Intersections of cycles in the combinatorial moduli space

A closed formula is obtained for the integral &smallint;H&d1; 1gk1y 2g-2 of tautological classes over the locus of hyperelliptic Weirstrauss points in the moduli space of curves. As a consequence, a relation between Hodge integrals is obtained.; The calculation utilizes the homeomorphism between the moduli space of curves Mg,1 and the combinatorial moduli space Mcombg,1 , a PL orbifold whose cells are enumerated by fatgraphs. A combinatorial cycle Hcombg⊂ Mcombg,1 representing the locus of hyperelliptic Weierstrass points is constructed by exploiting the symmetries of the hyperelliptic involution. The intersection of this cycle with the Witten cycle W1 is explicitly described at the chain level by analyzing the expansions of a 6-valent vertex. Using this description Hcombg∩ W1 , the duality between Witten cycles Wa and Mumford classes kappaa, and Kontsevich's scheme of integrating psi classes, the integral &smallint;H&d1; 1gk1y 2g-2 is reduced to a weighted sum over graphs and is evaluated by the enumeration of trees.