A Quantitative Look at Lagrangian Cobordisms
|
|
- Charlene Warren
- 5 years ago
- Views:
Transcription
1 A Quantitative Look at Lagrangian Cobordisms Lisa Traynor Bryn Mawr College Joint work with Joshua M. Sabloff, Haverford College December 2016 Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 1 / 36
2 Lagrangians and Legendrians y z y x x Symplectic Manifold (X 2n, ω) Contact Manifold (Y 2n+1, ξ) Lagrangian Submanifold L n : ω TL 0 Legendrian Submanifold Λ n : T Λ ξ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 2 / 36
3 Lagrangians and Legendrians y z y x x Symplectic Manifold (X 2n, ω) Exact Symplectic : ω = dλ Lagrangian Submanifold L n : ω TL 0 Exact Lagrangian: λ = df Contact Manifold (Y 2n+1, ξ) Legendrian Submanifold Λ n : T Λ ξ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 2 / 36
4 The Symplectization of a Contact Manifold Standard Contact Manifold: ( R 2n+1, ker α ) J 1 (R n ) = T R n R = R 2n+1, α = dz i y idx i Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 3 / 36
5 The Symplectization of a Contact Manifold Standard Contact Manifold: ( R 2n+1, ker α ) J 1 (R n ) = T R n R = R 2n+1, α = dz i y idx i Symplectization: ( R R 2n+1, d(e s α) ) R 2n+1 s Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 3 / 36
6 The Symplectization of a Contact Manifold Standard Contact Manifold: ( R 2n+1, ker α ) J 1 (R n ) = T R n R = R 2n+1, α = dz i y idx i Symplectization: ( R R 2n+1, d(e s α) ) R 2n+1 s There are no closed, exact Lagrangians (Gromov); Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 3 / 36
7 The Symplectization of a Contact Manifold Standard Contact Manifold: ( R 2n+1, ker α ) J 1 (R n ) = T R n R = R 2n+1, α = dz i y idx i Symplectization: ( R R 2n+1, d(e s α) ) R 2n+1 s There are no closed, exact Lagrangians (Gromov); For a Legendrian Λ, the cylinder R Λ is an exact Lagrangian. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 3 / 36
8 Lagrangian Cobordisms between Legendrians A Lagrangian cobordism from Λ to Λ + means: s + + R 2n+2 s s Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 4 / 36
9 Lagrangian Cobordisms between Legendrians A Lagrangian cobordism from Λ to Λ + means: s + + R 2n+2 s s Λ ± are Legendrian submanifolds in {s = s ± }; Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 4 / 36
10 Lagrangian Cobordisms between Legendrians A Lagrangian cobordism from Λ to Λ + means: s + + R 2n+2 s s Λ ± are Legendrian submanifolds in {s = s ± }; L is Lagrangian and cylindrical over Λ ± at ± : L = R Λ ± outside [s, s +]; Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 4 / 36
11 Lagrangian Cobordisms between Legendrians A Lagrangian cobordism from Λ to Λ + means: s + + R 2n+2 s s Λ ± are Legendrian submanifolds in {s = s ± }; L is Lagrangian and cylindrical over Λ ± at ± : L is embedded and exact: L = R Λ ± outside [s, s +]; e s α L = df, f = constant ± outside [s, s +]. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 4 / 36
12 Lagrangian Cobordisms between Legendrians A Lagrangian cobordism from Λ to Λ + means: s + + R 2n+2 s s Λ ± are Legendrian submanifolds in {s = s ± }; L is Lagrangian and cylindrical over Λ ± at ± : L is embedded and exact: L = R Λ ± outside [s, s +]; e s α L = df, f = constant ± outside [s, s +]. Arise in relative SFT (Eliashberg-Givental-Hofer) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 4 / 36
13 Qualitative Questions Given Λ, Λ + R 2n+1, does there exist a Lagrangian cobordism between them? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 5 / 36
14 Qualitative Questions E E Given Λ, Λ + R 2n+1, does there exist a Lagrangian cobordism between them? Non-symmetric relation! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 5 / 36
15 Qualitative Questions E E Given Λ, Λ + R 2n+1, does there exist a Lagrangian cobordism between them? Non-symmetric relation! How topologically rigid are Lagrangian cobordisms? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 5 / 36
16 Qualitative Questions E E Given Λ, Λ + R 2n+1, does there exist a Lagrangian cobordism between them? Non-symmetric relation! How topologically rigid are Lagrangian cobordisms? Fillings realize 4-ball genus! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 5 / 36
17 Qualitative Questions E E Given Λ, Λ + R 2n+1, does there exist a Lagrangian cobordism between them? Non-symmetric relation! How topologically rigid are Lagrangian cobordisms? Fillings realize 4-ball genus! A variety of qualitative questions have been studied by: Chantraine, Ekholm, Honda, Kálmán, Dimitroglou Rizell, Ghiggini, Golovko, Cornwell, Ng, Sivek, Bourgeois, Sabloff, Traynor, Capovilla-Searle, Hayden, Pan,... Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 5 / 36
18 Quantitative Questions (Length) Given Λ, Λ + R 2n+1, what is the minimal length" of any cobordism between them? h = s + 0 =s_ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 6 / 36
19 Quantitative Questions (Length) Given Λ, Λ + R 2n+1, what is the minimal length" of any cobordism between them? h = s + 0 =s_ (Width) Given a Lagrangian cobordism, what is its width"? B(c) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 6 / 36
20 Outline 1 Constructions of Lagrangian Cobordisms
21 Outline 1 Constructions of Lagrangian Cobordisms 2 Length of a Lagrangian cobordism
22 Outline 1 Constructions of Lagrangian Cobordisms 2 Length of a Lagrangian cobordism 3 Width of a Lagrangian Cobordism
23 Outline 1 Constructions of Lagrangian Cobordisms 2 Length of a Lagrangian cobordism 3 Width of a Lagrangian Cobordism Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 8 / 36
24 E Constructions of Lagrangian Concordances Isotopy Lemma (Eliashberg, Chantraine, Golovko, Ekholm-Honda-Kálmán,... ) Suppose Λ and Λ + are Legendrian isotopic. Then there exists a Lagrangian cobordism from Λ to Λ +. + R 3 s _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 9 / 36
25 E Constructions of Lagrangian Concordances Isotopy Lemma (Eliashberg, Chantraine, Golovko, Ekholm-Honda-Kálmán,... ) Suppose Λ and Λ + are Legendrian isotopic. Then there exists a Lagrangian cobordism from Λ to Λ +. + R 3 s _ Remark: The Lagrangian is not the trace of the isotopy. Most slices of the Lagrangian will not be Legendrian. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 9 / 36
26 Lagrangian Concordances from Isotopy E E Qualitatively Symmetric Concordances: _ 1 _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 10 / 36
27 Constructions of Lagrangian Cobordisms Theorem (Dimitroglou Rizell, Ekholm-Honda-Kálmán, Bourgeois-Sabloff-T ) If Λ + is obtained from Λ by a cusp-surgery", Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 11 / 36
28 Constructions of Lagrangian Cobordisms Theorem (Dimitroglou Rizell, Ekholm-Honda-Kálmán, Bourgeois-Sabloff-T ) If Λ + is obtained from Λ by a cusp-surgery", Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 11 / 36
29 Constructions of Lagrangian Cobordisms Theorem (Dimitroglou Rizell, Ekholm-Honda-Kálmán, Bourgeois-Sabloff-T ) If Λ + is obtained from Λ by a cusp-surgery", then there exists a Lagrangian cobordism from Λ to Λ +. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 11 / 36
