On Combinatorial Topology

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MATHEMATICS: A. W. TUCKER

PROC. N. A. S.

ON COMBINA TORIAL TOPOLOGY By A. W. TUCKER DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY

Communicated December 9, 1931

This note gives a preliminary account of some features of a paper on combinatorial topology to be published in more complete detail at a later date. The terminology and notation are patterned after that of Lefschetz,1 but the cells {EpI with which we shall deal are of the abstract type considered by Mayer.2 The advantage of using a general type of cell where the incidence numbers 77 may have any integral values, is described by Lefschetz (loc. cit., pp. 104-5). But there is one difficulty, viz., an Ep cancelling out of the chain-boundary of an Ep + i is not (combinatorially) distinguished from an Ep not on the boundary of Ep + 1 at all. This can be overcome by introducing a relation telling when an Ep is on the boundary of an E,(p < q). The only connection of this qualitative relation with the quantitative one of incidence numbers is that if Ep is not on the boundary of Ep + 1 the incidence number is 0. 1. Open and Closed Sub-Complexes.-A complex K is a set of cells such that the boundary-chain of a boundary-chain is always 0, i.e.,

F(F(Cp))

= 0

for an arbitrary chain Cp on the set. If the cells of K be separated into two sets L and J such that no cell of J is on the boundary of a cell of L, examination of incidence matrices shows that L and J are complementary sub-complexes called closed and open, respectively. By suitable manipulation of incidence matrices the inter-relations of K, L and J may be read off. In particular if K is "spherical" the identity of the homology characters of dimensions p and (p + 1) of L and J, respectively, may be obtained. (For a systematic treatment of incidence matrices it is found advantageous to adopt a canonical form in which the invariant factors appear in numerical order up the diagonal leading from the lower left corner, since then the incidence matrices of all dilmensions of a complex may be simultaneously reduced to canonical form.) 2. Blocks. Welds.-A p-block of K is defined to be a set of cells of K such that if a cell belongs to the set so do all cells of dimensions > p of its closure on K or . p of its star on K. (The closure on K of a cell of K consists of that cell and all cells of K on its boundary. The star on K of a cell of K consists of that cell and all cells of K on whose boundaries it lies.) By examining incidence matrices we find that a p-block of K is a subcomplex of K. Indeed it is the most general type of sub-set of K which can be directly shown to be a sub-complex from qualitative boundary

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MA THEMA TICS: A. W. TUCKER

relations alone. A 0-block is a closed sub-complex, an n-block an open sub-complex. Let the cells of K be divided into complementary sets 1K, 2K. Under what conditions can 2K be replaced by a cell Ep so that 'K = 'K + Ep is a complex, every sub-complex of which has the same homology characters as the corresponding sub-complex of K, and moreover is closed (open) if the latter is closed (open)? The answer, found by manipulating incidence matrices, is that it is necessary and sufficient that 2K be a p-cellblock of K, i.e., a p-block with the relative homology characters of a p-cell. Of course the incidence numbers of Ep with cells of 'K are determined by a choice of the basic relative p-cycle on 2K, and a cell of 'K on the boundary of a cell of 2K is supposed on the boundary of Ep and conversely. We call the passage from K to 'K a p-weld. It is a very general combinatorial operation. Subdivision is a special case of its inverse. (The concept of cellblock is a generalization of the idea, essentially due to Alexander, of "combinatorial cell"; cf. Lefschetz, loc. cit., p. 105.) Analogous considerations show that a necessary and sufficient condition that 2K may be dropped from K without affecting homology characters, etc., is that 2K be a block with the relative homology characters of the vacuous set. 3. Multiplication.-Comparison of products, joins and intersections leads to the conclusion that they can be obtained from a formal w-multi-

plication w

EpXEEq=Ep+q++

orO

by taking w = 0, 1 and - n, respectively. The single formula w

w

F(CP

X

CQ)

=

F(CP)

X

V

Cq + (1)P +w CP

X

F(CQ)

(3.1)

under suitable conventions yields the well-known formulae for the boundary-chains of products, joins and intersections. This unification allows a theorem giving the homology characters of a product complex immediate application to joins and some intersections. 4. Intersections.-As indicated in §3 we introduce intersections as a special case of w-multiplication, viz., by taking w =-n. A fundamental assumption is that if Ep is on the boundary of Eq, EpEr $ 0 implies Eq.Er $ 0. The intersection 'K.2K of complexes 'K and 2K is the set of non-vacuous cells 'Ep.2Eq. From (3.1) it follows that 1K.2K is a complex. We raise the question-is 1K.2K identifiable with aK(a = 1 or 2), i.e., is it possible to weld the non-vacuous intersection cells in which

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MA THEMA TICS: A. W. TUCKER

aEp figures into a p-cell

PROC. N. A. S.

