQjS W S ! S onto S . More precisely, we set M D Œ0; 2  C = ; where the equivalence relation  is generated by .2; c/  .0; fQ.c//

for 8c 2 C ;

and .; c/  .0; c/

for 8c 2 S; 8 2 Œ0; 2:

This completes the 2-nd step of our construction. It will be convenient to represent a point of M by Œ; y; r ( 2 Œ0; 2, y 2 ˙g , r 2 Œ0; 1) under the equivalence relation generated by (i) Œ; y; 0 D Œ; y 0 ; 0 iff .y/ D .y 0 /, (ii) Œ2; y; r D Œ0; fr .y/; r for r > 0, (iii) Œ; y; 0 D Œ0; y; 0. S can be identified with a subspace of M by the correspondence: S 3 .y/

! Œ0; y; 0 2 M :

Define @M to be fŒ; y; r 2 M j r D 1g, and M to be M  @M . Also define a map W M ! C by p .Œ; y; r/ D r exp. 1/: We set

D

jM .

Theorem 9.1. M admits a structure of complex manifold of complex dimension two such that W M ! D is a surjective proper holomorphic map satisfying 1

.0/ D S , jM  W M  ! D  is a smooth fiber bundle with fiber ˙g , where M  D M  1 .0/ and D  D D  f0g, and (iii) the topological monodromy coincides with f up to isotopy and conjugation. (i) (ii)

Proof. We shall construct for each point p 2 S a small complex coordinate neighborhood Wp . M / containing p. There are two cases to be considered: Case A. p is a node. Case B. p is a generic point.

202

9 Singular Open-Book

First we consider Case A. Suppose p is a node, and is an intersection point of two irreducible components, say 1 and 2 , of S. Let m1 and m2 be the multiplicities of 1 and 2 , respectively. Set m D gcd.m1 ; m2 / and n1 D m1 =m, n2 D m2 =m. Remember that a closed nodal neighborhood Np of p has been fixed (Chap. 3). We define a subspace Wp of M by Wp D fŒ; y; r j .y/ 2 Np g; where Np is the (open) nodal neighborhood corresponding to Np and  W ˙g ! S is the pinched covering. Set Wp .0 / D fŒ; y; r 2 Wp j  D 0 .const./g; then Wp .0/ is identified with C \ Wp . By the definition of generalized quotient (Chap. 3), the outer-most face of Wp .0/, i.e. @Wp .0/ WD fŒ0; y; r 2 Wp j r D 1g; consists of m annuli, A1 ; A2 ; : : : ; Am ; which are permuted by f cyclically: f .A˛ / D A˛C1 ;

˛ D 1; 2; : : : ; m  1;

and f .Am / D A1 . The m-th power f m W A˛ ! A˛ is a linear twist with the screw number 1=n1 n2 . Also by the definition of generalized quotient (Chap. 3), there exist parametrizations of the banks of Np , fz j jzj < 1g ! D1 ; fz j jzj < 1g ! D2 ; such that if we identify Di with fz j jzj < 1g through these parametrizations .i D 1; 2/ and define Ti W Di ! Di .i D 1; 2/ by

! p 1 .1  jzj/ ; T1 .z/ D z exp m2

9 Singular Open-Book

203

! p 1 T2 .z/ D z exp .1  jzj/ ; m1 then the following identities hold: Ti  D  f

on  1 .Di / \ @Wp .0/; i D 1; 2:

Consider the unit (open) polydisk Int in C2 : Int D f.z1 ; z2 / j jz1 j < 1; jz2 j < 1g: Let be a complex number (with 0  j j < 1) and let A denote the fiber of .z1 /m2 .z2 /m1 W Int ! C over : A D f.z1 ; z2 / 2 Int j .z1 /m2 .z2 /m1 D g: p Set e./ D exp. 1/ and take a small positive real ı. As we did in Chap. 7, we define a continuous family of homeomorphisms h W Aı ! Ae. /ı ; as follows:

0    2

      .1  t/ t z1 ; e z2 ; h .z1 ; z2 / D e m2 m1

where t D t.jz1 j; jz2 j/ is the real-valued function t W Int ! Œ0; 1 introduced in Chap. 7 (See Fig. 7.3): 8 1  jz2 j ˆ ˆ < 2.1  jz1 j/ tD 1  jz1 j ˆ ˆ :1  2.1  jz2 j/

if

jz1 j  jz2 j < 1;

if

1 > jz1 j  jz2 j:

(Note that " in the definition of t in Chap. 7 is here replaced by 1). The final stage of h is the monodromy f 0 D h2 W Aı ! Aı of .z1 /m2 .z2 /m1 W Int ! C: As we proved in Chap. 7, Aı consists of m annuli, permuted cyclically by f 0 W Aı ! Aı , and on each annulus, the m-th power of f 0 is a linear twist with the screw number 1=n1 n2 . (Theorem 7.3). Moreover, let  0 jAı W Aı ! A0

204

9 Singular Open-Book

denote the restriction, to Aı , of the map  0 W Int ! A0 defined in the proof of Theorem 7.4:  8 z2 2 jjz1 j < 0; jz1jz  2 D20 j jz j 1 2   0 .z1 ; z2 / D  : jz1 jjz2 j  z1 ; 0 2 D 0 1jz2 j

jz1 j

1

if jz1 j  jz2 j < 1; if 1 > jz1 j  jz2 j:

Here the " in Chap. 7 is replaced again by 1, and the two banks of A0 are denoted by D10 and D20 ( 0 jAı W Aı ! A0 is a pinched covering). If we identify these banks with fz j jzj < 1g through the parametrizations z 7! .z; 0/ and z 7! .0; z/, then we have . 0 jAı / f 0 D T1 . 0 jAı / on . 0 /1 .D10 / \ Aı and

. 0 jAı / f 0 D T2 . 0 jAı /

on . 0 /1 .D20 / \ Aı :

(See the proof of Theorem 7.4). Therefore, there exist homeomorphisms H j@Wp .0/ W @Wp .0/ ! Aı and H jNp W Np ! A0 satisfying 1.  0 .H j@Wp .0/ / D .H jNp / , 2. f 0 .H j@Wp .0/ / D .H j@Wp .0/ / f , 3. Ti .H jNp / D .H jNp / Ti ;

and

i D 1; 2.

Next, for each r with 0  r  1, let r0 W Aı ! Arı be the projection which satisfies the following conditions: .r/ .r/ If .z1 ; z2 / 2 Aı and r0 .z1 ; z2 / D .z1 ; z2 /, then .r/

.r/

(a) t.jz1 j; jz2 j/ D t.jz1 j; j z2 j/, .r/

.r/

(b) j.z1 /m2 .z2 /m1 j D rı, (c) arg.z1 / D

.r/ arg.z1 /,

and .r/

arg.z2 / D arg.z2 /.

