Maps between surfaces of genus n

Question

Could anyone help me find maps $f:\Sigma_{n}\longrightarrow\Sigma_{m}$ with $0<n<m\ \ $ NOT homotopic to a constant map or prove its existence without using K-theory?

Thanks

Answer 1

I asked a friend about this, and he gave a way to construct a lot of these. I suppose the easiest is the torus. We have the projection map $\Sigma_1 \rightarrow S^1$. Now pick any nontrivial loop in $\Sigma_n$ and let $\gamma: S^1 \rightarrow \Sigma_n$ be that loop. This gives a ton of examples. You can use the same idea to build maps $\Sigma_n \rightarrow \Sigma_m$ for pretty much any $n$, by collapsing the genus $n$ surface onto a loop formed by any particular generator of the fundamental group you like and mapping that into $\Sigma_m$. All such maps are not homotopic to constants because they send the generators of one fundamental group to nontrivial loop by construction. Similar arguments can build more complicated maps.