First steps in stable Hamiltonian topology

Abstract

In this paper we study topological properties of stable Hamiltonian structures. In particular, we prove the following results in dimension three: The space of stable Hamiltonian structures modulo homotopy is discrete; there exist stable Hamiltonian structures that are not homotopic to a positive contact structure; stable Hamiltonian structures are generically Morse-Bott (i.e. all closed orbits are Bott nondegenerate) but not Morse; the standard contact structure on the 3-sphere is homotopic to a stable Hamiltonian structure which cannot be embedded in 4-space. Moreover, we derive a structure theorem in dimension three and classify stable Hamiltonian structures supported by an open book. We also discuss implications for the foundations of symplectic field theory.

104 pages, 3 figures, reference added

References in corpus (7)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Geometry of contact transformations and domains: orderability versus squeezing
• Lectures on open book decompositions and contact structures
• The Weinstein conjecture for stable Hamiltonian structures
• Weak Symplectic Fillings and Holomorphic Curves
• Vanishing of the contact homology of overtwisted contact 3--manifolds
• Towards a good definition of algebraically overtwisted

Cited by in corpus (9)

• Weak Symplectic Fillings and Holomorphic Curves
• Rabinowitz Floer homology: A survey
• Invariance property of Morse homology on noncompact manifolds
• Pseudo-rotations with sufficiently Liouvillean rotation number are C^0-rigid
• On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification
• Reeb orbits and the minimal discrepancy of an isolated singularity
• Near-symplectic 2n-manifolds
• An overtwisted disk in a virtual contact structure and the Weinstein conjecture
• Periodic approximations of irrational pseudo-rotations using pseudoholomorphic curves