Collapsing a simple closed curve on a surface ($H_{\ast }({\Sigma }_g/A)$)

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.

For genus $g\ge 1$, collapse a nonseparating simple closed curve on a closed orientable surface. Its homology class is primitive, as a curve meeting it transversely once detects by the intersection pairing on an oriented surface. The long exact sequence in relative homology therefore gives $H_0=\mathbb{Z}$, $H_1={\mathbb{Z}}^{2g-1}$, $H_2=\mathbb{Z}$, and zero higher groups for the quotient. Geometrically it has the homotopy type ${\Sigma }_{g-1}\vee S^1$.

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 $0\to \mathbb{Z}\to H_2({\Sigma }_g,A)\to \mathbb{Z}\to 0$. The two collapsed sides retain independent fundamental classes.