30 Construction Example Lagrangian genus 1 filling of a Legendrian m(5 2 ): Legendrian isotopy and cusp pinches as you move up! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 12 / 36
31 Outline 1 Constructions of Lagrangian Cobordisms 2 Length of a Lagrangian cobordism 3 Width of a Lagrangian Cobordism Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 13 / 36
32 Length Question: Given Λ, Λ + R 2n+1, what is the minimal length" of any cobordism between them? h = s + 0 =s_ minimal length = inf{h : Lagrangian cobordism from Λ to Λ + that is cylindrical outside[0, h]}. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 14 / 36
33 Flexibility Theorem (Sabloff-T, 16: Selecta Mathematica) There exists an arbitrarily short Lagrangian cobordism between 1 a Legendrian and its vertical translate, h~0 + s 0 _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 15 / 36
34 Flexibility Theorem (Sabloff-T, 16: Selecta Mathematica) There exists an arbitrarily short Lagrangian cobordism between 1 a Legendrian and its vertical translate, h~0 + s 0 _ 2 a Legendrian and its horizontal translate, h~0 s 0 Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 15 / 36
35 Flexibility Theorem (Sabloff-T, 16: Selecta Mathematica) There exists an arbitrarily short Lagrangian cobordism between 1 a Legendrian and its vertical translate, h~0 + s 0 _ 2 a Legendrian and its horizontal translate, h~0 s 0 3 a Legendrian and its vertical expansion. h~0 + s 0 _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 15 / 36
36 Rigidity Theorem (Sabloff-T, 16) There exist obstructions to arbitrarily short Lagrangian cobordisms between 1 a Legendrian and its vertical contraction; h~ln s 0 2 _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 16 / 36
37 Rigidity Theorem (Sabloff-T, 16) There exist obstructions to arbitrarily short Lagrangian cobordisms between 1 a Legendrian and its vertical contraction; h~ln s 0 2 _ 2 vertically shifted Hopf links: h 1 v s 0 1 u h { ( ) ln 1 u 1 v, if u v, ln ( ) u v, if u v. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 16 / 36
38 Lower Bound to Length (Step 1) Assign capacities" to a Legendrian c(λ, ε, θ) R >0 { }, ε is an augmentation of the DGA A(Λ), ε : (A(Λ), ) (F 2, 0), θ LCH (Λ, ε). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 17 / 36
39 Lower Bound to Length (Step 1) Assign capacities" to a Legendrian c(λ, ε, θ) R >0 { }, ε is an augmentation of the DGA A(Λ), ε : (A(Λ), ) (F 2, 0), θ LCH (Λ, ε). Example: r 0 λ LCH 1 (U(r), ε); c(u(r), ε, λ) = r. Fundamental Class Fundamental Capacity Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 17 / 36
40 Lower Bound to Length (Step 1) Assign capacities" to a Legendrian c(λ, ε, θ) R >0 { }, ε is an augmentation of the DGA A(Λ), ε : (A(Λ), ) (F 2, 0), θ LCH (Λ, ε). Example: r 0 λ LCH 1 (U(r), ε); c(u(r), ε, λ) = r. Fundamental Class Fundamental Capacity For θ 0, c(λ, ε, θ) is always the height of a Reeb chord! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 17 / 36
41 Lower Bound to Length (Step 2) From ε, θ for Λ and Lagrangian cobordism L from Λ to Λ +, get induced ε +, θ + for Λ +. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 18 / 36
42 Lower Bound to Length (Step 2) From ε, θ for Λ and Lagrangian cobordism L from Λ to Λ +, get induced ε +, θ + for Λ +. [Ekholm-Honda-Kálmán] Λ + A(Λ + ) Φ(L) ε + LCH (Λ +, ε + ) θ + Ψ L,ε Λ A(Λ ) ε F 2 LCH (Λ, ε ) θ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 18 / 36
43 Lower Bound to Length (Step 2) From ε, θ for Λ and Lagrangian cobordism L from Λ to Λ +, get induced ε +, θ + for Λ +. [Ekholm-Honda-Kálmán] Λ + A(Λ + ) Φ(L) ε + LCH (Λ +, ε + ) θ + Ψ L,ε Λ A(Λ ) ε F 2 LCH (Λ, ε ) θ Question: How do capacities c(λ +, ε +, θ + ) and c(λ, ε, θ ) compare? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 18 / 36
44 Lower Bound to Length (Step 3) Relate capacities for ends of a Lagrangian cobordism. Length-Capacity Inequality (Sabloff-T) If L is a Lagrangian cobordism from Λ to Λ + that is cylindrical outside [0, h], then e 0 c(λ, ε, θ ) e h c(λ +, ε +, θ + ). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 19 / 36
45 Lower Bound to Length (Step 3) Relate capacities for ends of a Lagrangian cobordism. Length-Capacity Inequality (Sabloff-T) If L is a Lagrangian cobordism from Λ to Λ + that is cylindrical outside [0, h], then e 0 c(λ, ε, θ ) e h c(λ +, ε +, θ + ). Remember ε +, θ + are induced by L. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 19 / 36
46 Lower Bound to Length (Step 3) Relate capacities for ends of a Lagrangian cobordism. Length-Capacity Inequality (Sabloff-T) If L is a Lagrangian cobordism from Λ to Λ + that is cylindrical outside [0, h], then e 0 c(λ, ε, θ ) e h c(λ +, ε +, θ + ). Remember ε +, θ + are induced by L. Get lower bounds to length of a cobordism! ( ) c(λ, ε, θ ) ln h. c(λ +, ε +, θ + ) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 19 / 36
47 Lower Bound to Length of a Contraction h 1 + s 0 2 _ By Length-Capacity Inequality: ( ) ( ) 2 c(u(2)), ε, λ ) ln = ln 1 c(u(1)), ε +, λ + ) = ln ( ) c(λ, ε, λ ) h. c(λ +, ε +, λ + ) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 20 / 36
48 Lower Bound to Length of a Contraction h 1 + s 0 2 _ By Length-Capacity Inequality: ( ) ( ) 2 c(u(2)), ε, λ ) ln = ln 1 c(u(1)), ε +, λ + ) = ln ( ) c(λ, ε, λ ) h. c(λ +, ε +, λ + ) Question: Can we get arbitrarily close to h = ln 2? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 20 / 36
49 Lower Bound to Length of a Contraction h 1 + s 0 2 _ By Length-Capacity Inequality: ( ) ( ) 2 c(u(2)), ε, λ ) ln = ln 1 c(u(1)), ε +, λ + ) = ln ( ) c(λ, ε, λ ) h. c(λ +, ε +, λ + ) Question: Can we get arbitrarily close to h = ln 2? Answer: Yes! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 20 / 36
50 Upper Bound to Length of a Contraction Lagrangian cobordism from Λ = U(2) to Λ + = U(1) of length A: Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 21 / 36
51 Upper Bound to Length of a Contraction Lagrangian cobordism from Λ = U(2) to Λ + = U(1) of length A: 1 1/2 Legendrian isotopy: λ s (t) = (x(t), ρ(s)y(t), ρ(s)z(t)) 0 A Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 21 / 36
52 Upper Bound to Length of a Contraction Lagrangian cobordism from Λ = U(2) to Λ + = U(1) of length A: 1 1/2 Legendrian isotopy: λ s (t) = (x(t), ρ(s)y(t), ρ(s)z(t)) 0 A Lagrangian immersion: Γ(s, t) = (s, x(t), ρ(s)y(t), ρ(s)z(t) + ρ (s)z(t)) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 21 / 36
53 Upper Bound to Length of a Contraction Lagrangian cobordism from Λ = U(2) to Λ + = U(1) of length A: 1 1/2 Legendrian isotopy: λ s (t) = (x(t), ρ(s)y(t), ρ(s)z(t)) 0 A Lagrangian immersion: Embedding condition: Γ(s, t) = (s, x(t), ρ(s)y(t), ρ(s)z(t) + ρ (s)z(t)) 1 e A /2 e s /2 e s d ds (es ρ(s)) 0 0 A Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 21 / 36
54 Upper Bound to Length of a Contraction Lagrangian cobordism from Λ = U(2) to Λ + = U(1) of length A: 1 1/2 Legendrian isotopy: λ s (t) = (x(t), ρ(s)y(t), ρ(s)z(t)) 0 A Lagrangian immersion: Embedding condition: So, Γ(s, t) = (s, x(t), ρ(s)y(t), ρ(s)z(t) + ρ (s)z(t)) 1 e A /2 e s /2 e s d ds (es ρ(s)) 0 0 A embedded Lagrangian cobordism when 1 < e A /2 ln 2 < A. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 21 / 36
