In Bredon's Topology and Geometry it‘s postulated that for a $k$-fold covering projection $p\colon X \to Y$, $\deg(p)=k$, where $\deg(p)=k$ The example given is of $X$ the $3$-dimensional sphere $S^3$ and $Y$ the Lens space $L(k,m)$, both of which have $H_3 = \mathbb Z$.
So I'd like to know where this comes from. In the case of $X=Y=S^1$ it‘s clear, but what about the general case?
Thanks for putting my question into TEX, I have to find it out myself, so it's going to take some time to edit. Here the short answer: For a continuous function $f\colon M \to N$ with $M$ and $N$ compact oriented $n$-dimensional manifolds, and its induced homological homomorphism $f_\#$ following is true
$f_\#([M])=\pm deg(f)[N]$, $[M]$ and $[N]$ the generators of the n-th homological groups $H_n(M)= H_n(N) = \mathbb Z$.