Degree of maps between surfaces and $H_1$

Question

Suppose we have a degree $d$ map between two closed compact oriented surfaces $f : \Sigma \to \Sigma'$. What does knowing $d$ tell us about the map on $f_\star : H_1(\Sigma, \mathbb{Z}) \to H_1(\Sigma', \mathbb{Z})$?

Answer 1

Here's an example to think about. You can get a map of any degree from a surface of any genus $\Sigma_g$ to a sphere. Take $d$ disjoint disks in $\Sigma_g$ and crush the complement of these disks to a point. Map the disks onto the sphere in an orientation preserving way, with the boundary going to a fixed point on the sphere. This is a degree $d$ map, but the map on $H_1$ is trivial no matter what $d$. So in this case, the map's degree is not related to $H_1$ in any way.

Answer 2

denote by $I$ and $I'$ the intersection matrices on $\Sigma$ and $\Sigma'$, in standard basis having form $$ \begin{pmatrix} 0 & 1 & 0 & 0 & \ldots \\ -1 & 0 & 0 & 0 &\\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \\ \vdots & & & & \ddots \end{pmatrix}. $$

then for matrix $A$ of the homomorphism $f_*$ the following condition holds: $$ A\cdot I\cdot A^t=d\cdot I'. $$ here $A^t$ corresponds to the operator $f^*:H^1(\Sigma')\to H^1(\Sigma)$, and the equation follows from the fact that $f^*$ is a ring homomorphism.