Cobordism of singular maps

Abstract

Throughout this paper we consider smooth maps of positive codimensions, having only stable singularities (see Arnold, Guseĭn-Zade and Varchenko [Monographs in Math. 83, Birkhauser, Boston (1988)]. 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 $X_{\tau }$ giving homotopy representation of cobordisms of singular maps with a given list of allowed singularities as in work of Rimányi and the author [Topology 37 (1998) 1177–1191; Mat. Sb. (N.S.) 108 (150) (1979) 433–456, 478; Lecture Notes in Math. 788, Springer, Berlin (1980) 223–244].

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 (ie when the list of allowed singularities contains only corank $1$ maps) we compute these ranks explicitly.

We give a very transparent homotopical description of the classifying space $X_{\tau }$ 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 [Itogi Nauki i Tekniki, Moscow (1988) 5–257].)