Combinatorial Manifolds Research Articles

Cobordism of Combinatorial Manifolds

Cobordism of Combinatorial Manifolds

Obstructions to locally flat embeddings of bounded combinatorial manifolds

Obstructions to locally flat embeddings of bounded combinatorial manifolds

Structures on M × R

A central role in the theory of smoothing combinatorial manifolds is played by the Cairns–Hirsch Theorem, which may be expressed (in a weak form) as follows:If M is a combinatorial manifold and if M × R has a differentiable structure a, compatible with its combinatorial structure then M has a differentiable structure λ, such that (M × R)α is diffeomorphic with Mλ × R.

On the combinatorial Schoenflies conjecture

Introduction. The combinatorial version of the Schoenflies conjecture in dimension n states: A combinatorial (n 1)-sphere on a combinatorial n-sphere decomposes the latter into two combinatorial ncells. The cases n =1, 2 are obvious, the case n =3 was solved by J. W. Alexander [1], see also W. Graeub [5], and E. Moise [8]. Nothing is known for n > 3 (compare J. F. Hudson and E. C. Zeeman in [6, p. 729]). On the other hand the following form of the Hauptvermutung for spheres is proved: If a combinatorial n-dimensional manifold is homeomorphic an n-dimensional sphere, then it is a combinatorial n-sphere if n04. This was proved by S. Smale for n $4, 5, 7 in [9], and improved by E. C. Zeeman to n #4 (unpublished). I would like to thank Professor Zeeman for pointing out this result to me. Further a generalized Schoenflies theorem was proved by M. Brown in [3]. If this Hauptvermutung were true for all n, a simple induction on the dimension n would prove the combinatorial Schoenflies conjecture for all n (compare Theorem 6). In the following we show that either the combinatorial Schoenflies conjecture is true for all dimensions n, or it is false for all dimensions n> 3.

A Combinatorial Analogue of Poincaré's Duality Theorem

For a non-negative integer s and a finite simplicial complexK, letβS(K) denote thes-dimensional Betti number ofKand letfs(K) denote the number ofs-simplices ofK. Our theorem, like Poincaré's, applies to combinatorial manifoldsM, but it concerns the numbersfs(M) instead of the numbersβS(M). One of the formulae given below is used by the author in (5) to establish a sharp upper bound for the number of vertices ofn-dimensional convex poly topes which have a given numberiof (n— 1)-faces. This amounts to estimating the size of the computation problem which may be involved in solving a system ofilinear inequalities innvariables, and was the original motivation for our study.

Cobordism Exact Sequences for Differential and Combinatorial Manifolds

Several exact sequences have been established recently which relate cobordism groups of various types [1], [3], [8]. These have all been established in the differential case. The main object of this paper is to establish the same results in the combinatorial case. We first define a combinatorial analogue to Thom's notion of t-regularity, and prove some basic lemmas on submanifolds representing double coverings (Part I). After this, we find that the same proofs are valid as in the differential case, so we treat the two cases together. This gives us an opportunity to collect together all the geometrical arguments used in the proofs of these theorems, which have hitherto been somewhat scattered in the literature. We establish the three exact sequences in Theorem 1; the proofs are so simple that one could axiomatise a category of manifolds in which they work. The proof would be valid for topological manifolds if only some corresponding version of Lemma 7 could be established in that case. We then give a proof, in the same order of ideas, that one of our maps is a derivation. The behaviour of the others with respect to multiplication is rather more complicated, and we hope to return to it in a subsequent paper. We shall assume known the definitions of combinatorial and of differential manifolds (which may have boundary). All manifolds occurring in this paper will be compact. A compact manifold without boundary is called closed. To avoid logical difficulties concerning sets, we may regard all manifolds as imbedded in a finite dimensional subspace of a given Hilbert space; however, it will be more convenient to state our constructions for abstract manifolds. We shall usually denote manifolds by the symbols: V, W,M,N.

Open 3-manifolds which are simply connected at infinity

A triangulated open manifold M will be called 1-connected at infinity if each compact subset C of M is contained in a compact polyhedron P in M such that M-P is connected and simply connected. Stallings has shown that, if M is a contractible open combinatorial manifold which is 1-connected at infinity and is of dimension n> 5, then M is piecewise-linearly homeomorphic to Euclidean n-space En [5].

Stellar neighborhoods in polyhedral manifolds

Introduction. It is the purpose of this paper to give some weaker analogues for polyhedral manifolds of well-known properties of combinatorial manifolds. This is now possible primarily because of the recent work of Mazur [3; 4] and M. Brown [1]. A crucial issue in this area is the extent to which these analogues may be improved for arbitrary triangulated manifolds. We shall prove a theorem which will be applied later on, after making some preliminary definitions. The join of two spaces X and Y is represented by X o Y. A map f of X o YY on itself is called ray preserving if for each ray p o q-q, p in X, q in Y, f(p o q-q) Cp o q-q. A subset K of a cone X o p is called starlike if p K and each segment pox, xCX, meets K in a connected set; p is the center of K. Consider En to be the open cone over Sn-1 from the origin p, in the usual way. We coordinatize En by (x, t), XESn-1, t a real number with O0t< 0< ; (Sn-1, 0) =p. Let Dn be the unit n-ball p O (Sn-1, 1).

