A topological proof of Wolpert's formula for the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates

TL;DR

This paper provides a topological proof of Wolpert's formula for the Weil-Petersson symplectic form using a new cell decomposition and explicit cocycle construction related to pants decompositions of surfaces.

Contribution

It introduces a natural cell decomposition and a standard cocycle on Teichmüller space, enabling a topological proof of Wolpert's formula in Fenchel-Nielsen coordinates.

Findings

Topological proof of Wolpert's formula established.

Explicit groupoid cocycle representing points in Teichmüller space.

Cell decomposition associated with pants decomposition of surfaces.

Abstract

We introduce a natural cell decomposition of a closed oriented surface associated with a pants decomposition, and an explicit groupoid cocycle on the cell decomposition which represents each point of the Teichm\"uller space $\mathcal{T}_g$. We call it the {\it standard cocycle} of the point of $\mathcal{T}_g$. As an application of the explicit description of the standard cocycle, we obtain a topological proof of Wolpert's formula for the Weil-Petersson symplectic form in terms of the Fenchel-Nielsen coordinates associated with the pants decomposition.