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

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. 
