Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-22T21:09:28.218Z Has data issue: false hasContentIssue false
  • English
  • Français

On holomorphic maps with only fold singularities

Published online by Cambridge University Press:  22 January 2016

Yoshifumi Ando
Yoshifumi Ando*
Affiliation:
Department of Mathematics, Faculty of Science, Yamaguchi University, Yamaguchi, 753-8512, Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let f : N ≡ P be a holomorphic map between n-dimensional complex manifolds which has only fold singularities. Such a map is called a holomorphic fold map. In the complex 2-jet space J2(n,n;C), let Ω10 denote the space consisting of all 2-jets of regular map germs and fold map germs. In this paper we prove that Ω10 is homotopy equivalent to SU(n + 1). By using this result we prove that if the tangent bundles TN and TP are equipped with SU(n)-structures in addition, then a holomorphic fold map f canonically determines the homotopy class of an SU(n + 1)-bundle map of TN ⊕ θN to TP⊕ θP, where θN and θP are the trivial line bundles.

Keywords

58K1558A2058C10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 164 , December 2001 , pp. 147 - 184
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2001

References

[A1] Ando, Y., The homotopy type of the space consisting of regular jets and folding jets in J2(n,n), Japan. J. Math., 24 (1998), 169-181.CrossRefGoogle Scholar
[A2] Ando, Y., Folding maps and the surgery theory on manifolds, J. Math. Soc. Japan, 53 (2001), 357-382.Google Scholar
[Bo] Boardman, J. M., Singularities of smooth mappings, Publ. Math. I.H.E.S., 33 (1967), 21-57.CrossRefGoogle Scholar
[Br] Brieskorn, E., Beispiele zur Differentialtopologie von Singularitäten, Invent. Math., 2 (1966), 1-14.Google Scholar
[E] Eliashberg, J. M., On singularities of folding types, Math. USSR. Izv., 4 (1970), 1119-1134.Google Scholar
[H] Hirzebruch, F., Topological Methods in Algebraic Geometry, Springer-Verlag, 1966.Google Scholar
[K] Kodaira, K., Complex Manifolds and Deformation of Complex Structures, Springer-Verlag, 1986.Google Scholar
[L] Levine, H. I., Singularities of differentiable maps, Lecture Notes in Math., Springer-Verlag, 192 (1971), 1-89.Google Scholar
[Ma] Mather, J., Stability of C°° mappings: VI. The nice dimensions, Lecture Notes in Math., Springer-Verlag, 192 (1971), 207-253.Google Scholar
[Mi] Milnor, J., Singular Points of Complex Hypersurfaces, Princeton Univ. Press, Princeton, 1968.Google Scholar
[Sa] Saeki, O., Notes on the topology of folds, J. Math. Soc. Japan, 44 (1992), 551–566.Google Scholar
[St] Steenrod, N., The Topology of Fibre Bundles, Princeton Univ. Press, Princeton, 1951.Google Scholar
[W] Whitney, H., Complex Analytic Varieties, Addison-Wesley, 1972.Google Scholar