9 Singular Open-Book

205

Note that these conditions uniquely determine the projection r0 W Aı ! Arı : If 0 < r  1,

r0 W Aı ! Arı

is a homeomorphism (in particular, 10 D identity of Aı ), and if r D 0, 00 W Aı ! A0 coincides with

 0 jAı W Aı ! A0 :

It can be easily checked that . 0 jArı / r0 D . 0 jAı /: We set V D f.z1 ; z2 / 2 Int j j.z1 /m2 .z2 /m1 j  ıg and define V .0 / by V .0 / D f.z1 ; z2 / 2 V j arg.z1 /m2 .z2 /m1 D 0 g [ A0 : Let us extend .H j@Wp .0/ / [ .H jNp / W @Wp .0/ [ Np ! Aı [ A0 to H jWp .0/ W Wp .0/ ! V .0/ by

H.Œ0; y; r/ WD r0 .H.Œ0; y; 1//;

where H.Œ0; y; 1/ D .H j@Wp .0/ /.y/. This H jWp .0/ is a homeomorphism: Wp .0/ ! V .0/: Let

.t; r/ W Œ0; 1  .0; 1 ! Œ0; 1 be a real-valued function defined by

.t; r/ D

8 ˆ ˆ <0 1 ˆr

ˆ :1

 t

1r 2



0t  1r 2 1Cr 2

t

1r ; 2  1Cr ; 2

 t  1:

206

9 Singular Open-Book

Using this function, we define a homeomorphism hQ  W V .0/ ! V ./ as follows: If .z1 ; z2 / 62 A0 , set         .1  .t; r// z1 ; e

.t; r/ z2 ; hQ  .z1 ; z2 / D e m2 m1 where t D t.jz1 j; jz2 j/ and r D j.z1 /m2 .z2 /m1 j=ı > 0, and if .z1 ; z2 / 2 A0 , set hQ  .z1 ; z2 / D .z1 ; z2 /: It is not difficult to see that hQ  is in fact a homeomorphism. We are now in a position to define the desired homeomorphism H W Wp ! V which extends H jWp .0/ W W p .0/ ! V .0/ and gives complex coordinates to Wp .D Wp  @Wp /. The definition is as follows: H.Œ; y; r/ WD hQ  H.Œ0; y; r/: To check the well-definedness of H , we will use a lemma, in which we identify each bank Di0 (i D 1 or 2) of A0 with D D fz j jzj < 1g, and let Ir W D ! D denote the map Ir .z/ D rz. Lemma 9.1. Let .z1 ; z2 / be a point of Arı , and put z D  0 .z1 ; z2 / 2 A0 D D10 [D20 . If z 2 Di0 (i D 1 or 2), then ( z Q  h2 .z1 ; z2 / D Ir Ti Ir1 .z/ 0

if

r  jzj  1;

if

0  jzj  r:

Proof. Suppose z 2 D10 . Then t D t.jz1 j; jz2 j/  1=2, and by the definition of  0 .z1 ; z2 / we can check jzj D 2t  1. We have  hQ 2 .z1 ; z2 / D z e 0



2 .1  .t; r// m2



9 Singular Open-Book

D

8 ˆ
207



2 m2



if   1 1 1 t if  2 r 2

1Cr  t  1; 2 1 1Cr t  ; 2 2

8
t u

Now let us prove the well-definedness of H . It will be sufficient to check it on each of the three generating relations for Œ; y; r: (i) H.Œ; y; 0/ D H.Œ; y 0 ; 0/ if .y/ D .y 0 /. In fact, H.Œ; y; 0/ D hQ  H.Œ0; y; 0/ D hQ  00 .H j@Wp .0/ /.y/ D hQ .H jNp /.y/ D .H jNp / .y/; because .H jNp / .y/ 2 A0 . Thus (i) follows. (ii) H.Œ2; y; r/ D H.Œ0; fr .y/; r/ for r > 0. First we will prove  0 H.Œ2; y; r/ D  0 H.Œ0; fr .y/; r/

.2 A0 /:

( )

In fact, noting that  0 r0 H.Œ0; y; 1/ D  0 .H j@Wp .0/ / D .H jNp / .y/; and using Lemma 9.1, we have  0 H.Œ2; y; r/ D  0 hQ2 H.Œ0; y; r/ D  0 hQ2 r0 .H Œ0; y; 1/ ( .H jNp / .y/ if r  j.H jNp / .y/j  1 D 1 Ir Ti Ir .H jNp / .y/; .i D1 or 2/ if 0  j.H jNp / .y/j  r ( .H jNp / .y/ D .H jNp / Ir Ti Ir1 .y/

208

9 Singular Open-Book

D .H jNp /  fr .y/ D  0 .H j@Wp .0/ / fr .y/ D  0 r0 .H j@Wp .0/ / fr .y/ D  0 H.Œ0; fr .y/; r/: Thus ( ) is verified. Note that . 0 jArı / W Arı ! A0 is a covering map except over one point .0; 0/, and that the correspondences y 7! H.Œ2; y; r/ and y 7! H.Œ0; fr .y/; r/ give two homeomorphisms @Wp .0/ ! Arı which coincide when r D 1; H.Œ2; y; 1/ D e h2 H Œ0; y; 1 D f 0 .H j@Wp .0/ /.y/ D .H j@Wp .0/ / f .y/ D H.Œ0; f .y/; 1/: Moreover, these homeomorphisms continuously change, depending on r > 0. Therefore, the identity ( ) implies in fact that H.Œ2; y; r/ D H.Œ0; fr .y/; r/ for r .0 < r  1/. This proves (ii). (iii) H.Œ; y; 0/ D H.Œ0; y; 0/. This is trivial because both sides are equal to .H jNp / .y/. (See the proof of (i)). This completes the proof of well-definedness of H W Wp ! V . The proof that H is a homeomorphism is left to the reader. By the definition of

W Wp ! C, .Œ; y; r/ D r e./:

On the other hand, if H.Œ; y; r/ D .z1 ; z2 /; then j.z1 /m2 .z2 /m1 j D rı because H.Œ; y; r/ 2 Arı , and also we have arg.z1 /m2 .z2 /m1 D  because H.Œ; y; r/ D hQ  H.Œ0; y; r/ 2 V ./. Thus if we define z01 D ı 0 z1 and z02 D ı 0 z2 , where ı 0 D .ı/1=.m1 Cm2 / , then with these coordinates .z01 ; z02 / on , we have the commutative diagram: H

Wp ! Int ? ? ? ?.z0 /m2 .z0 /m1 y y 1 2 C ! C: D

9 Singular Open-Book

209

The complex coordinates .z01 ; z02 / pulled back to Wp by H are the desired ones. This completes the (somewhat lengthy) consideration of Case A in the proof of Theorem 9.1. Next, we consider Case B where p 2 S is a generic point of S . Suppose p is on an irreducible component 1 whose multiplicity is m1  1. Let U be a sufficiently small open disk-neighborhood of p in 1 . We may assume that U does not contain any node of S and intersects at most one closed nodal neighborhood Np0 in S . For a small positive real r0 > 0, define Wp Wp

.r0 /

.r0 /

. M / by

D fŒ; y; r j .y/ 2 U; r  r0 g:

.r0 /

.r0 /

j r D r0 g/ consists of m1 copies The boundary @Wp .D fŒ; y; r 2 Wp of U , denoted by UQ 1 , UQ 2 , : : :, UQ m , and f W ˙g ! ˙g permutes them cyclically. .r0 /

If r0 is sufficiently small, Wp is disjoint from the “twisting region” of Wp0 and f m1 jUQ˛ W UQ ˛ ! UQ ˛ is the identity (˛ D 1; 2; : : : ; m). See Fig. 9.1. From this, it .r /

.r0 /

.r0 /

follows that Wp 0 .D Wp  @Wp / is homeomorphic to an open 4-disk if r0 is small enough. In Case A we constructed complex coordinates .z01 ; z02 / in Wp0 such that jWp0 D 0 m2 0 m1 .z1 / .z2 / . (We assume that the bank of Np0 on the irreducible component 1 is given by z02 D 0.) Since 1 is a closed oriented surface, we can put a complex structure on 1 , and may assume z01 gives the complex coordinate on Wp0 \ 1 . Let z001 be a complex coordinate on U that is compatible with the complex structure of 1 . The coordinate change

