# The homeomorphism group of a compact Hilbert cube manifold is an ANR

### Abstract

In this paper, we prove that the homeomorphism group, $H(M)$, of a compact $Q$-manfiold is an ANR. Results of Geoghegan and Torunczyk then show that $H(M)$ is an $l_2$-manifold.

As by-products of the proof, we obtain a CE approximation theorem for $l_2$-manifolds, a Vietoris theorem for simple homotopy theory (generalizing the result that a CE map between complexes is simple), and a proof that the nerve of a suitably nice open cover of a complex is simple homotopy equivalent to the complex.

Steve Ferry