Given a continuous map $f:X\to Y$, the mapping cylinder of $f$ is defined as the space obtained from $(X\times I)\sqcup Y$ by identifying $(x, 1)$ with $f(x)$ for all $x$.
Let $f:S^n\to S^n$ be a degree $m$ map. Let the domain sphere we written $s^n_d$ and the target sphere be written $S^n_t$.
On pg. 148 of Hacther's Algebraic Topology, the author writes that $H_n(M_f, S^n_d)\cong \mathbf Z/m\mathbf Z$ but does not give a proof.
I am unable to prove this. I thought of using the fact that the mapping cylinder deformation retracts to $S^n_t$. But this only gives us that $H_n(M_f)\cong \mathbf Z$.