Diagonal complexes

Joint work with Joseph Gordon. Generalizing a construction of J.L. Harer we introduce and study diagonal complexes related to a (possibly bordered) oriented surface F equipped with a number of labeled fixed points. Investigation of some natural forgetful maps combined with length assignment proves homotopy equivalence of some of the complexes to the space of metric ribbon graphs $RG^{met}_{g,n}$, to the (introduced by M. Kontsevich) tautological $S^{-1}$-bundles $L_{i}$, and to a more sophisticated bundle whose fibers are homeomorphic to some surgery of the surface F. As an application, we compute the powers of the first Chern class of $L_{i}$ in combinatorial terms. The latter result is an application of N. Mnev and G. Sharygin local combinatorial formula.