Symplectic topology as the geometry of generating functions

Viterbo, Claude

doi:10.1007/bf01444643

Cited by 338 publications

(453 citation statements)

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“…Namely, let The idea behind the definition is due to C. Viterbo [50] who originally used it in the context of Morse homology and finite-dimensional generating functions for Hamiltonian symplectomorphisms of R 2n . It has taken a considerable work involving new techniques to extend this idea to the Floer theory with various versions of Floer homology viewed as infinite-dimensional analogs of Morse homology and with the action functional viewed as an "infinite-dimensional generating function".…”

Section: Spectral Numbersmentioning

confidence: 99%

Commutator length of symplectomorphisms

Entov¹

2004

Comment. Math. Helv.

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Abstract.Each element x of the commutator subgroup [G, G] of a group G can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of x. The commutator length of G is defined as the supremum of commutator lengths of elements of [G, G].We show that for certain closed symplectic manifolds (M, ω), including complex projective spaces and Grassmannians, the universal cover Ham (M, ω) of the group of Hamiltonian symplectomorphisms of (M, ω) has infinite commutator length. In particular, we present explicit examples of elements in Ham (M, ω) that have arbitrarily large commutator length -the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of (M, ω). By a different method we also show that in the case c 1 (M ) = 0 the group Ham (M, ω) and the universal cover Symp 0 (M, ω) of the identity component of the group of symplectomorphisms of (M, ω) have infinite commutator length. Mathematics Subject Classification (2000). 53D22, 53D05, 53D40, 53D45.

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Section: Spectral Numbersmentioning

confidence: 99%

Commutator length of symplectomorphisms

Entov¹

2004

Comment. Math. Helv.

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“…Theorem 3 (Viterbo,[20]). Let S 1 and S 2 be two generating functions quadratic at infinity generating the same embedded Lagrangian submanifold L = φ…”

Section: Addition Of a Constant:smentioning

confidence: 99%

Morse homology for generating functions of Lagrangian submanifolds

Milinković

1999

Trans. Amer. Math. Soc.

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Abstract. The purpose of the paper is to give an alternative construction and the proof of the main properties of symplectic invariants developed by Viterbo. Our approach is based on Morse homology theory. This is a step towards relating the "finite dimensional" symplectic invariants constructed via generating functions to the "infinite dimensional" ones constructed via Floer theory in Y.-G. Oh, Symplectic topology as the geometry of action functional.

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“…From Viterbo's uniqueness theorem [19,15], it follows that the integer ind d 2 S 1 (z, ξ) − ind Q ∞ does not depend on S 1 , but only on Φ 1 . We call it the gf-index of z, denoted by ind gf (z).…”

Section: Definition 32mentioning

confidence: 99%

“…We list some of their properties in the next statement (some inequalities differ from those of [19], because some sign conventions differ). In [7] (see also [6] …”

Section: The Torus Casementioning

confidence: 99%

“…In [19], Claude Viterbo constructed a symplectic capacity c gf (V ) for V an open set of R 2n , and used it to prove several interesting results in symplectic geometry, including the following Symplectic Camel Theorem. Here the subscript "gf" stands for "generating functions", because this is the tool used to define c gf (V ); we summarize in Appendix A the definition and basic properties of this symplectic capacity.…”

Section: Introductionmentioning

confidence: 99%

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