How can I prove that:
For the closed orientable surface $M$ of genus $g \geq 1$, show that for each nonzero $a \in H^1(M; \mathbb Z)$ there exists $b \in H^1(M; \mathbb Z)$ with $ab \neq 0$.
2026-04-04 13:02:50.1775307770
Hatcher's Problem 3.2.18
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Use the information on page 207 / 208 about generators for $H^1(M;\mathbb{Z})$ and their products to construct such a $b$.