I am currently working on a problem and I have solved all the differents points except for one, that I have reduced to the following exercise which I could not solve:
Show that two homeomorphisms between closed surfaces (in my case, between tori) are homotopic if and only if they induce the same map on the fundamental group(s).
I have found (in Hatcher's Algebraic Topology, Proposition 1B.9 page 90) a proof of what seems to be a more general fact, but since I don't know much about algebraic topology I wonder if one could give a more elementary proof in the special case of this exercise. Any help would be appreciated!