55 Outline 1 Constructions of Lagrangian Cobordisms 2 Length of a Lagrangian cobordism 3 Width of a Lagrangian Cobordism Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 22 / 36
56 Width of a Symplectic Manifold B 2n (c) := { (x 1, y 1,..., x n, y n ) : π i (x 2 i + y 2 i ) c } (R 2n, ω 0 ). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 23 / 36
57 Width of a Symplectic Manifold B 2n (c) := { (x 1, y 1,..., x n, y n ) : π i (x 2 i + y 2 i ) c } (R 2n, ω 0 ). Width of a symplectic manifold (X, ω): w(x) := sup{c : ψ : B 2n (c) X, ψ ω = ω 0 }. B(c) X Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 23 / 36
58 Width of a Symplectic Manifold B 2n (c) := { (x 1, y 1,..., x n, y n ) : π i (x 2 i + y 2 i ) c } (R 2n, ω 0 ). Width of a symplectic manifold (X, ω): w(x) := sup{c : ψ : B 2n (c) X, ψ ω = ω 0 }. B(c) X We are working in X = R J 1 M: w(r J 1 M) =. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 23 / 36
59 Width of a Lagrangian Given a Lagrangian submanifold L (X, ω), relative width is: { w(x, L) = sup c ψ : B 2n (c) X, ψ ω = ω 0, ψ 1 (L) = B 2n (c) R n}. B(c) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 24 / 36
60 Width of a Lagrangian Given a Lagrangian submanifold L (X, ω), relative width is: { w(x, L) = sup c ψ : B 2n (c) X, ψ ω = ω 0, ψ 1 (L) = B 2n (c) R n}. B(c) Introduced by Barraud and Cornea, 05. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 24 / 36
61 Widths of Lagrangian Cobordisms Given a Lagrangian cobordism L, for a < b, L b a := {(s, x, y, z) L : a < s < b} (a, b) J 1 M. b B(c) a Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 25 / 36
62 Widths of Lagrangian Cobordisms Given a Lagrangian cobordism L, for a < b, L b a := {(s, x, y, z) L : a < s < b} (a, b) J 1 M. b B(c) a Question: Can we calculate w(l b a), for some L, a, b? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 25 / 36
63 Widths of Lagrangian Cobordisms Given a Lagrangian cobordism L, for a < b, L b a := {(s, x, y, z) L : a < s < b} (a, b) J 1 M. b B(c) a Question: Can we calculate w(l b a), for some L, a, b? Answer: Yes! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 25 / 36
64 Infinite Width Lemma For any Lagrangian cobordism L, for any a, w (L a ) =. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 26 / 36
65 Infinite Width Lemma For any Lagrangian cobordism L, for any a, w (L a ) =. Proof Sketch: B(r) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 26 / 36
66 Infinite Width Lemma For any Lagrangian cobordism L, for any a, w (L a ) =. Proof Sketch: B(R) B(r) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 26 / 36
67 Infinite Width Lemma For any Lagrangian cobordism L, for any a, w (L a ) =. Proof Sketch: B(R) B(r) Chop off top! We will consider: a =, s + b < +. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 26 / 36
68 Width of Cylindrical Lagrangian Cobordisms Cylindrical Lagrangian Cobordisms: L = R Λ, for a Legendrian Λ. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 27 / 36
69 Width of Cylindrical Lagrangian Cobordisms Cylindrical Lagrangian Cobordisms: L = R Λ, for a Legendrian Λ. Question: Can we calculate w((r Λ) b ) for some Λ and for some b? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 27 / 36
70 Width of Cylindrical Lagrangian Cobordisms Cylindrical Lagrangian Cobordisms: L = R Λ, for a Legendrian Λ. Question: Can we calculate w((r Λ) b ) for some Λ and for some b? Suffices to understand b = 0: Lemma For any Legendrian Λ, ( ) w (R Λ) b = e b w ( ) (R Λ) 0. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 27 / 36
71 Width of Cylindrical Lagrangian Cobordisms Cylindrical Lagrangian Cobordisms: L = R Λ, for a Legendrian Λ. Question: Can we calculate w((r Λ) b ) for some Λ and for some b? Suffices to understand b = 0: Lemma For any Legendrian Λ, ( ) w (R Λ) b = e b w ( ) (R Λ) 0. Question: Can we calculate w((r Λ) 0 ) for some Λ? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 27 / 36
72 Width of Cylindrical Lagrangian Cobordisms Cylindrical Lagrangian Cobordisms: L = R Λ, for a Legendrian Λ. Question: Can we calculate w((r Λ) b ) for some Λ and for some b? Suffices to understand b = 0: Lemma For any Legendrian Λ, ( ) w (R Λ) b = e b w ( ) (R Λ) 0. Question: Can we calculate w((r Λ) 0 ) for some Λ? Answer: Yes! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 27 / 36
73 Width of Cylinder over Legendrian Unknot r 0 r Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 28 / 36
74 Width of Cylinder over Legendrian Unknot r 0 r Theorem (Sabloff-T) w((r U(r)) 0 ) = 2r. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 28 / 36
75 Upperbound to Width of a Legendrian w ( (R U(r)) 0 ) 2r follows from: Theorem (Sabloff-T) Suppose Λ is a Legendrian that admits an augmentation. Then ( ) w (R Λ) 0 2c(Λ), where c(λ) is the minimum fundamental capacity (for any augmentation). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 29 / 36
76 Upperbound to Width of a Legendrian w ( (R U(r)) 0 ) 2r follows from: Theorem (Sabloff-T) Suppose Λ is a Legendrian that admits an augmentation. Then ( ) w (R Λ) 0 2c(Λ), where c(λ) is the minimum fundamental capacity (for any augmentation). size of ball 2 fundamental Reeb chord height" in = Λ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 29 / 36
77 Obstructions to Embeddings: Proof Sketch: Suppose there is an embedding ψ of B(α). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 30 / 36
78 Obstructions to Embeddings: Proof Sketch: Suppose there is an embedding ψ of B(α). By property of the fundamental class λ LCH (Λ, ε), through ψ(0) L there is a J-holomorphic disk" of area A c(λ, ε, λ). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 30 / 36
79 Obstructions to Embeddings: Proof Sketch: Suppose there is an embedding ψ of B(α). By property of the fundamental class λ LCH (Λ, ε), through ψ(0) L there is a J-holomorphic disk" of area A c(λ, ε, λ). There exists a holomorphic disk in B(α) with boundary in B(α) R n of area B A c(λ, ε, λ). By analytic continuation, this extends to a holomorphic disk with boundary in B(α) of area 2B 2c(Λ, ε, λ). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 30 / 36
80 Obstructions to Embeddings: Proof Sketch: Suppose there is an embedding ψ of B(α). By property of the fundamental class λ LCH (Λ, ε), through ψ(0) L there is a J-holomorphic disk" of area A c(λ, ε, λ). There exists a holomorphic disk in B(α) with boundary in B(α) R n of area B A c(λ, ε, λ). By analytic continuation, this extends to a holomorphic disk with boundary in B(α) of area 2B 2c(Λ, ε, λ). Classical Isoperimetric Inequality shows α 2B 2c(Λ, ε, λ). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 30 / 36
81 Lowerbound to width of a Legendrian 2r w ( (R U(r)) 0 ) follows from: Theorem (Sabloff-T) Suppose Λ has a vertically extendable" Reeb chord of height r. Then ( ) 2r w (R Λ) 0. Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 31 / 36
82 Lowerbound to width of a Legendrian 2r w ( (R U(r)) 0 ) follows from: Theorem (Sabloff-T) Suppose Λ has a vertically extendable" Reeb chord of height r. Then ( ) 2r w (R Λ) 0. Vertically Extendable No Yes Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 31 / 36
