# cellular maps induced by a homomorphism

Consider the following situation: We have a countable discrete group $G$ with a finite index (not necessarily normal) subgroup $H$. It is possible to construct 3 $CW$ complexes and a covering diagram as usual. [I am skipping the details, see for instance Geoghegan's Top. Methods in Group Theory]

By considering the free $R$ modules generated by the cosets of $H$ in $G$ it is possible to relate the cellular homology of these spaces(with coefficients) with the homologies of the groups. [please again see the above book.]

Now, the question: Suppose $f\colon H\rightarrow G$ is a homomorphism. How does this $f$ enters the above story? For example, the inclusion homomorphism $\subset \colon H\subset G$ (if I don't lie to myself) appears as the convering of complexes. Does a general homomorphism have a topological meaning?
Thank you.

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Yes, your map $f : H \to G$ induces a map $Bf : BH \to BG$ and if $H$ is finite index you can realize $Bf$ as a finite-sheeted covering space. In particular $Bf$ is a map of the classifying spaces so it has an induced map on (co)homology, giving the relation with homology you seem to be seeking. –  Ryan Budney Apr 26 '11 at 21:46
Thank you Ryan, this is what I need. Could you also explain or give a reference that explains how to see $Bf$ as a finite sheeted covering? Thank you. –  niyazi Apr 26 '11 at 21:55
Ryan, also, is transfer a functor, i.e is there a version of transfer for $f$? Which book explains these things? Thank you very much. –  niyazi Apr 26 '11 at 22:00
General covering space theory says the covering spaces of $BG$ are in bijective correspondence with subgroups of $\pi_1 BG$, but $\pi_1 BG$ is canonically ismorphic to $G$. So $H$ corresponds to a covering space of $BG$, and by design it's isomorphic to $BH$. There are various natural ways you can set up explicit models for classifying spaces, but I think most ways of getting what you want more or less factor through this way of thinking of things. –  Ryan Budney Apr 27 '11 at 0:43