On symplectic manifolds with aspherical symplectic form. (English) Zbl 0973.55004

Rudyak, Yuli; Tralle, Aleksy

Topol. Methods Nonlinear Anal. 14, No. 2, 353-362 (1999).

Let \((M^{2m},\omega)\) be a closed symplectic manifold such that \(\omega|_{\pi_2 (M)}=0\) (see [A. Floer, Commun. Math. Phys. 120, No. 4, 575-611 (1989; Zbl 0755.58022)] for the introduction of this restriction on \(\omega)\). The authors prove the following
Theorem: Let \(\mathbb{C} P^n\to E\to M\) be a \(\mathbb{C} P^n\)-fibration over \(M\). Then the Lyusternik-Shnirel’man category of \(E\) satisfies \(\text{cat} (E)\geq 2m+n\).
The proof uses the determination of the category weight of \(\omega\) [Y. B. Rudyak and J. Oprea, Math. Z. 230, No. 4, 673-678 (1999; Zbl 0931.53039)] and the collapsing of the Serre spectral sequence (which works for a trivial action and needs an extra-argument in the general case). The authors use this lower bound for \(\text{cat}(E)\) in a theorem of E. Kerman [Int. Math. Res. Not. 1999, No. 17, 953-969 (1999; Zbl 0958.37041)] and get: Let \(H\) be a metric Hamiltonian on the cotangent bundle \(T^*(M)\to M\). Then for every \(\varepsilon\) small enough, the set \(H^{-1}(\varepsilon)\) contains at least \(3m\) closed trajectories of the vector field on \(T^*(M)\) classically associated to \(\omega\).

Reviewer: Daniel Tanré (Villeneuve d’ Ascq)

MSC:

55M30	Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
57R17	Symplectic and contact topology in high or arbitrary dimension
53D05	Symplectic manifolds (general theory)

Keywords:

LS-category; Lyusternik-Shnirelman category; category weight

Citations:

Zbl 0755.58022; Zbl 0931.53039; Zbl 0958.37041