Discrete Hodge star operator on 3-manifolds

Abstract

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 $\mathcal{K}$ of $M$ and a metric on the simplicial cochain complex of $\mathcal{K}$. Taking the discrete metric, we arrive at the notion of the discrete Hodge star operator for each pair $(M,\mathcal{K})$. We prove, when $M$ is a $3$-manifold, that the discrete Hodge star operators acting on the cuspidal cohomology groups $H_{\mathrm{cusp}}^i(M,\mathbb{Q})$ are independent of $\mathcal{K}$ for $i=1,2$. As an application, we construct a canonical positive definite symmetric quadratic form on $H_{\mathrm{cusp}}^i(M,\mathbb{Q})$ 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.

arXiv commentProof of main theorem is flawed. 19 pages, 5 figures