Algebraic & Geometric Topology

Cobordisms of maps with singularities of given class

Yoshifumi Ando

Full-text: Open access


Let P be a smooth manifold of dimension p. We will describe the group of all cobordism classes of smooth maps of n–dimensional closed manifolds into P with singularities of given class (including all fold singularities if np) in terms of certain stable homotopy groups by applying the homotopy principle on the existence level, which is assumed to hold for those smooth maps. It will enable us to construct an explicit classifying space for this cobordism group in the dimensions n<p and np2.

Article information

Algebr. Geom. Topol., Volume 8, Number 4 (2008), 1989-2029.

Received: 6 June 2006
Revised: 18 August 2008
Accepted: 10 September 2008
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R45: Singularities of differentiable mappings
Secondary: 57R90: Other types of cobordism [See also 55N22] 58A20: Jets

singularity map cobordism stable homotopy


Ando, Yoshifumi. Cobordisms of maps with singularities of given class. Algebr. Geom. Topol. 8 (2008), no. 4, 1989--2029. doi:10.2140/agt.2008.8.1989.

Export citation


  • Y Ando, Smooth maps with singularities of bounded $\mathcal{K}$–codimensions
  • Y Ando, The homotopy type of the space consisting of regular jets and folding jets in $J\sp 2(n,n)$, Japan. J. Math. $($N.S.$)$ 24 (1998) 169–181
  • Y Ando, Fold-maps and the space of base point preserving maps of spheres, J. Math. Kyoto Univ. 41 (2001) 693–737
  • Y Ando, Folding maps and the surgery theory on manifolds, J. Math. Soc. Japan 53 (2001) 357–382
  • Y Ando, Invariants of fold-maps via stable homotopy groups, Publ. Res. Inst. Math. Sci. 38 (2002) 397–450
  • Y Ando, Existence theorems of fold-maps, Japan. J. Math. $($N.S.$)$ 30 (2004) 29–73
  • Y Ando, Smooth maps having only singularities with Boardman symbol $(1,0)$, Topology Appl. 142 (2004) 205–226
  • Y Ando, Stable homotopy groups of spheres and higher singularities, J. Math. Kyoto Univ. 46 (2006) 147–165
  • Y Ando, A homotopy principle for maps with prescribed Thom–Boardman singularities, Trans. Amer. Math. Soc. 359 (2007) 489–515
  • Y Ando, The homotopy principle for maps with singularities of given $\mathcal K$–invariant class, J. Math. Soc. Japan 59 (2007) 557–582
  • M F Atiyah, Bordism and cobordism, Proc. Cambridge Philos. Soc. 57 (1961) 200–208
  • M F Atiyah, Thom complexes, Proc. London Math. Soc. $(3)$ 11 (1961) 291–310
  • J M Boardman, Singularities of differentiable maps, Inst. Hautes Études Sci. Publ. Math. (1967) 21–57
  • O Burlet, G de Rham, Sur certaines applications génériques d'une variété close à $3$ dimensions dans le plan, Enseignement Math. $(2)$ 20 (1974) 275–292
  • H Cartan, S Eilenberg, Homological algebra, Princeton University Press (1956)
  • 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 der Math. und ihrer Grenzgebiete, N. F. 33, Academic Press, New York (1964)
  • J Eliashberg, Singularities of folding type, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970) 1110–1126
  • J Eliashberg, Surgery of singularities of smooth mappings, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972) 1321–1347
  • 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
  • S D Feit, $k$–mersions of manifolds, Acta Math. 122 (1969) 173–195
  • M L Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969) 707–734
  • M L Gromov, A topological technique for the construction of solutions of differential equations and inequalities, from: “Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2”, Gauthier-Villars, Paris (1971) 221–225
