Homology after collapsing a separating surface curve ($H_1\cong {\mathbb{Z}}^{2g},H_2\cong {\mathbb{Z}}^2$)

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