We can define the (first) homology of a surface $S$ by working with graphs embedded in $S$. That is, we take any (oriented) graph which is 2-cell embedded in $S$, and take cycles modulo boundaries in the usual way. Here, I am talking about homology with coefficients in $\mathbb{Z}$. The group that we get is independent of the graph, so is indeed a topological invariant of the surface.
I work with group-labelled graphs, which are oriented graphs with their edges labelled from a finite abelian group $\Gamma$. Proceeding as above, group-labelled graphs allow us to define group-labelled surfaces. That is, let $G$ be a $\Gamma$-labelled graph 2-cell embedded in a surface $S$. If each face of the embedding has group-value zero (the labels of edges on the boundary of the face sum to zero), then this gives a well-defined map on homology. In fact, the embedding of $G$ in S induces a homomorphism from $H_1(S)$ to $\Gamma$. So, we can forget about the $\Gamma$-labelled graph and just study this homomorphism.
My question is: how does this construction relate to taking homology with coefficients from $\Gamma$?
Someone once told me that what I am really doing is working with cohomology with coefficients in $\Gamma$, but I didn't really get this. Can someone please clarify?
Best Answer
Hi Tony.
This is not really a homology-question, the core of it is the fundamental group. The homomorphism you are using is used in the study of Van Kampen diagrams. Consider a presentation $G=\langle A|R\rangle$. A Van Kampen diagram on $S$ is a labeled graph like you have defined it. The only difference is that in a Van Kampen diagram all labels are generators (or their inverses) $a^{\pm 1}$ and not arbitrary words (although you could define it in this general way without problems because of the Van Kampen lemma).
Then every path in this graph has a group word written on it and "reading the word along a path" is a homomorphism {Paths}$\to G$ with respect to composition of paths. It turns out, that this is compatible with homotopy of paths so this induces a homomorphism $\pi_1(S,x_0)\to G$.
This is the general version of your homomorphism: If your $G$ happens to be abelian, then this homomorphism factorizes through $\pi_1(S,x_0)^{ab}$ which is $H_1(S)$ by the Hurewicz theorem.
This point of view clarifies some connections between the geometry of Van Kampen diagrams and group theoretic questions.
For example the Van Kampen lemma tells you that a group word is trivial if and only if there is a Van Kampen diagram on this disk with this word written on the boundary.
Another fact is this one: If there are no nontrivial "reduced" Van Kampen diagrams on the torus, then every two commuting elements of $G$ generate a cyclic subgroup (i.e. $xyx^{-1}y^{-1}=1$ has only the trivial solutions $x=a^k, y=a^m$ for some $a\in G$.). In a similar spirit one can prove: If there are no nontrivial reduced Van Kampen diagrams on the real projective plane, then there are no involutions in $G$ (i.e. $x^2=1$ has only the trivial solution $x=1$), and if there are no nontrivial reduced Van Kampen diagrams on Klein's bottle, then the only element that is conjugated to its own inverse is the identity (i.e. $yxy^{-1}=x^{-1}$ has only the trivial solution $x=1$).
This connection between geometry and group properties becomes less obscure, if one knows the fundamental groups of the disk (1), the torus ($\langle x,y | xyx^{-1}y^{-1}=1\rangle$), the real projective plane ($\langle x | x^2=1\rangle$) and Klein's bottle ($\langle x,y | yxy^{-1}=x^{-1}\rangle$).