# Degree of covering map of surface and the number of sheets

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 $$f:M\to N$$?

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 $$H_2(M,M\setminus \{q_i\};\mathbb{Z})\to H(N,N\setminus \{p\};\mathbb{Z}).$$
• I have a question, the formula that degree equals to the sum of local degrees that I learned is from Hatcher, but he only defined on $S^n$. Is there a reference for the general closed surfaces?
• How do you know that each isomorphism $H(M,M-q_i) \to H(N,N-p)$ is induced by an orientation preserving homeomorphism? This is the most important part, but I cannot see it in this proof. Commented Apr 30 at 13:51