I'm try to understanding a survey chapter in Angeloni - Metzler -Sieradski "Two dimensional Homotopy and combinatorial group theory" namely the one about Stable classification of $4$-manifold (see link).
Now he claims that the classifying map for the universal cover of $M$ $$u\colon M \to K(\pi_1(M),1)$$ induces an isomorphism $H^1(\pi_1(M);\mathbb{Z}/2)\cong H^1(M; \mathbb{Z}/2)$ and gives as a reference the well - known book of Brown "Cohomology of groups". I tried proving the result directly but something missing. And I'm unable to find the exact position of the result inside the mentioned book.
Can someone point me out in which chapter is this result proved on Brown's book or giving me some hints to prove it?
By definition, $u$ induces an isomorphism on $\pi_1$. By universal coefficients theorem and the Hurewicz theorem, for any path-connected space $X$ and any abelian group $G$, there is a natural isomorphisms $H^1(X,G)\cong\operatorname{Hom}(H_1(X),G))\cong\operatorname{Hom}(\pi_1(X),G)$. Thus $u$ induces an isomorphism on $H^1$ with any coefficients.