A corollary of the Kan and Thurston theorem states that the space $X$ (path connected) can be recovered, up to homotopy, by applying the fiber-wise $\mathbb{Z}$-completion functor of Bousfield–Kan (notation: $\dot{\mathbb{Z}}_\infty$) to the fibration $$K(P, 1)\to TX = K(G, 1)\to K(G/P, 1)$$ where $TX$ is a Kan–Thurston construction of $X$ (see references), $G/P = \pi_1(X)$, $P$ is a perfect group (i. e. $H_1(P; \mathbb{Z}) = 0$, trivial coefficients) and groups $P$, $G$ and $G/P$ form the exact sequence $$1\to P \to G \to G/P \to 1.$$

For understanding why $\dot{\mathbb{Z}}_\infty K(G, 1)\cong X$, I want to use the Whitehead theorem:

Let $f: X\to Y$ be a map of pointed connected $CW$-complexes such that $f$ induces an isomorphism on fundamental groups and on homology with any local coefficient system. Then $f$ is a homotopy equivalence.

As $P$ is perfect, the space $\mathbb{Z}_\infty K(P, 1)$ is simply-connected. Hence, from the long exact sequence of fibration $$\mathbb{Z}_\infty K(P, 1)\to \dot{\mathbb{Z}}_\infty K(G, 1)\to K(G/P, 1)$$ we have that the Kan–Thurston map $t: TX=K(G, 1)\to X$ induces an isomorphism $\pi_1(\dot{\mathbb{Z}}_\infty K(G, 1))\cong \pi_1(X)$. Moreover, by the main Kan–Thurston result, $t$ induces isomorphisms $H_\ast(TX; \mathcal{A}) \cong H_\ast(X; \mathcal{A})$.

Now consider a commutative diagram $\require{AMScd}$ \begin{CD} K(P, 1) @>>> K(G, 1) @>>> K(G/P, 1) \\ @VVV @VVV @| \\ \mathbb{Z}_\infty K(P, 1) @>>> \dot{\mathbb{Z}}_\infty K(G, 1) @>>> K(G/P, 1). \end{CD}

The morphism of fibrations in the diagram induces a morphism of spectral sequences. And by Zeeman's comparison theorem it is sufficient to prove isomorphism $H_\ast(\mathbb{Z}_\infty K(P, 1); \mathcal{A})\cong H_\ast(K(P, 1); \mathcal{A})$ for any local coefficient system $\mathcal{A}$.

It is known that the Bousfield–Kan $R$-completion ($R\subset \mathbb{Q}$ or $R = \mathbb{Z}/p$) of a space $Y$ with $R$-perfect fundamental group ($R\otimes H_1(\pi_1; \mathbb{Z}) = 0$) is $R$-good, that is, $H_\ast(R_\infty Y; R)\cong H_\ast(Y; R)$. It implies the desired isomorphism in the case of $\mathcal{A} = \mathbb{Z}$. But I don't know why $H_\ast(\mathbb{Z}_\infty K(P, 1); \mathcal{A})\cong H_\ast(P; \mathcal{A})$ for any local coefficient system $\mathcal{A}$. Moreover, it looks weird that $\mathcal{A}$ on $\mathbb{Z}_\infty K(P, 1)$ is trivial (thanks to $\pi_1(\mathbb{Z}_\infty K(P, 1)) = 0$) but we can't say the same about the local system on $K(P, 1)$….

**References**

- D. Kan and W. Thurston, Every connected space has the homology of a $K(\pi, 1)$, Topology Vol. 15. pp. 253–258, 1976.
- A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer, 1972.