On the homology of certain smooth covers of moduli spaces of algebraic curves

We suggest a general method of computation of the homology of certain smooth covers Mˆg,1(C) of moduli spaces Mg,1(C)Mg,1(C) of pointed curves of genus g. Namely, we consider moduli spaces of algebraic curves with level m   structures. The method is based on the lifting of the Strebel–Penner stratification of Mg,1(C)Mg,1(C). We apply this method for g≤2g≤2 and obtain Betti numbers; these results are consistent with Penner and Harer–Zagier results on Euler characteristics.