Canonical extensions of homeomorphisms

Abstract

In this paper some homeomorphism extension theorems for infinite-dimensional manifolds are redone in a canonical fashion. As an example we cite a special case of our main result. Let the Hilbert cube Q be donated by Π i=1 ∞ I i , where each I i is the closed interval [−1, 1],and let s= Π i=1 ∞ I 0 i , where each I 0 i is the open interval (−1, 1).It is known that if K⊂ s is compact and f: K→ s is any embedding, then f can be extended to a homeomorphism f ̃ :Q→Q satisfying f ̃ (s)=s . As a special case of Theorem 5.1 it follows that f̃ can be continuously chosen, i.e. chosen to depend continuously on f. We also give a number of applications of this result to spaces of embeddings of compacta and homeomorphism groups of infinite-dimensional manifolds.