Combinatorial Manifolds Research Articles

Neighborly 4-Polytopes and Neighborly Combinatorial 3-Manifolds with Ten Vertices

A combinatorial n-sphere is a simplicial n-complex whose body (i.e., the union of its members) is homeomorphic to the topological n-sphere Sn. A combinatorial n-manifold is a simplicial n-complex M such that M is connected, and for every vertex x in M the complex linker, M), the link of x in M, is a combinatorial (n — 1)-sphere. For more details the reader should consult Alexander [1] and Grünbaum [16]. All the spheres and manifolds to which we refer are combinatorial.

Vector fields on polyhedra

This paper presents a bundle theory for studying vector fields and their integral flows on polyhedra ∗ _ \ast and applications. Every polyhedron has a tangent object in the category of simplicial bundles in much the same way as every smooth manifold has a tangent object in the category of smooth vector bundles. One can show that there is a correspondence between piecewise smooth flows on a polyhedron P and sections of the tangent object of P (i.e., vector fields on P); using this result one can prove existence results for piecewise smooth flows on polyhedra. Finally an integral formula for the Euler characteristic of a closed, oriented, even-dimensional combinatorial manifold is given; as a consequence of this result one obtains a representation of Euler classes of such combinatorial manifolds in terms of piecewise smooth forms.

An enumeration of combinatorial 3-manifolds with nine vertices

A complete classification is given for non-neighborly combinatorial 3-manifolds with nine vertices. It is found that there are 1246 such types, and that they all are spheres. It is shown that 1057 of those spheres are polytopal. i.e. can be realized as boundary complexes of convex 4-polytopes. 115 spheres are non-polytopal, and 74 spheres remain undecided.

Classifications of simplicial triangulations of topological manifolds

In this note we announce theorems which classify simplicial (not necessarily combinatorial) triangulations of a given topological «-manifold M, n > 7 (> 6 if dM = 0 ) , in terms of homotopy classes of lifts of the classifying map r: M —• BTOP for the stable topological tangent bundle of M to a classifying space BTRIn which we introduce below. The (homotopic) fiber of the natural map ƒ: BTRIn —• BTOP is described in terms of certain groups of PL homology 3spheres. We also give necessary and sufficient conditions for a closed topological «-manifold M, n> 6, to possess a simplicial triangulation. The proofs of these results incorporate recent geometric results of F. Ancel and J. Cannon [1] , J. Cannon [2], R. D. Edwards [4] , and D. Galewski and R. Stern [5]. In [8], R. Kirby and L. Siebenmann show that in each dimension greater than four there exist closed topological manifolds which admit no piecewise linear manifold structure and hence cannot be triangulated as a combinatorial manifold. Also, R. D. Edwards [3] has recently shown that the double suspension of the Mazer homology 3-sphere is homeomorphic to S, thus showing that a simplicial triangulation of a topological manifold need not be combinatorial. But it is still unknown whether or not every topological manifold can be triangulated as a simplical complex. Our classification theorems for simplicial triangulations on a given topological manifold take the following forms: Let BTOP denote the classifying space for stable topological block bundles.

Neighborly combinatorial 3-manifolds with 9 vertices

A complete classification is given for neighborly combinatorial 3-manifolds with 9 vertices. It is found that there are 51 types, only one of which is not a sphere.

Combinatorial 3-manifolds with few vertices

It is proved that every combinatorial 3-manifold with at most eight vertices is a combinatorial sphere.

On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. I

Let K K be a proper rectilinear triangulation of a 2 2 -simplex S S in the plane and L ( K ) L(K) be the space of all homeomorphisms of S S which are linear on each simplex of K K and are fixed on Bd ( S ) \text {Bd}(S) . The author shows in this paper that L ( K ) L(K) with the compact open topology is simply-connected. This is a generalization of a result of S. S. Cairns in 1944 that the space L ( K ) L(K) is pathwise connected. Both results will be used in Part II of this paper to show that π 0 ( L 2 ) = π 1 ( L 2 ) = 0 {\pi _0}({L_2}) = {\pi _1}({L_2}) = 0 where L n {L_n} is a space of p.l. homeomorphisms of an n n -simplex, a space introduced by R {\mathbf {R}} . Thom in his study of the smoothings of combinatorial manifolds.

