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Write the closed oriented surface of genus $g$ as ${\Sigma }_g=A\cup B$, where $B$ is a closed disk, $A$ is a slightly enlarged once-punctured surface, and $A\cap B$ is an annulus. Thus $B$ is contractible, $A\cap B$ is homotopy equivalent to the circle, and $A$ is homotopy equivalent to the wedge sum of $2g$ circles. The boundary circle represents the product of the $g$ commutators in the fundamental group of $A$, so its image in the first homology group, which is the abelianization of that group, is zero.

The relevant part of the Mayer–Vietoris sequence is consequentlyThe degree-zero part says that the connected space ${\Sigma }_g$ has $H_0({\Sigma }_g)\cong \mathbb{Z}$, while the groups above degree two vanish. Hence