Commutator length of symplectomorphisms

Entov, Michael

doi:10.1007/s00014-001-0799-0

Cited by 32 publications

(40 citation statements)

References 49 publications

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“…there exists a constant C 0 such that jr .g 1 g 2 / r .g 1 / r .g 2 /j Ä C; for every g 1 ; g 2 2 G: A quasi-morphism r is called homogeneous if r .g n / D nr .g/ for all g 2 G and n 2 ‫.ޚ‬ The existence of homogeneous quasi-morphisms on the group of Hamiltonian diffeomorphisms and/or its universal cover is known for some classes of closed symplectic manifolds (see e.g. Barge-Ghys [4], Entov [9], Gambaudo-Ghys [13] and Givental [14]). In a recent work [11], Entov and Polterovich showed -by using Floer and Quantum homology -that for the class of symplectic manifolds which are monotone and whose quantum homology algebra is semi-simple, e…”

Section: Introduction and Resultsmentioning

confidence: 99%

Calabi quasi-morphisms for some non-monotone symplectic manifolds

Ostrover

2006

Algebr. Geom. Topol.

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In this work we construct Calabi quasi-morphisms on the universal cover eHam.M / of the group of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast to their work, we show that these quasimorphisms descend to non-trivial homomorphisms on the fundamental group of Ham.M /.53D05, 53D12, 53D45; 20F69

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Section: Introduction and Resultsmentioning

confidence: 99%

Calabi quasi-morphisms for some non-monotone symplectic manifolds

Ostrover

2006

Algebr. Geom. Topol.

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“…Then the "shift of the spectrum" trick of Y.Ostrover [38] (cf. [19], [20]) yields that for a certain E ∈ R, depending on φ,…”

Section: A Partial Symplectic Quasi-statementioning

confidence: 99%

Quasi-states and symplectic intersections

Entov¹,

Polterovich²

2006

Comment. Math. Helv.

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136

257

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We establish a link between symplectic topology and a recently emerged branch of functional analysis called the theory of quasi-states and quasi-measures (also known as topological measures). In the symplectic context quasi-states can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced recently by Yong-Geun Oh. As a consequence we prove a number of new results on rigidity of intersections in symplectic manifolds. This work is a part of a joint project with Paul Biran.

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“…Finally we would like to point out that the spectrality axiom, or (1.3), is a crucial ingredient in Entov's work [4] in his applications of spectral invariants to the study of the quasimorphisms and the commutator length of Hamiltonian diffeomorphisms. A proof of the spectrality axiom for the nondegenerate case is outlined in [4,Section 3].…”

Section: ρ(H; A) ∈ Spec(h)mentioning

confidence: 99%

“…A proof of the spectrality axiom for the nondegenerate case is outlined in [4,Section 3]. (See Part 4 of the page 76 of [4].)…”

Section: ρ(H; A) ∈ Spec(h)mentioning

confidence: 99%

Floer Mini-Max Theory, the Cerf Diagram, and the Spectral Invariants

Oh¹

2009

Journal of the Korean Mathematical Society

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Abstract. The author previously defined the spectral invariants, denoted by ρ(H; a), of a Hamiltonian function H as the mini-max value of the action functional A H over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant ρ(H; a) states that the mini-max value is a critical value of the action functional A H . The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, ω). We also prove that the spectral invariant function ρa : H → ρ(H; a) can be pushed down to a continuous function defined on the universal (étale) covering space ]Ham(M, ω) of the group Ham(M, ω) of Hamiltonian diffeomorphisms on general (M, ω). For a certain generic homotopy, which we call a Cerf homotopy H = {H s } 0≤s≤1 of Hamiltonians, the function ρa • H : s → ρ(H s ; a) is piecewise smooth away from a countable subset of [0, 1] for each non-zero quantum cohomology class a.The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer mini-max theory.

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Commutator length of symplectomorphisms

Cited by 32 publications

References 49 publications

Calabi quasi-morphisms for some non-monotone symplectic manifolds

Calabi quasi-morphisms for some non-monotone symplectic manifolds

Quasi-states and symplectic intersections

Floer Mini-Max Theory, the Cerf Diagram, and the Spectral Invariants

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