Lagrangian intersections, critical points and \(Q\)-category (Q1434180)

Let \(\text{crit}(M)\) be the least number of critical points of a function \(f:M \to\mathbb{R}\) on a closed manifold \(M\), then \(\text{crit}(M) \geq\text{cat} (M)+1\) due to the familiar Lusternik-Schnirelman theorem. The authors prove a ``stable'' version of this result: \[ \widetilde {\text{crit}}(M) \geq\inf_{n\in \mathbb{N}} \text{cat}(M\times D^{n+1}, M\times S^n)=Q\text{cat}(M)+1, \] where \(\widetilde{\text{crit}} (M)\) is the least number of critical points of a function defined on a vector bundle over \(M\) that are quadratic at infinity and the right hand side is expressed in terms of the relative \(Q\)-category introduced by \textit{E. Fadell} and \textit{S. Y. Husseini} [Rend. Semin. Mat. Fis. Milano 64(1994), 99--115 (1996; Zbl 0860.55011)]. The result is applied in symplectic geometry: for any Hamiltonian diffeomorphism \(\psi\) with compact support of \(T^*M\), the inequality \[ \#\bigl(\psi(M)\cap M\bigr)\geq Q\text{cat}(M)+1 \] holds. In this connection, some results concerning the Arnold conjecture \[ \#\bigl(\psi(M) \cap M\bigr) \geq\text{crit}(M) \] are mentioned.