Relative Hurewicz theorem extension

Question

Is there a relationship between $H_1(X,A)$ and $\pi_1(X,A)$ when $A$ intersects every path component of $X$, i.e. when $(X,A)$ is $0$-connected?

Answer 1

This is a lacuna in the literature on which I wrote a paper available here (this is with a recent correction). We interpret $\pi_1(X,A)$ as the fundamental groupoid on the set $A$ of base points. The aim is to give circumstances in which the canonical morphism $$\pi_1(X,A) \to H_1(X,A) $$can be interpreted as abelianisation of a groupoid. We clearly need $A$ meets each path component of $X$. But on the right hand side $A$ is considered as a subspace of $X$. So we have also to assume that as a subspace $A$ is totally disconnected. Finally, the abelianisation is what we call the total abelianisation, i.e. the left adjoint to the inclusion of the category of abelian groups into that of groupoids.

We then use covering morphisms of groupoids to relate this result to Crowell’s notion of derived module and show its relevance to covering spaces.