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Function spaces of CW homotopy type are Hilbert manifolds. (English) Zbl 1163.54010

Let \(X\) be a countable CW complex with \(\dim X\geq 1\), and \(Y\) be a separable completely metrizable ANR without isolated points. Let \(Y^X\) denote the space of all continuous mappings from \(X\) into \(Y\) with the compact open topology. Let \(\ell^2\) denote the separable real Hilbert space of square summable sequences. The authors main result says that the following properties are equivalent: (i) \(Y^X\) is an \(\ell^2\)-manifold; (ii) \(Y^X\) is an ANR; (iii) \(Y^X\) has the homotopy type of a CW complex. Several applications are given where further assumptions are imposed on \(X\) and \(Y\).

MSC:

54C35 Function spaces in general topology
55M15 Absolute neighborhood retracts
57N20 Topology of infinite-dimensional manifolds
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