Suppose $S$ be a closed orientable genous $g$ surface. Let $f$,$g$ be homeomorphis from $S$ to itself. Assume they induce the same map on 1st homology $H\_1(S, \mathbb Z).$


My question is; does this imply $f$, $g$ are homotopic or isotopic?


Any help is appreciated. Thank you in advance.



Homology and homotopy of a surface
Andy Putman's comment (Dehn twist on null-homologous curve; google for "Torelli group") is a concise and fairly complete answer to this question. I'm posting this CW answer so that the question does not resurface later.


