scholarly journals Contact homology of Hamiltonian mapping tori

2010 ◽  
pp. 203-241 ◽  
Author(s):  
Oliver Fabert
Keyword(s):  
Contact Homology ◽  
Mapping Tori ◽  
Hamiltonian Mapping

scholarly journals Higher algebraic structures in Hamiltonian Floer theory

2020 ◽  
Vol 20 (2) ◽  
pp. 179-215
Author(s):  
Oliver Fabert
Keyword(s):  
Algebraic Structures ◽  
Floer Theory ◽  
Manifold Structure ◽  
Lie Bracket ◽  
Witten Theory ◽  
Mapping Tori ◽  
Symplectic Field Theory ◽  
The Rich ◽  
Algebraic Formalism ◽  
Hamiltonian Mapping

AbstractIn this paper we show how the rich algebraic formalism of Eliashberg–Givental–Hofer’s symplectic field theory (SFT) can be used to define higher algebraic structures in Hamiltonian Floer theory. Using the SFT of Hamiltonian mapping tori we define a homotopy extension of the well-known Lie bracket and discuss how it can be used to prove the existence of multiple closed Reeb orbits. Furthermore we define the analogue of rational Gromov–Witten theory in the Hamiltonian Floer theory of open symplectic manifolds. More precisely, we introduce a so-called cohomology F-manifold structure in Hamiltonian Floer theory and prove that it generalizes the well-known Frobenius manifold structure in rational Gromov–Witten theory.

CYLINDRICAL CONTACT HOMOLOGY OF A DEHN TWIST

2009 ◽  
Vol 20 (12) ◽  
pp. 1479-1525
Author(s):  
MEI-LIN YAU
Keyword(s):  
Dehn Twist ◽  
Open Book ◽  
Contact Homology

We use open book representations of contact 3-manifolds to compute the cylindrical contact homology of a Stein-fillable contact 3-manifold represented by the open book whose monodromy is a positive Dehn twist on a torus with boundary.

scholarly journals Legendrian contact homology and topological entropy

2019 ◽  
Vol 11 (01) ◽  
pp. 53-108 ◽  
Author(s):  
Marcelo R. R. Alves
Keyword(s):  
Growth Rate ◽  
Topological Entropy ◽  
Infinite Family ◽  
Contact Structures ◽  
Legendrian Knots ◽  
Contact Homology ◽  
Contact Manifolds ◽  
3 Dimensional ◽  
Reeb Flows ◽  
Legendrian Contact Homology

In this paper we study the growth rate of a version of Legendrian contact homology, which we call strip Legendrian contact homology, in 3-dimensional contact manifolds and its relation to the topological entropy of Reeb flows. We show that: if for a pair of Legendrian knots in a contact 3-manifold [Formula: see text] the strip Legendrian contact homology is defined and has exponential homotopical growth with respect to the action, then every Reeb flow on [Formula: see text] has positive topological entropy. This has the following dynamical consequence: for all Reeb flows (even degenerate ones) on [Formula: see text] the number of hyperbolic periodic orbits grows exponentially with respect to the period. We show that for an infinite family of 3-manifolds, infinitely many different contact structures exist that possess a pair of Legendrian knots for which the strip Legendrian contact homology has exponential growth rate.

scholarly journals Legendrian contact homology and nondestabilizability

2011 ◽  
Vol 9 (1) ◽  
pp. 33-44 ◽  
Author(s):  
Clayton Shonkwiler ◽  
David Shea Vela-Vick
Keyword(s):  
Contact Homology ◽  
Legendrian Contact Homology
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