graph theory - Homology with Coefficients

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 $G=langle A|Rrangle$. 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 $ain G$.). 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 ($langle x,y | xyx^{-1}y^{-1}=1rangle$), the real projective plane ($langle x | x^2=1rangle$) and Klein's bottle ($langle x,y | yxy^{-1}=x^{-1}rangle$).