Does anyone know a good description of homotopy classes of continuous functions $\Sigma\_g \longrightarrow \mathbb{R}P^2$, where $\Sigma\_g$ is the closed oriented surface of genus $g > 1$.


Thanks for the attention!



Homotopy classes of continuous functions $\Sigma_g \longrightarrow \mathbb{R}P^2$
This  [mathstackexchange answer](https://math.stackexchange.com/questions/36488/how-to-compute-homotopy-classes-of-maps-on-the-2-torus)  gives a nice description of the set of homotopy class of maps from $T^2=\Sigma\_1$ to any space $X$, and includes a particular mention of the example $X={\mathbb R}P^2$. Answers to this  [mathoverflow question](https://mathoverflow.net/questions/110444/determining-homotopy-classes-t2-rp2)  give a couple of other ways to calculate the set $[T^2, {\mathbb R}P^2]$. It seems to me that all these methods can be extended to general orientable surfaces. The answer is as follows: there are ${\mathbb N}$ homotopy classes of maps that induce zero on $H\_1$, plus $(2^{2g}-1)\cdot 2$ maps that are not zero on $H\_1$ (there are two homotopy classes for each possible non-zero homomorphism on $H\_1$).


