Legendrian contact homology of conormal lifts Tobias Ekholm Uppsala University, Sweden Berkeley, April 4, 2011
Plan of the talk Background Legendrian contact homology Computations for links in R 3 Application to transverse links in ( R 3 , ξ std ) Remarks about other dimensions and relations to string topology
Background – basic observations Diff top. A smooth submanifold K ⊂ M .
Background – basic observations Diff top. A smooth submanifold K ⊂ M . Sympl geom. The Lagrangian conormal of K in T ∗ M , p ∈ T ∗ M : π ( p ) ∈ K, p | TK = 0 ⊂ T ∗ M, � � L K = with ideal Legendrian boundary Λ K = L K ∩ S ∗ M .
Background – basic observations Diff top. A smooth submanifold K ⊂ M . Sympl geom. The Lagrangian conormal of K in T ∗ M , p ∈ T ∗ M : π ( p ) ∈ K, p | TK = 0 ⊂ T ∗ M, � � L K = with ideal Legendrian boundary Λ K = L K ∩ S ∗ M . Isotopy. An isotopy K t of K induces a Lagrangian (resp. a Legendrian) isotopy of L K t (resp. Λ K t ).
Background – Invariants Symplectic deformation invariants of Λ K give isotopy invariants of K ⊂ M . We will study holomorphic curve invariants on the symplectic side.
Background – Invariants Symplectic deformation invariants of Λ K give isotopy invariants of K ⊂ M . We will study holomorphic curve invariants on the symplectic side. For most of the talk we restrict attention to the case M = R 3 and K ⊂ R 3 a smooth link. In this case Λ K ⊂ S ∗ R 3 is a union of Legendrian tori. We will compute its Legendrian contact homology (the most basic part of SFT).
Background – Survey of results For knots, the knot contact homology of L. Ng is a combinatorial invariant that was inspired by this construction and conjectured to agree with the Legendrian contact homology of Λ K . The work surveyed in this talk proves that conjecture.
Background – Survey of results For knots, the knot contact homology of L. Ng is a combinatorial invariant that was inspired by this construction and conjectured to agree with the Legendrian contact homology of Λ K . The work surveyed in this talk proves that conjecture. Ng derived many properties of his knot contact homology. In particular, it follows from his work that the Legendrian contact homology of Λ K is an unknot detector that contains the A -polynomial.
Background – Survey of results For knots, the knot contact homology of L. Ng is a combinatorial invariant that was inspired by this construction and conjectured to agree with the Legendrian contact homology of Λ K . The work surveyed in this talk proves that conjecture. Ng derived many properties of his knot contact homology. In particular, it follows from his work that the Legendrian contact homology of Λ K is an unknot detector that contains the A -polynomial. In this talk we focus on describing the holomorphic curves needed to compute the Legendrian homology. This description lead to new invariants of transverse knots in R 3 and also constitutes an important model for computations in other manifolds and in higher dimensions.
Background – some refs In the case under study, the technical details of Legendrian contact homology were worked out in
Background – some refs In the case under study, the technical details of Legendrian contact homology were worked out in Ekholm, T.; Etnyre, J.; Sullivan, M. Legendrian contact homology in P × R . Trans. Amer. Math. Soc. 359 (2007), no. 7, 3301–3333 Ekholm, T.; Etnyre, J.; Sullivan, M. Orientations in Legendrian contact homology and exact Lagrangian immersions , Internat. J. Math. 16 (2005), no. 5, 453–532.
Legendrian contact homology – set up The Reeb vector field R of a contact form α on a contact manifold Y is characterized by dα ( R, · ) = 0 and α ( R ) = 1 . For Y = S ∗ M identified with the unit conormal bundle, and α = p dq , the flow of R is the geodesic flow. A flow line of R beginning and ending on a Legendrian submanifold Λ K ⊂ Y is a Reeb chord of Λ . For Λ = Λ K Reeb chords correspond to binormal geodesic chords on K .
Legendrian contact homology – set up The Reeb vector field R of a contact form α on a contact manifold Y is characterized by dα ( R, · ) = 0 and α ( R ) = 1 . For Y = S ∗ M identified with the unit conormal bundle, and α = p dq , the flow of R is the geodesic flow. A flow line of R beginning and ending on a Legendrian submanifold Λ K ⊂ Y is a Reeb chord of Λ . For Λ = Λ K Reeb chords correspond to binormal geodesic chords on K . The symplectization of Y is X = Y × R with symplectic form d ( e t α ) where t ∈ R . Fix an almost complex structure J on X such that J ( ∂ t ) = R , J (ker α ) = ker α , and dα ( v, Jv ) > 0 .