83 Constructing Embeddings Proof Sketch: Ψ : R J 1 M T R + T M (s, x, y, z) ((e s, z), (x, e s y)) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 32 / 36
84 Constructing Embeddings Proof Sketch: Ψ : R J 1 M T R + T M (s, x, y, z) ((e s, z), (x, e s y)) r r 0 1 x Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 32 / 36
85 Constructing Embeddings Proof Sketch: Ψ : R J 1 M T R + T M (s, x, y, z) ((e s, z), (x, e s y)) r r 0 1 x Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 32 / 36
86 Constructing Embeddings Proof Sketch: Ψ : R J 1 M T R + T M (s, x, y, z) ((e s, z), (x, e s y)) r r 0 1 x Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 32 / 36
87 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36
88 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Example: Question: Can we still find a symplectic embedding of B(2)? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36
89 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Example: Question: Can we still find a symplectic embedding of B(2)? Answer: Yes! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36
90 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Example: Question: Can we still find a symplectic embedding of B(2)? Answer: Yes! Question: Can we embed a bigger ball? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36
91 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Example: Question: Can we still find a symplectic embedding of B(2)? Answer: Yes! Question: Can we embed a bigger ball? Answer: No! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36
92 Width of Non-Cylindrical Lagrangian Cobordisms What if L is a non-cylindrical cobordism? Example: Question: Can we still find a symplectic embedding of B(2)? Answer: Yes! Question: Can we embed a bigger ball? Answer: No! Width does not see the negative end! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 33 / 36
93 Upper Bound for Width of Lagrangian Cobordisms Theorem (Sabloff-T) If L is a Lagrangian cobordism from Λ to Λ + and Λ is fillable, then ( ) w L 0 2c(Λ + ), where c(λ + ) is the minimum fundamental capacity (for any augmentation). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 34 / 36
94 Upper Bound for Width of Lagrangian Cobordisms Theorem (Sabloff-T) If L is a Lagrangian cobordism from Λ to Λ + and Λ is fillable, then ( ) w L 0 2c(Λ + ), where c(λ + ) is the minimum fundamental capacity (for any augmentation). Proof is similar in spirit to the proof when L = R Λ: Use Seidel Isomorphism to get the existence of a J-holomorphic disk through ψ(0) ψ(b(α)). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 34 / 36
95 Length-Width Connection Can reprove our earlier length result between Λ = U(2) and Λ + = U(1): h ln 2 h 1 + s 0 2 _ Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 35 / 36
96 Length-Width Connection Can reprove our earlier length result between Λ = U(2) and Λ + = U(1): h ln 2 h 1 + s 0 2 _ Corollary Suppose L is a Lagrangian cobordism from Λ = U(2) to Λ + = U(1) that is cylindrical outside [ h, 0]. Then ( ) ( ) 2 c(u(2)) ln 2 = ln = ln h. 1 c(u(1)) Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 35 / 36
97 Length-Width Connection Can reprove our earlier length result between Λ = U(2) and Λ + = U(1): h ln 2 h 1 + s 0 2 _ Corollary Suppose L is a Lagrangian cobordism from Λ = U(2) to Λ + = U(1) that is cylindrical outside [ h, 0]. Then ( ) ( ) 2 c(u(2)) ln 2 = ln = ln h. 1 c(u(1)) Proof: 2e h c(u(2))=e h w((r Λ ) 0 ) = w((r Λ ) h ) w(l0 ) 2c(U(1)). Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 35 / 36
98 Questions Much to be understood about length and width! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 36 / 36
99 Questions Much to be understood about length and width! Calculate widths of other Lagrangian cobordisms when Λ admits an augmentation/filling! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 36 / 36
100 Questions Much to be understood about length and width! Calculate widths of other Lagrangian cobordisms when Λ admits an augmentation/filling! Can we calculate the width or length of a Lagrangian cobordism when Λ does not admit an augmentation/filling? For example, when Λ is stabilized or loose? Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 36 / 36
101 Questions Much to be understood about length and width! Calculate widths of other Lagrangian cobordisms when Λ admits an augmentation/filling! Can we calculate the width or length of a Lagrangian cobordism when Λ does not admit an augmentation/filling? For example, when Λ is stabilized or loose? Thank you! Lisa Traynor (Bryn Mawr) Lagrangian Cobordisms Tech Topology 36 / 36
The Minimal Length of the Lagrangian Cobordism Between Legendrians
Bryn Mawr College Scholarship, Research, and Creative Wor at Bryn Mawr College Mathematics Faculty Research and Scholarship Mathematics 2016 The Minimal Length of the Lagrangian Cobordism Between Legendrians
More informationBasic Objects and Important Questions
THE FLEXIBILITY AND RIGIDITY OF LAGRANGIAN AND LEGENDRIAN SUBMANIFOLDS LISA TRAYNOR Abstract. These are informal notes to accompany the first week of graduate lectures at the IAS Women and Mathematics
More informationDuality for Legendrian contact homology
2351 2381 2351 arxiv version: fonts, pagination and layout may vary from GT published version Duality for Legendrian contact homology JOSHUA M SABLOFF The main result of this paper is that, off of a fundamental
More informationEmbedded contact homology and its applications
Embedded contact homology and its applications Michael Hutchings Department of Mathematics University of California, Berkeley International Congress of Mathematicians, Hyderabad August 26, 2010 arxiv:1003.3209
More informationarxiv: v1 [math.gt] 20 Dec 2017
SYMPLECTIC FILLINGS, CONTACT SURGERIES, AND LAGRANGIAN DISKS arxiv:1712.07287v1 [math.gt] 20 Dec 2017 JAMES CONWAY, JOHN B. ETNYRE, AND BÜLENT TOSUN ABSTRACT. This paper completely answers the question
More informationarxiv: v2 [math.sg] 3 Jun 2016
arxiv:1510.08838v2 [math.sg] 3 Jun 2016 THE STABLE MORSE NUMBER AS A LOWER BOUND FOR THE NUMBER OF REEB CHORDS GEORGIOS DIMITROGLOU RIZELL AND ROMAN GOLOVKO Abstract. Assume that we are given a closed
More informationApplications of embedded contact homology
Applications of embedded contact homology Michael Hutchings Department of Mathematics University of California, Berkeley conference in honor of Michael Freedman June 8, 2011 Michael Hutchings (UC Berkeley)
More informationKnot Contact Homology, Chern-Simons Theory, and Topological Strings
Knot Contact Homology, Chern-Simons Theory, and Topological Strings Tobias Ekholm Uppsala University and Institute Mittag-Leffler, Sweden Symplectic Topology, Oxford, Fri Oct 3, 2014 Plan Reports on joint
More informationTHE EXISTENCE PROBLEM
THE EXISTENCE PROBLEM Contact Geometry in High Dimensions Emmanuel Giroux CNRS ENS Lyon AIM May 21, 2012 Contact forms A contact form on a manifold V is a non-vanishing 1-form α whose differential dα at
More informationWe thank F. Laudenbach for pointing out a mistake in Lemma 9.29, and C. Wendl for detecting a large number errors throughout the book.
