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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
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A thing to notice about your question is that homotopic homeomorphisms are automatically isotopic. –  Francesco Lin Jun 13 '12 at 21:46
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1 Answer

up vote 9 down vote accepted

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.

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Wouldn't you prefer a simplicial answer? (What a stupid joke!) –  Fernando Muro Jun 14 '12 at 6:19
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Perhaps -- the possibilities for the answer were manifold. (Another bad joke.) –  Kevin Walker Jun 14 '12 at 13:47
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