Handle attaching in symplectic homology and the Chord Conjecture

Kai Cieliebak

Handle attaching in symplectic homology and the Chord Conjecture

Kai Cieliebak

Journal of the European Mathematical Society (2002)

Volume: 004, Issue: 2, page 115-142
ISSN: 1435-9855

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http://dx.doi.org/10.1007/s100970100036

Abstract

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Arnold conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a haracteristic chord with respect to any contact form. I confirm this conjecture if the know has Thurston-Bennequin invariant

- 1

. More generally, existence of chords is proved for a standard Legendrian unknot on the boundary of a subcritical Stein manifold of any dimension. There is also a multiplicity result which implies in some situations existence of infinitely many chords. The proof relies on the behaviour of symplectic homology under handle attaching. The main observation is that symplectic homology only changes in the presence of chords.

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Cieliebak, Kai. "Handle attaching in symplectic homology and the Chord Conjecture." Journal of the European Mathematical Society 004.2 (2002): 115-142. <http://eudml.org/doc/277310>.

@article{Cieliebak2002,
abstract = {Arnold conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a haracteristic chord with respect to any contact form. I confirm this conjecture if the know has Thurston-Bennequin invariant $−1$. More generally, existence of chords is proved for a standard Legendrian unknot on the boundary of a subcritical Stein manifold of any dimension. There is also a multiplicity result which implies in some situations existence of infinitely many chords. The proof relies on the behaviour of symplectic homology under handle attaching. The main observation is that symplectic homology only changes in the presence of chords.},
author = {Cieliebak, Kai},
journal = {Journal of the European Mathematical Society},
keywords = {contact form; Legendrian submanifold; Reeb flow; Legendrian knot; contact form; Legendrian submanifold; Reeb flow; Legendrian knot},
language = {eng},
number = {2},
pages = {115-142},
publisher = {European Mathematical Society Publishing House},
title = {Handle attaching in symplectic homology and the Chord Conjecture},
url = {http://eudml.org/doc/277310},
volume = {004},
year = {2002},
}

TY - JOUR
AU - Cieliebak, Kai
TI - Handle attaching in symplectic homology and the Chord Conjecture
JO - Journal of the European Mathematical Society
PY - 2002
PB - European Mathematical Society Publishing House
VL - 004
IS - 2
SP - 115
EP - 142
AB - Arnold conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a haracteristic chord with respect to any contact form. I confirm this conjecture if the know has Thurston-Bennequin invariant $−1$. More generally, existence of chords is proved for a standard Legendrian unknot on the boundary of a subcritical Stein manifold of any dimension. There is also a multiplicity result which implies in some situations existence of infinitely many chords. The proof relies on the behaviour of symplectic homology under handle attaching. The main observation is that symplectic homology only changes in the presence of chords.
LA - eng
KW - contact form; Legendrian submanifold; Reeb flow; Legendrian knot; contact form; Legendrian submanifold; Reeb flow; Legendrian knot
UR - http://eudml.org/doc/277310
ER -

Citations in EuDML Documents

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Kai Cieliebak, Urs Frauenfelder, Alexandru Oancea, Rabinowitz Floer homology and symplectic homology

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