Coisotropic characteristic classes
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Coisotropic characteristic classes Jean-Philippe Chassé1 Received: 19 April 2019 / Accepted: 9 October 2019 © Fondation Carl-Herz and Springer Nature Switzerland AG 2019
Abstract In this paper, we introduce coisotropic characteristic classes in order to study coisotropic immersions in Cn , and prove that they actually are the isotropic classes introduced by Lalonde (Math Ann 285:343–351, 1989). We conclude with some remarks on the differences between the h-principles for coisotropic immersions, and the one for isotropic immersions that the coisotropic classes fail to capture. Keywords Symplectic topology · Characteristic classes · Coisotropic immersions Mathematics Subject Classification 57R17 · 57R42 Résumé Dans cet article, nous introduisons des classes caractéristiques coisotropes afin d’étudier les immersions coisotropes dans Cn et démontrons qu’elles sont en réalité les classes isotropes introduites par Lalonde (1989). Nous concluons par une discussion sur les différences entre le h-principe pour les immersions isotropes et celui pour les immersions coisotropes que les classes coisotropes n’arrivent pas à détecter.
1 Introduction Symplectic topology has long been done through the study of Lagrangian submanifolds, as many natural questions arising from physics can be rephrased in terms of these special submanifolds. However, the study of coisotropic submanifolds, of which Lagrangian submanifolds are a special case, have seen a rise in popularity in recent years. For example, their hypothetical role in homological mirror symmetry has been explored by Kapustin and Orlov [5], and their deformation theory has led to some interesting results by Oh and Park [10]. Furthermore, their link with Hamiltonian dynamics has made possible some results of Kerman and Lalonde [6] on the group of Hamiltonian diffeomorphisms of a symplectically aspherical manifold. This gives the motivation to revisit old ideas about isotropic submanifolds, and see if they can be applied in the coisotropic context.
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Jean-Philippe Chassé [email protected] Department of Mathematics and Statistics, Université de Montréal, PO Box 6128, Centre-ville Station, Montreal, Quebec, Canada
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J.-P. Chassé
On the other hand, characteristic classes have played an important role in symplectic topology since the work of Arnol’d [1] on the now-called Maslov class. From the homological point of view, it is only a degree 1 cohomology class of the Lagrangian Grassmannian L (n) = U(n)/O(n). However, in the context of symplectic topology, this class can be seen as an obstruction to the transversality of a pair (L 0 , L 1 ) of Lagrangian subbundles of some fixed symplectic bundle E over a finite dimensional CW-complex B. Indeed, one can construct a map d(L 0 , L 1 ) : B → L (n) such that the pullback of the Maslov class by d(L 0 , L 1 ) is zero whenever L 0 and L 1 are transversal. Therefore the calculation of the cohomological ring of L (n) and its limit U/O, done from the cohomological point of view by Fuks [4] and from the Riemannian point of view b
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