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

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2025年5月26日

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Nariya Kawazumi

摘要

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 ${\mathcal {T}}_g$ $T_{g}$ . We call it the standard cocycle of the point of ${\mathcal {T}}_g$ $T_{g}$ . 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.