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Assumption: $X$ is a connected CW complex, and $H_{\ast}(X;\mathbb{Z})=\bigoplus_{n \geq 0} H_n (X; \mathbb{Z})$ is finitely generated.

Question: does there exist a finite CW complex $Y$ and a map $f:Y \to X$ inducing an isomorphism in integral homology?

Remarks:

  1. The answer is almost trivially ''yes'' if $\pi_1 (X)=1$.

  2. P. Vogel gave a positive answer to my question (''Un Theoreme de Hurewicz homologique'') if $H_1 (X; \mathbb{Z})=0$ and if $\pi_1 (X)$ satisfies an additional finiteness hypothesis (it has to be ''locally perfect'', i.e. each element is contained in a finitely generated perfect subgroup; a stronger condition that being perfect).

  3. I am happy about a positive answer even under the restrictive assumption $H_1 (X; \mathbb{Z})=0$.

  4. By the Kan-Thurston theorem, one might assume that $X=BG$ for a discrete group.

  5. It is easy to produce an $f:Y \to X$, $Y$ finite, such that $f$ is surjective in homology: $X$ is the union of all its finite subcomplexes, and each homology class of $X$ is supported on one of them. Take a finite generating system for $H_{\ast}(X)$, pick a finite subcomplex for each generator, and take the union of these subcomplexes. It is also no big deal to achieve, by attaching cells to $X$, that $f$ is injective on $H_0$ and $H_1$ as well.

EDIT: the last part of the last sentence is wrong; thus I deleted my answer.

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  • $\begingroup$ You might want to look at Wall's work math.uchicago.edu/~shmuel/tom-readings/… $\endgroup$ Feb 12, 2014 at 13:17
  • $\begingroup$ This was my first idea, but I did not find anything that is really close to what i am looking for. $\endgroup$ Feb 12, 2014 at 13:22
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    $\begingroup$ @BenjaminSteinberg Wall's finiteness obstruction only applies to finitely dominated spaces, no? $\endgroup$ Feb 12, 2014 at 14:07
  • $\begingroup$ yes: in order to apply Walls theory, one needs a finite domination and in particular finitely presented fundamental group. Moreover, I do seek for a homology isomorphism, not a homotopy equivalence; and I have no control on the groups, so no chance to rule out the finiteness obstruction. I think it is really a different problem. $\endgroup$ Feb 12, 2014 at 14:11
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    $\begingroup$ When $H_1(Y)=0$ by arxiv.org/abs/1401.2554 and your 1. you can get a zig-zag $X\rightarrow X'\leftarrow Y$ with $Y$ finite and both maps inducing isomorhpishms in homology. $\endgroup$ Feb 13, 2014 at 9:18

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