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Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Michael Usher

Geometry & Topology 15 (2011) 1313–1417

We develop a family of deformations of the differential and of the pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold (M,ω) which upon passing to homology yields ring isomorphisms with the big quantum homology of M. By studying the properties of the resulting deformed version of the Oh–Schwarz spectral invariants, we obtain a Floer-theoretic interpretation of a result of Lu which bounds the Hofer–Zehnder capacity of M when M has a nonzero Gromov–Witten invariant with two point constraints, and we produce a new algebraic criterion for (M,ω) to admit a Calabi quasimorphism and a symplectic quasistate. This latter criterion is found to hold whenever M has generically semisimple quantum homology in the sense considered by Dubrovin and Manin (this includes all compact toric M), and also whenever M is a point blowup of an arbitrary closed symplectic manifold.

Hamiltonian Floer theory, spectral invariant, quasimorphism, semisimple quantum homology
Mathematical Subject Classification 2010
Primary: 53D40, 53D45
Received: 19 July 2010
Revised: 5 April 2011
Accepted: 13 June 2011
Published: 1 August 2011
Proposed: Leonid Polterovich
Seconded: Danny Calegari, Yasha Eliashberg
Michael Usher
Department of Mathematics
University of Georgia
Athens GA 30602