Three approaches towards Floer homology of cotangent bundles. (English) Zbl 1109.53080

Weber, Joa

J. Symplectic Geom. 3, No. 4, 671-701 (2005).

Summary: Consider the cotangent bundle of a closed Riemannian manifold and an almost complex structure close to the one induced by the Riemannian metric. For Hamiltonians which grow, for instance, quadratically in the fibers outside a compact set, one can define Floer homology and show that it is naturally isomorphic to singular homology of the free loop space. We review the three isomorphisms constructed by C. Viterbo [Functors and computations in Floer homology with applications, Part II, Preprint October 1966,
http://math.polytechnique.fr/cmat/viterbo/viterbo.thml D. A. Salamon and J. Weber [Geom. Funct. Anal. 16, No. 5, 1050-1138 (2006; Zbl 1118.53056)] and A. Abbondandolo and M. Schwarz [Commun. Pure Appl. Math. 59, No. 2, 254–316 (2006; Zbl 1084.53074)]. The theory is illustrated by calculating Morse and Floer homology in case of the Euclidean \(n\)-torus. Applications include existence of non-contractible periodic orbits of compactly supported Hamiltonians on open unit disc cotangent bundles which are sufficiently large over the zero section.

MSC:

53D40	Symplectic aspects of Floer homology and cohomology
57R58	Floer homology

Keywords:

Riemannian manifold; almost complex structure; Hamiltonians

Citations:

Zbl 1084.53074; Zbl 1118.53056