#### Volume 15, issue 3 (2011)

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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 $\left(M,\omega \right)$ 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 $\left(M,\omega \right)$ 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
##### 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