Homology after collapsing a nonseparating surface curve ($H_1\cong {\mathbb{Z}}^{2g-1},H_2\cong \mathbb{Z}$)

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 ${\Sigma }_{g-1}\vee S^1$.