Manifolds with few cells and the stable homotopy of spheres

Let f: Sn+k1 -+ Sn and form the complex V(f) = Sn V Sk Uf+[ifn,ik] en+k where it E jrT(SI) is the canonical generator and [ , ] denotes Whitehead product. The complex V(f) is a Poincare duality complex. Under the assumption that f is in the stable range we show that V(f) has the homotopy type of a smooth, combinatorial or topological manifold iff the mapf lies in the image of the 0, PL or Top J-homomorphism respectively. Let VrS denote the stable homotopy groups of spheres and suppose that e E 77s We may represent p by a map f: Sn+k-1 Sn, and form the complex X(f) = Sn uf en+k. By studying the complex X(f) we may often detect that p $ 0. For example in this way one may study the Hopf invariant, ec-invariant, etc. [1], [2], [9]. We may also form the complex V(f) = Sn V Sk Uf+[i ,ik3 en+k where it E 77t(St) is the canonical generator. The complex V(f) is in fact a Poincare duality complex, and it is therefore to be hoped that by asking questions about smoothing V( f) we may discover information about the element p and vice versa. The present note is devoted to a small stab in this direction. Our main result is the following: THEOREM. Suppose that p E 77e 1. Represent p by a mapf: Sn+k-_ S n >> k, andform the complex V(f) = Sn V Sk Uf?[if,ik] en+k. Then V( f ) has the homotopy type of a closed topological, combinatorial or smooth manifold, if 9 lies in the image of the homomorphism JTOP, JPL or JO respectively. We make no claim for originality for the preceding theorem. The result should be well known on the basis of the results of [5], but it does Received by the editors November 6, 1970. AMS 1969 subject classifcations. Primary 5540, 5560.

Manifolds with Few Cells and the Stable Homotopy of Spheres

Let $f:{S^{n + k - 1}} \to {S^n}$ and form the complex $V(f) = {S^n} \vee {S^k}{ \cup _{f + [{i_n},{i_k}]}}{e^{n + k}}$ where ${i_t} \in {\pi _t}({S^t})$ is the canonical generator and [ , ] denotes Whitehead product. The complex $V(f)$ is a Poincaré duality complex. Under the assumption that $f$ is in the stable range we show that $V(f)$ has the homotopy type of a smooth, combinatorial or topological manifold iff the map $f$ lies in the image of the $O$, PL or Top $J$-homomorphism respectively.

Regular neighbourhoods and mapping cylinders

It is well known that a regular neighbourhood of a polyhedron in a piecewise linear manifold may be regarded as a simplicial mapping cylinder. The aim of this paper is to show that if the polyhedron is a locally unknotted submanifold of the interior then the class of maps giving rise to such regular neighbourhoods has a simple characterization. At the same time, it is possible to answer the question: Given a simplicial map f defined on a combinatorial manifold, when is the image of f also a combinatorial manifold? Marshall Cohen has answered this question when the image is required to be isomorphic to the domain; the methods used here are those developed in (1), to which the reader is referred for definitions and notation.

SURGERY OF POINCARÉ COMPLEXES

In the paper one studies the problem of extending the method of Morse surgery for the study of smooth manifolds to the case of the category of Poincare complexes. In the latter case we compute an obstruction to the surgery of a Poincare complex up to homotopy equivalence which, as in the classical case, is in the Wall group. In contrast to the cases of smooth or combinatorial manifolds we are able to handle only complexes with vanishing boundary.Bibliography: 3 items.

Noncombinatorial triangulations and the Poincaré conjecture

Kirby and Siebenmann have recently proved that every boundaryless topological n-manifold (n ?5) having trivial 4-dimensional Z2cohomology can be given a combinatorial triangulation; they have also given an example of a topological manifold which supports no combinatorial triangulation. It is quite possible, however, that the manifold constructed might be triangulable by a complex which is not a combinatorial manifold. The main purpose of this note is to show that if this could be done in such a way that the open simplexes are locally flat, then either the 3or the 4-dimensional Poincare conjecture would be false. Let us call such a triangulation locally flat; note that we do not require that the closed simplexes be locally flat. Then our main result is:

Construction of Transverse Fields

In this paper we give local conditions for a rectilinear embedding of a non-bounded combinatorial manifold,Mn, in Euclidean space, which are sufficient to prove thatMnhas a transverse field (see 1.1 and 1.2, definitions).In a sequel to this paper (6), we will show how with this transverse field we can construct a normal microbundle for the embedded manifoldMn.Our object in this research was only to obtain an existence theorem for normal microbundles. However, the method of proof via the construction of a transverse field yields as corollaries by Cairns (1), Whitehead (9), or Tao (8), results on smoothing. Earlier smoothing results achieved by the construction of transverse fields in the special case of (global) codimension 1 were obtained by Noguchi (5), and Tao (7; 8).After the research for this paper was completed, a paper of Davis (2) came to our attention.

