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In this paper we will let Q = 11^.x [-1, 1] denote the Hubert cube, l2 = {(Xf)lSlXfl < °°} denote separable Hubert space, C(X> X) denote the continuous functions from X to X with the sup norm topology with respect to a suitable metric on X, H(X) C C(X, X) denote the self-homeomorphisms of X, and H(X) denote the closure of H(X) in C(X, X). By a manifold, we will mean a separable metric space modelled on R", Q, or l2. The following question is of interest to both infinite dimensional and finite dimensional topologists. Compare [A—B, p. 792]. Ql. If M is a manifold modelled on R or Q, is H(M) an /2-manifold? General results which are known include: (1) Geoghegan [G] has shown that HA(M) x l2 TM HA(M). Here, HA(M) is the space of homeomorphisms of M which fix a proper closed subset A of M (2) Toruncyzk [T] has shown that the product of l2 with a complete separable metric ANR is an l2-manifold. These results reduce Ql to the question: Ql'. If M is a compact manifold, is H{M) an ANR? (3) Mason and Luke and Mason [M], [L-M] have answered Ql' affirmatively when M is a two-dimensional manifold. Using completely different techniques, we have proven the analogous result for Ö-manifolds. Recall that a closed set A CM is called a Z-set if there are maps of M into M ~ A which are arbitrarily close to the identity.

The homeomorphism group of a compact 𝑄-manifold is an ANR