The Euler Characteristic of a Generic Wavefront in a 3-Manifold

Abstract

We give a relation between Euler characteristics of a generic closed Legendrian surface and its wavefront. 0. INTRODUCTION In this note we shall compute the Euler characteristic of a generic wave front in a 3-manifold. Let N be a (2n + l)-dimensional smooth manifold and K be a contact structure on N (i.e., K is a nondegenerate tangent hyperplane field on N ). An immersion i: L N is said to be Legendrian if dimL = n and di,(TxL) c Kx for any x E L. We say that a smooth fibre bundle t: E -4 M is Legendrian if its total space E is furnished with a contact structure and its fibres are Legendrian submanifolds. For a Legendrian immersion i: L -E, 7t o i : L -* M is called a Legendrian map and the image of the Legendrian map Xt o i is called the wavefront of i. It is denoted by W(i). From now on, we only consider the case of n = 2. Then it is known that a generic wavefront has (semicubic) cuspidal edges (A2), swallowtails (A3), and points of transversal self-intersection (A1A1, A1A2, AIAIAI) as singularities [1] (see Figure 1 on the next page). We shall refer to the AI AI A I-type point as a triple point of i. If L is a closed surface, then the number of swallowtails and triple points are finite. Our main result is the following: Theorem. Let i : L -? E be a generic Legendrian immersion of a closed surface. Then we have X(W(i)) = X(L) + T(i) + s() 2' where x (X) is the Euler characteristic of X, T(i) is the number of triple points on W(i), and S(i) is the number of swallowtails. We remark that the corresponding result for a generic wavefront in a 2manifold is easily verified and that X(W(i)) = -d(i), where d(i) denotes the number of double points on W(i). Received by the editors December 23, 1991. 1991 Mathematics Subject Classification. Primary 58C27.