Using flows to construct Hilbert space factors of function spaces

Keesling, James

doi:10.1090/S0002-9947-1971-0283751-8

Using flows to construct Hilbert space factors of function spaces
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Trans. Amer. Math. Soc. 161 (1971), 1-24 Request permission

Abstract:

Let X and Y be metric spaces. Let $G(X)$ be the group of homeomorphisms of X with the compact open topology. The main result of this paper is that if X admits a nontrivial flow, then $G(X)$ is homeomorphic to $G(X) \times {l_2}$ where ${l_2}$ is separable infinite-dimensional Hilbert space. The techniques are applied to other function spaces with the same result. Two such spaces for which our techniques apply are the space of imbeddings of X into Y, $E(X,Y)$, and the space of light open mappings of X into (or onto) Y, LO (X, Y). Some applications of these results are given. The paper also uses flows to show that if X is the $\sin (1/x)$-curve, then $G(X)$ is homeomorphic to ${l_2} \times N$, where N is the integers.

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