Legendrian contact homology in 𝑃×ℝ

Ekholm, Tobias; Etnyre, John; Sullivan, Michael

doi:10.1090/S0002-9947-07-04337-1

Legendrian contact homology in $P \times \mathbb {R}$
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by Tobias Ekholm, John Etnyre and Michael Sullivan PDF

Trans. Amer. Math. Soc. 359 (2007), 3301-3335 Request permission

Abstract:

A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form $P\times \mathbb {R}$, where $P$ is an exact symplectic manifold, is established. The class of such contact manifolds includes 1-jet spaces of smooth manifolds. As an application, contact homology is used to provide (smooth) isotopy invariants of submanifolds of $\mathbb {R}^n$ and, more generally, invariants of self transverse immersions into $\mathbb {R}^n$ up to restricted regular homotopies. When $n=3$, this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng.

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