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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