Characteristic classes and existence of singular maps

Kalmár, Boldizsár; Terpai, Tamás

doi:10.1090/S0002-9947-2012-05544-9

Characteristic classes and existence of singular maps
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by Boldizsár Kalmár and Tamás Terpai PDF

Trans. Amer. Math. Soc. 364 (2012), 3751-3779 Request permission

Abstract:

The existence of a corank one map of negative codimension puts strong restrictions on the topology of the source manifold. It implies many vanishing theorems on characteristic classes and often even vanishing of the cobordism class of the source manifold. Most of our results lie deeper than just vanishing of Thom polynomials of the higher singularities. We blow up the singular map along the singular set and then perturb the arising nongeneric corank one map.

References

Yoshifumi Ando, On the elimination of Morin singularities, J. Math. Soc. Japan 37 (1985), no. 3, 471–487. MR 792988, DOI 10.2969/jmsj/03730471
Yoshifumi Ando, On the elimination of Morin singularities, J. Math. Soc. Japan 37 (1985), no. 3, 471–487. MR 792988, DOI 10.2969/jmsj/03730471
Yoshifumi Ando, Folding maps and the surgery theory on manifolds, J. Math. Soc. Japan 53 (2001), no. 2, 357–382. MR 1815139, DOI 10.2969/jmsj/05320357
Yoshifumi Ando, Folding maps and the surgery theory on manifolds, J. Math. Soc. Japan 53 (2001), no. 2, 357–382. MR 1815139, DOI 10.2969/jmsj/05320357
Yoshifumi Ando, A homotopy principle for maps with prescribed Thom-Boardman singularities, Trans. Amer. Math. Soc. 359 (2007), no. 2, 489–515. MR 2255183, DOI 10.1090/S0002-9947-06-04326-1
M. F. Atiyah, Immersions and embeddings of manifolds, Topology 1 (1962), 125–132. MR 145549, DOI 10.1016/0040-9383(65)90020-0
Augustin Banyaga and David E. Hurtubise, A proof of the Morse-Bott lemma, Expo. Math. 22 (2004), no. 4, 365–373. MR 2075744, DOI 10.1016/S0723-0869(04)80014-8
J. M. Boardman, Singularities of differentiable maps, Inst. Hautes Études Sci. Publ. Math. 33 (1967), 21–57. MR 231390
Daniel S. Chess, A note on the classes $[S^{k}_{1}(f)]$, Singularities, Part 1 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 221–224. MR 713061
Albrecht Dold, Erzeugende der Thomschen Algebra ${\mathfrak {N}}$, Math. Z. 65 (1956), 25–35 (German). MR 79269, DOI 10.1007/BF01473868
A. M. Gabrièlov, Combinatorial formulas for Pontrjagin classes and $\textrm {GL}$-invariant chains, Funktsional. Anal. i Prilozhen. 12 (1978), no. 2, 1–7, 95 (Russian). MR 498893
J. W. L. Glaisher, On the residue of a binomial-theorem coefficient with respect to a prime modulus, Quart. J. Pure App. Math. 30 (1899), 150–156.
M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518
Hansjörg Geiges and Federica Pasquotto, A formula for the Chern classes of symplectic blow-ups, J. Lond. Math. Soc. (2) 76 (2007), no. 2, 313–330. MR 2363418, DOI 10.1112/jlms/jdm061
G. Mikhalkin, Blowup equivalence of smooth closed manifolds, Topology 36 (1997), no. 1, 287–299. MR 1410476, DOI 10.1016/0040-9383(95)00062-3
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
Toru Ohmoto, Osamu Saeki, and Kazuhiro Sakuma, Self-intersection class for singularities and its application to fold maps, Trans. Amer. Math. Soc. 355 (2003), no. 9, 3825–3838. MR 1990176, DOI 10.1090/S0002-9947-03-03345-2
Piotr Pragacz, Thom polynomials and Schur functions: towards the singularities $A_i(-)$, Real and complex singularities, Contemp. Math., vol. 459, Amer. Math. Soc., Providence, RI, 2008, pp. 165–178. MR 2444400, DOI 10.1090/conm/459/08969
V. A. Rohlin, Intrinsic definition of Pontryagin’s characteristic cycles, Doklady Akad. Nauk SSSR (N.S.) 84 (1952), 449–452 (Russian). MR 0050276
F. Ronga, Le calcul de la classe de cohomologie entière duale à $\overline \sum {}^k$, Proceedings of Liverpool Singularities—Symposium, I (1969/70), Lecture Notes in Mathematics, Vol. 192, Springer, Berlin, 1971, pp. 313–315 (French). MR 0293648
F. Ronga, Le calcul des classes duales aux singularités de Boardman d’ordre deux, Comment. Math. Helv. 47 (1972), 15–35 (French). MR 309129, DOI 10.1007/BF02566786
Rustam Sadykov, The Chess conjecture, Algebr. Geom. Topol. 3 (2003), 777–789. MR 1997337, DOI 10.2140/agt.2003.3.777
Rustam Sadykov, Osamu Saeki, and Kazuhiro Sakuma, Obstructions to the existence of fold maps, J. Lond. Math. Soc. (2) 81 (2010), no. 2, 338–354. MR 2602999, DOI 10.1112/jlms/jdp072
Osamu Saeki, Notes on the topology of folds, J. Math. Soc. Japan 44 (1992), no. 3, 551–566. MR 1167382, DOI 10.2969/jmsj/04430551
Osamu Saeki and Kazuhiro Sakuma, Maps with only Morin singularities and the Hopf invariant one problem, Math. Proc. Cambridge Philos. Soc. 124 (1998), no. 3, 501–511. MR 1636580, DOI 10.1017/S0305004197002478
C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292–311. MR 120654, DOI 10.2307/1970136