On Homeomorphisms of Infinite Dimensional Bundles. III

Abstract

ABSTRACT. In this paper we continue the study of homeomorphisms and prove an analogue of the homeomorphism extension theorem for bundles modeled on Hilbert cube manifolds; thus we generalize previous results for Ç-manifolds Hypothesis. Throughout the following let M denote a connected Q-manifold and, for matter of convenience, assume that all spaces considered are metrizable.Our notation and definitions follow that of [7] and [8].2. Separation of sets. We recall that for a bundle E = (E, p, B) and for a pair (F, U) denoting an open set of E and an open cover of F, a (F, ll)-z'soropy ifil on E is a map p = {¡lA: E x I -* E such that pQ -id, each pt is an isomorphism with support in F, and i^.lyl is limited by ll. A closed set K in the product space X x Y is an X-projective Z-set provided the projection of K into X is contained in a closed Z-set of X. A closed subset K of E is a fibre Zset provided K O p~lib) is a Z-set in p~lib) for each b e B. The purpose of this section is to prove the following theorem.

Delooping controlled pseudo-isotopies of Hilbert cube manifolds

“…For fiber preserving Q-manifold results, including sliced Z-set unknotting, see [7], [8] or [ll]. We will need to use results about approximate fibrations from [ 151 and [ 161.…”

On homeomorphisms of infinite-dimensional bundles. I

“…In this paper we extend the results on infinite-dimensional separable .) In a later paper the author and Chapman [19] are able to use results of this paper to extend all the results to bundles modeled on s-manifolds, and in a certain appropriate manner, to bundles modeled on Q-manifolds [20],…”