= Relative homology of a surface modulo disjoint circles
{title2=$H_2(\Sigma_g,A)\cong\mathbb Z^{c},\quad H_1(\Sigma_g,A)\cong\mathbb Z^{2g+c-2}$}
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.
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