Is it true that if the fundamental groups of two spaces are isomorphic, then their first homology groups are isomorphic? I think the answer is yes;
Let $f:\pi_1(X)\to\pi_1(Y)$ be an isomorphism and $p:\pi_1(X)\to \pi_1(X)^*, q:\pi_1(Y)\to \pi_1(Y)^*$ be the quotient maps, where $\pi_1(X)^*, \pi_1(Y)^*$ are the abelianizations of the fundamental groups. I need to show that $p\circ f$ and $q\circ f^{-1}$ are isomorphisms. But I don't how to do that.