A generalisation of the Morse inequalities (Q2747266)

A family of variational families for Legendrian embeddings, into the 1-jet bundle of a closed manifold is constructed, that can be obtained from the zero section through Legendrian embeddings, by discretising the action functional. The second variation of a generating function obtained as above is computed at a nondegenerate critical point and a formula is proved relating the signature of the second variation to the Maslov index as the mesh goes to zero. This generalises a theorem of J. W. Robbin and D. A. Salamon for quadratic Hamiltonians. Also, it strengthens a theorem of C. Viterbo which states that, in the symplectic case, the difference of the signature of the second variation of the discretisation \(\Phi^N\) at two different critical points is independent of \(N\), but at the cost of having to take \(N\) sufficiently large. This is used to prove a generalisation of the Morse inequalities thus refining a theorem of \textit{Yu. V. Chekanov} stating that, in the nondegenerate case, the number of critical points is bounded from below by the sum of the Betti numbers. A proof of M. Chekanov's theorem had also been given by Chaperon, based on a result by D. Théret; the present proof is independent of this work.