Homotopy class of maps between orientable surfaces

Question

Let $\Sigma_g$ denote the orientable surface of genus $g$. I am interested in the set of maps $\Sigma_g \to \Sigma_h$ considered up to homotopy $[\Sigma_g, \Sigma_h]$. I know when $g=h=0$ that these maps are just classified by degree. Is there a concrete set of things that I can compute that tells me wether two maps $f,g : \Sigma_g \to \Sigma_h$ are homotopic?

I would also be interested in finding (based) maps $f,g: \Sigma_g \to \Sigma_h$ that are not homotopic but such that $\pi_1(f) = \pi_1(g)$.

Answer 1

Surfaces of genus $>0$ are classifying spaces of discrete groups (because their universal covers are contractible). Hence, two (based) maps between them are homotopic iff they induce the same homomorphism on fundamental groups.

Answer 2

If $h\neq 0$ then for any map $f$ with $\pi_1(f)=0$ can be lifted to the universal cover ( which is contractible) and hence can be homotoped into a single point.

If $h=0$, then it is a sphere, and in case of sphere there is Hopf degree theorem https://en.m.wikipedia.org/wiki/Hopf_theorem .

Answer 3

Have a look at the paper

Ellis, G.~J. "Homotopy classification the J.H.C. Whitehead way". Exposition. Math. 6 (2) (1988) 97--110.

Here is a pdf.