Let $W$ be a closed orientable surface of genus 3. Prove that any continuous map $g: \mathbb{R} P^2 \rightarrow W$ must be homotopic to a constant map.
How do I prove this statement? Do I have to use some information about the universal cover? What should be my method?
Let $g:\mathbb{R}P^2\rightarrow W$, the map induced on fundamental groups $g_1:\pi_1(\mathbb{R}^2)\rightarrow \pi_1(W)$ is trivial since $\pi_1(W)$ does not have torsion. This implies that $g$ can be lifted to $\hat g:\mathbb{R}P^2\rightarrow \mathbb{R}^2$.
Fix the origin of $\mathbb{R}_2$ and define $\hat g_t(x)=t\hat g(x)$. Let $p:\mathbb{R}^2\rightarrow W$ be the covering map, $p\circ\hat g_t$ define an homotopy between $g$ and a constant map.