Nagoya Mathematical Journal

On holomorphic maps with only fold singularities

Yoshifumi Ando

Full-text: Open access


Let $f:N\to P$ 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 $J^2(n, n; \mathbf{C})$, let $\Omega^{10}$ denote the space consisting of all 2-jets of regular map germs and fold map germs. In this paper we prove that $\Omega^{10}$ 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 $TN\oplus\theta_N$ to $TP\oplus\theta_P$, where $\theta_N$ and $\theta_P$ are the trivial line bundles.

Article information

Nagoya Math. J., Volume 164 (2001), 147-184.

First available in Project Euclid: 27 April 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58K15: Topological properties of mappings
Secondary: 32S70: Other operations on singularities 57R45: Singularities of differentiable mappings


Ando, Yoshifumi. On holomorphic maps with only fold singularities. Nagoya Math. J. 164 (2001), 147--184.

Export citation


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