Discrete Hodge star operator on 3-manifolds

Scott Wilson introduced the notion of combinatorial Hodge star operators on a compact oriented triangulated manifold $M$, which act on the singular cohomology ring of $M$. Such an operator depends on both a triangulation $\mathscr K$ of $M$ and a metric on the simplicial cochain complex of $\mathscr K$. Taking the discrete metric, we arrive at the notion of the discrete Hodge star operator for each pair $(M,\mathscr K)$. We prove, when $M$ is a $3$-manifold, that the discrete Hodge star operators acting on the cuspidal cohomology groups $H^i_\mathrm{cusp}(M,\mathbb Q)$ are independent of $\mathscr K$ for $i=1,2$. As an application, we construct a canonical positive definite symmetric quadratic form on $H^i_\mathrm{cusp}(M,\mathbb Q)$ for $i=1,2$. On the other hand, we will interpret our result from a point of view on the Langlands Program; we provide a supporting evidence for the conjecture of Prasanna and Venkatesh, which predicts a degree-shifting action of a motivic cohomology group on the cohomology ring of an arithmetic group.