# A Morse-Bott approach to contact homology

Abstract : Contact homology was introduced by Eliashberg, Givental and Hofer. In this theory, we count holomorphic curves in the symplectization of a contact manifold, which are asymptotic to periodic Reeb orbits. These closed orbits are assumed to be nondegenerate and, in particular, isolated. This assumption makes practical computations of contact homology very difficult.
In this thesis, we develop computational methods for contact homology in Morse-Bott situations, in which closed Reeb orbits form
submanifolds of the contact manifold. We require some Morse-Bott type assumptions on the contact form, a positivity property for
the Maslov index, mild requirements on the Reeb flow, and
$c_1(\xi) = 0$.
We then use these methods to compute contact homology for several examples, in order to illustrate their efficiency. As an application of these contact invariants, we show that $T^5$ and $T^2 \times S^3$ carry infinitely many pairwise non-isomorphic
contact structures in the trivial formal homotopy class.
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Document type :
Theses
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Cited literature [35 references]

https://tel.archives-ouvertes.fr/tel-00002421
Contributor : Frederic Bourgeois <>
Submitted on : Tuesday, February 18, 2003 - 3:47:29 PM
Last modification on : Thursday, February 25, 2021 - 9:46:03 AM
Long-term archiving on: : Friday, April 2, 2010 - 6:31:47 PM

### Identifiers

• HAL Id : tel-00002421, version 1

### Citation

Frederic Bourgeois. A Morse-Bott approach to contact homology. Mathematics [math]. Stanford University, 2002. English. ⟨tel-00002421⟩

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