= Homology after collapsing a nonseparating surface curve
{title2=$H_1\cong\mathbb Z^{2g-1},\quad H_2\cong\mathbb Z$}
For genus $g\geq1$, 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> $\Sigma_{g-1}\vee S^1$.
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