On homeomorphism spaces of Hilbert manifolds

Abstract

Abstract. Let M be a Hubert manifold modeled on the separable Hubert space l2.We prove the following Fibred Homeomorphism Extension Theorem: Let M X B" -B" be the product bundle over the n-ball B", then any (fibred) homeomorphism on M X S"~ ' extends to a homeomorphism on all M X B" if and only if it extends to a mapping on all M X B". Moreover, the size of the extension may be restricted by any open cover on M X B". The result is then applied to study the space of homeomorphisms %(M) under various topologies given on %(M). For instance, if M = l2 and %(l2) has the compact-open topology, the %(l2) is an absolute extensor for all metric spaces. A counterexample is provided to show that the statement above may not be generalized to arbitrary manifold M. [To] showed that %(M) is, in fact, an ANR.Let M be a separable Hubert manifold modeled on the Hubert space l2. The function space, G(M), can be equipped with the c-o or majorant topology. Here the topology plays an important role since %(M) may fail to be locally contractible in the c-o topology. The following example illustrates that point. Example 1. We remove from the open interval (0,1) a sequence of disjoint open intervals which have their diameters tend to zero as they approach 1. Denote the resulting space by X. Let Y denote the space obtained by attaching circles to the end points in X as illustrated in the figure below. According to [We], M = Y X l2 is a Hubert manifold. But, by switching the upper and lower half of smaller and smaller