CONTRACTIBLE NEIGHBORHOODS IN THE GROUP OF HOMEOMORPHISMS OF A MANIFOLD

Abstract

Let M be a comEact pi~cewise linear manifold and H(M) the space of all homeomorphisms of M onto itself with the supremum topology: p (f,g) = sup{d(f(x), g(x)}. For some xEM years there has been considerable interest in the question of whether H(M) is an ~2-manifold; i.e., a separable metric space which is locally homeomorphic to ~2 ' the Hilbert space of square-summable sequences. It is known that H(M) is locally contractible [1,3]. There are other partial results along these lines including the fact that PLH(M), the subspace of H(M) consisting of all piecewise linear homeomorphisms, is an ~~-manifold, where ~ ~ denotes the subspace of ~2 consisting of those sequences having only finitely many nonzero entries [5]. Let Ho(Bn) and PLHo(Bn) denote the subspaces of H(Bn) and PLH(Bn), respectively