Legendrian contact homology – set up The Reeb vector field R of a contact form α on a contact manifold Y is characterized by dα ( R, · ) = 0 and α ( R ) = 1 . For Y = S ∗ M identified with the unit conormal bundle, and α = p dq , the flow of R is the geodesic flow. A flow line of R beginning and ending on a Legendrian submanifold Λ K ⊂ Y is a Reeb chord of Λ . For Λ = Λ K Reeb chords correspond to binormal geodesic chords on K . The symplectization of Y is X = Y × R with symplectic form d ( e t α ) where t ∈ R . Fix an almost complex structure J on X such that J ( ∂ t ) = R , J (ker α ) = ker α , and dα ( v, Jv ) > 0 . If c is a Reeb chord of Λ then c × R ⊂ X is a J -holomorphic strip with boundary on the Lagrangian submanifold Λ × R .
Legendrian contact homology – DGA The Legendrian contact homology algbera of Λ is the free unital (non-commutative) algebra � � A (Λ) = Z [ H 1 (Λ)] Reeb chords (For general Y , A (Λ) is an algebra over the contact homology algebra Q ( Y ) of Y generated by closed Reeb orbits. In this talk we work with contact forms for which there are no closed orbits in Y and for which c 1 (ker α ) = 0 )
Legendrian contact homology – DGA The Legendrian contact homology algbera of Λ is the free unital (non-commutative) algebra � � A (Λ) = Z [ H 1 (Λ)] Reeb chords (For general Y , A (Λ) is an algebra over the contact homology algebra Q ( Y ) of Y generated by closed Reeb orbits. In this talk we work with contact forms for which there are no closed orbits in Y and for which c 1 (ker α ) = 0 ) A (Λ) is a DGA. The grading | c | of a Reeb chord is defined by a Maslov index. The differential ∂ : A (Λ) → A (Λ) is linear, satisfies Leibniz rule, and is defined on generators through a holomorphic curve count. The DGA ( A ( λ ) , ∂ ) is invariant under deformations up to quasi-isomorphism.
The Legendrian contact homology – the differential � ∂a = # M A ( a, b ) A b , | a |−| A |−| b | =1 where M A ( a ; b ) is the moduli space of holomorphic disks u : D → X , dim( M A ( a ; b )) = | a | − | A | − | b | .
Computations – strategy To compute the Legendrian contact homology of Λ K for a link K , we braid K around the unknot U . Then Λ K ⊂ J 1 (Λ U ) ⊂ S ∗ R 3 .
Computations – strategy To compute the Legendrian contact homology of Λ K for a link K , we braid K around the unknot U . Then Λ K ⊂ J 1 (Λ U ) ⊂ S ∗ R 3 . In the limit as K → U , holomorphic disks with boundary on Λ K admits a description in terms of holomorphic disks with boundary on Λ U with flow trees determined by Λ K ⊂ J 1 (Λ) attached along their boundaries.
Computations – strategy To compute the Legendrian contact homology of Λ K for a link K , we braid K around the unknot U . Then Λ K ⊂ J 1 (Λ U ) ⊂ S ∗ R 3 . In the limit as K → U , holomorphic disks with boundary on Λ K admits a description in terms of holomorphic disks with boundary on Λ U with flow trees determined by Λ K ⊂ J 1 (Λ) attached along their boundaries. Our computation thus has three main steps: Describe holomorphic disks for Λ U . Describe flow trees for Λ K ⊂ J 1 (Λ U ) . Count the resulting rigid disks with flow trees.
Computations – some refs The computation is joint work: Ekholm, T.; Etnyre, J.; Ng, L.; Sullivan, M. Knot contact homology , 105 pages, in preparation.
Computations – some refs The computation is joint work: Ekholm, T.; Etnyre, J.; Ng, L.; Sullivan, M. Knot contact homology , 105 pages, in preparation. the techniques are based on Ekholm, T. Morse flow trees and Legendrian contact homology in 1-jet spaces , Geom. Topol. 11 (2007), 1083–1224. Ekholm, T.; Etnyre, J.; Sabloff, J. A duality exact sequence for Legendrian contact homology , Duke Math. J. 150 (2009), no. 1, 1–75.
Computations – the unknot To describe holomorphic disks of ( S ∗ R 3 × R , Λ U × R ) we use the contactomorphism Ψ: S ∗ R 3 = J 1 ( S 2 ) = T ∗ S 2 × R , � � Ψ( x, y ) = y, x − � x, y � y, � x, y � , and the correspondence between holomorphic disks in T ∗ S 2 with boundary on the projection π (Λ K ) and 1 -parameter families of holomorphic disks in J 1 ( S 2 ) × R with boundary on Λ K × R .
Computations – the unknot The front of the unknot in J 0 ( S 2 ) ≈ S 2 × R can be drawn as follows: � � Recall Ψ( x, y ) = y, x − � x, y � y, � x, y � , the first and last coordinates are depicted.
Computations – the unknot In order for the disk/tree relation to hold, Λ U must be front generic, i.e. in general position with respect to the fibers in J 1 ( S 2 ) . This holds outside neighborhoods of the poles and near the poles we get the following after perturbation:
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