© The London Mathematical Society 2006
MORSE THEORY AND GLOBAL COEXISTENCE OF SINGULARITIES ON WAVE FRONTS
Institut Fourier, CNRS UMR 5582, 100 rue des Maths, Domaine Universitaire, BP 74, 38402 Saint Martin d'Hères cedex, France emmanuel.ferrand{at}ujf-grenoble.fr
Independent Moscow University and Moscow State University Bolshoi Vlasevky Pereulok, Dom 11, Moscow 119002, Russia pushkar{at}mccme.ru
Received 4 November 2005.
We produce examples of Legendrian isotopy classes in ST*
n (the space of cooriented contact elements of n, endowed with its standard contact structure) such that any element of those classes has a singular (non-immersed) wave front, that is, the projection to the base of the fibration ST*n 
n, when restricted to such a Legendrian embedding, has local singularities. Furthermore, our examples are such that the underlying Legendrian homotopy classes contain some Legendrian embeddings whose wave fronts are immersed. These examples are motivated by a question of Arnold about the removability of singularities of wave fronts by means of Legendrian isotopy. The two key points, which are of independent interest, are as follows.
(1) A duality argument which allows the question to be translated into a problem about Legendrian submanifolds of the one-jets space of the sphere, and hence to use the technology of generating families.
(2) Several independent constructions of families of functions on compact manifolds such that the ‘critical sets’ of these families are connected. For example, we prove that given two compact, connected manifolds M and N of positive dimension, there exists
 |
f/y)(x,y) = 0 is regular and defines a smooth and connected submanifold of M x N.