Compact space homotopy equivalent to a CW complex

Question

Assume that $X$ is a compact Hausdorff space homotopy equivalent to some CW complex. Does it follow that it is homotopy equivalent to a compact CW complex?

Answer 1

As I understand it, the answer is no. The following facts are what I have pieced together from Mislin's article Wall's Finiteness Obstruction, which appears as chapter 26 of The Handbook of Algebraic Topology. In summary, there are indeed compact spaces with CW homotopy type which are not of the homotopy type of a finite CW complex.

A connected space $X$ is said to be finitely dominated if there is a finite complex $P$ and maps $\phi:X\rightarrow P$, $\psi:P\rightarrow X$ with $\psi\circ\phi\simeq id_X$.

A finitely dominated space has the homotopy type of a CW complex of finite dimension.

However, examples produced by Lyra, On a Conjecture in Homotopy Theory, and later Wall, Finiteness Conditions for CW-Complexes, I, II, showed that there are finitely dominated spaces which are not of the homotopy type of a finite CW complex. Moreover Mather showd the remarkable fact that

A connected space $X$ is finitely dominated if and only if $X\times S^1$ has the homotopy type of a finite CW complex.

It follows that $X\times S^1$ is compact, and therefore so is $X$, as it is the image of the continuous projection.

In general, given a finitely dominated space $X$ there is Wall's finiteness obstruction, which is a class $\sigma(X)\in\tilde K_0(\mathbb{Z}[\pi_1(X)])$ which vanishes if and only if $X$ is homotopy equivalent to a finite CW complex.