Transformations of Manifolds with a Boundary

VOL. 12, 1926

MA THEMA TICS: S. LEFSCHETZ

737

least four inherent trends in mutation present in the family (tendencies (a) to postponing the division of the body after nuclear division, (b) to delay in completing mitotic division of the nucleus when once begun, (c) to flattening of the body, (d) to elongation of the body). By their conjunction or disjunction in inheritance these four trends give rise to the several subfamilies and genera and to some of the subgenera mentioned in the taxonomic chart. But this treatment of a phylogeny as a shuffling of trends is too unfamiliar for condensed discussion. * Metcalf-The Opalinid Ciliate Infusorians, Museum, pp. 244-246.

Bulletin

120, United States National

TRANSFORMATIONS OF MANIFOLDS WITH A BOUNDARY By S. LE1VSCHrTZ DEPARTMUNT OP MATHEIATICS, PRINCETON UNIVGRSITY Communicated October 22, 1926

1. The object of this note is to outline the extension of the coincidence and fixed point formulas which I have already obtained for transformations of manifolds without boundary.' The notations will be those of the papers just quoted. By means of matrices we shall first greatly simplify the formulas already given. Any matrix of elements ai, or ajj will be denoted by a (i.e., like the elements with positional indices dropped), its transverse by a' and if it is square the sum of the main diagonal elements will be called the trace. We also set C;, = ||(v^.Ys*,/) 11 where the y's are cycles of fundamental sets for the operation ,, C,, is square of order R,, and its determinant 1 (Veblen). Then the first formula, p. 43, of the Transactions paper gives at once: (1) (- 1)+ CIAEn_= a,A ; e-;, = ( 1)(+)C, a By applying (53.3), loc. cit., to the first formula on p. 47, replacing the d's by the y's, then making use of (1) and of the fact that transposed matrices have equal traces we find: (2) (rn . r") = 2(-1)u trace C.. ., C,,71a}, whose form is as advantageous as it is condensed. 2. We now consider a manifold Mn with a boundary F,,n1 It is found necessary to demand that F,,1 be itself a manifold and that if M. is a copy of M,, when we match the corresponding boundary points then Vn = Mn - M,, is also a manifold (without boundary, of course). It amounts to certain homogeneity requirements along F-,,. --

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MA THEMA TICS: S. LEFSCHETZ

PROC. N. A. S.

Manageable fundamental sets for the operation Z and each dimensionality are found by associating four types of cycles: (a) Cycles of M. independent of boundary cycles. (b) Cycles of M. dependent upon boundary cycles. (c) Skew-symmetric cycles (difference between a complex on M. and its M. image) that meet F,,.-- in cycles O. . (d) Skewsymmetric cycles that meet F.,_ in non-bounding cycles. It is found convenient to take in the sets for ,u cycles the types in the order: a c d b when p < '/2n, a c b d when A 2 '/2n. Then by a proper choice of the cycles of each type Cl, will have the form 0 1 1 0 0 1 A 1 0 B= ;1 0 -1 0 0 1 B 0 10Oi The terms are all matrices, the units corresponding to unit-matrices of various orders. For , < '/2n the sign is + and B is of the first form. For IA = l/2n even, the sign is also + and B of the second form, while for A = 1/2n odd the sip is - and B of the third form. The diagonal matrices are square and the sequence of their orders is the same for pu and n - u. This implies very interesting duality theorems. Let namely r, be the number of cycles in each group (in the order named), then: r1 = r1_-> r3 = r3 r2 = =

3. The transformations are defined this time by means of complexes K, on M. X M, with their boundary on that of the product manifold. However, if T, T' are two transformations, K,, K' the related complexes, the index (K,.K4) is not a class-invariant unless T and T' are rather special. In the last analysis it is found advisable to choose them so that T may be approximated as closely as desired by a transformation that transforms M. into the interior of itself and T' by one that assigns no transform to the boundary. T, T', being so chosen, we associate with them transformations U, U' of V. defined thus: Let A be the M. image of a point A of M,. Then UA = UA = TA. Let also A' be any TA, then the pairs A', A', constitute all points U'A. If r,r,' are the cycles which represent U, U' on V. X V', then (K,, .K') = (r. r'). Everything is then reduced to a coincidence problem for V". 4. As regards U and U'. what matters chiefly is that the U transform of a skew-symmetric cycle is O 0, while the U' transform of any cycle is skew-symmetric. Hence the corresponding a's and P's can be exhibited as very simple matrices similar to Cp. When this is done and (2) applied it is found that

(Kn.Kn)

=

2(-1)"

trace

p' -,,P,(.

(3)

Voi. 12, 1926

MA THEMA TICS.- P. SLA VENA S

739

Here P,, is the transformation matrix for T and the cycles of types (a) and (b), that is, for those of M. itself; p,. - is the transformation matrix for U' and the skew-symmetric cycles. The fixed-points formula is, obtained by making in (3) all p's equal to one for T, all P's equal to one for T'. Transformations of the class of the identity are special cases of either types considered, and strikingly enough the number of signed fixed points is found again to be equal to the Euler characteristic.2' Very similar results can be obtained for transformations of one M. into another. 1 These PROCODINGs, 2, 90 (1923); 11, 290 (1925); Trans. Amer. Math. Soc., 28, 1 (1926). 'See these PROCOUDINGS,

11, 290 (1925); also Hopf, Math. Ann., 96, 225 (1926).

A POSSIBLE WA Y TO DISCUSS THE FUNDAMENTAL PRINCIPLES OF RELATIVITY By PAUL SLAVZNAS1

YAuX UNIVERSITY OBSERVATORY Communicated November 11, 1926

Let us consider, as the element of the space-time, the point Ai, at which is installed a clock indicating a variable quantity ti. In order to denote the different points and the corresponding readings of the clocks installed at these points we will use different values of the index i. Suppose an electromagnetic signal were sent from A, at the moment t; indicated by its clock. We denote by means of the symbol [tjij, the moment of reception of this signal at the point Aj recorded by the clock installed at Aj. It is evident that

[t4],

= function of

t,.

(1)

We will not make any assumption regarding the physical nature. of the clocks. However, each variable ti should, necessarily, increase (or decrease) continually in the course of time. The knowledge of the functions of the type (1), given for all possible combinations of indices i and j,- will supply us the complete idea of relative positions of the points A, in the space-time.2 Since no assumption has been made about physical nature of the clocks, we have right to transform-if needed-a variable ti into another variable 0,. We can write t;- pi(0;) (2)