Annulus Conjecture and Stability of Homeomorphisms in Infinite-Dimensional Normed Linear Spaces

Abstract

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