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].)