ERRATA TO THE BOOK FROM STEIN TO WEINSTEIN AND BACK K. CIELIEBAK AND Y. ELIASHBERG We thank F. Laudenbach for pointing out a mistake in Lemma 9.29, and C. Wendl for detecting a large number errors throughout
More informationCONORMAL BUNDLES, CONTACT HOMOLOGY, AND KNOT INVARIANTS
CONORMAL BUNDLES, CONTACT HOMOLOGY, AND KNOT INVARIANTS LENHARD NG Abstract. Blah. 1. Introduction String theory has provided a beautiful correspondence between enumerative geometry and knot invariants;
More informationNON-ISOTOPIC LEGENDRIAN SUBMANIFOLDS IN R 2n+1
NON-ISOTOPIC LEGENDRIAN SUBMANIFOLDS IN R 2n+1 TOBIAS EKHOLM, JOHN ETNYRE, AND MICHAEL SULLIVAN Abstract. In the standard contact (2n+1)-space when n > 1, we construct infinite families of pairwise non-legendrian
More informationA Note on V. I. Arnold s Chord Conjecture. Casim Abbas. 1 Introduction
IMRN International Mathematics Research Notices 1999, No. 4 A Note on V. I. Arnold s Chord Conjecture Casim Abbas 1 Introduction This paper makes a contribution to a conjecture of V. I. Arnold in contact
More informationarxiv: v2 [math.sg] 21 Nov 2018
GEOMETRIC CRITERIA FOR OVERTWISTEDNESS ROGER CASALS, EMMY MURPHY, AND FRANCISCO PRESAS arxiv:1503.06221v2 [math.sg] 21 Nov 2018 Abstract. In this article we establish efficient geometric criteria to decide
More informationarxiv: v4 [math.sg] 28 Feb 2016
SUBCRITICAL CONTACT SURGERIES AND THE TOPOLOGY OF SYMPLECTIC FILLINGS arxiv:1408.1051v4 [math.sg] 28 Feb 2016 PAOLO GHIGGINI, KLAUS NIEDERKRÜGER, AND CHRIS WENDL Abstract. By a result of Eliashberg, every
More informationLEGENDRIAN CONTACT HOMOLOGY IN THE BOUNDARY OF A SUBCRITICAL WEINSTEIN 4-MANIFOLD
LEGENDRIAN CONTACT HOMOLOGY IN THE BOUNDARY OF A SUBCRITICAL WEINSTEIN 4-MANIFOLD TOBIAS EKHOLM AND LENHARD NG Abstract. We give a combinatorial description of the Legendrian contact homology algebra associated
More informationKnots and Mirror Symmetry. Mina Aganagic UC Berkeley
Knots and Mirror Symmetry Mina Aganagic UC Berkeley 1 Quantum physics has played a central role in answering the basic question in knot theory: When are two knots distinct? 2 Witten explained in 88, that
More information1. Introduction M 3 - oriented, compact 3-manifold. ο ρ TM - oriented, positive contact structure. ο = ker(ff), ff 1-form, ff ^ dff > 0. Locally, ever
On the Finiteness of Tight Contact Structures Ko Honda May 30, 2001 Univ. of Georgia, IHES, and USC Joint work with Vincent Colin and Emmanuel Giroux 1 1. Introduction M 3 - oriented, compact 3-manifold.
More informationSymplectic folding and non-isotopic polydisks
Symplectic folding and non-isotopic polydisks R. Hind October 10, 2012 1 Introduction 1.1 Main results Our main result demonstrates the existence of different Hamiltonian isotopy classes of symplectically
More informationarxiv: v2 [math.sg] 27 Jul 2018
GEOMETRIC GENERATION OF THE WRAPPED FUKAYA CATEGORY OF WEINSTEIN MANIFOLDS AND SECTORS arxiv:1712.09126v2 [math.sg] 27 Jul 2018 BAPTISTE CHANTRAINE, GEORGIOS DIMITROGLOU RIZELL, PAOLO GHIGGINI, AND ROMAN
More informationKnotted Legendrian surfaces with few Reeb chords
U.U.D.M. Report 2010:20 Knotted Legendrian surfaces with few Reeb chords Georgios Dimitroglou Rizell Filosofie licentiatavhandling i matematik som framläggs för offentlig granskning den 26 januari, kl
More informationArea preserving isotopies of self transverse immersions of S 1 in R 2
U.U.D.M. Project Report 2010:17 Area preserving isotopies of self transverse immersions of S 1 in R 2 Cecilia Karlsson Examensarbete i matematik, 30 hp Handledare och examinator: Tobias Ekholm December
More informationarxiv:math/ v3 [math.sg] 20 Sep 2004
arxiv:math/0404267v3 [math.sg] 20 Sep 2004 PLANAR OPEN BOOK DECOMPOSITIONS AND CONTACT STRUCTURES JOHN B. ETNYRE Abstract. In this note we observe that while all overtwisted contact structures on compact
More informationOn growth rate and contact homology
On growth rate and contact homology Anne Vaugon To cite this version: Anne Vaugon. On growth rate and contact homology. Algebraic and Geometric Topology, Mathematical Sciences Publishers, 2015, 15 (2),
More informationConvex Symplectic Manifolds
Convex Symplectic Manifolds Jie Min April 16, 2017 Abstract This note for my talk in student symplectic topology seminar in UMN is an gentle introduction to the concept of convex symplectic manifolds.
