Degree of covering map of surface and the number of sheets

Question

Given a covering map of closed orientable surfaces $f:M\to N$, it induces a homomorphism $f_*:\mathbb{Z}\cong H_2(M)\to H_2(N)\cong\mathbb{Z}$, suppose $f_*$ is of degree $n$. Then how to see $n$ is just the number of the sheets of the covering map $f:M\to N$?

Answer 1

Hint: given $p\in N$, suppose we have $f^{-1}(p)=\{q_1,\ldots, q_m\}$. Take a trivializing open neighborhood $U$ of $p$, so that $f^{-1}(U)\cong \coprod_{i=1}^m V_i$, where $V_i$ is a small neighborhood of $q_i$. What is the local degree of the restricted map $f|_{V_i}:V_i\to U$?

Edit: I think this answers the question posed in the comments. Note that we have a map of pairs $(M,M\setminus \{q_1,\ldots, q_n\})\to (N,N\setminus \{p\})$ given by $f$. Indeed, $f$ sends $M\setminus \{q_1\ldots, q_n\}$ into $N\setminus \{p\}$. We get a commutative diagram where the top square comes from the long exact sequence of pairs:

$$\require{AMScd} \begin{CD} H_2(M;\mathbb{Z})@>{f_*}>> H_2(N;\mathbb{Z})\\ @VVV @VVV \\ H_2(M,M\setminus\{q_1,\ldots, q_n\};\mathbb{Z}) @>{}>> H(N,N\setminus \{p\};\mathbb{Z})\\ @V{\cong}VV@VV{=}V\\ \bigoplus_{i}H_2(M,M\setminus\{q_i\};\mathbb{Z})@>{\sum}>>H(N,N\setminus \{p\};\mathbb{Z}). \end{CD}$$ If you trace the path of the fundamental class $[M]\in H_2(M;\mathbb{Z})$ through the diagram, the result should follow. The bottom arrow is the sum of the individual maps $$H_2(M,M\setminus \{q_i\};\mathbb{Z})\to H(N,N\setminus \{p\};\mathbb{Z}).$$