R. B. Sher

Realizing cell-like maps in Euclidean space

Abstract Suppose X and Y are locally compact finite dimensional metric spaces and f:X ↠ Y is a map. Then we say that the map f can be realized in Euclidean space of dimension k if there exist closed embeddings g:X → Ek and g :Y → E k and a closed map h: E k ↠ E k such that hg = gf and h|Ek⧹g(X) is a homeomorphism of Ek⧹g(X) onto E k ⧹ g (Y) . Our main result is that closed cell-like maps can be realized. In particular we show, for a closed embedding g:X → En, how to obtain a realization of f for k = n + dim Y + 1. This yields an extension theorem for cell-like maps on closed subsets X of Euclidean space, where we require that the extension be a homeomorphism on the complement of X. Some applications are given in shape theory, the primary of which is that cell-like maps on finite dimensional metric compacta are shape preserving. As a tool, we use the notion of completely regular mappings as applied to upper semicontinuous decompositions of manifolds. Some of the results obtained here may be of interest in their own right.

Tame polyhedra in wild cells and spheres

It is shown that each arc on a disk D in E4 can be homeomorphically approximated by an arc in D which is tame in E4. Some applications of this are given. Also, we construct an everywhere wild (n-1)-sphere in En, n2_3, each of whose arcs is tame in En.

Triangulating neighborhoods in topological manifolds

Suppose X is a compact subset of the topological q-manifold Q, q ≥ 5, ∂Q = . It is shown that some neighborhood of X supports a piecewise linear structure provided the inclusion- induced homomorphism i∗:H4(Q;Z2)→H4X;Z2)is zero. Thus methods for studying embeddings of compacta in piecewise linear manifolds can often be applied without assuming piecewise linearity. As an example of such an application, it is pointed out that McMillans criterion for cellularity in piecewise linear manifolds of dimension five or more also holds in topological manifolds. Examples are given to show that piecewise linear neighborhoods may fail to exist stably, even in the case when X is a piecewise linear manifold embedded as a locally flat submanifold of Q.

Shape properties of the Stone-Čech compactification

Abstract In this paper it is shown that if X is a connected space which is not pesudocompact, then β X is not movable and does not have metric shape. In particular β X cannot have trivial shape. It is also shown that if X is Lindelof and KχβX − X is a continuum, then K cannot be movable or have metric shape unless it is a point.

Finiteness results in n-homotopy theory

Abstract We establish some finiteness results in the n -homotopy category. These results, unlike the corresponding ones in the “ordinary” homotopy category, do not involve algebraic obstructions.

Extending cell-like maps on manifolds

Let X be a closed subset of a manifold M and Go be a cell. like upper semicontinuous decomposition of X. We consider the problem of extending G to a cell-like upper semicontinuous decomposition G of M such that M/G t M. Under fairly weak restrictions (which vanish if M = En or Sn and n ; 4) we show that such a G exists if and only if the trivial extension of Go obtained by adjoining to Go the singletons of M X, has the desired property. In particular, the nondegenerate elements of Bings dogbone decomposition of E3 are not elements of any cell-like upper semicontinuous decomposition G of E3 such that E3/G = E3. Call a cell-like upper semicontinuous decomposition G of a metric space X simple if X/G : X and say that the closed set Y is simply embedded in X if each simple decomposition of Y extends trivially to a simple decomposition of X. We show that tame manifolds in E3 are simply embedded and, with some additional restrictions, obtain a similar result for a locally flat k-manifold in an m-manifold (k, m f 4). Examples are given of an everywhere wild simply embedded simple closed curve in E3 and of a compact absolute retract which embeds in E3 yet has no simple embedding in E3.

A complement theorem for shape concordant compacta

Let X and Y be compacta of polyhedral shape lying in the manifold M. Under suitable conditions, it is shown that if X and Y are shape concordant, then M X is homeomorphic to M Y.

A complement theorem in the universal Menger compactum

A. Chigogidze has shown that two Z-sets in the universal Menger compactum of dimension k + 1 have the same fc-shape if and only if their complements are homeomorphic. We show that this result holds for weak Zsets. The class of weak Z-sets, defined herein and analogous to the weak Z-sets in Q , contains but is larger than the class of Z-sets. We give some examples of weak Z-sets in the universal Menger compactum and in Q that are not Z-sets.