More informationLectures on the topology of symplectic fillings of contact 3-manifolds
Lectures on the topology of symplectic fillings of contact 3-manifolds BURAK OZBAGCI (Last updated on July 11, 2016) These are some lectures notes for my course in the Summer School "Mapping class groups,
More informationThe Geometry of Symplectic Energy
arxiv:math/9306216v1 [math.dg] 12 Jun 1993 The Geometry of Symplectic Energy Introduction François Lalonde UQAM, Montréal June 23, 2011 Dusa McDuff SUNY, Stony Brook One of the most striking early results
More informationEMBEDDING ALL CONTACT 3 MANIFOLDS IN A FIXED CONTACT 5 MANIFOLD
EMBEDDING ALL CONTACT 3 MANIFOLDS IN A FIXED CONTACT 5 MANIFOLD JOHN B. ETNYRE AND YANKI LEKILI ABSTRACT. In this note we observe that one can contact embed all contact 3 manifolds into a Stein fillable
More informationarxiv: v1 [math.sg] 21 Oct 2016
LEGENDRIAN FRONTS FOR AFFINE VARIETIES ROGER CASALS AND EMMY MURPHY arxiv:1610.06977v1 [math.sg] 21 Oct 2016 Abstract. In this article we study Weinstein structures endowed with a Lefschetz fibration in
More informationCONTACT SURGERY AND SYMPLECTIC CAPS
CONTACT SURGERY AND SYMPLECTIC CAPS JAMES CONWAY AND JOHN B. ETNYRE Abstract. In this note we show that a closed oriented contact manifold is obtained from the standard contact sphere of the same dimension
More informationAn integral lift of contact homology
Columbia University University of Pennsylvannia, January 2017 Classical mechanics The phase space R 2n of a system consists of the position and momentum of a particle. Lagrange: The equations of motion
More informationScalar curvature and the Thurston norm
Scalar curvature and the Thurston norm P. B. Kronheimer 1 andt.s.mrowka 2 Harvard University, CAMBRIDGE MA 02138 Massachusetts Institute of Technology, CAMBRIDGE MA 02139 1. Introduction Let Y be a closed,
More informationMaking cobordisms symplectic
Making cobordisms symplectic arxiv:1504.06312v3 [math.sg] 7 Apr 2017 Yakov Eliashberg Stanford University Abstract Emmy Murphy MIT We establish an existence h-principle for symplectic cobordisms of dimension
More informationLECTURES ON THE TOPOLOGY OF SYMPLECTIC FILLINGS OF CONTACT 3-MANIFOLDS
LECTURES ON THE TOPOLOGY OF SYMPLECTIC FILLINGS OF CONTACT 3-MANIFOLDS BURAK OZBAGCI ABSTRACT. These are some lecture notes for my mini-course in the Graduate Workshop on 4-Manifolds, August 18-22, 2014
More information2 LEGENDRIAN KNOTS PROBLEM SESSION. (GEORGIA TOPOLOGY CONFERENCE, MAY 31, 2001.) behind the examples is the following: We are Whitehead doubling Legen
LEGENDRIAN KNOTS PROBLEM SESSION. (GEORGIA TOPOLOGY CONFERENCE, MAY 31, 2001.) Conducted by John Etnyre. For the most part, we restricted ourselves to problems about Legendrian knots in dimension three.
More informationSymplectic Geometry versus Riemannian Geometry
Symplectic Geometry versus Riemannian Geometry Inês Cruz, CMUP 2010/10/20 - seminar of PIUDM 1 necessarily very incomplete The aim of this talk is to give an overview of Symplectic Geometry (there is no
More informationTHE CARDINALITY OF THE AUGMENTATION CATEGORY OF A LEGENDRIAN LINK
THE CARDINALITY OF THE AUGMENTATION CATEGORY OF A LEGENDRIAN LINK LENHARD NG, DAN RUTHERFORD, VIVEK SHENDE, AND STEVEN SIVEK ABSTRACT. We introduce a notion of cardinality for the augmentation category
More informationSymplectic topology and algebraic geometry II: Lagrangian submanifolds
Symplectic topology and algebraic geometry II: Lagrangian submanifolds Jonathan Evans UCL 26th January 2013 Jonathan Evans (UCL) Lagrangian submanifolds 26th January 2013 1 / 35 Addendum to last talk:
More information1 Motivation and Background
1 Motivation and Background Figure 1: Periodic orbits of the Reeb vector field on S 3 parametrized by S 2, aka the Hopf fibration (credit: Niles Johnson). My research is in symplectic and contact geometry
More informationON KNOTS IN OVERTWISTED CONTACT STRUCTURES
ON KNOTS IN OVERTWISTED CONTACT STRUCTURES JOHN B. ETNYRE ABSTRACT. We prove that each overtwisted contact structure has knot types that are represented by infinitely many distinct transverse knots all
More informationExistence of Engel structures. Thomas Vogel
Existence of Engel structures Thomas Vogel Existence of Engel structures Dissertation zur Erlangung des Doktorgrades an der Fakultät für Mathematik, Informatik und Statistik der Ludwig Maximilians Universität
More information2 JOHN B. ETNYRE AND KO HONDA (M;ο) (= one satisfying ; ffi (M;ο)) cannot be symplectically cobordant to an overtwisted contact structure, but the opp
ON SYMPLECTIC COBORDISMS JOHN B. ETNYRE AND KO HONDA Abstract. In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely
More informationInvertible Dirac operators and handle attachments
Invertible Dirac operators and handle attachments Nadine Große Universität Leipzig (joint work with Mattias Dahl) Rauischholzhausen, 03.07.2012 Motivation Not every closed manifold admits a metric of positive
More informationAUGMENTATIONS ARE SHEAVES
AUGMENTATIONS ARE SHEAVES LENHARD NG, DAN RUTHERFORD, VIVEK SHENDE, STEVEN SIVEK, AND ERIC ZASLOW ABSTRACT. We show that the set of augmentations of the Cheanov Eliashberg algebra of a Legendrian lin underlies
More informationLECTURE 15: COMPLETENESS AND CONVEXITY
LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other
More informationJ-holomorphic curves in symplectic geometry
J-holomorphic curves in symplectic geometry Janko Latschev Pleinfeld, September 25 28, 2006 Since their introduction by Gromov [4] in the mid-1980 s J-holomorphic curves have been one of the most widely
More informationSymplectic 4-manifolds, singular plane curves, and isotopy problems
Symplectic 4-manifolds, singular plane curves, and isotopy problems Denis AUROUX Massachusetts Inst. of Technology and Ecole Polytechnique Symplectic manifolds A symplectic structure on a smooth manifold
More informationHIGHER GENUS KNOT CONTACT HOMOLOGY AND RECURSION FOR COLORED HOMFLY-PT POLYNOMIALS
HIGHER GENUS KNOT CONTACT HOMOLOGY AND RECURSION FOR COLORED HOMFLY-PT POLYNOMIALS TOBIAS EKHOLM AND LENHARD NG Abstract. We sketch a construction of Legendrian Symplectic Field Theory (SFT) for conormal
More informationarxiv: v4 [math.gt] 14 Oct 2013
OPEN BOOKS AND EXACT SYMPLECTIC COBORDISMS MIRKO KLUKAS arxiv:1207.5647v4 [math.gt] 14 Oct 2013 Abstract. Given two open books with equal pages we show the existence of an exact symplectic cobordism whose
