# Can the Kan-Thurston theorem be turned into some kind of equivalence between groups and spaces?

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).

• 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. Aug 19, 2015 at 14:19
• I'm afraid I don't understand what your specific question is that is not answered by the statement of the Kan-Thurston theorem. Aug 19, 2015 at 14:20
• @MarkGrant I added "simply connected" to avoid your remark. Aug 19, 2015 at 14:28
• @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. Aug 19, 2015 at 14:30
• 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. Aug 19, 2015 at 14:33

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.
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 $M$.