Fig. 9.1 Construction of complex coordinates

210

9 Singular Open-Book

z001 D z001 .z01 /

on

U \ Wp 0

is bi-holomorphic. .r / Let us define complex coordinates .z001 ; z002 / on Wp 0 as follows: .r / If q 2 .Wp 0  Wp0 /, then put .z001 .q/; z002 .q// D .z001 ..q//;

.q/1=m1 /;

.r /

and if q 2 Wp0 \ Wp 0 , then put .z001 .q/; z002 .q// D .z001 .z01 ; z02 /;

.q/1=m1 /;

where z001 .z01 ; z02 / stands for the composition q D .z01 ; z02 / 7! .z01 ; 0/ 7! z01 7! z001 .z01 /

( )

where the first arrow stands for ”projection” and the third arrow stands for “coordinate change”. Since the composition ( ) converges to the projection  W q 7! .q/, as .q/ .2 Np0 \ 1 / approaches the boundary of the bank Np0 \ 1 , the above two .r / cases continuously match up to give complex coordinates .z001 ; z002 / on Wp 0 . The .r0 / 0 0 00 00 coordinate change between .z1 ; z2 / (on Wp0 ) and .z1 ; z2 / (on Wp ) is easily seen to be biholomorphic. The function jWp is given by D .z002 /m1 in terms of the coordinates .z001 ; z002 /. Case B is completed. In this way we have constructed complex coordinates on a neighborhood of each point of S . The coordinate changes among them are easily checked to be biholomorphic. Thus we have constructed a complex structure on an open neighborhood W of S in M with which jW W W ! C is holomorphic. Evidently, there is a (shrinking) embedding h W M ! W into W such that the diagram commutes (with a small real  > 0): hW,!

M ! ? ? y

W ? ?. y

jW /

D ! D: 

Pulling back the complex structure of W to M , we can make M a complex manifold of complex dimension two. The conditions (i), (ii) and (iii) in the statement of Theorem 9.1 are readily verified. This completes the proof of Theorem 9.1. u t

9.1 Completion of the Proof of Theorem 7.2

211

9.1 Completion of the Proof of Theorem 7.2 We will define a map

O W Pg ! SOg ;

which is to be the inverse to the monodromy correspondence O W SOg ! Pg : Let hf i be any element of Pg . By Theorem 3.1, f W ˙g ! ˙g is isotopic to an

f 0 W ˙g ! ˙g

in superstandard form. Let  W ˙g ! S be its minimal quotient. By the singular open-book construction starting from f 0 and , we obtain a degenerating family .Mf ; Df ; f / whose monodromy coincides with f up to isotopy and conjugation (Theorem 9.1). This .Mf ; Df ; f / is normally minimal because  W ˙g ! S is the minimal quotient (Corollary 8.1). Suppose hf1 i D hf2 i, i.e. f1 coincides with f2 up to isotopy and conjugation. Then by Theorem 4.2, there exist a homeomorphism h W ˙g ! ˙g and a numerical homeomorphism H W S Œf1  ! S Œf2  such that the diagram commutes: ˙g ? ? 1 y

h

! ˙g ? ? y 2

S Œf1  ! S Œf2  H

In other words, their minimal quotients 1 W ˙g ! S Œf1 

212

9 Singular Open-Book

and 2 W ˙g ! S Œf2  are topologically equivalent. If f10 and f20 are the corresponding superstandard forms, then by Theorem 4.2 (iii), f10 D h1 f20 h: Since the construction of a singular open-book made in this chapter is purely topological and has no topological ambiguity once the initial data f 0 W ˙g ! ˙g (in superstandard form) and  W ˙g ! S are given, we conclude that the degenerating families .M1 ; D1 ; 1 / and .M2 ; D2 ; 2 / resulting from f10 and f20 respectively are topologically equivalent. Thus sending hf i 2 Pg to the topological equivalence class ŒMf ; Df ; f  of .Mf ; Df ; f /, we obtain a well-defined map O W Pg ! SO g : Lemma 9.2. O O D id. This is clear because the topological monodromy of .Mf ; Df ; hf i (Theorem 9.1). To prove the bijectivity of , O it remains to show

f

/ is equal to

Lemma 9.3. O O D id. Proof. Let ŒM; D;  be any element of SOg . By Theorem 7.4, the monodromy homeomorphism f W Fı ! Fı ; constructed in Chap. 7, is a pseudo-periodic map in superstandard form and there exist a pinched covering  W Fı ! F0 which is a generalized quotient of f . This  W Fı ! F0 is actually the minimal quotient because F0 is normally minimal (cf. Corollary 8.1.). By essentially the same argument as was done in the proof of Theorem 9.1 (especially in Case A), M is shown to have the structure of a singular open-book (minus the boundary) which is constructed starting from the data f W Fı ! Fı and  W Fı ! F0 :

9.2 Characterization of the Triples .S; Y; c/ That Come from Pseudo-periodic Maps

213

Therefore, ŒM; D;  coincides with ŒMf ; Df ;

f

 D O .hf i/ D O .ŒM; O D;

This proves O O D id.

/: t u

As we remarked at the beginning of Chap. 9, the bijectivity of O W SOg ! Pg (just proved) implies the bijectivity of  W Sg ! Pg . Thus Theorem 7.2 is completely proved. t u Recall the diagram b  SO g ! Pg ? ? ? ?D ˇy y Sg ! Pg 

from Chap. 8. Now we have proved that O and  are bijective maps. This readily implies Proposition 9.1. The map ˇ W SOg ! Sg is bijective.

9.2 Characterization of the Triples .S; Y; c/ That Come from Pseudo-periodic Maps Let S be a connected numerical chorizo space which satisfies the minimality condition (Chap. 4); Y the decomposition diagram of S (Chap. 6), namely, a weighted graph whose vertices (resp. edges) are in one-to-one correspondence to the bodies (resp. arches) of S . Let c 2 HW1 .Y / be a class in the weighted cohomology group of Y (Chap. 6). In this paragraph, we will give a necessary and sufficient condition under which the triple .S; Y; c/ comes from a pseudo-periodic map f W ˙g ! ˙g of negative twist, of a surface of a given genus g  2. There are several necessary conditions for this that we have already proved: 1. Let 0 be any irreducible component of S , fp1 ; p2 ; : : : ; pk g the set of the intersection points among 0 and the other irreducible components. Let mi be the multiplicity of the irreducible component which intersects 0 at pi .i D 1; 2; : : : ; k/, and let m0 be the multiplicity of 0 . Then m0 divides m1 C m2 C    C mk (see Proposition 3.1). 2. Let e be an edge of Y . Then the weight of e is equal to the gcd of the successive multiplicities on the arch corresponding to e (see Corollary 6.2. (i)). 3. Let v be a vertex of Y . Suppose the core part P0 of the body corresponding to v is contained in an irreducible component 0 . Let m0 be the multiplicity of 0 . Let m1 ; m2 ; : : : ; mk be the multiplicities that have the same meaning as in (1). If P0 has genus 0, then the weight of v is equal to gcd .m0 ; m1 ; : : : ; mk /. If P0 has genus  1, then the weight of v divides gcd .m0 ; m1 ; : : : ; mk / (see Corollary 6.2 (ii) and (iii).)

