Volume 15, issue 3 (2011)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 21
Issue 6, 3191–3810
Issue 5, 2557–3190
Issue 4, 1931–2555
Issue 3, 1285–1930
Issue 2, 647–1283
Issue 1, 1–645

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Subscriptions
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

Michael Usher

Geometry & Topology 15 (2011) 1313–1417
Abstract

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.

Keywords
Hamiltonian Floer theory, spectral invariant, quasimorphism, semisimple quantum homology
Mathematical Subject Classification 2010
Primary: 53D40, 53D45
References
Publication
Received: 19 July 2010
Revised: 5 April 2011
Accepted: 13 June 2011
Published: 1 August 2011
Proposed: Leonid Polterovich
Seconded: Danny Calegari, Yasha Eliashberg
Authors
Michael Usher
Department of Mathematics
University of Georgia
Athens GA 30602
USA
http://math.uga.edu/~usher