Characterizing closed curves on Riemann surfaces via homology groups of coverings

TL;DR

This paper links geometric intersection properties of closed curves on hyperbolic Riemann surfaces to homology groups of their coverings, providing new characterizations and algebraic tools for understanding surface topology.

Contribution

It introduces a homological characterization of simple closed curves and a $p$-adic Reidemeister pairing to detect geometric intersections on Riemann surfaces.

Findings

Homology groups of $p$-coverings detect geometric intersections.

Characterization of simple closed curves via homology.

Proof that surface groups are conjugacy $p$-separable.

Abstract

Let $S$ be a hyperbolic oriented Riemann surface of finite type. The main purpose of this paper is to show that non-trivial geometric intersection between closed curves on $S$ is detected by some symplectic submodules they naturally determine in the homology groups of the compactifications of unramified $p$-coverings of $S$, for $p\geq 2$ a fixed prime. In particular, this gives a characterization of simple closed curves on $S$ in terms of homology groups of $p$-coverings. We then define a $p$-adic Reidemeister pairing on the fundamental group of $S$ and show that the free homotopy classes of two loops have trivial geometric intersection if and only if they are orthogonal with respect to this pairing. As an application, we give a geometric argument to prove that oriented surface groups are conjugacy $p$-separable (a combinatorial proof of this fact was recentely given by Paris).