214

9 Singular Open-Book

4. Let W0 be the gcd of the weights of all vertices of Y . Then the homomorphism c W H1 .Y W Z/ ! Z=W0 determined by c 2 HW1 .Y / is onto. (See Lemma 6.6.) 5. Let S D m1 1 Cm2 2 C  Cms s be the expression of S as a (formal) divisor. Let Pi be a part of i (cf. Chap. 3). Then 2  2g D

s X

mi .Pi /

.< 0/

i D1

where .Pi / is the Euler characteristic of Pi . Theorem 9.2. Given a triple .S; Y; c/ satisfying the above conditions .1/  .5/, there exists a pseudo-periodic map f W ˙g ! ˙g of negative twist whose minimal quotient S Œf  and its decomposition diagram are topologically equivalent to S and Y in such a way that the cohomology classes cŒf  and c naturally correspond. More precisely, let f W S Œf  ! Y Œf  and  W S ! Y denote the collapsing maps. Then there exist a numerical homeomorphism H W S Œf  ! S and a weighted isomorphism W Y Œf  ! Y such that the diagram f

S Œf  ! Y Œf  ? ? ? ? Hy y S

! 

Y

commutes and  .c/ D cŒf . The proof is indirect in the sense that it applies our results of Part II on degenerating families of Riemann surfaces of genus g. We will start with Winters’ existence theorem restated in our context. Theorem 9.3 (Winters [70, Corollary 4.3]). Let S D m1 1 C m2 2 C    C ms s be a numerical chorizo space satisfying the above condition (1). Then there exist a complex surface M and a proper holomorphic map WM !D onto an open unit disk D such that

1

.0/ has normal crossings and such that

m1 1 C m2 2 C    C ms s is the divisor expression of

1

.0/.

9.2 Characterization of the Triples .S; Y; c/ That Come from Pseudo-periodic Maps

215

Outline of the proof. From topological viewpoint, Winters’ argument may be interpreted as follows: First one embeds i (considered as a Riemann surface) in a certain complex surface Ni . In case i has no self-intersection points, the description of Ni is easy; Ni is a holomorphic bundle over i whose fiber is biholomorphically homeomorphic to C and which admits a cross section (this is the embedded image of i ). The self-intersection number of i in Ni is 0 1 1 @X mj j  i A ;  mi j ¤i

where j i denotes the number of the nodes of S which are contained in i \j . The above number is an integer because of Condition (1). Also this condition assures the existence of such an Ni that admits a holomorphic map i W Ni ! C whose zero-set i1 .0/ (as a divisor) is expressed as mi i C

r.i / X

.i / .i /

mk ck ;

kD1 .i / where c1.i / , c2.i / , : : :, cr.i / are the fibers of the bundle Ni ! i over the intersection .i /

.i /

.i /

.i /

points, p1 , p2 , : : :, pr.i / , of i with the other irreducible components, mk .i /

being the multiplicity of the irreducible component which meets i at pk (k D 1; 2; : : : ; r.i /). See Fig. 9.2. In case i has self-intersection points, Ni is an immersed image of a holomorphic C-bundle NQ i over the normalized Q i (D i with the self-intersections separated). NQ i admits a cross section whose self-intersection number is equal to

(i)

C1

(i) C2 . . .

(i )

Cr (i)

Θi

Fig. 9.2 Other components intersect i transversely

Ni

216

9 Singular Open-Book

Fig. 9.3 i has self-intersections

0 1 1 @X  mj j  i A  2Ki ; mi j ¤i

Ki denoting the number of the self-intersection points of i . The reason for the appearance of 2Ki is that each (transverse) self-intersection point of i contributes 2 to the self-intersection number i  i in Ni . (See Fig. 9.3) Similarly to the case of a non-singular i , one can adjust the construction of the Ni so that it has a holomorphic map i W Ni ! C whose divisor i1 .0/ consists of .i / .i / .i / .i / .i / .i / mi i and m1 c1 C    C mr.i /cr.i / , where c1 , : : :, cr.i / have the same meaning as before. At a self-intersection point of i , i is locally expressed as i

D .z1 z2 /mi

with appropiate coordinates .z1 ; z2 /. Now we have obtained the parts (one for each i ) to construct W M ! D. To get M , one has only to do “plumbing” with these parts N1 ; N2 ; : : : ; Ns ; i.e. glue Ni and Nj at each intersection point of i and j (in S ) so that the fiber and the base coordinates are interchanged at the time of gluing. At an intersection point of i and j , i

W Ni ! C

is locally expressed as i

D .z1 /mj .z2 /mi

with appropiate coordinates .z1 ; z2 /. Thus one can make the plumbing so that i ’s are patched together to give a holomorphic map W M ! C. We can easily modify

9.2 Characterization of the Triples .S; Y; c/ That Come from Pseudo-periodic Maps

217

M and to obtain a surjective, proper and holomorphic map W M ! D whose fiber over 0, 1 .0/, coincides with the numerical chorizo space S (as a divisor). This completes the outline of the proof of Theorem 9.3. t u Addendum 1 In the above construction, we may assume that the number of the connected components of the general fiber of i W Ni ! C is equal to .i /

.i /

.i /

gcd .mi ; m1 ; m2 ; : : : ; mr.i / /; for each i D 1; 2; : : : ; s. N.B. The general fiber of

i

W Ni ! C is non-compact if r.i /  1.

Proof. Let 0 be any irreducible component of S ; m0 its multiplicity. Suppose 0 intersects the other irreducible components in r points p1 ; p2 ; : : : ; pr : Let m 1 ; m 2 ; : : : ; mr be the multiplicities of the irreducible components which intersect 0 in p1 ; p 2 ; : : : ; p r ; respectively. First suppose 0 has no self-intersection points, in which case, N0 can be constructed as follows. Take an open disk D0 on 0 having coordinate z1 and containing p1 ; p2 ; : : : ; pr : The values of the z1 -coordinate for p1 ; p2 ; : : : ; pr in D0 are supposed to be a1 ; a2 ; : : : ; ar : Let D1 be another open disk on 0 such that D1 D0 . Suppose C has coordinate z2 . The bundle N0 is constructed by holomorphically pasting D1  C and .0  D0 /  C along .D1  D0 /  C so that the holomorphic maps 0

D .z1  a1 /m1 .z1  a2 /m2    .z1  ar /mr .z2 /m0 W D1  C ! C

218

and

9 Singular Open-Book

00

D .z2 /m0 W .0  D0 /  C ! C

coincide on .D1  D0 /  C. (The condition that m0 divides m 1 C m 2 C    C mr assures that the fibers of

0

W D1  C ! C

make a trivial foliation in .D1  D0 /  C, each leaf covering D1  D0 just once.) If N0 is so constructed, the general fiber of the resulting holomorphic map 0

.D

0

[

00

/ W N0 ! C

has gcd .m0 ; m1 ; : : : ; mr / number of components. (cf. Proposition 4.2.) This completes the proof when 0 has no self-intersection points. If it has selfintersection points, the proof is essentially the same. t u

9.3 Completion of the Proof of Theorem 9.2 Note that the general fiber of W M ! D constructed by Theorem 9.3 is not necessarily connected. But just as in the proof of Theorem 7.4, we can construct a pinched covering  W Fı ! F0 D S; from the general fiber Fı D 1 .ı/ to the central fiber F0 D 1 .0/ (identified with S ), where ı .2 D/ is a small positive real. Let ARCH 0 be an arch of S . Then by the argument before Lemma 4.2,  1 .ARCH 0 / consists of m annuli, m being the gcd of the multiplicities of the successive irreducible components on ARCH 0 . By Condition (2), this number m is equal to the weight of the edge .ARCH 0 / of Y . Let BDY 0 be a body of S . Let P0 be the core part of BDY 0 , 0 the irreducible component containing P0 . Let m0 ; m 1 ; : : : ; m k have the same meaning as in Condition (3). By Addendum 1, we may assume that the number of the connected components of  1 .BDY 0 / is equal to gcd .m0 ; m1 ; : : : ; mk /: If P0 has genus 0, this number coincides with the weight of the vertex .BDY 0 / of Y . (Condition (3).) If P0 has genus  1, the weight of the vertex .BDY 0 / divides

