# Homotopy class of maps between orientable surfaces

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 $\pi_1(f) = \pi_1(g)$.

• look at a primer on mapping class groups
– Max
Oct 11 '16 at 3:36
• @max I don't think OP needs to go for primer , because he is not asking for clarifying all homeomorphism... He is only interested when two maps induced same map into fundamental level. Oct 11 '16 at 3:48
• primer includes in particular a criterion for determining when two self-homeomorphisms are homotopic. one selects a suitable (finite) collection of curves and must check if the induced action on the curves coincides.
– Max
Oct 11 '16 at 3:55
• also the dehn-nielsen-baer theorem for based surfaces is discussed
– Max
Oct 11 '16 at 3:57
• @max read the last paragraph of that question, for answering that you don't need anything. Oct 11 '16 at 7:30

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.
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 .