# Homology and homotopy of a surface

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.

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No -- a Dehn twist about a nontrivial separating simple closed curve acts as the identity on homology, but is not nullhomotopic. The keyword to search for is "Torelli group". – Andy Putman Jun 13 '12 at 20:45
A thing to notice about your question is that homotopic homeomorphisms are automatically isotopic. – Francesco Lin Jun 13 '12 at 21:46