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

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 Teichmüller 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.

arXiv commentA new section is added in order to determine the proportion between the Weil-Petersson and the Atiyah-Bott-Goldman symplectic forms