Lagrangian intersections, critical points and Qcategory

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 [27]. The key homotopical result is that Qcat(M) can be identified with the relative LS-category of Fadell and Husseini [9] of the pair (M×D ⁿ⁺¹ ,M×S ⁿ ) for n big enough. Combining this result with the work of Laudenbach and Sikorav [19], we obtain that if M is closed, for any hamiltonian diffeomorphism with compact support of T ^* M, #((M)M)Qcat(M)+1, which improves all previously known homotopical estimates of this intersection number. Mathematics Subject Classification (2000):53D12, 55M30, 57R70.