= Collapsing a simple closed curve on a surface
{title2=$H_*(\Sigma_g/A)$}
A simple closed curve in a <closed orientable surface> has an annular <collar neighbourhood>, so its collapse is governed by the <collapsing a pair theorem>. In the <long exact sequence in relative homology>, the decisive map is $H_1(A;\mathbb Z)\to H_1(\Sigma_g;\mathbb Z)$, sending the <circle> generator to its <homology class>. That class is zero for a separating curve and a <primitive homology class> for a nonseparating curve. The two possibilities lead to <homology after collapsing a separating surface curve> and <homology after collapsing a nonseparating surface curve>.
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