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 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 –torsion parts for big primes coincide with those of the homology groups of the corresponding Kazarian space. (A prime 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 maps) we compute these ranks explicitly.
We give a very transparent homotopical description of the classifying space 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].)
"Cobordism of singular maps." Geom. Topol. 12 (4) 2379 - 2452, 2008. https://doi.org/10.2140/gt.2008.12.2379