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 $H_n$?