Annulus conjecture and stability of homeomorphisms in infinite-dimensional normed linear spaces

Abstract

If E E is an arbitrary infinite-dimensional normed linear space, it is shown that if all homeomorphisms of E E onto itself are stable, then the annulus conjecture is true for E E . As a result, this confirms that the annulus conjecture for Hilbert space is true. A partial converse is that for those spaces E E which have some hyperplane homeomorphic to E E , if the annulus conjecture is true for E E and if all homeomorphisms of E E onto itself are isotopic to the identity, then all homeomorphisms of E E onto itself are stable.