Induced homomorphism of a covering space How can I determine what's the induced homology homomorphism of a covering $S^{n} \rightarrow RP^{n}$? I suppose that a Hurewicz homomorpism would be pretty effective, but since I know nothing about higher homotopy groups of spheres and their generators I'd rather avoid it.

All of the homology groups of $\mathbb{S}^n$ are trivial, except of top and bottom one. The induced map $H_0(\mathbb{S}^n) \to H_0(\mathbb{R}\mathbb{P}^n)$ will always be isomorphism (this is very easy to calulate). The top homology group $H_n(\mathbb{R}\mathbb{P}^n)$ is either trivial or $\mathbb{Z}$, depending on whether $\mathbb{R}\mathbb{P}^n$ is orientable or not. 
To calculate the map $\mathbb{Z} \to \mathbb{Z}$ between the top homology groups in the orientable case, you can use the local degree method, described e.g. in Hatcher's book on page 136. In case of covering, since the map is local homeomorphism, the local degrees will all be 1 (or -1, depending on the choice of orientations), so  degree will be the (minus) multiplicity of the covering, which is 2 in this case.