More informationThe A-B slice problem
The A-B slice problem Slava Krushkal June 10, 2011 History and motivation: Geometric classification tools in higher dimensions: Surgery: Given an n dimensional Poincaré complex X, is there an n manifold
More informationA MORSE-BOTT APPROACH TO CONTACT HOMOLOGY
A MORSE-BOTT APPROACH TO CONTACT HOMOLOGY a dissertation submitted to the department of mathematics and the committee on graduate studies of stanford university in partial fulfillment of the requirements
More informationThe Minimal Element Theorem
The Minimal Element Theorem The CMC Dynamics Theorem deals with describing all of the periodic or repeated geometric behavior of a properly embedded CMC surface with bounded second fundamental form in
More informationCombinatorial Heegaard Floer Theory
Combinatorial Heegaard Floer Theory Ciprian Manolescu UCLA February 29, 2012 Ciprian Manolescu (UCLA) Combinatorial HF Theory February 29, 2012 1 / 31 Motivation (4D) Much insight into smooth 4-manifolds
More informationNON-ISOTOPIC LEGENDRIAN SUBMANIFOLDS IN R 2n+1. Tobias Ekholm, John Etnyre & Michael Sullivan. Abstract
j. differential geometry 71 (2005) 85-128 NON-ISOTOPIC LEGENDRIAN SUBMANIFOLDS IN R 2n+1 Tobias Ekholm, John Etnyre & Michael Sullivan Abstract In the standard contact (2n +1)-space when n>1, we construct
More informationCLASSIFICATION OF TIGHT CONTACT STRUCTURES ON SURGERIES ON THE FIGURE-EIGHT KNOT
CLASSIFICATION OF TIGHT CONTACT STRUCTURES ON SURGERIES ON THE FIGURE-EIGHT KNOT JAMES CONWAY AND HYUNKI MIN ABSTRACT. Two of the basic questions in contact topology are which manifolds admit tight contact
More information1 Motivation and Background
1 Motivation and Background Figure 1: Periodic orbits of the Reeb vector field on S 3 parametrized by S 2, aka the Hopf fibration (credit: Niles Johnson). My research is in symplectic and contact geometry
More informationLagrangian Intersection Floer Homology (sketch) Chris Gerig
Lagrangian Intersection Floer Homology (sketch) 9-15-2011 Chris Gerig Recall that a symplectic 2n-manifold (M, ω) is a smooth manifold with a closed nondegenerate 2- form, i.e. ω(x, y) = ω(y, x) and dω
More informationCOMPUTABILITY AND THE GROWTH RATE OF SYMPLECTIC HOMOLOGY
COMPUTABILITY AND THE GROWTH RATE OF SYMPLECTIC HOMOLOGY MARK MCLEAN arxiv:1109.4466v1 [math.sg] 21 Sep 2011 Abstract. For each n greater than 7 we explicitly construct a sequence of Stein manifolds diffeomorphic
More informationarxiv:math/ v1 [math.gt] 14 Dec 2004
arxiv:math/0412275v1 [math.gt] 14 Dec 2004 AN OPEN BOOK DECOMPOSITION COMPATIBLE WITH RATIONAL CONTACT SURGERY BURAK OZBAGCI Abstract. We construct an open book decomposition compatible with a contact
More informationLecture XI: The non-kähler world
Lecture XI: The non-kähler world Jonathan Evans 2nd December 2010 Jonathan Evans () Lecture XI: The non-kähler world 2nd December 2010 1 / 21 We ve spent most of the course so far discussing examples of
More informationContact Structures and Classifications of Legendrian and Transverse Knots
Contact Structures and Classifications of Legendrian and Transverse Knots by Laura Starkston lstarkst@fas.harvard.edu Advisor: Andrew Cotton-Clay Harvard University Department of Mathematics Cambridge,
More informationAPPLICATIONS OF HOFER S GEOMETRY TO HAMILTONIAN DYNAMICS
APPLICATIONS OF HOFER S GEOMETRY TO HAMILTONIAN DYNAMICS FELIX SCHLENK Abstract. We prove that for every subset A of a tame symplectic manifold (W, ω) the π 1 -sensitive Hofer Zehnder capacity of A is
More informationMath 225A: Differential Topology, Final Exam
Math 225A: Differential Topology, Final Exam Ian Coley December 9, 2013 The goal is the following theorem. Theorem (Hopf). Let M be a compact n-manifold without boundary, and let f, g : M S n be two smooth
More informationLecture III: Neighbourhoods
Lecture III: Neighbourhoods Jonathan Evans 7th October 2010 Jonathan Evans () Lecture III: Neighbourhoods 7th October 2010 1 / 18 Jonathan Evans () Lecture III: Neighbourhoods 7th October 2010 2 / 18 In
More informationTransversal torus knots
ISSN 1364-0380 (on line) 1465-3060 (printed) 253 Geometry & Topology Volume 3 (1999) 253 268 Published: 5 September 1999 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Transversal torus knots
More informationAIM Workshop - Contact topology in higher dimensions - (May 21-25, 2012): Questions and open problems
AIM Workshop - Contact topology in higher dimensions - (May 21-25, 2012): Questions and open problems Warning The following notes, written by Sylvain Courte with the help of Patrick Massot, describe the
More information1 Contact structures and Reeb vector fields
1 Contact structures and Reeb vector fields Definition 1.1. Let M be a smooth (2n + 1)-dimensional manifold. A contact structure on M is a maximally non-integrable hyperplane field ξ T M. Suppose that
More informationA SHORT GALLERY OF CHARACTERISTIC FOLIATIONS
A SHORT GALLERY OF CHARACTERISTIC FOLIATIONS AUSTIN CHRISTIAN 1. Introduction The purpose of this note is to visualize some simple contact structures via their characteristic foliations. The emphasis is
More informationLECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in
More informationTIGHT PLANAR CONTACT MANIFOLDS WITH VANISHING HEEGAARD FLOER CONTACT INVARIANTS
TIGHT PLANAR CONTACT MANIFOLDS WITH VANISHING HEEGAARD FLOER CONTACT INVARIANTS JAMES CONWAY, AMEY KALOTI, AND DHEERAJ KULKARNI Abstract. In this note, we exhibit infinite families of tight non-fillable
More informationarxiv: v1 [math.gt] 5 Aug 2015
HEEGAARD FLOER CORRECTION TERMS OF (+1)-SURGERIES ALONG (2, q)-cablings arxiv:1508.01138v1 [math.gt] 5 Aug 2015 KOUKI SATO Abstract. The Heegaard Floer correction term (d-invariant) is an invariant of
More informationarxiv: v4 [math.sg] 30 Nov 2010
WEAK SYMPLECTIC FILLINGS AND HOLOMORPHIC CURVES KLAUS NIEDERKRÜGER AND CHRIS WENDL arxiv:1003.3923v4 [math.sg] 30 Nov 2010 Abstract. English: We prove several results on weak symplectic fillings of contact
More informationarxiv: v2 [math.gt] 19 Oct 2014
VIRTUAL LEGENDRIAN ISOTOPY VLADIMIR CHERNOV AND RUSTAM SADYKOV arxiv:1406.0875v2 [math.gt] 19 Oct 2014 Abstract. An elementary stabilization of a Legendrian knot L in the spherical cotangent bundle ST
More informationB B. B g ... g 1 g ... c c. g b
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000 000 S 0002-9939(XX)0000-0 CONTACT 3-MANIFOLDS WITH INFINITELY MANY STEIN FILLINGS BURAK OZBAGCI AND ANDRÁS I.
