I not really familiar with these subjects. I read this question and I was really surprised by the answer. My question is probably vague (so please do bear with me). The cited question/answer suggests that any integral homology of a simply connected space can be realized as integral homology of some Eilenberg-MacLane space. Does it mean that connected spaces are the same thing (up to ?) as groups in some sense ? My question is vague, very vague. I will be happy to see a correct formulation of my question (if it makes sense).

  • $\begingroup$ Groups up to isomorphism are the same thing as path-connected aspherical complexes (i.e. $K(G,1)$'s) up to homotopy equivalence. This is much more elementary than the Kan-Thurston theorem, however. $\endgroup$
    – Mark Grant
    Aug 19, 2015 at 14:19
  • $\begingroup$ I'm afraid I don't understand what your specific question is that is not answered by the statement of the Kan-Thurston theorem. $\endgroup$ Aug 19, 2015 at 14:20
  • $\begingroup$ @MarkGrant I added "simply connected" to avoid your remark. $\endgroup$
    – google
    Aug 19, 2015 at 14:28
  • $\begingroup$ @JeremyRickard That is why I was apologizing for my vague question. I was asking for a correct formulation. I thought I have been clear about the intuition. May be I was wrong, sorry. $\endgroup$
    – google
    Aug 19, 2015 at 14:30
  • 1
    $\begingroup$ I think the question you really want to ask is : does there exist an equivalence (up to some localization) between the category of spaces and groups behind Kan-Thurston? And I guess the answer is no, Kan-Thurston construction not being functorial. $\endgroup$
    – user43326
    Aug 19, 2015 at 14:33

1 Answer 1


Here's something that might be considered an answer.

In Section 11 of Baumslag, Dyer, Heller, "The topology of discrete groups", the following theorem is proved, basically building on the ideas of the Kan-Thurston theorem.

A perfect homomorphism of groups $G\to H$ is a surjective group homomorphism such that the kernel is a perfect group. Let $\mathcal{C}$ be the category whose objects are perfect homomorphisms, and whose maps are commutative squares.

The theorem is:

There exists a class $\mathcal{F}$ of morphisms in $\mathcal{C}$ such that the category of fractions $\mathcal{C}[\mathcal{F}^{-1}]$ is equivalent to the homotopy category of pointed connected CW complexes.

The functor from $\mathcal{C}$ to spaces is defined by the Quillen plus construction to $BG$ with respect to $P=\mathrm{Ker}(G\to H)$, and the class $\mathcal{F}$ is just the maps that become equivalences after applying the plus construction; the class $\mathcal{F}$ can also be characterized as the maps $\phi\colon (G'\to H')\to (G\to H)$ such that $H'\to H$ is an isomorphism and the induced maps $H_*(G',\phi^*M)\to H_*(G,M)$ in group homology are isos for every $H$-module $M$.


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