Quotient of surface of genus 2

Question

Let $g$ be the quotient map from $M_2$ to $M_1$ (namely torus). Why $g$ induces isomorphism on $H_2$?

Also, Hatcher talks about the degree of $f$ from sphere to sphere in his book. Can it be generalized to any map between topological spaces $X$ and $Y$, if the highest homology groups of $X$ and $Y$ are $Z$? How?

Answer 1

It depends on the map, sometimes you will not get an isomorphism of $H_2$: For instance, suppose that $f: M_2\to M_1$ is a branched cover of degree $2$, then $$ |H_2(M_1):f_*(H_2(M_2))|=2. $$ More generally, you can realize any finite index.

See here for a geometric interpretation of degree in the smooth setting. This should answer many questions that you have. In general, one can define $n$-th degree for a map $f: X\to Y$, it is simply the index $$ |H_n(Y): f_*(H_n(X))|. $$ However, this notion is not very geometric once you leave the realm of topological $n$-manifolds (or pseudo-manifolds).

Answer 2

For your first question, the simplest is probably to see that this is already true at the 2-cell level...

For you second question, see this page on the degree.

Answer 3

Let me update what I've learned now. We can actually define the degree of a map $f$ between two connected closed orientable n-manifolds $M$ and $N$, which is problem 3.3.7 in Hatcher's book on Page 258. Also it equals to the sum of local degree of $f$, which is problem 3.3.8.

The idea to prove 3.3.8 is similar to Hatcher's proof for spheres in section 2.2 (Proposition 2.30)

Just use the following commutative diagram. (I use two arrows to represent a diagonal arrow, since amscd doesn't support diagonal arrows) $$ \require{AMScd} \begin{CD} @<\cong<< H_n(U_i,U_i-x_i) @>f_*>> H_n(V, V - y)\\ @VVV @VV k_i V @VV\cong V \\ H_n(M , M - x_i) @<p_i<< H_n(M , M - f^{-1}y) @>f_* >> H_n(N, N - y)\\ @AAA @AA j A @AA\cong A \\ @<\cong<< H_n(M) @>f_*>> H_n(N) \end{CD} $$