Lagrangian intersections, critical points and Qcategory

Abstract

For a manifold $M$, we prove that any function defined on a vector bundle of basis $M$ and quadratic at infinity has at least $Qcat(M)+1$ critical points. Here $Qcat(M)$ is a homotopically stable version of the LS-category defined by Scheerer, Stanley and Tanré. The key homotopical result is that $Qcat(M)$ can be identified with the relative LS-category of Fadell and Husseini of the pair $(M\times D^{n+1}, M\times S^n)$ for $n$ big enough. Combining this result with the work of Laudenbach and Sikorav, we obtain that if $M$ is closed, for any hamiltonian diffeomorphism with compact support $\psi$ of $T^{\ast}M$, $\# (\psi (M) \cap M)\geq Qcat(M)+1$, which improves all previously known homotopical estimates of this intersection number.