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 [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 ψ of T * M, #(ψ(M) ∩ M) ≥ Qcat (M) + 1, which improves all previously known homotopical estimates of this intersection number. (2000): 53D12, 55M30, 57R70. Mathematics Subject Classification

The Lusternik-Schnirelmann category of 𝑆𝑝(3)

“…Our proof shows that the category of the space Sp(3) coincides with the cone length of Sp(3) and with a stabilized version of the category, denoted Qcat(Sp(3)); see [17,25]. From the main theorem of P.-M. Moyaux and L. Vandembroucq in [15] we know that Crit(Sp(3)) − 1 is less than the cone length and is bounded below by Qcat. Theorem 1 now implies that Crit(Sp(3)) = 6.…”

Q-topological complexity

“…(4) As mentioned in [15,Remark 7], using [23,Theorem 15] (or Theorem 3.12 above for the cofibration * → X), we can see that, for any path-connected finite dimensional CW-complex X, the product X × T p , where T p is the p-fold product of p circles S 1 , satisfies Qcat(X × T p ) = cat(X × T p ) as soon as p ≥ dim(X) + 3. In a similar way, considering p-fold products of the sphere S 2 , we have:…”

“…This invariant, defined using a fibrewise extension of a functor Q k equivalent to Ω k Σ k , has been in particular used in the study of critical points (see [2,Chap. 7], [15]) and in the study of the Ganea conjecture. More precisely, after N. Iwase [12] showed that the Ganea conjecture, which asserted that cat(X × S n ) = catX + 1 for any n ≥ 1, was not true in general, although it was known to be true for many classes of spaces (e.g.…”