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Throughout this paper we consider smooth maps of positive codimensions, having only stable singularities (see Arnold, Guseȋn-Zade and Varchenko [3, Section 1.4, 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 X giving homotopy representation of cobordisms of singular maps with a given list of allowed singularities as in work of Rimányi and the author [29; 34; 35].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 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 [4, page 212].) 57R45, 55P42; 57R42, 55P15In this paper all smooth maps will have positive codimensions and stable singularities. The aim of the present paper is to establish a close relationship between Kazarian's homological characteristic spectral sequence (generalising Vassiliev's complex) and the classifying space for cobordisms of maps having singularities only from a given fixed list (see the papers of the author [34] and the author with Rimányi [29]). Such a relationship was conjectured by M Kazarian, and it can be expressed as follows (in our formulation): There is a spectrum with a filtration such that the arising homological spectral sequence gives Kazarian's characteristic spectral sequence while the homotopy groups of the spectrum give the corresponding cobordism groups of singular maps (with a shift of the dimension). Hence the Hurewicz map for this spectrum induces a rational isomorphism from the cobordism groups of singular maps to the homology groups of Kazarian's space. This allows us to extend quite a few classical theorems of cobordism theory to cobordisms of singular maps. In particular (at least modulo torsion): b) We extend the definition of Conner and Floyd of the characteristic numbers of bordism classes and the statement that these numbers form a complete set of invariants. We give a complete and computable set of obstructions to elimination of singularities by cobordism.c) For any set of corank one singularities we determine explicitly the ranks of the corresponding cobordism groups.d) We give some general results beyond Morin maps, and also on the torsion groups.e) We show a "Postnikov like tower" that produces the classifying space for cobordisms of singular maps as an iterat...

We give three formulas expressing the Smale invariant of an immersion f of a (4k − 1)-sphere into (4k + 1)-space. The terms of the formulas are geometric characteristics of any generic smooth map g of any oriented 4k-dimensional manifold, where g restricted to the boundary is an immersion regularly homotopic to f in (6k − 1)-space.The formulas imply that if f and g are two non-regularly homotopic immersions of a (4k − 1)-sphere into (4k + 1)-space then they are also non-regularly homotopic as immersions into (6k − 1)-space. Moreover, any generic homotopy in (6k − 1)-space connecting f to g must have at least a k (2k − 1)! cusps, where a k = 2 if k is odd and a k = 1 if k is even.1991 Mathematics Subject Classification. 57R42; 57R45.

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