For context, I am doing Hatcher's exercise 3.3.9, which aims to show if $p: M\to N$ is a finite sheeted covering map of two closed, connected orientable manifolds, then the degree of $p$ (as defined in Hatcher) matches the number of sheets of the cover (up to changing fundamental class to not deal with signs).
I know this is possible by summing local degrees (i.e using the previous exercise of Hatcher). I am wondering if this can be done using transfer maps.
My thought so far are as follows: Denote by $\tau: C_*(N)\to C_*(M)$ which sends a singular chain to the sum of its lifts. Then $\tau$ induces a map on homology $H_n(N)\to H_n(M)$, and we can see that $p\circ \tau$ is multiplication by $k$ the number of sheets of the cover. Each map individually is defined by $p_*([M])=a[N]$ and $\tau_*([N])=b[M]$, for some $a,b\in \mathbb{Z}$, and so we have $ab=k$, and we want to show $a=k$, this is thus equivalent to showing $b=1$. After this I don't really know if there is anything that can be done? I wasn't able to do any more than "setting the stage" for this theoretical alternative proof.
Thank youuu