I would like to know how much is to be expected from the relation of the homology of a $n$-sheeted covering manifold $M$ and its base $N$ (let's say $\pi:M\mapsto N, deg \space\pi=n$).
I'm interested in this question at some different layers, first of all, what is the relation of the complexes $C_M$ and $C_N$ (other than sometimes a surjection induced by $\pi$)? Does this map induced by $\pi$ commutes with the respective differentials in $C_M$ and $C_N$? And how much information can we gain at the homology level? (in any sense, information of $N$ known $M$, and viceversa).
Any help is welcome, I'm interested in this question for three specific homologies ranging form less difficult to more difficult: singular homology, Morse homology and Embedded Contact Homology (ECH).