Let $X$ be a (connected) closed $n$-manifold and $G=\pi\_1(X)$ be the fundamental group of $X$. There is a classifying map $f: X \rightarrow K(G, 1)$ which induces an isomorphism on $\pi\_1$. I would like know when the map $f\_\*: H\_n(X, \mathbb{Z}) \rightarrow H\_n(K(G,1), \mathbb{Z})$ is injective, or even when $f\_\*: H\_n(X, \mathbb{Q}) \rightarrow H\_n(K(G,1), \mathbb{Q})$ is injective. (Here $K(G, 1)$ is not assumed to be a manifold.)


For example, if $X$ is simply connected, the induced map $f\_\*$ is the zero map on $H\_n$ and when $X$ is the n-torus $T^n$, the induced map $f\_\*$ is an isomorphism. Is there any condition on $\pi\_1(X)$ that will imply injectivity of $f\_\*$ on $H\_n$?



comparing homology of a space and homology of the classifying space of its fundamental group
There can not be such a condition.


For any finitely presented group $G$, we can find a closed 4-manifold $N$ with fundamental group $G$. In dimension $n \geq 6$, we can now take $M = N \times S^{n-4}$, a $n$-manifold with fundamental group $G$. Then the classifying map $f\colon M \to BG$ of the universal cover $\tilde{M} \to M$ factors through $N$, hence
$$f\_{\ast}\colon \ H\_n(M;\mathbb Z) \to H\_n(BG;\mathbb Z)$$ is zero.


