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.
Categories: math.GT