More informationLegendrian Approximations
U.U.D.M. Project Report 2014:22 Legendrian Approximations Dan Strängberg Examensarbete i matematik, 30 hp Handledare och examinator: Tobias Ekholm Juni 2014 Department of Mathematics Uppsala University
More informationGeometry of generating function and Lagrangian spectral invariants
Geometry of generating function and Lagrangian spectral invariants Yong-Geun Oh IBS Center for Geometry and Physics & POSTECH & University of Wisconsin-Madsion Paris, Alanfest, July 18, 2012 Yong-Geun
More informationStein fillings of contact 3-manifolds obtained as Legendrian sur
Stein fillings of contact 3-manifolds obtained as Legendrian surgeries Youlin Li (With Amey Kaloti) Shanghai Jiao Tong University A Satellite Conference of Seoul ICM 2014 Knots and Low Dimensional Manifolds
More informationReeb dynamics in dimension 3
Reeb dynamics in dimension 3 Dan Cristofaro-Gardiner Member, Institute for Advanced study Institute for Advanced Study September 24, 2013 Thank you! Thank you for inviting me to give this talk! Reeb vector
More informationContact Geometry of the Visual Cortex
Ma191b Winter 2017 Geometry of Neuroscience References for this lecture Jean Petitot, Neurogéométrie de la Vision, Les Éditions de l École Polytechnique, 2008 John B. Entyre, Introductory Lectures on Contact
More informationKNOT AND BRAID INVARIANTS FROM CONTACT HOMOLOGY I
KNOT AND BRAID INVARIANTS FROM CONTACT HOMOLOGY I LENHARD NG Abstract. We introduce topological invariants of knots and braid conjugacy classes, in the form of differential graded algebras, and present
More informationSqueezing Lagrangian tori in dimension 4. R. Hind E. Opshtein. January 9, 2019
Squeezing Lagrangian tori in dimension 4 R. Hind E. Opshtein January 9, 2019 arxiv:1901.02124v1 [math.sg] 8 Jan 2019 Abstract Let C 2 be the standard symplectic vector space and L(a, b) C 2 be the product
More informationarxiv: v1 [math.sg] 11 Mar 2013
SYMPLECTIC RATIONAL BLOW-UP TATYANA KHODOROVSKIY arxiv:1303.581v1 [math.sg] 11 Mar 013 Abstract. Fintushel and Stern defined the rational blow-down construction [FS] for smooth 4-manifolds, where a linear
More informationLEGENDRIAN CONTACT HOMOLOGY IN P R
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 7, July 2007, Pages 3301 3335 S 0002-9947(07)04337-1 Article electronically published on January 26, 2007 LEGENDRIAN CONTACT HOMOLOGY
More informationLEGENDRIAN SOLID-TORUS LINKS
LEGENDRIAN SOLID-TORUS LINKS LENHARD NG AND LISA TRAYNOR Abstract. Differential graded algebra invariants are constructed for Legendrian links in the 1-jet space of the circle. In parallel to the theory
More informationCOBORDISM OF DISK KNOTS
ISRAEL JOURNAL OF MATHEMATICS 163 (2008), 139 188 DOI: 10.1007/s11856-008-0008-3 COBORDISM OF DISK KNOTS BY Greg Friedman Department of Mathematics, Texas Christian University 298900 Fort Worth, TX 76129
More informationExistence of Engel structures
Annals of Mathematics, 169 (2009), 79 137 Existence of Engel structures By Thomas Vogel Abstract We develop a construction of Engel structures on 4-manifolds based on decompositions of manifolds into round
More informationLiouville and Weinstein manifolds
Liouville and Weinstein manifolds precourse for SFT 8, Berlin, August 6-7, 2016 Sylvain Courte Contents 1 Liouville and Weinstein manifolds 2 1.1 Symplectic convexity........................... 2 1.2 Liouville
More informationBRAIDS AND OPEN BOOK DECOMPOSITIONS. Elena Pavelescu. A Dissertation. Mathematics
BRAIDS AND OPEN BOOK DECOMPOSITIONS Elena Pavelescu A Dissertation in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree
More informationCONTACT GEOMETRY AND TOPOLOGY
CONTACT GEOMETRY AND TOPOLOGY D. MARTÍNEZ TORRES Two general remarks: (i) Why topology in the title?: Differential topology studies smooth manifolds. Manifolds of the same dimension look locally the same,
More informationTHE CONTACT HOMOLOGY OF LEGENDRIAN SUBMANIFOLDS IN R 2n+1
THE CONTACT HOMOLOGY OF LEGENDRIAN SUBMANIFOLDS IN R 2n+1 TOBIAS EKHOLM, JOHN ETNYRE, AND MICHAEL SULLIVAN Abstract. We define the contact homology for Legendrian submanifolds in standard contact (2n +
More informationAn Ozsváth-Szabó Floer homology invariant of knots in a contact manifold
An Ozsváth-Szabó Floer homology invariant of knots in a contact manifold Matthew Hedden Department of Mathematics, Massachusetts Institute of Technology Abstract Using the knot Floer homology filtration,
More informationLefschetz pencils and the symplectic topology of complex surfaces
Lefschetz pencils and the symplectic topology of complex surfaces Denis AUROUX Massachusetts Institute of Technology Symplectic 4-manifolds A (compact) symplectic 4-manifold (M 4, ω) is a smooth 4-manifold
More informationarxiv:math/ v1 [math.gt] 26 Sep 2005
EXPLICIT HORIZONTAL OPEN BOOKS ON SOME PLUMBINGS arxiv:math/0509611v1 [math.gt] 26 Sep 2005 TOLGA ETGÜ AND BURAK OZBAGCI ABSTRACT. We describe explicit open books on arbitrary plumbings of oriented circle
More informationSCIENnIFIQUES d L ÉCOLE. SUPÉRIEUkE. Weak symplectic fillings and holomorphic curves. Klaus NIEDERKRÜGER & Chris WENDL SOCIÉTÉ MATHÉMATIQUE DE FRANCE
ISSN 0012-9593 ASENAH quatrième série - tome 44 fascicule 5 septembre-octobre 2011 annales SCIENnIFIQUES d L ÉCOLE hormale SUPÉRIEUkE Klaus NIEDERKRÜGER & Chris WENDL Weak symplectic fillings and holomorphic
More informationThe topology of symplectic four-manifolds
The topology of symplectic four-manifolds Michael Usher January 12, 2007 Definition A symplectic manifold is a pair (M, ω) where 1 M is a smooth manifold of some even dimension 2n. 2 ω Ω 2 (M) is a two-form
More informationBordism and the Pontryagin-Thom Theorem
Bordism and the Pontryagin-Thom Theorem Richard Wong Differential Topology Term Paper December 2, 2016 1 Introduction Given the classification of low dimensional manifolds up to equivalence relations such
More informationCosmetic crossings and genus-one knots
Cosmetic crossings and genus-one knots Cheryl L. Balm Michigan State University Saturday, December 3, 2011 Joint work with S. Friedl, E. Kalfagianni and M. Powell Crossing changes Notation Crossing disks
More informationBroken pencils and four-manifold invariants. Tim Perutz (Cambridge)
Broken pencils and four-manifold invariants Tim Perutz (Cambridge) Aim This talk is about a project to construct and study a symplectic substitute for gauge theory in 2, 3 and 4 dimensions. The 3- and
More informationarxiv: v1 [math.sg] 27 Nov 2017
On some examples and constructions of contact manifolds Fabio Gironella arxiv:1711.09800v1 [math.sg] 27 Nov 2017 Abstract The first goal of this paper is to construct examples of higher dimensional contact
More informationOn Mahler s conjecture a symplectic aspect
Dec 13, 2017 Differential Geometry and Differential Equations On Mahler s conjecture a symplectic aspect Hiroshi Iriyeh (Ibaraki University) Contents 1. Convex body and its polar 2. Mahler s conjecture
More information