Approximating certain cell-like maps by homeomorphisms

TL;DR

This paper proves that certain cell-like maps from high-dimensional manifolds to ANRs with the disjoint disc property can be approximated by homeomorphisms, offering an alternative proof of Siebenmann's theorem.

Contribution

It establishes that maps with cell-like point-inverses are approximable by homeomorphisms when the target has the disjoint disc property, in dimensions five and higher.

Findings

Maps with cell-like inverses are approximable by homeomorphisms in high dimensions.

The disjoint disc property of the target space is crucial for approximation.

Provides an alternative, self-contained proof of Siebenmann's Approximation Theorem.

Abstract

Given a proper map f : M $\rightarrow$ Q, having cell-like point-inverses, from a manifold-without-boundary M onto an ANR Q, it is a much-studied problem to find when f is approximable by homeomorphisms, i.e., when the decomposition of M induced by f is shrinkable (in the sense of Bing). If dimension M $\geq$ 5, J. W. Cannon's recent work focuses attention on whether Q has the disjoint disc property (which is: Any two maps of a 2-disc into Q can be homotoped by an arbitrarily small amount to have disjoint images; this is clearly a necessary condition for Q to be a manifold, in this dimension range). This paper establishes that such an f is approximable by homeomorphisms whenever dimension M $\geq$ 5 and Q has the disjoint disc property. As a corollary, one obtains that given an arbitrary map f : M $\rightarrow$ Q as above, the stabilized map f $\times$ id($\mathbb{R}^{2}$) : M $\times$ $\mathbb{R}^{2}$ -> Q $\times$ $\mathbb{R}^{2}$ is approximable by homeomorphisms. The proof of the theorem is different from the proofs of the special cases in the earlier work of myself and Cannon, and it is quite self-contained. This work provides an alternative proof of L. Siebenmann's Approximation Theorem, which is the case where Q is given to be a manifold.