definition of first homology group

Question

I am using a definition of the first homology group by Miranda (Algebraic Curves and Riemann surfaces), which is as follows,

The fundamental group $\pi_1(X,\alpha)$ is the group that consists of homotopy classes of closed paths starting and ending at $\alpha \in X$. Let $[\pi_1,\pi_1]$ be its commutator subgroup. We define the first homology group of $X$ to be $H_1(X)=\pi_1(X,\alpha)/[\pi_1,\pi_1]$.

I am wondering now, is in this definition $H_1(X)$ the same as the integral homology $H_1(X,\mathbb{Z})$?

Answer 1

This is not the same definition, of course, but the result is isomorphic to the usual definition if $X$ is path-connected. This is called the Hurewicz theorem.