Let $\phi:G\to H$ a group homomorphism. I'm interested in knowing what can be said about $H^*(X;G)$ and $H^*(X;H)$ (singular cohomology) in terms of $\phi$.
One can define $\phi^*:C^n(X;G)\to C^n(X;H)$ on singular chain complexes via composition $\phi\circ -$ and gets a chain map $\phi^*:C^*(X;G)\to C^*(X;H)$.
I'd like to know if this chain map (or the induced map on cohomology) has properties related to $\phi$, in the sense that injectivity or surjectivity of $\phi$ might imply something on the induce maps, not only as group homomorphisms but as ring homomorphisms in the case of the cohomology ring (with cup product), since it is easy to see that $\phi$ also induces a ring homomorphism.
Best Answer
Suppose that $G$ is an injective object. Then injectivity of $\phi$ may be rephrased as the existence of a $\psi: H \to G$ such that $\psi \circ \phi = 1_G$. This morphism gos through all the constructions, preserving composition (although reversing it in the dual construction), so that we end up with surjectivity of the resulting morphism.
Dually, the same holds if $H$ is projective.