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 $${\mathcal {T}}_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> . We call it the standard cocycle of the point of $${\mathcal {T}}_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> . 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.