Geometry & Topology

Cobordism of singular maps

András Szűcs

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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 Xτ 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 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 [Itogi Nauki i Tekniki, Moscow (1988) 5–257].)

Article information

Geom. Topol., Volume 12, Number 4 (2008), 2379-2452.

Received: 8 September 2006
Revised: 27 June 2008
Accepted: 26 July 2008
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R45: Singularities of differentiable mappings 55P42: Stable homotopy theory, spectra
Secondary: 57R42: Immersions 55P15: Classification of homotopy type

cobordism singular map Pontrjagin–Thom construction Kazarian spectral sequence


Szűcs, András. Cobordism of singular maps. Geom. Topol. 12 (2008), no. 4, 2379--2452. doi:10.2140/gt.2008.12.2379.

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