9.3 Completion of the Proof of Theorem 9.2

219

gcd .m0 ; m1 ; : : : ; mk /: We will change M and so that the number of the connected components of  1 .BDY 0 / is equal to the weight, say w, of .BDY 0 /: We can take a pair of simple closed curves C1 and C2 on P0 cutting each other transversely once, because P0 has genus  1. Cut open N0 . M / along C1  C (which is the bundle over C1 induced from the C-bundle N0 ! 0 ) then reglue the two copies .C1  C/0 and .C1  C/1 by the following rotation of the fibers: p .z1 ; z2 / 7! .z1 ; exp.2w 1=m0 /z2 /; Since w divides m0 and is locally expressed as holomorphic map

.z1 ; z2 / 2 C1  C:

WM !D D .z2 /

m0

at a generic point of 0 , 0

naturally induces a

W M 0 ! D0

of the modified complex surface M 0 . (NB. M and M 0 are diffeomorphic.) It is easy to see that, in M 0 ,  1 .BDY 0 / has w number of connected components. Proceeding in this way on each body of S , we may assume that the number of connected components of  1 .BDY  / is equal to the weight of the vertex .BDY  / of Y , for each . Number the connected components of  1 .BDY  / cyclically for each BDY  . Let ARCH 0 be any arch of S . Let BDY 0 and BDY 1 be two bodies which ARCH 0 connects (Possibly BDY 0 D BDY 1 ). Now a cochain c representing the cohomology class c 2 HW1 .Y / indicates which connected components of  1 .BDY 0 / should be joined to which ones of  1 .BDY 1 / by the annuli over ARCH 0 . (Cf. Lemma 6.5). We can realize this joining by modifying M and as follows: Let l be a simple arc on ARCH 0 joining the two end-circles of ARCH 0 , and let C1 be a simple closed curve on an irreducible component of ARCH 0 (or on an attaching bank) which cuts l transversely once. Then we do the same process as before of cutting open M along C1  C and re-glue the two copies .C1  C/0 and .C1  C/1 through appropiate rotation of the “fibers” C. Proceeding in this way on each arch in S , we will get the modified complex surface, again denoted by M , and a proper, surjective, and holomorphic map W M ! D having the desired aspect of the joints among the connected components of  1 .ARCH  / and  1 .ARCH  / indicated by the cochain c. The general fiber Fı is now connected because of Condition (4). (See Lemma 6.6). The fiber Fı has genus g because of Condition (5). Let f W Fı ! Fı be the monodromy homeomorphism associated to the degenerating family .M; D; /. Then f is a pseudo-periodic map of negative twist with the desired properties. This completes the proof of Theorem 9.2. t u

220

9 Singular Open-Book

9.4 Concluding Remark Let .S; Y; c/ be a triple consisting of a compact, connected, and numerical chorizo space S satisfying the minimality condition; a weighted graph Y (with a collapsing map  W S ! Y ); and a weighted cohomology class c 2 HW1 .Y /. Two such triples .S1 ; Y1 ; c1 / and .S2 ; Y2 ; c2 / are equivalent if there exist a numerical homeomorphism H W S1 ! S2 and a weighted isomorphism W Y1 ! Y2 such that the diagram 1 S1 ! Y1 ? ? ? ? Hy y S2 ! Y2 2

commutes and such that  .c2 / D c1 . Let Tg denote the set of all equivalence classes of such triples .S; Y; c/ that satisfy Conditions (1)(5) stated before Theorem 9.2. By sending each pseudo-periodic map f W ˙g ! ˙g .g  2/ of negative twist to the triple .S Œf ; Y Œf ; cŒf /, defined in Chaps. 4 and 6, we obtain a map

W Pg ! Tg : Theorems 4.2 and 6.3 ensure well-definedness and injectivity of this map, and Theorem 9.2 ensures its surjectivity (thus bijectivity). Therefore, we have proved that every map in the following diagram is bijective (if g  2): ! SOg Tg

 b  & Î b  ˇ   Pg ! #

Tg  % Î  ! Sg Tg

 Map ˇ: Theorem 8.1 and Proposition 9.1. Map b  (with inverse b  ): Lemmas 9.2 and 9.3. Map : Theorem 7.2. Map : Theorems 4.2, 6.3 and 9.2.

Appendix A

Periodic Maps Which Are Homotopic

The purpose of this appendix is to give a proof of Theorem 2.2. We will state the theorem again. Theorem A.1. Let f and f 0 be (orientation-preserving) periodic maps of a compact surface ˙ each component of which has negative Euler characteristic. Suppose f and f 0 W .˙; @˙/ ! .˙; @˙/ are homotopic as maps of pairs. Then there exists a homeomorphism h W ˙ ! ˙ isotopic to the identity, such that f D h1 f 0 h. Proof. First we consider the case when ˙ is connected. Note that the quotient space M D ˙=f is an orbifold with negative orbifold-Euler characteristic, (cf. [62]). Case 1. The underlying space jM j of M is not S 2 nor D 2 . Let DM D M [ M be the double of M . (If @M D ;, set DM D M .) We put a hyperbolic metric on DM so that DM admits a decomposition by a finite number of simple closed geodesics G 1 ; G 2 ; : : : ; Gl which satisfy the following conditions: 1. No Gi passes through a cone point (of course, this can never happen, because the cone angle is less than 2), 2. the boundary curves of M are closed geodesics, and are members of fG1 ; G2 ; : : : ; Gl g; 3. the intersections of G1 ; G2 ; : : : ; Gl are only double points, S 4. each component of DM  li D1 Gi is an open cell whose closure is a polygon with more than 3 edges, and Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5, © Springer-Verlag Berlin Heidelberg 2011

221

222

A Periodic Maps Which Are Homotopic

Fig. A.1 Hyperbolic parts (“annulus with a cone” and “pants”)

Fig. A.2 Decomposing the double DM by simple closed geodesics

S 5. each component of DM  li D1 Gi contains at most one cone point. Such a decomposition is certainly possible. In fact, one can start with two kinds of hyperbolic parts; annuli with one cone point and pants, both having geodesic boundaries of length 1. See Fig. A.1. Glue them along the boundaries and construct a hyperbolic orbifold homeomorphic to DM . Add further simple closed geodesics to get a desired decomposition. See Fig. A.2. Let D˙ D ˙ [ ˙ denote the double of ˙. The periodic map f W ˙ ! ˙ symmetrically extends to a periodic map D˙ ! D˙, which will be denoted by f again. Lift the metric on DM just constructed, to D˙ to make the latter a (smooth) hyperbolic surface. The periodic map f W D˙ ! D˙ preserves this metric . Let  1 ;  2 ; : : : ; m be the lifts to D˙ of the simple closed geodesics G1 ; G2 ; : : : ; Gl : The total number m of the i ’s might be different from the total number l of the Gi0 s. The simple closed geodesics  1 ;  2 ; : : : ; m

A Periodic Maps Which Are Homotopic

223

satisfy the following conditions: (i) No i passes a multiple point of f , (ii) the boundary curves of ˙ are members of f1 ; 2 ; : : : ; m g; (iii) the intersections of 1 ;  2 ; : : : ; m are only double points, S (iv) each component of D˙  m i Di i is an open cell whose closure is a polygon with more than 3 edges, and S (v) each component of D˙  m i D1 i contains at most one cone point. Moreover, we have the following: (vi) No pair i , j .i ¤ j / are freely homotopic (because they are distinct closed geodesics. Cf. [16].) (vii) i and j .i ¤ j / have minimal intersection. Cf. [16]. (viii) f preserves the configuration 1 [ 2 [    [ m I f.

