# Finiteness properties of general topological spaces

It is known that all metric compact ANR have the homotopy type of finite CW complexes. Which spaces are homotopy equivalent or finitely dominated by CW complexes of finite type?

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Let $X$ be a space, which we may as well assume connected. If $X$ is an ANR, then Milnor gives that $X$ is homotopy equivalent to a countable CW-complex $X'$. The question of when $X'$ is homotopy equivalent to a complex of finite type is addressed in Theorem A of

Wall, C. T. C. Finiteness conditions for CW-complexes. Ann. of Math. (2) 81 1965 56–69.

and discussed further in

Wall, C. T. C. Finiteness conditions for CW complexes. II. Proc. Roy. Soc. Ser. A 295 1966 129–139.

Note the necessary condition that $\pi_1(X')$ be finitely presented. I suspect there are more modern references building on Wall's work, but I couldn't find any.

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Bestvina and Brady famously demonstrated the necessity of the finitely presented hypothesis. They constructed a group of type FP_2 which is not finitely presented. –  HJRW Oct 11 '12 at 10:31
(Or have I got this mixed up with the D(2) problem, which is still open? Anyway, the Bestvina--Brady paper is a good modern reference.) –  HJRW Oct 11 '12 at 10:41