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Abstract

A smooth mapping $f:L^n → (M^{2n},ω)$ of a smooth n-dimensional manifold L into a smooth 2n-dimensional symplectic manifold (M,ω) is called isotropic if f*ω vanishes. In the last ten years, the local theory of singularities of isotropic mappings has been rapidly developed by Arnol'd, Givental' and several authors, while it seems that the global theory of their singularities has not been well studied except for the work of Givental' [G1] in the case of dimension 2 (cf. [A], [Au], [I2], [I-O]). In the present paper, we are concerned with typical singularities with corank 1 of isotropic maps $f:L^n → (M^{2n},ω)$ (arbitrary dimension n), so-called open Whitney umbrellas of higher order, investigated by Givental' [G2], Ishikawa [I1] and Zakalyukin [Z], and our purpose is to give their topological invariants from the viewpoint of "Thom polynomial theory" (cf. [T], [P], [K], [AVGL]). These are obtained as a variant of Porteous' formulae on Thom polynomials for $A_k$-singularities [P]. Throughout this paper, manifolds are assumed to be paracompact Hausdorff spaces and of class $C^{∞}$, and maps are also of class $C^{∞}$.