Self-intersection class for singularities and its application to fold maps

Ohmoto, Toru; Saeki, Osamu; Sakuma, Kazuhiro

doi:10.1090/S0002-9947-03-03345-2

Self-intersection class for singularities and its application to fold maps
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by Toru Ohmoto, Osamu Saeki and Kazuhiro Sakuma PDF

Trans. Amer. Math. Soc. 355 (2003), 3825-3838 Request permission

Abstract:

Let $f :M \to N$ be a generic smooth map with corank one singularities between manifolds, and let $S(f)$ be the singular point set of $f$. We define the self-intersection class $I(S(f)) \in H^*(M; \mathbf {Z})$ of $S(f)$ using an incident class introduced by Rimányi but with twisted coefficients, and give a formula for $I(S(f))$ in terms of characteristic classes of the manifolds. We then apply the formula to the existence problem of fold maps.

References

P. M. Akhmetiev and R. R. Sadykov, A remark on elimination of singularities for mappings of $4$-manifolds into $3$-manifolds, preprint, 2002.
Y. Ando, Existence theorems of fold-maps, preprint, 2001.
Dynamical systems. VI, Encyclopaedia of Mathematical Sciences, vol. 6, Springer-Verlag, Berlin, 1993. Singularity theory. I; A translation of Current problems in mathematics. Fundamental directions, Vol. 6 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988 [ MR1088738 (91h:58010a)]; Translation by A. Iacob; Translation edited by V. I. Arnol′d. MR 1230637
M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
J. M. Boardman, Singularities of differentiable maps, Inst. Hautes Études Sci. Publ. Math. 33 (1967), 21–57. MR 231390, DOI 10.1007/BF02684585
Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706, DOI 10.1007/978-1-4612-0647-7
A. Dold, Lectures on algebraic topology, Die Grundlehren der mathematischen Wissenschaften, Band 200, Springer-Verlag, New York-Berlin, 1972 (German). MR 0415602, DOI 10.1007/978-3-662-00756-3
S. F. Il′jukevič, On the solution of the problem of the diffraction of a plane wave by two parallel nonconducting circular cylinders, Vescī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 1968 (1968), no. 5, 45–52 (Russian). MR 0273904
Ja. M. Èliašberg, Singularities of folding type, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 1110–1126 (Russian). MR 0278321
Ja. M. Èliašberg, Surgery of singularities of smooth mappings, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1321–1347 (Russian). MR 0339261
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
L. Fehér and R. Rimányi, Calculation on Thom polynomials for group actions, preprint, 2000, arXiv:math.AG/0009085.
L. Fehér and R. Rimányi, Thom polynomials with integer coefficients, to appear in Illinois J. Math.
Takuo Fukuda, Topology of folds, cusps and Morin singularities, A fête of topology, Academic Press, Boston, MA, 1988, pp. 331–353. MR 928406
M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518, DOI 10.1007/978-1-4615-7904-5
K. Győry, On the Diophantine equation $\binom nk=x^l$, Acta Arith. 80 (1997), no. 3, 289–295. MR 1451415, DOI 10.4064/aa-80-3-289-295
André Haefliger and A. Kosiński, Un théorème de Thom sur les singularités des applications différentiables, Séminaire Henri Cartan; 9e année: 1956/57. Quelques questions de topologie, Exposé no. 8, Secrétariat mathématique, Paris, 1958, pp. 6 (French). MR 0124063
Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 119214, DOI 10.1090/S0002-9947-1959-0119214-4
Shigeki Kikuchi and Osamu Saeki, Remarks on the topology of folds, Proc. Amer. Math. Soc. 123 (1995), no. 3, 905–908. MR 1283556, DOI 10.1090/S0002-9939-1995-1283556-7
Harold I. Levine, Elimination of cusps, Topology 3 (1965), no. suppl, suppl. 2, 263–296. MR 176484, DOI 10.1016/0040-9383(65)90078-9
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, DOI 10.1515/9781400881826
Anthony Phillips, Submersions of open manifolds, Topology 6 (1967), 171–206. MR 208611, DOI 10.1016/0040-9383(67)90034-1
I. R. Porteous, Simple singularities of maps, Proceedings of Liverpool Singularities—Symposium, I (1969/70), Lecture Notes in Mathematics, Vol. 192, Springer, Berlin, 1971, pp. 286–307. MR 0293646
Richárd Rimányi, Thom polynomials, symmetries and incidences of singularities, Invent. Math. 143 (2001), no. 3, 499–521. MR 1817643, DOI 10.1007/s002220000113
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, Studying the topology of Morin singularities from a global viewpoint, Math. Proc. Cambridge Philos. Soc. 117 (1995), no. 2, 223–235. MR 1307077, DOI 10.1017/S0305004100073072
O. Saeki, Fold maps on $4$-manifolds, preprint.
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
Kazuhiro Sakuma, On special generic maps of simply connected $2n$-manifolds into $\mathbf R^3$, Topology Appl. 50 (1993), no. 3, 249–261. MR 1227553, DOI 10.1016/0166-8641(93)90024-8
Kazuhiro Sakuma, A note on nonremovable cusp singularities, Hiroshima Math. J. 31 (2001), no. 3, 461–465. MR 1870989
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335