I am reading a paper which is asking me to view the homology group $H_1(G;\mathbb{R})$ of a (presentation of a) group as a vector space. Now, my knowledge of homology is basically non-existent, but I do know that $H_1(G;\mathbb{R})$ "is" the abelianisation of $G$.
From the context of the paper the following seems reasonable. However, having searched around a bit I am having trouble verifying it. So I thought that I would ask here.
Is the following true:
The homology group $H_1(G;\mathbb{R})$ is the real vector space consisting of maps $\phi: G\rightarrow \mathbb{R}$ (possibly also $\phi$ has finite support).
But I cannot think how this relates to the abelianisation definition. (And also I am pretty sure that these maps do not form a vector space!)
It's $H_1(G, \mathbb{Z})$ that's the abelianization of $G$. $H_1(G, \mathbb{R})$ is the tensor product of the abelianization of $G$ with $\mathbb{R}$, which is in particular a real vector space. It is much smaller than what you said.