Let $f:X \rightarrow Y$ be a finite covering map between compact oriented surfaces and let $K$ be the kernel of the induced map $f\_\ast: H\_1(X) \rightarrow H\_1(Y)$. Here homology has rational coefficients.


Question: must $K$ be nondegenerate with respect to the symplectic algebraic intersection pairing $\omega$ on $H\_1(X)$? In other words, is it true for all nonzero $k \in K$, there exists some $k’ \in K$ with $\omega(k,k’) \neq 0$?



Nondegeneracy of kernel of map on homology induced by covering of surfaces
The answer is yes. Let me work with $H^1$, which is canonically isomorphic to $H\_1$ by Poincaré duality. I claim that $K=\operatorname{Ker} f\_\*$ is the orthogonal of $\ \operatorname{Im}f^\* $: this is because $(\beta \cdot f^\*\alpha )=(f\_\*\beta \cdot \alpha )$ for $\alpha $ in $H^1(Y)$ and $\beta $ in $H^1(X)$.


Now for $\alpha ,\beta \in H^1(Y)$ we have $(f^\*\alpha \cdot f^\*\beta )=d(\alpha \cdot \beta )$, where $d$ is the degree of the covering. Thus $\ \operatorname{Im}f^\* $ is a non-degenerate subspace of $H^1(X)$, hence its orthogonal $K$ is also non-degenerate.