A geometric condition for smoothability of bounded combinatorial manifold

A geometric condition for smoothability of bounded combinatorial manifold

On suspending homotopy spheres

It should be noted that M. Hirsch [4] has proved both theor-ems for the case in which the manifolds are smooth by the methods of differential topology. The case for piecewise linear (combinatorial) manifolds follows immediately as a result of their admitting compatible smooth structures. We prove Theorem 2 first as follows: Let F4 be a fake 4-cell and h: Bd F4X [0, I)-*F4acollarforBdF4in F4. PutX=Cl(F4-h[0, 1/2]). Note that X is homeomorphic to F4. Let N5= F4X [-2, 2]. We claim that Int N5 is a contractible open 5-manifold which is 1-connected at infinity, admits a piecewise linear triangulation and thus by Stallings [6] is homeomorphic to El. Obviously the interior of N5 is contractible and since by Van Kampen's theorem Bd N5 is simply connected (it is in fact a homotopy 4-sphere) Int N5 is 1-connected at infinity. It is easily seen that the double 2N5 of N6 is a homotopy 5-sphere and thus by Poincare's conjecture for topological n-manifolds (n ?5), which was proved by E. H. Connell and can be found in M. H. A. Newman [5], a 5-sphere. Hence Int N5 is homeomorphic to an open subset of S5. Consequently, it admits a piecewise-linear triangulation and is homeomorphic to E5. Clearly XX (-1) and XX 1 are cellular in Int N5, for they have arbitrarily small neighborhoods homeomorphic to Int N5. It follows (Theorem 1 of [1]) that there is a mapping F: N5-*N5 such that Fl Bd N5 = id and the only nondegenerate inverse sets of F are XX(-1) and XX1. Let g: Bd XX [-1, 1] --S(Bd X) be the natural mapping. Fg-' is a homeomorphism of S(BdX) onto F(BdXX[-1, 1]). Hence F(Bd XX [-1, 1]) is a 4-sphere. Clearly F(Bd X X [-1, 1 ]) is locally flat in Int N5 except

Almost combinatorial manifolds and the annulus conjecture.

Almost combinatorial manifolds and the annulus conjecture.

Shorter Notes: Local Flatness of Combinatorial Manifolds in Codimension One

Shorter Notes: Local Flatness of Combinatorial Manifolds in Codimension One

Local flatness of combinatorial manifolds in codimension one

PROOF. Let x be any point of K and let v be a vertex of K containing x in the interior of its star, St(v, K). Without loss of generality, we may assume that v is the origin in Rn+l. The radial projection r of the link, Lk(v, K), of v in K on Sn is a combinatorial (n-1)-sphere in Sn whose cells are geodesic simplexes on Sn. By the main theorem of [2], there is a homeomorphism h of Sn (onto itself) taking r(Lk(v, K)) onto Sn-1. Let h* denote the radial extension of h to a homeomorphism of Rn+'. Then h* maps St(v, K) into Rn. Thus K is locally flat in R+1.

Obstructions to locally flat embeddings of combinatorial manifolds

Obstructions to locally flat embeddings of combinatorial manifolds

Some normal subgroups of homomorphisms

lement of a euclidean neighborhood, and for B the closure of a euclidean neighborhood. They are able to show that the set of h which satisfy (a) is just the group P(X) generated by the elements of H(X) which agree with the identity on some open set, provided X has what they call a stable structure. In case X is a combinatorial manifold, the existence of a stable structure is equivalent to orientability. Their methods rely heavily on recent results concerning locally flat embeddings of spheres and the generalized Schoenflies theorem. In the present paper, we extend some of their results to a more general class of topological spaces which includes locally normed linear spaces. The definition of stable structure readily extends to the latter class of spaces, and we are able to show that the existence of a stable structure is equivalent to the condition that some h other than the identity should satisfy (a). However, we are unable to identify P(X) as the set of h satisfying (a) in the infinite dimensional case except whenX is something like a sphere in a normed linear space. 2. Bridging homeomorphisms. Let X be a topological space, and I1(X) the group of all homeomorphisms h of X onto itself. The restriction of h to a subset A of X will be denoted by h I A, and g I A = h I A will be shortened to g = h I A. The identity of H(X) is e. We define P(X) to be the set of elements g e H(X) satisfying