Function spaces of CW homotopy type are Hilbert manifolds

Smrekar, Jaka; Yamashita, Atsushi

doi:10.1090/S0002-9939-08-09584-1

Function spaces of CW homotopy type are Hilbert manifolds
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by Jaka Smrekar and Atsushi Yamashita PDF

Proc. Amer. Math. Soc. 137 (2009), 751-759 Request permission

Abstract:

Let $X$ be a countable CW complex and $Y$ an ANR (for metric spaces) and let $Y^X$ denote the space of continuous maps from $X$ to $Y$ with the compact-open topology. We show that, under mild restrictions, the following are equivalent: (1) $Y^X$ is an $\ell ^2$-manifold, (2) $Y^X$ is an ANR, (3) $Y^X$ has the homotopy type of a CW complex. We also give a few interesting examples and applications.

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