CW type of inverse limits and function spaces

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.

💡 Research Summary

The paper undertakes a systematic study of when the mapping space map(X,Y) — the set of continuous maps from a CW complex X to a CW complex Y equipped with the compact‑open topology — has the homotopy type of a CW complex. It begins by recalling that, in general, map(X,Y) need not be of CW type; classical counter‑examples are presented to illustrate the failure of CW structure when either the source or the target possesses “wild” homotopy groups or infinite dimensional cells.

The first major contribution is a collection of necessary conditions. By analysing the homotopy fiber sequence associated to the evaluation map ev: map(X,Y)×X→Y, the author shows that if map(X,Y) is of CW type then X must be of finite dimension and Y must have controlled homotopy groups: either Y is simply connected with all higher homotopy groups finite, or at least the groups that appear in the Postnikov tower of Y must stabilize after finitely many stages. In particular, when Y has infinite‑dimensional rational homotopy (e.g. an infinite wedge of spheres), the mapping space cannot be CW.

The second major contribution is a set of sufficient conditions that extend all previously known results. The author proves that if Y is elliptic in the sense of rational homotopy theory (i.e. Y has finite-dimensional rational homotopy and cohomology, equivalently all its homotopy groups are finite in each degree), then for any CW complex X, the mapping space map(X,Y) is of CW type. The proof uses a careful cell‑by‑cell construction of a CW model for map(X,Y) via the adjunction between smash products and mapping spaces, together with the finiteness of the Postnikov stages of Y.

A striking novelty is the converse of this sufficient condition: the paper shows that if there exists a CW source X (for instance a sphere Sⁿ) such that map(X,Y) has CW type, then Y must be elliptic. This equivalence (elliptic ⇔ CW type of all mapping spaces) closes a long‑standing gap in the literature and provides a new homotopy‑theoretic characterization of elliptic spaces.

The author then connects these findings to the geometric Moore conjecture. The conjecture asserts that a simply connected finite complex admits an eventual geometric exponent at a prime p if and only if it is elliptic. By restating the conjecture in terms of mapping spaces—specifically, that map(Sⁿ,Z) has CW type for all n if and only if Z satisfies the conjecture—the paper translates a deep algebraic‑topological problem into a question about the homotopy type of function spaces. Using p‑completion techniques and Bousfield–Kan spectral sequences, the author demonstrates that a positive answer to the mapping‑space formulation would imply the Moore conjecture, and conversely, known cases of the conjecture yield new CW‑type results for the corresponding function spaces.

The final part of the work treats inverse limits of fibrations. The author considers towers

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