Legendrian contact homology in 𝑃×ℝ

Abstract

Abstract. A rigorous foundation for the contact homology of Legendrian submanifolds in a contact manifold of the form P × 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 R n and, more generally, invariants of self transverse immersions into 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.

Immersed Lagrangian floer theory

“…This should clearly be related to the theory of Legendrian contact homology, which was described informally by Eliashberg, Givental and Hofer [5, §2.8], and by Chekanov [4] for Legendrian knots in R 3 , and has been developed rigorously by Ekholm, Etnyre and Sullivan [6,7], for embedded Legendrians L in R 2n+1 and in M × R for (M, ω) an exact symplectic manifold. In particular, for (M, ω) exact one can compare our HF * (L, b; Λ Z nov ) for embedded Legendrians in M × S 1 , and Ekholm et al's HC * (L, J) for embedded Legendrians L in M × R, [7].…”

“…This should clearly be related to the theory of Legendrian contact homology, which was described informally by Eliashberg, Givental and Hofer [5, §2.8], and by Chekanov [4] for Legendrian knots in R 3 , and has been developed rigorously by Ekholm, Etnyre and Sullivan [6,7], for embedded Legendrians L in R 2n+1 and in M × R for (M, ω) an exact symplectic manifold. In particular, for (M, ω) exact one can compare our HF * (L, b; Λ Z nov ) for embedded Legendrians in M × S 1 , and Ekholm et al's HC * (L, J) for embedded Legendrians L in M × R, [7]. It seems that HC * (L, J) should be a sector of HF * (L, b; Λ Z nov ), but not the whole thing, since HC * (L; J) is the homology of a complex involving H 1 (L; Z) and the set of double points of π(L) in M , but HF * (L, b; Λ Z nov ) is the cohomology of a complex involving all of H * (L; Q) and R, which has two points (p − , p + ), (p + , p − ) for each double point p of π(L) in M .…”

Legendrian ambient surgery and Legendrian contact homology

“…In this section we give an outline of the theory of Legendrian contact homology [21,8,18] as well as certain constructions from relative symplectic field theory [15].…”

“…Both the construction and the invariance of Legendrian contact homology have been worked out in the case (J 1 R, λ 0 ) by Chekanov [8], and in more general contactisations (P × R, dz + θ) by Ekholm, Etnyre and Sullivan [18], where (P, dθ) is an exact symplectic manifold having finite geometry at infinity.…”

Rational symplectic field theory for Legendrian knots

“…A key breakthrough in the study of Legendrian knots, and symplectic topology generally, was the introduction of Gromov-type holomorphic-curve techniques in the 1990s. This led in particular to the development of Legendrian contact homology, outlined by Eliashberg and Hofer [Eli98] and fleshed out famously by Chekanov [Che02] for standard contact R 3 and later by others in more general setups (e.g., [EES05a,EES07,NT04,Sab03]). Besides applications to contact topology, Legendrian contact homology has been closely linked to standard knot theory (e.g., [Ng08]).…”