Homotopy characterization of ANR mapping spaces

TL;DR

This paper characterizes when the space of continuous maps from a CW complex or certain other spaces to an ANR is itself an ANR, linking this property to metrizability and CW homotopy type.

Contribution

It establishes a precise homotopy-theoretic criterion for the ANR property of mapping spaces from CW complexes and hemicompact spaces.

Findings

Mapping space is an ANR iff it is metrizable and has CW homotopy type for CW complexes.

The characterization extends to compactly generated hemicompact spaces without metrizability assumptions.

Provides a homotopy-theoretic condition for the ANR property of function spaces.

Abstract

Let Y be an absolute neighborhood retract (ANR) for the class of metric spaces and let X be a Hausdorff space. Let map(X,Y) denote the space of continuous maps from X to Y with the compact open topology. It is shown that if X is a CW complex then map(X,Y) is an ANR for the class of metric spaces if and only if map(X,Y) is metrizable and has the homotopy type of a CW complex. The same holds also when X is a compactly generated hemicompact space (metrizability assumption is void in this case).