A homomorphism on $\pi_1(X)$ descending to a homomorphism on its abelianization

2011-01-28T18:08:30Z

Let $X$ be a path-connected topological space. Denote $\pi(X)$ as the fundamental group (note this doesn't need a basepoint up to isomorphism, as $X$ is path-connected) and denote $H_1(X)$ as the 1st singular homology group. Let:

$\phi : \pi_1(X) \to H_1(X)$

be the function that sends the homotopy class of a loop $f$ to the homology class of the singular 1-cycle $f$. It can be checked that $\phi$ is well-defined and a homomorphism. I also have reason to believe that it naturally descends to a homomorphism:

$\overline{\phi}: \pi_1(X)_{ab} \to H_1(X)$

on the abelianization of the fundamental group. Could anyone shed a little more light on why this is the case? In particular, is $\overline{\phi}$ uniquely determined by $\phi$ (if so, how it uniquely determined), and why is it a homomorphism?

A homomorphism on $\pi_1(X)$ descending to a homomorphism on its abelianization - MathOverflow [closed]

A homomorphism on $\pi_1(X)$ descending to a homomorphism on its abelianization