Fredric D. Ancel

Proper hereditary shape equivalences preserve property C

Abstract In a recent paper [6], van Mill and Mogilski prove that a proper hereditary shape equivalence preserves property C, if its domain is σ-compact. In this note, the same result is established without the hypothesis of σ-compactness.

The shrinkability of bing-whitehead decompositions

THIS PAPER is a study of a special class of toroidal decompositions of 3-manifolds called Bing-Whitehead decompositions. It is well known that a Bing-Whitehead decomposition of a 3-manifold is shrinkable if all successive stages are Bing nested; but it is not shrinkable if all successive stages are Whitehead nested. (See Figs 1 and 2.) Consider a Bing-Whitehead decomposition of a 3-manifold which is defined by h, successive Bing nested stages. followed by 1 Whitehead nested stage, followed by h, successive Bing nested stages, followed by 1 Whitehead nested stage, . . The principal result of this paper is that this decomposition is shrinkable if and only if

Z-COMPACTIFICATIONS OF OPEN MANIFOLDS

Abstract Suppose an open n -manifold M n may be compactified to an ANR M n so that M n −M n is a Z -set in M n . It is shown that (when n ⩾5) the double of M n along its “ Z -boundary” is an n -manifold. More generally, if M n and N n each admit compactifications with homeomorphic Z -boundaries, then their union along this common boundary is an n -manifold. This result is used to show that in many cases Z -compactifiable manifolds are determined by their Z -boundaries. For example, contractible open n -manifolds with homeomorphic Z -boundaries are homeomorphic. As an application, some special cases of a weak Borel conjecture are verified. Specifically, it is shown that closed aspherical n -manifolds ( n ≠4) having isomorphic fundamental groups which are either word hyperbolic or CAT (0) have homeomorphic universal covers.

Resolving wild embeddings of codimension-one manifolds in manifolds of dimensions greater than 3

Abstract For n ⩾4, every embedding of an ( n −1)-manifold in an n -manifold has a δ-resolution for each δ >0. Consequently, for n ⩾4, every embedding of an ( n -1)-manifold in an n -manifold can be approximated by tame embeddings.

Cones that are cells, and an application to hyperspaces☆

Abstract Let Y be a compact metric space that is not an ( n −1)-sphere. If the cone over Y is an n-cell, then Y ×[0,1] is an n-cell; if n ≤4, then Y is an ( n −1)-cell. Examples are given to show that the converse of the first part is false (for n ≥5) and that the second part does not extend beyond n =4. An application concerning when hyperspaces of simple n-ods are cones over unique compacta is given, which answers a question of Charatonik.

Rigid 3-dimensional compacta whose squares are manifolds

A space is rigid if its only self-homeomorphism is the identity. In response to a question of Jan van Mill, Ancel and Singh have given examples of rigid «-dimensional compacta, for each n > 4, whose squares are manifolds. We construct a rigid 3-dimensional compactum whose square is the manifold S3 X S3. In fact, we construct uncountably many topologically distinct compacta with these properties.

Rigid finite-dimensional compacta whose squares are manifolds

A space is rigid if its only sclf-homcomorphism is the identity. We answer questions of Jan van Mill by constructing for each n. 4 « n < oo. a rigid «-dimensional compactum whose square is homogeneous because it is a manifold. Moreover, for each n. 4 « n < x. we give uncountably many topologically distinct such examples. Infinite-dimensional examples are also given.

COLLECTIONS OF PATHS AND RAYS IN THE PLANE WHICH FIX ITS TOPOLOGY

Abstract A collection F of proper maps into a locally compact Hausdorff space X fixes the topology of X if the only locally compact Hausdorff topology on X which makes each element of F continuous and proper is the given topology. In I 2 =[-1, 1]×[-1, 1], neither the collection of analytic paths nor the collection of regular twice differentiable paths fixes the topology. However, in I 2 , both the collection of C ∞ arcs and the collection of regular C 1 arcs fix the topology. In I 2 =[−1,1]×[−1,1], the collection of polynomial rays together with any collection of paths does not fix the topology. However, in R 2 , the collection of regular injective entire rays together with either the collection of C ∞ arcs or the collection of regular C 1 arcs fixes the topology.

On the Sternfeld-Levin counterexamples to a conjecture of Chogoshvili-Pontrjagin

Abstract An inconclusive proof in a 1937 paper by G. Chogoshvili spawned an interesting dimensiontheoretic conjecture which we call the Chogoshvili-Pontrjagin Conjecture. In 1991, Y. Sternfeld found an ingenious counterexample to this conjecture which he and M. Levin greatly generalized in 1995. In this note we point out a previously unobserved property of the Sternfeld-Levin examples, and we reinterpret their significance in light of this property. Also, we present a version of the Levin-Sternfeld proof which is more “topological” and less “lattice-theoretic” than the original.

Topologies on Rn induced by smooth subsets

Abstract If L is a collection of subsets of R n , let T L denote the largest topology on R n which restricts to the standard topology on each element of L , and let H L denote the homeomorphism group of R n with the topology T L . Let T std denote the standard topology on R n and let H std denote the homeomorphism group of R n with the standard topology. 1. Theorem 1. If L is any collection of subsets of R n which contains all C 1 regular 1- manifolds, then T L = T std . A natural collection of subsets of R n called smooth sets is defined which includes the zero set of every nonconstant polynomial and every C 2 regular submanifold of R n of dimension n . 2. Theorem 2. If L is the collection of all smooth subsets of R n , then T L is strictly larger than T std and H L is strictly smaller than H std . 3. Theorem 3. There is an injective function f : R n → R n which is discontinuous at each point of a countable dense subset of R n , and whose restriction to each smooth subset of R n is continuous .