# definition of first homology group

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 $H_1(X,\mathbb{Z})$?

• Yes, it is..... – Randall May 17 '18 at 11:42

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.
• Thank you. I can't really see the intuitive difference in the definitions of $H_1(X)$ and $H_1(X,\mathbb{Z})$, can you help me with that? – TheBeiram May 17 '18 at 12:45