A geometric condition for smoothability of combinatorial manifolds

A geometric condition for smoothability of combinatorial manifolds

A Representation of Orientable Combinatorial 3-Manifolds

The following question has been posed by Bing [1]: Which compact, connected 3-manifolds can be obtained from S3 as follows: Remove a finite collection of mutually exclusive (but perhaps knotted and linking) polyhedral tori T1, T2, * *, To from S3, and sew them back. This paper answers that question by showing that every closed, connected, orientable, 3-manifold is obtainable from S3 in the above way. Whereas this fact can now be deduced from general theorems of differential topology, the combinatorial proof given here is direct and elementary; while, in the proof, a study is made of a certain type of homeomorphism of a two dimensional manifold that is of interest in itself. Having obtained the above mentioned result on 3-manifolds, it is then easy to deduce the well known result (Theorem 3) that the combinatorial cobordism group for orientable 3-manifolds is trivial.

On Contractible Open Manifolds

By an open manifold we mean a non-compact space, that is triangulable by a countable complex which is a combinatorial manifold without boundary (see next section).

Singularities of piecewise linear mappings. I Mappings into the real line

In this note we consider piecewise linear mappings of combinatorial manifolds into the real line. We define—in a purely combinatorial way—a certain class of mappings called nondegenerate and we prove that every continuous mapping may be approximated by a nondegenerate one. Nondegenerate mappings on a combinatorial manifold behave like differentiable nondegenerate mappings on a differentiate manifold. In particular, the index of a singularity can be defined and one can prove the Morse inequalities and an analogue of the Reeb theorem about a function with only two nondegenerate singularities. The approximation theorem can also be extended to height functions on combinatorial submanifolds of Euclidean space. Detailed proofs will be published later. We give here descriptions of the singularities, a statement of the main theorems and a very brief sketch of the approximation theorem.

Generalized Poincare's Conjecture in Dimensions Greater Than Four

Poincare has posed the problem as to whether every simply connected closed 3-manifold (triangulated) is homeomorphic to the 3-sphere, see [18] for example. This problem, still open, is usually called Poincare's conjecture. The generalized Poincare conjecture (see [11] or [28] for example) says that every closed n-manifold which has the homotopy type of the nsphere S is homeomorphic to the n-sphere. One object of this paper is to prove that this is indeed the case if n > 5 (for differentiable manifolds in the following theorem and combinatorial manifolds in Theorem B).

Imbedding of Manifolds in Euclidean Space

This chapter discusses the piecewise linear imbeddings in R q of compact, n -dimensional, combinatorial manifolds that are (m – 1)-connected, where 0 m ≤ n . The condition m > 0 means that such a manifold is connected. If a closed, that is, compact, unbounded, n -manifold M is (m – l)-connected and 2 m > n , then it follows from the Poincare duality that M has the homotopy type of a w -sphere. Therefore, if it turns out that every such manifold is a combinatorial n -sphere, or even if it can be piecewise linearly imbedded in R n+1 , then the theorem proved in the chapter is valid for 0 m ≤ n . The chapter presents the proof of the theorem that states that if 0 m ≤ n , then every closed, (m – l)-connected n-manifold can be imbedded in R 2n–m+1 .

An exact sequence in differential topology

AN EXACT SEQUENCE IN DIFFERENTIAL TOPOLOGY MORRIS W. HIRSCH Communicated by Deane Montgomery, May 12, 1960 1. Introduction. The purpose of this note is to describe an exact sequence relating three series of abelian groups: T w , defined by Thorn [3]; 0 n , defined by Milnor [ l ] ; and A n , defined below. The sequence is written .^T n ^6 n -^A n -^T n - 1 -^ We now describe these groups briefly. T o obtain T w , divide the group of diffeomorphisms of the n — 1 sphere S 71 1 by the normal subgroup of those diffeomorphisms t h a t are extendable to the w-ball. See [2] for details. The set 0 n is the set of /-equivalence classes of closed, oriented, differentiable w-manifolds t h a t are homotopy spheres. If M is an oriented manifold, let —If be the oppositely oriented manifold. Two closed oriented w-manifolds M and N are J-equivalent if there is an oriented w + 1-manifold X whose boundary is the disjoint union of M and — N 9 and which admits both M and N as deformation retracts. We denote the /-equivalence class of M by [M]. If [M] and [N] are elements of 0 n , their sum is defined to be [M f N], where [ M # N] is obtained by removing the interior of an w-ball from M and N and identifying the boundaries in a suitable way. Details may be found in [1]. The group A n is defined analogously using combinatorial manifolds. Instead of the interior of an w-ball, the interior of an ^-simplex is removed. If M is a combinatorial manifold, we write (M) for its / - equivalence class. 2. The sequence. To define k: 6 n —>A n , we observe that every differ- entiable manifold M defines a combinatorial manifold IF, unique up to combinatorial equivalence, by means of a smooth triangulation of M [4]. We define k [M] = (M). Let g: S n ~~ 1 —^S n ~ 1 represent an element 7 of T n . According to J. Munkres [2], there is a unique (up to diffeomorphism) differentiable manifold V y corresponding to 7, such that F 7 = 5 n . To obtain V y , identify two copies of R n — 0 by the diffeomorphism x - + ( l / | x | ) g ( x / | x | ) . Here R n is Euclidean w-space and \x is the usual norm. The diffeomorphism class of V y depends only on 7, and