  • M L Gromov, Partial differential relations, Ergebnisse der Math. und ihrer Grenzgebiete [Results in Math. and Related Areas] (3) 9, Springer, Berlin (1986)
  • M W Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959) 242–276
  • K Ikegami, O Saeki, Cobordism group of Morse functions on surfaces, J. Math. Soc. Japan 55 (2003) 1081–1094
  • B Kalmár, Cobordism invariants of fold maps
  • B Kalmár, Cobordism of fold maps, stably framed manifolds and immersions
  • B Kalmár, Cobordism group of Morse functions on unoriented surfaces, Kyushu J. Math. 59 (2005) 351–363
  • B Kalmár, Cobordism group of fold maps of oriented $3$–manifolds into the plane, Acta Math. Hungar. 117 (2007) 1–25
  • M Kazarian, Multisingularities, cobordisms, and enumerative geometry, Uspekhi Mat. Nauk 58 (2003) 29–88
  • M Kazarian, Thom polynomials, from: “Singularity theory and its applications”, Adv. Stud. Pure Math. 43, Math. Soc. Japan, Tokyo (2006) 85–135
  • S Kobayashi, K Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers John Wiley & Sons, New York-London (1963)
  • J A Lees, On the classification of Lagrange immersions, Duke Math. J. 43 (1976) 217–224
  • H I Levine, Singularities of differentiable maps, from: “Proceedings of Liverpool Singularities Symposium, (1969/1970)”, Lecture Notes in Math. 192, Springer, Berlin (1971) 1–85
  • J Martinet, Déploiements versels des applications différentiables et classification des applications stables, from: “Singularités d'applications différentiables (Sém., Plans-sur-Bex, 1975)”, Lecture Notes in Math. 535, Springer, Berlin (1976) 1–44
  • J N Mather, Stability of $C\sp{\infty }$ mappings. III. Finitely determined mapgerms, Inst. Hautes Études Sci. Publ. Math. (1968) 279–308
  • 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 N Mather, On Thom–Boardman singularities, from: “Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971)”, Academic Press, New York (1973) 233–248
  • A Phillips, Submersions of open manifolds, Topology 6 (1967) 171–206
  • A du Plessis, Maps without certain singularities, Comment. Math. Helv. 50 (1975) 363–382
  • A du Plessis, Contact-Invariant regularity conditions, from: “Singularités d'applications différentiables (Sém., Plans-sur-Bex, 1975)”, Lecture Notes in Math. 535, Springer, Berlin (1976) 205–236
  • A du Plessis, Homotopy classification of regular sections, Compositio Math. 32 (1976) 301–333
  • R Rimányi, A Szücs, Pontrjagin–Thom–type construction for maps with singularities, Topology 37 (1998) 1177–1191
  • R Sadykov, Bordism groups of solutions to differential relations
  • R Sadykov, Bordism groups of special generic mappings, Proc. Amer. Math. Soc. 133 (2005) 931–936
  • O Saeki, Topology of special generic maps into ${\bf R}\sp 3$, Mat. Contemp. 5 (1993) 161–186 Workshop on Real and Complex Singularities (São Carlos, 1992)
  • O Saeki, Topology of singular fibers of differentiable maps, Lecture Notes in Math. 1854, Springer, Berlin (2004)
  • O Saeki, K Sakuma, On special generic maps into ${\bf R}\sp 3$, Pacific J. Math. 184 (1998) 175–193
  • J-P Serre, Groupes d'homotopie et classes de groupes abéliens, Ann. of Math. $(2)$ 58 (1953) 258–294
  • S Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. $(2)$ 69 (1959) 327–344
  • E H Spanier, Function spaces and duality, Ann. of Math. $(2)$ 70 (1959) 338–378
  • E H Spanier, Algebraic topology, McGraw-Hill Book Co., New York (1966)
  • N Steenrod, The topology of fibre bundles, Princeton Math. Series 14, Princeton University Press (1951)
  • R E Stong, Notes on cobordism theory, Math. notes, Princeton University Press (1968)
  • A Sz\Hucs, Cobordism of singular maps, Geom. Topol. (2008) 2379–2452
  • R Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954) 17–86