## Geometry & Topology

### Cobordism of singular maps

András Szűcs

#### 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 $Xτ$ 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

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

Dates
Revised: 27 June 2008
Accepted: 26 July 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800129

Digital Object Identifier
doi:10.2140/gt.2008.12.2379

Mathematical Reviews number (MathSciNet)
MR2443969

Zentralblatt MATH identifier
1210.57028

#### Citation

Szűcs, András. Cobordism of singular maps. Geom. Topol. 12 (2008), no. 4, 2379--2452. doi:10.2140/gt.2008.12.2379. https://projecteuclid.org/euclid.gt/1513800129

#### References

• Y Ando, Cobordism of maps without prescribed singularities
• D Arlettaz, Exponents for extraordinary homology groups, Comment. Math. Helv. 68 (1993) 653–672
• V I Arnol$'$d, S M Guseĭn-Zade, A N Varchenko, Singularities of differentiable maps. Vol. I. The classification of critical points, caustics and wave fronts, Monographs in Math. 82, Birkhäuser, Boston (1985) Translated from the Russian by I Porteous and M Reynolds
• V I Arnol$'$d, V A Vasil$'$ev, V V Goryunov, O V Lyashko, Singularities. I. Local and global theory, from: “Current problems in mathematics. Fundamental directions, Vol. 6”, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1988) 5–257
• M F Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961) 200–208
• M G Barratt, P J Eccles, $\Gamma \sp{+}$–structures. I. A free group functor for stable homotopy theory, Topology 13 (1974) 25–45
• O Burlet, Cobordismes de plongements et produits homotopiques, Comment. Math. Helv. 46 (1971) 277–288
• D S Chess, Singularity theory and configuration space models of $\Omega\sp nS\sp n$ of nonconnected spaces, Topology Appl. 25 (1987) 313–338
• P E Conner, E E Floyd, Differentiable periodic maps, Ergebnisse series 33, Springer, New York (1964)
• T Ekholm, A Sz\Hucs, T Terpai, Cobordisms of fold maps and maps with a prescribed number of cusps, Kyushu J. Math. 61 (2007) 395–414
• J Eliashberg, Cobordisme des solutions de relations différentielles, from: “South Rhone seminar on geometry, I (Lyon, 1983)”, Travaux en Cours, Hermann, Paris (1984) 17–31
• R Godement, Topologie algébrique et théorie des faisceaux, Actualit'es Sci. Ind. 1252, Publ. Math. Univ. Strasbourg 13, Hermann, Paris (1958)
• M Gromov, Partial differential relations, Ergebnisse series 9, Springer, Berlin (1986)
• A Haefliger, A Kosiński, Un théorème de Thom sur les singularités des applications différentiables, from: “Séminaire Henri Cartan; 9e année: 1956/57. Quelques questions de topologie, Exposé no. 8”, Secrétariat mathématique, Paris (1958) 6
• A Hatcher, Algebraic topology, Cambridge University Press (2002)
• D Husemoller, Fibre bundles, McGraw-Hill, New York (1966)
• K Ikegami, Cobordism group of Morse functions on manifolds, Hiroshima Math. J. 34 (2004) 211–230
• K Ikegami, O Saeki, Cobordism group of Morse functions on surfaces, J. Math. Soc. Japan 55 (2003) 1081–1094
• K Jänich, Symmetry properties of singularities of $C\sp{\infty }$–functions, Math. Ann. 238 (1978) 147–156
• B Kalmár, Cobordism group of Morse functions on unoriented surfaces, Kyushu J. Math. 59 (2005) 351–363
• M Kazarian, private communication
• M Kazarian, Classifying spaces of singularities and Thom polynomials, from: “New developments in singularity theory (Cambridge, 2000)”, NATO Sci. Ser. II Math. Phys. Chem. 21, Kluwer Acad. Publ., Dordrecht (2001) 117–134
• M Kazarian, Characteristic classes in singularity theory, Habilitation thesis, Moscow (2003) Available at \setbox0\makeatletter\@url http://www.mi.ras.ru/~kazarian/publ {\unhbox0
• U Koschorke, Vector fields and other vector bundle morphisms–-a singularity approach, Lecture Notes in Math. 847, Springer, Berlin (1981)
• G Lippner, A Sz\H ucs, Multiplicative properties of Morin maps
• J N Mather, Stability of $C\sp{\infty }$ mappings. IV. Classification of stable germs by $R$–algebras, Inst. Hautes Études Sci. Publ. Math. (1969) 223–248
• J W Milnor, J D Stasheff, Characteristic classes, Annals of Math. Studies 76, Princeton University Press (1974)
• R E Mosher, M C Tangora, Cohomology operations and applications in homotopy theory, Harper & Row Publishers, New York (1968)
• R Rimányi, A Sz\H ucs, Pontrjagin–Thom–type construction for maps with singularities, Topology 37 (1998) 1177–1191
• C Rourke, B Sanderson, The compression theorem. I, Geom. Topol. 5 (2001) 399–429
• R Sadykov, Bordism groups of solutions of differential relations
• O Saeki, Topology of singular fibers of differentiable maps, Lecture Notes in Math. 1854, Springer, Berlin (2004)
• J-P Serre, Groupes d'homotopie et classes de groupes abéliens, Ann. of Math. $(2)$ 58 (1953) 258–294
• A Sz\Hucs, Analogue of the Thom space for mappings with singularity of type $\Sigma \sp{1}$, Mat. Sb. $($N.S.$)$ 108 (150) (1979) 433–456, 478
• A Sz\Hucs, Cobordism of maps with simplest singularities, from: “Proc. Sympos., Univ. Siegen, 1979)”, Lecture Notes in Math. 788, Springer, Berlin (1980) 223–244
• A Sz\Hucs, Cobordism of immersions and singular maps, loop spaces and multiple points, from: “Geometric and algebraic topology”, Banach Center Publ. 18, PWN, Warsaw (1986) 239–253
• A Sz\Hucs, On the cobordism groups of immersions and embeddings, Math. Proc. Cambridge Philos. Soc. 109 (1991) 343–349
• A Sz\Hucs, Topology of $\Sigma\sp {1,1}$–singular maps, Math. Proc. Cambridge Philos. Soc. 121 (1997) 465–477
• A Sz\Hucs, On the cobordism group of Morin maps, Acta Math. Hungar. 80 (1998) 191–209
• A Sz\Hucs, Elimination of singularities by cobordism, from: “Real and complex singularities”, Contemp. Math. 354, Amer. Math. Soc. (2004) 301–324
• A Sz\Hucs, Cobordism of singular maps
• T Terpai, Cobordisms of fold maps of $(2K+2)$–manfolds into $\BR^{3K+2}$, to appear in Banach Center Publications
• R Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954) 17–86
• C T C Wall, A second note on symmetry of singularities, Bull. London Math. Soc. 12 (1980) 347–354
• R Wells, Cobordism groups of immersions, Topology 5 (1966) 281–294