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Generating families for images of Lagrangian submanifolds and open swallowtails

Published online by Cambridge University Press:  24 October 2008

Stanisław Janeczko
Stanisław Janeczko
Affiliation:
Institute of Mathematics, Technical University of Warsaw, Polan
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Summary

In this paper we study the symplectic relations appearing as the generalized cotangent bundle liftings of smooth stable mappings. Using this class of symplectic relations the classification theorem for generic (pre) images of lagrangian submanifolds is proved. The normal forms for the respective classified puilbacks and pushforwards are provided and the connections between the singularity types of symplectic relation, mapped lagrangian submanifold and singular image, are established. The notion of special symplectic triplet is introduced and the generic local models of such triplets are studied. We show that the open swallowtails are canonically represented as pushforwards of the appropriate regular lagrangian submanifolds. Using the SL2(ℝ) invariant symplectic structure of the space of binary forms of n appropriate dimension we derive the generating families for the open swallowtails and the respective generating functions for its regular resolutions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 100 , Issue 1 , July 1986 , pp. 91 - 107
Copyright
Copyright © Cambridge Philosophical Society 1986

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