Characterizing closed curves on Riemann surfaces via homology groups of coverings

Details Characterizing closed curves on Riemann surfaces via homology groups of coverings Characterizing closed curves on Riemann surfaces via homology groups of coverings

2011-11-10

arxiv.org/abs/1111.2373v1

Mathematics

Algebraic Topology

11 pages

Scientific paper

Let $S_{g,n}$, for $2g-2+n>0$, be a closed oriented Riemann surface of genus $g$ from which $n$ points have been removed. The purpose of this paper is to show that closed curves on $S_{g,n}$ are characterized by the submodules they determine in the homology groups of finite unramified coverings of $S_{g,n}$. More precisely, for a given finite unramified covering $\pi: S\rightarrow S_{g,n}$, let us denote by $\ol{S}$ the closed Riemann surface obtained filling in the punctures of $S$. Then, for a given closed curve $\gamma$ on $S_{g,n}$, the cycles supported on the irreducible components of $\pi^{-1}(\gamma)$ span a submodule $V_\gamma$ of the homology group $H_1(\ol{S},\Z)$. The main result of the paper is a characterization of simple closed curves on $S_{g,n}$. A non-power closed curve $\gamma$ on $S_{g,n}$ is homotopic to a simple closed curve if and only if, for a fixed prime $p$, every finite unramified $p$-covering $\pi: S\rightarrow S_{g,n}$ is such that the associated submodule $V_\gamma$ of $H_1(\ol{S},\Z)$ is isotropic for the standard intersection pairing on the closed Riemann surface $\ol{S}$. We then prove that, if $\gamma$ and $\gamma'$ are two non homotopic simple closed curves on $S_{g,n}$, then there is a finite unramified $p$-covering $\pi: S\rightarrow S_{g,n}$ such that $V_\gamma\neq V_{\gamma'}$ in the homology group $H_1(\ol{S},\Z)$. 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 recently given by Paris).

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