CW type of inverse limits and function spaces

TL;DR

This paper investigates conditions under which the space of continuous functions between CW complexes has the homotopy type of a CW complex, extending previous results and relating to fundamental problems in algebraic topology.

Contribution

It provides new necessary and sufficient conditions for function spaces to have CW homotopy type, extending prior work and connecting to major open problems like the Moore conjecture.

Findings

Established new criteria for CW homotopy type of function spaces.

Extended known results to broader classes of spaces.

Provided solutions to problems involving inverse limits of fibrations.

Abstract

Given CW complexes X and Y, let map(X,Y) denote the space of continuous functions from X to Y with the compact open topology. The space map(X,Y) need not have the homotopy type of a CW complex. Here the results of an extensive investigation of various necessary and various sufficient conditions for map(X,Y) to have the homotopy type of a CW complex are exhibited. The results extend all previously known results on this topic. Moreover, appropriate converses are given for the previously known sufficient conditions. It is shown that this difficult question is related to well known problems in algebraic topology. For example, the geometric Moore conjecture (asserting that a simply connected finite complex admits an eventual geometric exponent at any prime if and only if it is elliptic) can be restated in terms of CW homotopy type of certain function spaces. Spaces of maps between CW complexes are a particular case of inverse limits of systems whose bonds are Hurewicz fibrations between spaces of CW homotopy type. Related problems concerning CW homotopy type of the limit space of such a system are also studied. In particular, an almost complete solution to a well known problem concerning towers of fibrations is presented.