m [

i / D

i D1

m [

i

i D1

due to the construction of 1 ; 2 ; : : : ; m : So far we have only considered the periodic map f . Now we consider the other f 0 W ˙ ! ˙: Extend f 0 symmetrically to the double D˙ and denote the resulting periodic map by f 0 W D˙ ! D˙ also. Put a hyperbolic metric  0 on D˙ which is invariant under f 0 and such that the boundary curves of ˙ are closed geodesics. The simple closed curves 1 ;  2 ; : : : ; m are no longer geodesics with respect to  0 , in general. But they are freely homotopic to simple closed geodesics 10 ; 20 ; : : : ; m0 with respect to  0 . [16, Lemma 2.3.] These curves are distinct thanks to property (vi) of the i ’s. We will construct an isotopy g W D˙ ! D˙;

0    1;

224

A Periodic Maps Which Are Homotopic

such that (a) g0 D idD˙ , (b) if i D i0 (for instance, a boundary curve of ˙), then g .i / D i0 , and (c) g1 .i / D i0 , i D 1; 2; : : : ; m. The construction is only a mimic of Lemmmas 2.4, 2.5 of Casson’s lecture notes, [16], where the case m D 2 is treated, and is done essentially by an innermost arc argument. We proceed by induction. Suppose i D i0 .i D 1; 2; : : : ; k/ for some k < m. We will find an isotopy which starts with the identity, setwise preserves 0 i .D i0 / for i D 1; 2; : : : ; k, and at the final stage sends kC1 to kC1 . (It will be helpful to consider 10 ; 20 ; : : : ; m0 as patterns drawn on a floor, that remain fixed, and 1 ;  2 ; : : : ;  m 0 as loops laid on it, that can move.) First we isotop kC1 to separate it from kC1 , 0 0 and then using the annulus between kC1 and kC1 move kC1 onto kC1 , as we will sketch now. A typical move is performed through the shaded disk in Fig. A.3. The geodesics 10 ; : : : ; k0 .1 ; : : : ; k / 0 (cf. [16, Lemma 2.5]). Also they have minimal intersection with the geodesic kC1 have minimal intersection with kC1 because of property (vii) of i ’s. Thus as in Casson’s Lemma 2.5, i0 \ .the shaded disk/ is a family of arcs passing “through” the disk from “top to bottom”, for i D 1; 2; : : : ; k. But if there were a situation as shown in Fig. A.4, we would have an obstacle; we could not move kC1 “along” 10 , : : :, k0 . This “bad” situation is, however, prohibited by property (iv) of the i ’s. Therefore, we have the situation of Fig. A.3, and can move kC1 through the shaded disk “along” 10 ; 20 ; : : : ; k0 .D 1 ; 2 ; : : : ; k /:

This completes the inductive step, and we have obtained an isotopy g W D˙ ! D˙;

0 1

satisfying conditions (a), (b), (c) stated above. Γ k+1

Fig. A.3 A move of kC1

Γ′k+1

Γ′1 ,

...,

Γ′k

A Periodic Maps Which Are Homotopic Fig. A.4 A situation in which the move of kC1 is obstructed

225 Γ k+1

Γ′k+1

Γ′1 , . . . ,

Γ′k+1 (= Γ 1 , . . . , Γ k )

Let g W D˙ ! D˙ be the final stage g1 of the isotopy. Then g.1 [ 2 [    [ m / D 10 [ 20 [    [ m0 : Lemma A.1. f 0 .10 [ 20 [    [ m0 / D 10 [ 20 [    [ m0 . Proof. First note that f 0 .i0 / ' f 0 .i /

.because i0 ' i /

' f .i / .because f 0 ' f / D j

.for some j , because of property (viii) of the i ´s/

' j0

.because j ' j0 /;

where “'” denotes “is freely homotopic to”. The metric  0 on D˙ is invariant under f 0 , so f 0 .i0 / is a simple closed geodesic as well as i ’s. In a free homotopy class of simple closed curves, there is only one simple closed geodesic, (cf. [16, Lemma 2.4]). Therefore, f 0 .i0 / D j0 . t u Let H2 be the hyperbolic plane which is the universal covering of .D˙;  0 /. Let D D H [ S1 be its compactification to the unit disk. Let gQ  W H2 ! H2 ;

0 1

be the lifted isotopy of g starting from gQ 0 D idH2 . Then gQ WD gQ 1 W H2 ! H2 is a lift of g W D˙ ! D˙: By Nielsen [52], the isotopy gQ  W H2 ! H2

226

A Periodic Maps Which Are Homotopic

extends to an isotopy .gQ  /O W D ! D; and being a lift of an isotopy of a compact surface D˙, the restriction of .gQ  /O to S1 is constant because gQ  varies equivariantly with respect to the group of superpositions of the covering H2 ! D˙, and the variation is determined by the restriction of gQ  to a compact set (because D˙ is compact). Since the euclidean size of a fundamental domain for D˙ tends to zero if it goes to infinity, then gQ  at S1 does not depend on . Thus .gQ  /OjS1 D id jS1 . Let fQ W H2 ! H2 be a lift of f W D˙ ! D˙; and

.fQ/O W D ! D

its extention. Since

f ' f 0 W D˙ ! D˙;

fQ is homotopic, through a lifted homotopy, to a lift fQ0 of f 0 . For the same reason as above, we have .fQ/OjS1 D .fQ0 /OjS1 . Now let us regard  1 [ 2 [    [  m and

10 [ 20 [    [ m0

as finite graphs  and  0 , respectively, drawn on D˙. They decompose D˙ into finite cell complexes. Consider their lifts Q , Q 0 in H2 . Q consists of lifts of the i ’s, and Q 0 of lifts of the i ’s. (Each lift of i0 is a geodesic line.) Q and Q 0 decompose H2 into locally finite cell complexes. gQ W H2 ! H2 preserves these patterns: g. Q Q / D Q 0 . .0/ Let Q and Q 0.0/ be the set of vertices of Q and Q 0 , respectively. Lemma A.2. gQ j Q .0/ W Q .0/ ! Q 0.0/ is equivariant with respect to fQ j Q .0/ and fQ0 j Q 0.0/ . Proof. Take a vertex vQ 2 Q .0/ . vQ is an intersection point of two infinite curves Qi and Qj , which are lifts of i and j . By property (iii) of the i ’s, the pair fQi ; Qj g is uniquely determined by vQ . We have gQ fQ.Qi / D fQ0 g. Q Qi /

and

gQ fQ.Qj / D fQ0 g. Q Qj /:

(Proof. Let inf.Qi / denote the pair of the two “infinite” points of Qi in S1 . Then Q O.fQ/O.inf.Qi // D .fQ0 / O.g/O.inf. Q Qi // D inf.fQ0 g. Q Qi // inf.gQ fQ.Qi // D .g/

A Periodic Maps Which Are Homotopic

227

because .g/O Q j S1 D id and .fQ/O j S1 D .fQ0 /O j S1: Q Qi / are geodesic lines in H2 with the same pair of infinite points. gQ fQ.Qi / and fQ0 g. Then they coincide. Similarly, gQ fQ.Qj / D fQ0 g. Q Qj /.) Therefore, fgQ fQ.Qv/g D gQ fQ.Qi / \ gQ fQ.Qj / D fQ0 g. Q Qi / \ fQ0 g. Q Qj / D ffQ0 g.Q Q v/g; Q v/. which proves gQ fQ.Qv/ D fQ0 g.Q

t u

Remember that the Qi ’s are geodesic lines with respect to the lifted metric Q of . Using Lemma A.2, we can find an isotopy gQ .1/ W H2 ! H2 ;

