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Cobordism of Morse maps and its application to map germs

Published online by Cambridge University Press:  01 July 2009

KAZUICHI IKEGAMI  and
OSAMU SAEKI
KAZUICHI IKEGAMI
Affiliation:
Faculty of Engineering, Hiroshima Institute of Technology, 2-1-1 Miyake, Saeki-ku, Hiroshima 731-5193, Japan. e-mail: suuich@wish.ocn.ne.jp
OSAMU SAEKI
Affiliation:
Faculty of Mathematics, Kyushu University, Hakozaki, Fukuoka 812-8581, Japan. e-mail: saeki@math.kyushu-u.ac.jp
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Abstract

Let f: M → S1 be a Morse map of a closed manifold M into the circle, where a Morse map is a smooth map with only nondegenerate critical points. In this paper, we classify such maps up to fold cobordism. In the course of the classification, we get several fold cobordism invariants for such Morse maps. We also consider a slightly general situation where the source manifold M has boundary and the map f restricted to the boundary has no critical points. Let g: (Rm, 0) → (R2, 0), m ≥ 2, be a generic smooth map germ, where the target R2 is oriented. Using the above-mentioned fold cobordism invariants, we show that the number of cusps with a prescribed index appearing in a C∞ stable perturbation of g, counted with signs, gives a topological invariant of g.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 147 , Issue 1 , July 2009 , pp. 235 - 254
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

REFERENCES

[1]Callahan, J.Singularities and plane maps. Amer. Math. Monthly 81 (1974), 211–240.CrossRefGoogle Scholar
[2]Conner, P. E. and Floyd, E. E.Differentiable Periodic Maps. Ergebnisse der Math. und ihrer Grenzgebiete, Band 33 (Springer–Verlag, 1964).Google Scholar
[3]Dudziński, P., Łecki, A., Nowak–Przygodzki, P. and Szafraniec, Z.On topological invariance of the Milnor number mod 2. Topology 32 (1993), 573–576.CrossRefGoogle Scholar
[4]Fukuda, T.Local topological properties of differentiable mappings. I. Invent. Math. 65 (1981/82), 227–250.CrossRefGoogle Scholar
[5]Fukuda, T.Local topological properties of differentiable mappings. II. Tokyo J. Math. 8 (1985), 501–520.CrossRefGoogle Scholar
[6]Fukuda, T. Topology of folds, cusps and Morin singularities. A Fête of Topology. Edited by Matsumoto, Y., Mizutani, T. and Morita, S. (Academic Press, Inc., 1988), pp. 331–353.CrossRefGoogle Scholar
[7]Fukuda, T. and Ishikawa, G.On the number of cusps of stable perturbations of a plane-to-plane singularity. Tokyo J. Math. 10 (1987), 375–384.CrossRefGoogle Scholar
[8]Fukui, T., Nuño Ballesteros, J. J. and Saia, M. J.On the number of singularities in generic deformations of map germs. J. London Math. Soc. (2) 58 (1998), 141–152.CrossRefGoogle Scholar
[9]Ikegami, K.Cobordism group of Morse functions on manifolds. Hiroshima Math. J. 34 (2004), 211–230.CrossRefGoogle Scholar
[10]Ikegami, K. and Saeki, O.Cobordism group of Morse functions on surfaces. J. Math. Soc. Japan 55 (2003), 1081–1094.CrossRefGoogle Scholar
[11]Kalmár, B.Cobordism group of Morse functions on unoriented surfaces. Kyushu J. Math. 59 (2005), 351–363.CrossRefGoogle Scholar
[12]Kalmár, B. Pontrjagin–Thom construction for singular maps with negative codimension. Master Thesis (Eötvös Loránd University, 2005), http://www.cs.elte.hu/math/diploma/math/.Google Scholar
[13]Kalmár, B. Pontryagin–Thom type construction for negative codimensional singular maps. arXiv:math/0612116v1 [math.GT].Google Scholar
[14]Kazarian, M. Thom polynomials. Singularity Theory and its Applications. Edited by Izumiya, S., Ishikawa, G., Tokunaga, H., Shimada, I. and Sano, T., Adv. Stud. Pure Math., Vol. 43 (Math. Soc. Japan, 2006), pp. 85–135.CrossRefGoogle Scholar
[15]Levine, H. I.Elimination of cusps. Topology 3 (1965), suppl. 2, 263–296.CrossRefGoogle Scholar
[16]Massey, D. B.Lê Cycles and Hypersurface Singularities, Lecture Notes in Math., Vol. 1615 (Springer–Verlag, 1995).CrossRefGoogle Scholar
[17]Milnor, J.Singular Points of Complex Hypersurfaces. Ann. Math. Stud., Vol. 61 (Princeton Univ. Press, 1968).Google Scholar
[18]Nishimura, T.Topologically ∞–determined map germs are topologically cone-like. Compositio Math. 64 (1987), 117–129.Google Scholar
[19]Nuño Ballesteros, J. J. and Saia, M. J.Multiplicity of Boardman strata and deformations of map germs. Glasgow Math. J. 40 (1998), 21–32.CrossRefGoogle Scholar
[20]Ohsumi, M.Whitney's umbrellas in stable perturbations of a map germ (Rn, 0) → (R2n−1, 0). Tokyo J. Math. 29 (2006), 475–493.CrossRefGoogle Scholar
[21]Ohsumi, M. Singularities appearing in a stable perturbation of a map-germ. Real and complex singularities Proceedings of the Australian–Japanese Workshop (World Scientific, 2007), pp. 323–341.CrossRefGoogle Scholar
[22]Quine, J. R.A global theorem for singularities of maps between oriented 2-manifolds. Trans. Amer. Math. Soc. 236 (1978), 307–314.CrossRefGoogle Scholar
[23]Rimányi, R. and Szűcs, A.Pontrjagin–Thom-type construction for maps with singularities. Topology 37 (1998), 1177–1191.CrossRefGoogle Scholar
[24]Sadykov, R. Bordism groups of solutions to differential relations. preprint, June 2004, revised form in arXiv:math/0608460v1 [math.AT].Google Scholar
[25]Saeki, O.Topology of Singular Fibers of Differentiable Maps. Lecture Notes in Math., Vol. 1854 (Springer–Verlag, 2004).CrossRefGoogle Scholar
[26]Saeki, O.Cobordism of Morse functions on surfaces, the universal complex of singular fibers and their application to map germs. Algebr. Geom. Topol. 6 (2006), 539–572.CrossRefGoogle Scholar
[27]Saeki, O. and Yamamoto, T.Singular fibers and characteristic classes. Topology Appl. 155 (2007), 112–120.CrossRefGoogle Scholar
[28]Szűcs, A. Analogue of the Thom space for mappings with singularity of type Σ1 (in Russian). Mat. Sb. (N.S.) 108 (150) (1979), 433–456, 478; English translation in Math. USSR-Sb. 36 (1980), 405–426.Google Scholar
[29]Szűcs, A.Cobordism of singular maps. Geom. Topol. 12 (2008), 2379–2452.CrossRefGoogle Scholar
[30]Wall, C. T. C.Finite determinacy of smooth map-germs. Bull. London Math. Soc. 13 (1981), 481–539.CrossRefGoogle Scholar
[31]Wall, C. T. C.Topological invariance of the Milnor number mod 2. Topology 22 (1983), 345–350.CrossRefGoogle Scholar