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 {l_2} onto itself, are stable in the sense of Brown-Gluck. These facts can be used to show that all homeomorphisms of X X onto itself are isotopic to the identity mapping where X X is either the Hilbert cube or l 2 {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 X modeled on Q Q or s s , every homeomorphism of X X onto itself is stable.