Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs

We give a new and simple proof for the computation of the oriented and the unoriented fold cobordism groups of Morse functions on surfaces. We also compute similar cobordism groups of Morse functions based on simple stable maps of 3-manifolds into the plane. Furthermore, we show that certain cohomology classes associated with the universal complexes of singular fibers give complete invariants for all these cobordism groups. We also discuss invariants derived from hypercohomologies of the universal homology complexes of singular fibers. Finally, as an application of the theory of universal complexes of singular fibers, we show that for generic smooth map germs g: (R^3, 0) --> (R^2, 0) with R^2 being oriented, the algebraic number of cusps appearing in a stable perturbation of g is a local topological invariant of g.Comment: This is the version published by Algebraic & Geometric Topology on 7 April 200

An elementary approach to dessins d'enfants and the Grothendieck-Teichm\"uller group

We give an account of the theory of dessins d'enfants which is both elementary and self-contained. We describe the equivalence of many categories (graphs embedded nicely on surfaces, finite sets with certain permutations, certain field extensions, and some classes of algebraic curves), some of which are naturally endowed with an action of the absolute Galois group of the rational field. We prove that the action is faithful. Eventually we prove that this absolute Galois group embeds into the Grothendieck-Teichm\"uller group $GT_0$ introduced by Drinfel'd. There are explicit approximations of $GT_0$ by finite groups, and we hope to encourage computations in this area. Our treatment includes a result which has not appeared in the literature yet: the Galois action on the subset of regular dessins - that is, those exhibiting maximal symmetry -- is also faithful.Comment: 58 pages, about 30 figures. Corrected a few typos. This version should match the published paper in L'enseignement Mathematiqu

The homotopy type of the cobordism category

The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d=2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one.Comment: 40 pages. v2 has improved notation, added explanations, and minor mistakes fixed. v3 has minor corrections and improvements. Final submitted versio

Crossed simplicial groups and structured surfaces

We propose a generalization of the concept of a Ribbon graph suitable to provide combinatorial models for marked surfaces equipped with a G-structure. Our main insight is that the necessary combinatorics is neatly captured in the concept of a crossed simplicial group as introduced, independently, by Krasauskas and Fiedorowicz-Loday. In this context, Connes' cyclic category leads to Ribbon graphs while other crossed simplicial groups naturally yield different notions of structured graphs which model unoriented, N-spin, framed, etc, surfaces. Our main result is that structured graphs provide orbicell decompositions of the respective G-structured moduli spaces. As an application, we show how, building on our theory of 2-Segal spaces, the resulting theory can be used to construct categorified state sum invariants of G-structured surfaces.Comment: 86 pages, v2: revised versio