Approximating certain cell-like maps by homeomorphisms
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 dimensio...
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Main Author | |
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Format | Journal Article |
Language | English |
Published |
27.07.2016
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Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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DOI: | 10.48550/arxiv.1607.08270 |