A note on stable homeomorphisms of infinite-dimensional manifolds

Abstract:

In papers by R. D. Anderson and R. Wong, respectively, it is shown that all homeomorphisms of the Hilbert cube onto itself, or of the infinite dimensional separable Hilbert space ${l_2}$ onto itself, are stable in the sense of Brown-Gluck. These facts can be used to show that all homeomorphisms of $X$ onto itself are isotopic to the identity mapping where $X$ is either the Hilbert cube or ${l_2}$. It follows that some versions of the infinite-dimensional annulus conjecture are true. In this note we give a simple proof of Anderson’s result. It follows from Brown-Gluck’s technique that for any connected manifold $X$ modeled on $Q$ or $s$, every homeomorphism of $X$ onto itself is stable.