Intersection Theory of Manifolds With Operators with Applications to Knot Theory

Let 9N be an oriented combinatorial manifold with boundary, let 9)1, be any of its covering complexes, and let G be any free abelian group of covering transformations. The homology groups of 9N* are R-modules, where R denotes the integral group ring of the (multiplicative) group G. Each such homology module H has a well-defined torsion sub-module T, and the corresponding Betti module is B = H/T. The automorphism y -> -1 of the group G extends to a unique automorphism a -a & of the ring R = R. Following Reidemeister [4] there is defined an intersection S which is a pairing of the homology modules of dual dimension to the ring R, and also a linking V which is a pairing of dual torsion sub-modules to Ro/R, where Ro is the quotient field of R. Two duality theorems are proved: (1) S is a primitive pairing to R/rm of dual Betti modules with coefficients modulo tm and Tm respectively, where or is zero or a prime element of R and m is any positive integer. (2) V is a primitive pairing of dual torsion modules to Ro/R. These theorems are analogous to the Burger duality theorems [1]; in case R is the ring of integers, they specialize to the Poincar6-Lefschetz duality theorems for manifolds with boundary. Although dual modules are not, in general, isomorphic, it is demonstrated that if one torsion module has elementary divisors A0 c Al c ... then its dual has elementary divisors Ao C Al C * ... . (The numbering differs from that of [2] in that we begin with the first non-zero ideal.) This result is applied to the maximal abelian covering of a link in a closed 3-manifold. It is proved that the elementary divisors Ai of the 1-dimensional torsion module are symmetric in the sense that Ai = At . For the case of a knot or link in Euclidean 3-space, the ideal A0 is generated by the Alexander polynomial, and the of A0 has been proved previously by Seifert [6] for knots and Torres [7] for links. The symmetry of the Alexander polynomial was proved by Seifert and Torres in a slightly more precise form [8, Cors. 2 and 3]. The problem of similar extra precision in the more general case of a link in an arbitrary closed 3-manifold remains open.

The Existence of Periodic Points

A periodic point of a transformation of a set into itself is a point which is carried back to its original position by some iterate of the transformation. The purpose of this paper is to demonstrate topological conditions which ensure the existence of periodic points. THEOREM 1. Let T be a mapping of an orientable combinatorial manifold M into itself, and suppose that the corresponding homomorphism T* of rational p-cycle classes is an isomorphism onto. Let u be a rational p-cycle class and let v be any other rational cycle class of M. Let M be given a definite orientation. Then if the intersection cycle classes (T u, v), (T'u, v), , (T u, v) are all zero, where B is the p-dimensional Betti number of M, it follows that the intersection cycle class (u, v) is zero. PROOF. The rational p-cycle classes of M form a vector space of dimension B. Hence the B + 1 cycle classes u, T*u, ... , T*u must be related by a nontrivial dependency c0u + cjT*u + *. + CBT*U = 0. Let Ck, be the first nonzero coefficient. By hypothesis, T* has an inverse. Operating on the dependency by 71k gives CkU + Ck+lT*u + * + cBT* u = 0. Intersecting this combination of cycle classes with v and dividing by Ck gives the conclusion (u, v) = 0. THEOREM 2. Let T be a homeomorphism of an orientable combinatorial manifold M onto itself. If the Euler characteristic of M is not zero, then T has a periodic point. PROOF. Theorem 2 is a consequence of Theorem 1 applied to the orientable combinatorial product manifold M X M. Let M be given a definite orientation, thereby inducing an orientation of M X M. Let D be the cycle class in M X Ml which is the image of the basic cycle on M under the diagonal mapping x -* (x, x) of M into M X M. The homeomorphism T determines a homeomorphism 1 X T: (x, y) -+ (x, Ty) of M X M onto itself. If T has no periodic point, then the intersection classes ((1 X T)* D, D), ((1 X T) D, D), * a must all vanish. Hence by Theorem 1 the intersection class (D, D) must vanish. But a known result asserts that the algebraic value of the 0-cycle class (D, D) is equal within sign to the Euler characteristic of M. Since the Euler characteristic of M was assumed to be different from zero, T must have a periodic point. If Theorems 1 and 2 are established for manifolds with boundary then Theorem 2 can be extended to complexes by the imbedding method used by Lefschetz to extend his fixed-point formula. It is simpler, however, to make direct use of the Lefschetz formula. The method below was used by J. Nielsen in an investigation of transformations of surfaces. THEOREM 3. Let T be a mapping of a complex K into itself which induces auto-