0    1;

.1/ .1/ .1/ such that gQ 0 D g, Q gQ  .Q / D Q 0 , gQ  is equivariant with respect to the group of .1/ covering translations of H2 ! D˙, and the final stage gQ 1 is “linear” from each edge of Q to an edge of Q 0 with respect to Q and Q 0 (the lifted metric of  0 ). Then .1/ gQ 1 jQ W Q ! Q 0

is equivariant with respect to fQjQ and fQ0 jQ 0 . .1/ Finally by the Alexander trick, we can deform gQ 1 within each cell and obtain an isotopy gQ .2/ W H2 ! H2 ; 0    1; .2/ .1/ .2/ .1/ .2/ such that gQ 0 D gQ 1 , gQ  jQ D gQ  jQ , gQ  is equivariant with respect to the group of covering translations of H2 ! D˙;

and the final stage .2/

gQ 1 W H2 ! H2 is equivariant with respect to fQ and fQ0 . This homeomorphism gQ 1 projects down to a homeomorphism h W D˙ ! D˙: .2/

The restricted homeomorphism hj˙ W ˙ ! ˙ is the one whose existence is asserted by Theorem 2.2. This completes the proof of Case 1.

228

A Periodic Maps Which Are Homotopic

Case 2. The underlying space jM j of M is S 2 . The proof will be accompanied by several lemmas. Lemma A.3. The orders of f and f 0 W ˙ ! ˙ are equal. Proof. Let L.g/ denote the Lefschetz number of a homeomorphism g W ˙ ! ˙. Then the order of f is equal to the smallest positive integer n such that L.f n / < 0 because if f n ¤ id˙ , L.f n / is equal to the number of the fixed points of f n (see [23, pp. 130, 121]) which is non-negative, while L.id˙ / D .˙/ < 0. Thus our assumption f ' f 0 implies that their orders coincide. t u Lemma A.4. There is a bijective correspondence between the set of multiple points of f and the same set of f 0 which preserves the valencies. Proof. We impose a hyperbolic metric on ˙ and identify the universal covering ˙Q with H2 , which is compactified to D D H2 [ S1 as before. Let F .f / denote the set of fixed points of f . Take a point p0 2 F .f / and lift it to pQ0 2 H2 . Let fQ W H2 ! H2 be a lift of f which fixes pQ0 . By Nielsen [50, Sect. 2], fQ extends to a homeomorphism .fQ/O W D ! D: The lift

fQ W H2 ! H2

is a periodic map because it is a lift of a periodic map and it has a fixed point. Then .fQ/O W D ! D is also a periodic map. By [31], an orientation-preserving periodic map of a 2-disk D is conjugate to a rotation. In particular, .fQ/OjS1 has no fixed points. By our assumption f ' f 0 W ˙ ! ˙; fQ is homotopic, through a lifted homotopy, to a lift fQ0 of f 0 . Then by the same argument as in Case 1, .fQ/OjS1 D .fQ0 /OjS1 . By Brouwer’s fixed point theorem, .fQ0 /O W D ! D has a fixed point. But .fQ0 /jS1 .D .fQ/OjS1 / has no fixed points, so the fixed point of fQ0 is in H2 . Then the same argument as for fQ can apply to fQ0 , and .fQ0 /O W D ! D is conjugate to a rotation. Thus the fixed point pQ00 of fQ0 is uniquely determined in H2 , which projects to a fixed point p00 of f 0 . The correspondence

A Periodic Maps Which Are Homotopic

229

' W F .f / ! F .f 0 / is defined by sending p0 to p00 . ' is independent of the choice of the lift pQ0 , but it might depend on the homotopy between f and f 0 . We will fix the homotopy throughout the argument. Clearly ' is bijective because the roles of f and f 0 are symmetric. The valency of p0 (resp. p00 ) with respect to f (resp. f 0 ) is the same as the valency of pQ0 (resp. pQ00 ) with respect to fQ (resp. fQ0 ), which can be read off from the action of .fQ/O (resp. .fQ0 /O) on S1 . But .fQ/OjS1 D .fQ0 /OjS1 . Thus the valency of p0 is equal to the valency of p00 D '.p0 /. Similarly for each factor m of the order of f we can construct a bijective correspondence between F .f m / and F ..f 0 /m /. This correspondence preserves the valencies exactly as above. This completes the proof of Lemma A.4. t u Lemmmas A.3, A.4 and Nielsen’s theorem (Theorem 1.2) imply that f W ˙ ! ˙ and f 0 W ˙ ! ˙ are conjugate. (Remember that we are considering a closed ˙ in Case 2.) In particular, we have Lemma A.5. M D ˙=f and M 0 D ˙ 0 =f 0 are homeomorphic as orbifolds. Let p 1 ; p2 ; : : : ; p s ;

s  3;

be the cone points of M with valencies .m1 ; 1 ; 1 /; .m2 ; 2 ; 2 /; : : : ; .ms ; s ; s /; respectively. Consider a polygon P (s-gon) in H2 whose angles are =1 ; =2 ; : : : ; =s : See Fig. A.5. (Such a P exists because orb .M / < 0.) M can be considered as a hyperbolic “bi-hedron” having two faces, each congruent with P .

Fig. A.5 A hyperbolic polygon

230

A Periodic Maps Which Are Homotopic

Let  be the hyperbolic metric on ˙ obtained by lifting the bi-hedron metric on M through the projection ˙ ! M . We lift this metric  farther to the universal covering ˙Q to make it a hyperbolic plane H2 ./. The plane H2 ./ is tessellated by tiles, each congruent with P . The fundamental region for the action of 1orb .M / ([62, Sect. 13]) is F D P [ P , where P is a flipped P having an edge in common with it. Let ri be the rotation (in H2 ./) of angle 2=i centered at vertex pi .2 P /, i D 1; 2; : : : ; s. It is known that the rotations r1 , r2 , : : :, rs generate the group 1orb .M /, the orientation-preserving automorphism group of the tessellation, (cf. Milnor [45] or [41]). Moreover, from our construction, ri is a lift of f i mi W ˙ ! ˙;

i D 1; 2; : : : ; s:

Let f W ˙ ! ˙;

0    1;

0

be a homotopy between f and f W f0 D f , f1 D f 0 . Then, through a lifted homotpy .fi mi /Q W H2 ./ ! H2 ./; ri is homotopic to a homeomorphism ri0 W H2 ./ ! H2 ./: This ri0 is a lift of .f 0 /i mi , satisfies .ri0 /OjS1 D .ri /OjS1 , and is topologically equivalent to a rotation. (See Proof of Lemma A.4.) Let pi0 be the center of the “rotation” ri0 . We denote the cone point of M 0 to which pi0 projects by the same notation pi0 . Thus we have obtained the following correspondence: M 3 pi

! pi0 2 M 0 ;

i D 1; 2; : : : ; s:

