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Negligible Subsets in the Space Of Homeomorphisms Raymond Y Wong

Summary: 1
Negligible Subsets in the Space Of Homeomorphisms
Raymond Y Wong
We show that for various compact metric spaces X, the space of
homeomorphisms H(X) is homeomorphic to H(X)\K, where K
= i >0 Ki H(X) with each Ki is either (1) closed and equi-
uniformly continuous or (2) topologically complete.
Our motivation for the study of negligible subsets is that, for a compact piecewise
linear n-manifold M, it has been a long standing problem as to whether the space of
homeomorphisms H(M) is an absolute neighborhood retract (The Homeomorphism Group
Problem). It turns out ([GH]) that there is a dense G subset G H(M) which is
homeomorphic to a s-manifold, where s is the countable infinite product of open interval
(-1, 1), and such that the complement H(M)\G is a countable union of closed sets {Ki}
each of which is a Z-set in the sense of [An] ( it means that, for any homotopically trivial
open set U in H(M), U\ Ki remains homotopically trivial). It is therefore natural to ask
whether the union i >0 Ki may be deleted from H(M). If the answer is yes, then H(M) is
homeomorphic to G.
Notation. For a compact metric space (X, d), let C(X) denote the space of
continuous functions of X into X. The metric defined on C(X) is (f, g) = sup{d(f(x),
g(x)) | x X}. Without lost of generality, we may assume (f, g) < 1. Let H(X) denote


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics