Cobordism of singular maps

Throughout this paper we consider smooth maps of positive codimensions, having only stable singularities (see [Ar2], §1.4 in Chapter 1). We prove a conjecture due to M. Kazarian, connecting two classifying spaces in singularity theory for this type of singular maps.. These spaces are: 1) Kazarian’s space (generalising Vassiliev’s algebraic complex and) showing which cohomology classes are represented by singularity strata. 2) The space Xτ giving homotopy representation of cobordisms of singular maps with a given list of allowed singularities [R–Sz], [Sz1], [Sz2]. Our results are: a) We obtain that the ranks of cobordism groups of singular maps with a given list of allowed stable singularities, and also their p-torsion parts for big primes p coincide with those of the homology groups of the corresponding Kazarian space. (A prime p is “big” if it is greater than half of the dimension of the source manifold.) For all types of Morin maps (i.e. when the list of allowed singularities contains only corank 1 maps) we compute these ranks explicitly. b) We give a very transparent homotopical description of the classifying space Xτ as a fibration. Using this fibration we solve the problem of elimination of singularities by cobordisms. (This is a modification of a question posed by Arnold, [Ar1], page 212.) AMS Classification numbers Primary: 57R45, 55P42 Secondary: 57R42, 55P15