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Morse Theory and Global Coexistence of Singularities on Wave Fronts
Author(s) -
Ferrand Emmanuel,
Pushkar Petr E.
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610706023088
Subject(s) - isotopy , mathematics , submanifold , pure mathematics , gravitational singularity , embedding , morse theory , homotopy , morse code , mathematical analysis , projection (relational algebra) , space (punctuation) , combinatorics , linguistics , philosophy , algorithm , artificial intelligence , computer science , electrical engineering , engineering
We produce examples of Legendrian isotopy classes in ST * ℝ n (the space of cooriented contact elements of ℝ 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 * ℝ n → ℝ 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 existsf : N × M → ℝ , ( x , y ) → f ( x , y ) , x ∈ N , y ∈ Msuch that the equation ∂ f /∂ y )( x,y ) = 0 is regular and defines a smooth and connected submanifold of M × N .