Cobordism groups of special generic functions and groups of homotopy spheres

Osamu SAEKI

doi:10.4099/math1924.28.287

Osamu SAEKI

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Keywords: Cobordism group, special generic map, Morse function, homotopy sphere

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2002 Volume 28 Issue 2 Pages 287-297

DOI https://doi.org/10.4099/math1924.28.287

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Published: 2002 Received: February 08, 2002 Available on J-STAGE: December 16, 2008 Accepted: - Advance online publication: - Revised: -
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Date of correction: December 16, 2008 Reason for correction: - Correction: CITATION Details: Wrong : [1] Y. Ando, Folding maps and the surgery theory on manifolds, J. Math. Soc. Japan, 53 (2001), 357-382.
[2] Y. Ando, Fold-maps and the space of base point preserving maps of spheres, J. Math. Kyoto Univ., 41 (2001), 693-737.
[3] O. Burlet and G. de Rham, Sur certaines applications génériques d'une variété close à 3 dimensions dans le plan, Enseign. Math., 20 (1974), 275-292.
[4] E. Calabi, Quasi-surjective mappings and a generalization of Morse theory, Proc. U. S. -Japan Seminar in Differential Geometry, Kyoto, 1965, pp. 13-16.
[5] J. Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ4=0), Lecture Notes in Math., 53, Springer-Verlag,1968.
[6] J. Cerf, La stratification naturelles des espaces de fonctions difféerentiables réelles et la thèoreme de la pseudo-isotopie, Publ. Math. Inst. HES, 39 (1970), 5-173.
[7] J. Eliashberg, Cobordisme des solutions de relations différentielles, South Rhone seminar on geometry, I (Lyon, 1983), Travaux em Cours, Hermann, Paris, 1984, pp. 17-31.
[8] M. Gromov, Partial differential relations, Ergebnisse der Mathematik and ihrer Grenzgebi-ete (3), Vol. 9, Springer-Verlag, Berlin, 1986.
[9] M. Kervaire and J. W. Milnor, Groups 504-537.
[10] J. W. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math., 64 (1956), 399-405.
[11] P. Porto and Y. K. S. Furuya, On special generic maps from a closed manifold into the plane, Topology Appl., 35 (1990), 41-52.
[12] G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées, Actualités Scien-tifiques et Industrielles 1183, Hermann, Paris, 1952, pp. 91-154.
[13] R. Rimanyi and A. Szücs, Pontrjagin-Thorn-type construction for maps with singularities, Topology, 37 (1998), 1177-1191.
[14] O. Saeki, Topology of special generic maps of manifolds into Euclidean spaces, Topology Appl., 49 (1993), 265-293.
[15] O. Saeki, Topology of special generic maps into R³, in “Workshop on Real and Complex Singularities”, Matemática Contemporânea, 5 (1993), 161-186.
[16] O. Saeki, Global theory of singularities of differentiable mappings: present status and prospects (translation of Sugaku, 48 (1996), 385-399), Sugaku Expositions, 12 (1999), 21-37.
[17] O. Saeki and K. Sakuma, On special generic maps into R³, Pacific J. Math., 184 (1998), 175-193.
[18] S. Smale, Generalized Poincaré conjecture in dimensions greater than four, Ann. of Math., 74 (1961), 391-406.
[19] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv., 28 (1954), 17-86.
[20] R. Wells, Cobordism groups of immersions, Topology, 5 (1966), 281-294.

Right : [1] Y. Ando, Folding maps and the surgery theory on manifolds, J. Math. Soc. Japan, 53 (2001), 357-382.
[2] Y. Ando, Fold-maps and the space of base point preserving maps of spheres, J. Math. Kyoto Univ., 41 (2001), 693-737.
[3] O. Burlet and G. de Rham, Sur certaines applications génériques d'une variété close à 3 dimensions dans le plan, Enseign. Math., 20 (1974), 275-292.
[4] E. Calabi, Quasi-surjective mappings and a generalization of Morse theory, Proc. U. S. -Japan Seminar in Differential Geometry, Kyoto, 1965, pp. 13-16.
[5] J. Cerf, Sur les difféomorphismes de la sphère de dimension trois (Γ₄=0), Lecture Notes in Math., 53, Springer-Verlag,1968.
[6] J. Cerf, La stratification naturelles des espaces de fonctions difféerentiables réelles et la théorème de la pseudo-isotopie, Publ. Math. Inst. HES, 39 (1970), 5-173.
[7] J. Eliashberg, Cobordisme des solutions de relations différentielles, South Rhone seminar on geometry, I (Lyon, 1983), Travaux em Cours, Hermann, Paris, 1984, pp. 17-31.
[8] M. Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 9, Springer-Verlag, Berlin, 1986.
[9] M. Kervaire and J. W. Milnor, Groups of homotopy spheres: I, Ann. of Math., 77 (1963), 504-537.
[10] J. W. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math., 64 (1956), 399-405.
[11] P. Porto and Y. K. S. Furuya, On special generic maps from a closed manifold into the plane, Topology Appl., 35 (1990), 41-52.
[12] G. Reeb, Sur certaines propriétés topologiques des variétés feuilletées, Actualités Scientifiques et Industrielles 1183, Hermann, Paris, 1952, pp. 91-154.
[13] R. Rimányi and A. Szücs, Pontrjagin-Thorn-type construction for maps with singularities, Topology, 37 (1998), 1177-1191.
[14] O. Saeki, Topology of special generic maps of manifolds into Euclidean spaces, Topology Appl., 49 (1993), 265-293.
[15] O. Saeki, Topology of special generic maps into R³, in “Workshop on Real and Complex Singularities”, Matemática Contemporânea, 5 (1993), 161-186.
[16] O. Saeki, Global theory of singularities of differentiable mappings: present status and prospects (translation of Sugaku, 48 (1996), 385-399), Sugaku Expositions, 12 (1999), 21-37.
[17] O. Saeki and K. Sakuma, On special generic maps into R³, Pacific J. Math., 184 (1998), 175-193.
[18] S. Smale, Generalized Poincaré conjecture in dimensions greater than four, Ann. of Math., 74 (1961), 391-406.
[19] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv., 28 (1954), 17-86.
[20] R. Wells, Cobordism groups of immersions, Topology, 5 (1966), 281-294.

Date of correction: December 16, 2008 Reason for correction: - Correction: PDF FILE Details: -

Abstract

Morse functions with only minima and maxima as their critical points are called special generic functions. We show that for n greater than or equal to 6, the n-dimensional oriented cobordism group of such functions is naturally isomorphic to the h-cobordism group of oriented homotopy n-spheres. We also study the unoriented cobordism group of special generic functions.

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