Relative homology of a surface modulo disjoint circles ($H_2({\Sigma }_g,A)\cong {\mathbb{Z}}^c,H_1({\Sigma }_g,A)\cong {\mathbb{Z}}^{2g+c-2}$)

Let $A$ be a nonempty union of disjoint embedded circles in a closed oriented topological surface ${\Sigma }_g$, and let $c$ be the number of components after cutting along them. Excision and Poincare-Lefschetz duality identify the relative homology with the complementary-degree cohomology of the cut surface. Its components have boundary, so the relative groups are free, with the displayed ranks; $H_0$ and higher groups vanish. The rank of the circle classes in $H_1({\Sigma }_g)$ is $|{\pi }_0(A)|+1-c$. For a single circle these groups detect whether its class is zero, but for genus at least two cannot distinguish a disk boundary from an essential separating circle.