= Homology after collapsing a separating surface curve
{title2=$H_1\cong\mathbb Z^{2g},\quad H_2\cong\mathbb Z^2$}
Collapse a separating simple closed curve on a <closed orientable surface> of genus $g$. If its two sides have genera $g_1,g_2$, the quotient is homeomorphic to $\Sigma_{g_1}\vee\Sigma_{g_2}$. Its <integral homology> is $\mathbb Z$ in degree zero, $\mathbb Z^{2g}$ in degree one, $\mathbb Z^2$ in degree two, and zero above degree two. In <relative homology>, the <circle> maps to zero in $H_1(\Sigma_g)$, and the second relative group fits into a split sequence $0\to\mathbb Z\to H_2(\Sigma_g,A)\to\mathbb Z\to0$. The two collapsed sides retain independent <fundamental classes>.
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