On holomorphic maps with only fold singularities

Abstract

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.