(a)Ep, for each aEp, so that the set I (a)Ep I forms

a complex isomorphic with aKj The answer is-a necessary and sufficient condition that 1K.2K be identifiable with aK is that for each aEp the cells of bK (b = 2 or 1) with which aEp has non-vacuous intersections form an n-cellblock whose basic relative n-cycle can be taken so as to coincide on the cellblock with a fixed n-chain of bK. 5. Manifolds.-Consider the intersection complex K.K* of a complex K and its dual complex K*, the latter being defined abstractly in an obvious manner (cf. Mayer, loc. cit., p. 18). We assume thatEpE p only for i = j. Then, by the theorem stated at the end of § 4, K.K* is identifiable with K* if, and only if, K has star uniformity, i.e., the star of each cell of K is an n-cellblock whose basic relative n-cycle can be taken so as to coincide on the celiblock with a fixed n-chain of K. If in addition to star unformity K has closure uniformity (= star uniformity for K*, by definition) K.K* is identifiable with both K and K*, and so possesses absolute duality relations for Betti numbers and torsion coefficients.

RO (K) =R"_ v (K)

Gt (K)

= o7Pi (K)

Is such a K an absolute manifold? Absolute duality relations alone do not justify calling it a manifold. One expects "invariance under subdivision" as well. Investigation of the latter leads to three results: (a) a necessary and sufficient condition that K.K* have star uniformity is that K have star and closure uniformity; (b) a necessary and sufficient condition that K.K* have closure uniformity is that K have intercept uniformity, i.e., that for all pairs of distinct cells Ep,Eq of K the common part of the star of Ep on K and the closure of Eq on K have the relative homology characters of the vacuous set; (c) a necessary and sufficient condition that K.K* have intercept uniformity is that K have intercept uniformity-from which we conclude that star, closure and intercept uniformity make K an absolute manifold. By using "modular" star and closure uniformity apropos to K.K* taken modulo a suitable sub-complex, relative duality relations are obtained analogous to the Fundamental Duality Theorem of Lefschetz (1cc. cit., p. 142). The investigation of "invariance under subdivision" in this case follows without much added complication the lines laid down in the last paragraph, and so contributes to the criteria determining a relative manifold. A supplementary contribution is furnished by a discussion of conditions on the modular cells, which indicates that a relative manifold is most tractable and informative when it has a "regular boundary." It is interesting to note that the dual of the intersection complex K.K*

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can be considered as arising from K and K* by a special 0-multiplication which could equally well have been used to develop the above manifold theory. Of course between a manifold MI and its dual M* will automatically exist an intersection theory, viz., that given by M.M*. But a theory for M alone may also be constructed by first constructing a theory of looping coefficients for a "spherical" M (cf. Lefschetz, loc. cit., p. 216). In the latter connection the result stated in §1 for a "spherical" K enables us to obtain the analogue of Alexander's Duality Theorem. 6. Elementary Subdivision.-In dealing with joins (1-multiplication) it is convenient to introduce an E -1 (suggested by the "1" of Alexander3) having an incidence number of 1 with each suitably oriented 0-cell and such that the join EpE_ 1 = Ep for an arbitrary Ep. If E 1 may be added to a complex so as to get another complex the latter is called the augmented complex and the former is said to be augmentable. For us elementary subdivision means the inverse of a welding operation, which (inverse) is obtained by joining an augmented 0-cell to the boundary, either augmented or not, of the cell to be subdivided. A necessary and sufficient condition that a cell of a complex be elementarily subdivisible is that the cell have a closure, augmented or not as the case may be, with the homology characters of the vacuous set. We call a complex derivable if all of its cells are elementarily subdivisible. The property of being derivable is invariant under elementary subdivision. By subdividing first the 1-cells, then the 2-cells, and so on, of a derivable K we get a simplicial complex K' which can always be geometrically realized. From our standpoint a (closed) simplicial complex may be defined as a set of cells such that (a) if Ep is on the boundary of El there exists a unique Eq_ p 1 on the boundary of Eq such that the join EpEq_ p 1 = 4 Eq, and (b) if Ep is on the boundary of Eq there exists an E7, for each p < r < q, on the boundary of Eq and on whose boundary Ep lies. Such a complex has closure and intercept uniformity. It turns out that any K with closure uniformity is augmentable and derivable, and conversely. Intercept uniformity in K ties up with star uniformity in K'. Hence our manifolds may be defined in terms of elementary subdivision. 1 Lefschetz,

"Topology," Amer. Math. Soc. Colloquium Publications, 12, New York

(1930). 2 Mayer, "Uber abstrakte Topologie," Monatsh. f. Math. u. Phys., 36, 1-42, 219-258 (1929). 3

Alexander, "The Combinatorial Theory of Complexes," Ann. Math., 31, 294-322

(1930).