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Floer theory and reduced cohomology on open manifolds

Yoel Groman

Geometry & Topology 27 (2023) 1273–1390
Abstract

We construct Hamiltonian Floer complexes associated to continuous, and even lower semicontinuous, time-dependent exhaustion functions on geometrically bounded symplectic manifolds. We further construct functorial continuation maps associated to monotone homotopies between them, and operations which give rise to a product and unit. The work rests on novel techniques for energy confinement of Floer solutions as well as on methods of non-Archimedean analysis. The definition for general Hamiltonians utilizes the notion of reduced cohomology familiar from Riemannian geometry, and the continuity properties of Floer cohomology. This gives rise, in particular, to local Floer theory. We discuss various functorial properties as well as some applications to existence of periodic orbits and to displaceability.

Keywords
symplectic cohomology, geometrically bounded manifolds, tame symplectic manifolds
Mathematical Subject Classification 2010
Primary: 53D40
Secondary: 32P05
References
Publication
Received: 20 September 2017
Revised: 2 November 2021
Accepted: 30 November 2021
Published: 15 June 2023
Proposed: Yakov Eliashberg
Seconded: Leonid Polterovich, Gang Tian
Authors
Yoel Groman
Einstein Institute of Mathematics
The Hebrew University of Jerusalem - Givat Ram
Jerusalem
Israel

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