Coisotropic submanifolds, leaf-wise fixed points, and presymplectic embeddings. (English) Zbl 1208.57011
It is proved that the number of leafwise fixed points of a Hamiltonian diffeomorphism \(\phi\) of a geometrically bounded symplectic manifold is bounded below by the sum of the \(\mathbb Z_2\)-Betti numbers of the closed, regular coisotropic submanifold \(N\), provided that the Hofer distance between \(\phi\) and the identity is small enough and the pair \((N,\phi)\) is non-degenerate. Moreover, the bound is optimal if there exists a \(\mathbb Z_2\)-perfect Morse function on \(N\). As an application, a presymplectic non-embedding result is proved.
A version of the Arnold-Givental conjecture for coisotropic submanifolds is analyzed.
A version of the Arnold-Givental conjecture for coisotropic submanifolds is analyzed.
Reviewer: Gheorghe Pitiş (Braşov)
MSC:
57R17 | Symplectic and contact topology in high or arbitrary dimension |
53D12 | Lagrangian submanifolds; Maslov index |
53C40 | Global submanifolds |
55M20 | Fixed points and coincidences in algebraic topology |
58C30 | Fixed-point theorems on manifolds |
57R50 | Differential topological aspects of diffeomorphisms |