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