By Lemma A.4, this correspondence preserves the valencies. We impose a 0 structure of a hyperbolic bi-hedron on M 0 whose faces P 0 , P are congruent with P , P preserving the above correspondence of the vertices. Let  0 be the hyperbolic metric on ˙ obtained by lifting the bi-hedron metric on M 0 through the projection ˙ ! M 0 . We lift  0 to ˙Q to make it a hyperbolic plane H2 . 0 /, in which the polygon P 0 is inscribed with vertices p10 ; p20 ; : : : ; ps0 : In H2 ./, the topological rotation ri0 is a genuine rotation with center pi0 of angle 2=i , i D 1, 2, : : :, s. The plane H2 . 0 / is tessellated by tiles, each congruent with P 0 . The rotations r10 , r20 , : : :, rs0 generate the group 1orb .M 0 /, the orientationpreserving automorphism group of the tessellation. Since P 0 is congruent with P , the group 1orb .M 0 / is isomorphic to 1orb .M / via the correspondence

A Periodic Maps Which Are Homotopic

ri

! ri0 ;

231

i D 1; 2; : : : ; s:

Note that H2 ./ and H2 . 0 / are merely different “pictures” on the same space ˙Q , Q the group so 1orb .M / and 1orb .M 0 / are considered as subgroups of Homeo.˙/, Q Q of all self-homeomorphisms of ˙ . If we fix the action of 1 .˙/ on ˙ , then 1 .˙/ Q contained in  orb .M / \  orb .M 0 /. Moreover, is also a subgroup of Homeo.˙/, 1 1 the action of 1 .˙/ is isometric, with respect to H2 ./ and at the same time with respect to H2 . 0 /. Lemma A.6. The isomorphism between 1orb .M / and 1orb .M 0 / given by the correspondence ri ! ri0 .i D 1; 2; : : : ; s/ restricts to the identity on 1 .˙/. Proof. Take an element g 2 1 .˙/. Since 1 .˙/ < 1orb .M 0 /, g can be written as a product of r10 ; r20 ; : : : ; rs0 : gD

.r10 ; r20 ; : : : ; rs0 /:

gD

.r1 ; r2 ; : : : ; rs /:

We will show that For this, compactify H ./ to D D H2 ./ [ S1 as before. Then 2

gjS O 1D Since both g and

.r10 ; r20 ; : : : ; rs0 /OjS1 D

.r1 ; r2 ; : : : ; rs /OjS1 :

.r1 ; r2 ; : : : ; rs / are isometries of H2 ./, we have gD

.r1 ; r2 ; : : : ; rs /: t u

Let us construct a homeomorphism hQ W ˙Q ! ˙Q which is equivariant with respect to the actions of 1orb .M / and 1orb .M 0 /. The construction is obvious. First, map the fundamental region F D P [ P “isometrically” to the fundamental region 0 F 0 D P [ P . Then extend it equivariantly to the whole space. By Lemma A.6, we have hQ g D g hQ for all g 2 1 .˙/. Thus hQ projects to a homeomorphism h W ˙ ! ˙. Lemma A.7. h W ˙ ! ˙ satisfies f D h1 f 0 h. Proof. Let fQ W ˙Q ! ˙ be a lift of f W ˙ ! ˙. Since fQ preserves the tessellation, fQ is written as a product of r1 , r2 , : : :, rs : fQ D '.r1 ; r2 ; : : : ; rs /: Then

'.r10 ; r20 ; : : : ; rs0 /

232

A Periodic Maps Which Are Homotopic

is a lift of f 0 ; denote it by fQ0 . Since hQ is equivariant with respect to the actions of 1orb .M / and 1orb .M 0 /, we have Q hQ fQ D hQ '.r1 ; r2 ; : : : ; rs / D '.r10 ; r20 ; : : : ; rs0 / hQ D fQ0 h; so h f D f 0 h as asserted.

t u

Lemma A.8. h W ˙ ! ˙ is isotopic to the identity. Q Proof. It will be sufficient to prove .h/OjS 1 D id:, because then h W ˙ ! ˙ preserves the free homotopy class of every simple closed curve. Let g be any element of 1 .˙/ different from 1. Then g W H2 ! H2 is a hyperbolic transformation, and for any point x 2 H2 , gn .x/ converges (in D) to a definite point Vg 2 S1 as n ! C1. Also g n .x/ converges to another definite point Ug 2 S1 as n ! 1. Nielsen [50, Sect. 1] calles Ug , Vg the negative and the positive fundamental points of g. Then, in D, we have n n Q Q Q n Q .h/O.V g / D .h/O. lim g .x// D lim .h/.g .x// D lim g .h/.x/ D Vg : n!1

n!1

n!1

Q Also .h/O.U g / D Ug . But the set of fundamental points fUg j g 2 1 .˙/g [ fVg j g 2 1 .˙/g Q is dense is S1 ([53, Sect. 1 Case b)]). This proves .h/OjS 1 D id .

t u

The proof is completed in Case 2. Case 3. The underlying space jM j of M is D 2 . Making the double DM , the proof is reduced to Case 2. Now we have completed the proof in the case when ˙ is connected. In the general case when ˙ is not necessarily connected, divide the set of the components of ˙ into cycles under the permutation caused by f . Since f 0 is homotopic to f , f 0 causes the same permutation. Then in each cycle we can argue just as in the proof of Theorem 2.3(ii). (Beware that we need here the “homotopy implies isotopy” theorem, [10, 21].) This completes the proof of Theorem A.1. u t

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•

Index

-sequence (asterisk sequence), 136 i -boundary, 179 j -boundary, 179 s-number, 137

general fiber, 173 generalized quotient, viii, 55

irreducible component, 53, 175 achiral, xi admissible coordinates, 175 admissible neighborhood, 195 admissible system, 4 amphidrome, viii, 6 amphidrome annulus, 35 arch, 96 atomic singular fiber, xii

bank, 54 binding, 200 black annulus, 63 blowing down, 137 body, 96

chain diagram, 136 chiral, xi chorizo space, xi, 53 closed part, 54 collapsing map, 149, 154 cone point, 9 conjugate, 9 contraction, 137 cut curve, vii, 3

deformation, 54 degenerating family, x, 173

linear twist, 17 linearization, 20, 30

mapping cylinder, 200 Milnor fiber, 193 minimal, 173 minimal quotient, ix, 60, 119 minimality conditions, 60 monodromy exponent, 11 monodromy homeomorphism, 173 multiple point, 8 multiplicity, 54, 70, 82, 90, 175

negative twist, viii, 53 nodal neighborhood, 54 node, 53 non-amphidrome annulus, 20 normal crossing, 175 normally minimal, 189 numerical chorizo space, 55 numerical homeomorphism, 93

open-book, 200 orbifold, 9 ordinary arch, 113 ordinary body, 113

Y. Matsumoto and J.M. Montesinos-Amilibia, Pseudo-periodic Maps and Degeneration of Riemann Surfaces, Lecture Notes in Mathematics 2030, DOI 10.1007/978-3-642-22534-5, © Springer-Verlag Berlin Heidelberg 2011

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238 page, 200 part, 53 partition graph, viii, 7, 149 periodic map, 8 periodic part, 20 pinched covering, 54 plumbing, 216 precise system, 6 pseudo-periodic map, vii, 3 refined partition graph, 151 renormalization of a linear twist, 71 renormalization of a rotation, 61 renormalization of a special twist, 83 Riemann surface with nodes, 53 screw number, vii, 5, 22 simple point, 8 singular fiber, 173 singular open-book, 200 special body, 113 special tail, 113

Index special twist, 18 specialization, 35, 44 spinnable structure, 200 standard form, 19 subchorizo space, 96 superstandard form, 65, 77 superstandard parametrization, 101, 105, 112

tail, 96 topological monodromy, 174 topologically equivalent, 174

valency, 8 valency of a boundary curve, 9 valency of a multiple point, 9

weighted cohomology, 165 weighted graph, 152 weighted isomorphism, 152 white annulus, 63

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