# Penner's formula for volume of the Moduli Space

In his paper "Weil-Petersson Volumes" Penner gives the following formula for the integral of a top-dimensional cohomology class $\omega$ on the moduli space $\mathcal M_g^s$ of $s$-punctured riemann surfaces of genus $g$:

$$\int_{\mathcal M_g^s}\omega=\sum_G\frac{1}{|Aut(G)|}\int_{D(G)}\phi^*\omega$$

where the sum is taken over all isomorphism classes of ribbon graphs $G$, $\phi:\widetilde{\mathcal T_g^s}\to\mathcal M_g^s$ is the natural map from the decorated Teichmüller space to the moduli space, and $D(G)$ is more or less the pre-image under $\phi$ of the orbicell in $\mathcal M_g^s$ corresponding to $G$.

My question is to what extent can this formula be generalized. Specifically,

1) Is a similar formula for evaluating forms of arbitrary degree?

2) Is this formula the shadow'' of a general formula for evaluating cohomology classes on (smooth) orbifolds?

I myself cannot understand Penner's paper well enough to see where the top-dimensional assumption on $\omega$ enters the picture. References to the literature would be much appreciated as well.

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(of course, neither side makes sense if $\omega$ is not top dimensional) –  John Pardon Jan 28 '12 at 18:56
Right. By similar formula of arbitrary degree'' I'm asking for the integral over a cycle of the appropriate dimension. –  Steve Jan 29 '12 at 2:27
Isn't this formula essentially tautological, once you believe that this triangulation exists? i.e. a similar formula exists for any triangulated manifold (with no need for the 1/|Aut| factor for an honest triangulation, of course). –  Tom Church Jan 29 '12 at 6:06