Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?

Question

Milnor proved that any paracompact Hausdorff space which is equi-locally convex (and hence in particular locally contractible) is homotopy equivalent to a CW complex. However, unlike being paracompact and Hausdorff, the property of being equi-locally convex seems slightly arbitrary here, while the weaker property of being locally contractible is more conceptual. Does anyone know of a reference for this possible strengthening of Milnor's result, or, possibly, a counterexample?

Answer 1

After some more digging I found a (somewhat non-explicit) counterexample to the original question. In his paper "un espace metrique lineaire qui n’est pas un retracte absolu" Cauty constructs a metric vector space $V$ which is not an absolute neighborhood retract. According to the characterization established in "une caracterisation des retractes absolus de voisinage" (also Cauty) a metric space $X$ is an absolute neighborhood retract if and only if each open subset of $X$ is homotopy equivalent to a CW complex. It follows that the metric vector space $V$ above contains an open subset $U \subset V$ which is not homotopy equivalent to a CW. Since $U$ is metrizable it is paracompact and Hausdorff, and since it $V$ is locally contractible so is $U$. Hence $U$ is a counterexample to the original question.

Answer 2

I think Milnor answers your question in the paper where he proves the theorem you refer to ("On spaces having the homotopy type of a CW-complex", Trans. AMS, 90 (1959), 272-280):

Whitehead had observed that any compact space with the homotopy type of a CW-complex is dominated by a finite complex. It therefore has integral homology in only finitely many degrees. But Borsuk had constructed a locally contractible compact metric space $C$ such that $H_n(C,\mathbb{Z})\neq 0$ for all $n\geq0$.

Answer 3

It seems that at least a partial answer can be given using the formalism of $\infty$-topoi: if $X$ is a paracompact Hausdorff space which is locally contractible (in the strong sense discussed above) and such that the $\infty$-topos $Sh(X)$ of sheaves (of $\infty$-groupoids) on $X$ is hypercomplete, then $X$ is homotopy equivalent to a CW complex. This happens, for example, when $X$ has finite covering dimension.

To see this, recall Lurie's HTT section 7.1. Given a suitable basis $B$ for the topology on $X$, Lurie constructs a model structure on the category $Top_{/X}$ of spaces over $X$ which is a model for the $\infty$-category $Sh(X)$. If $X$ is locally contractible then one can construct a hypercovering of the terminal sheaf on $X$, given concretely by a split simplicial object $U_\bullet$ in $Top_{/X}$ such that each $U_n$ is a coproduct of contractible open subsets of $X$. The geometric realization $|U_\bullet| \in Top_{/X}$ encodes a sheaf which is $\infty$-connective. When $Sh(X)$ is hypercomplete the sheaf $|U_\bullet|$ admits a section (which in this case will be realized by an actual map $X \to |U_\bullet|$ in $Top_{/X}$ since in this model structure the terminal object is cofibrant and every object is fibrant) and so $X$ is a retract of $|U_\bullet|$. Since $U_\bullet$ is a split simplicial object the realization $|U_\bullet|$ coincides with the corresponding homotopy colimit, and since each $U_n$ is contractible this means that $|U_\bullet|$ is homotopy equivalent to a CW complex. Since $X$ is a retract of $|U_\bullet|$ it follows that $X$ is homotopy equivalent to CW complex as well.

This answer is not unsatisfying, but it would be more satisfying if the assumptions above were met by every space which is actually a CW complex. This leads to the following natural question: is $Sh(X)$ hypercomplete for every CW complex $X$?