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Lagrangian Floer Homology Lagrangian Floer Homology John Gabriel P. Pelias Introduction Action Functional Holomorphic Strips Lagrangian Floer Homology The Floer Complex Applications John Gabriel P. Pelias Department of Mathematics University of California, Santa Cruz Motivation Lagrangian Floer Homology John Gabriel P. Motivation: Floer's proof of an Arnold-type Conjecture: Pelias Introduction Action Functional Theorem (Floer) Holomorphic Strips Let (M; !) be a closed symplectic manifold, L a compact Lagrangian The Floer Complex submanifold of M, and a Hamiltonian diffeomorphism of (M; !). Applications Assume that the symplectic area of any topological disc in M with boundary in L vanishes. Assume moreover that (L) and L intersect transversely. Then the number of intersection points of L and (L) satisfies the lower bound X i #( (L) \ L) ≥ dim H (L; Z=2): i Floer's Approach Lagrangian Floer Homology John Gabriel P. Idea: Given a tranversely intersecting pair of Lagrangian Pelias submanifolds L0;L1, construct a chain complex CF (L0;L1) freely Introduction generated by the intersection points 2 L0 \ L1, satisfying: Action Functional 2 Holomorphic @ = 0, so that the (Lagrangian) Floer (co)homology Strips The Floer Complex HF (L0;L1) := ker @= im @ Applications is well-defined 0 if L1 and L1 are Hamiltonian isotopic, then 0 HF (L0;L1) ' HF (L0;L1) if L1 is Hamiltonian isotopic to L0, then (with suitable coefficients) ∗ HF (L0;L1) ' H (L0) Smooth Trajectories Between Two Lagrangians Lagrangian Floer Homology John Gabriel P. Pelias Let L0;L1 be compact Lagrangian submanifolds of symplectic Introduction manifold (M; !) that intersect transversely (and hence at finitely Action Functional many points). Holomorphic Strips The Floer The space of smooth trajectories from L to L , endowed with Complex 0 1 1 Applications the C -topology: P(L0;L1) := fγ : [0; 1] ! M j γ smooth, γ(0) 2 L0; γ(1) 2 L1g constant paths 2 P(L0;L1) ! intersection points 2 L0 \ L1 For a fixed γ^ 2 P(L0;L1), let P(^γ) denote the connected component of P(L0;L1) containing γ^. Action Functional Lagrangian Floer Homology John Gabriel P. The universal cover of P(^γ) is Pelias Introduction P](^γ)= f(γ; [Γ]) : γ 2 P(^γ); Γ: smooth homotopy from γ^ to γg Action Functional Holomorphic Strips (i.e. Γ : [0; 1] × [0; 1] ! M smooth with Γ(s; ·) 2 P(^γ) for all The Floer s 2 [0; 1], Γ(0; ·) =γ ^, and Γ(1; ·) = γ.) Complex Applications The action functional A is defined on P](^γ) by Z A(γ; [Γ]) := − ! Γ Z = − Γ∗! [0;1]×[0;1] Z 1 Z 1 @Γ @Γ = − ! ; ds ^ dt 0 0 @s @t Action Functional Lagrangian Floer Homology WARNING: Actually, I lied: A is not well-defined! John Gabriel P. Pelias Introduction We impose the following assumption (which will also avoid other Action Functional complications later): Holomorphic Strips The Floer Complex Assume that for each i = 0; 1, Applications [!] · π2(M; Li) = 0; i.e. any topological disk in M with boundary in Li has vanishing symplectic area. Note that, in particular, this implies that M is symplectically aspherical, i.e. [!] · π2(M) = 0: Action Functional Lagrangian Floer Homology John Gabriel P. 0 Pelias To see that A is well-defined: Let γ 2 P](^γ) and Γ, Γ be homotopic Introduction smooth homotopies from γ^ to γ. Action Functional We have the cylinder Γ#Γ0 : S1 × [0; 1] ! M, where Γ#Γ0(s; i) Holomorphic Strips is a loop in Li, for i = 0; 1. The Floer 0 0 Complex Γ; Γ are homotopic ) Γ(s; i); Γ (s; i) are homotopic in Li ) Applications there exist topological disks whose boundaries are 0 Γ#Γ (s; i) ⊂ Li Li Lagrangian ) these disks have symplectic area 0 symplectic asphericity ) symplectic area of the cylinder is also 0: Z Z Z 0 = ! = ! − ! Γ#Γ0 Γ Γ0 Riemannian Metric Lagrangian Floer Homology To do Morse theory, we need a gradient flow, and hence we need a John Gabriel P. Pelias Riemannian metric on P](^γ). Introduction Recall: Let J 2 End(TM) be an almost complex structure on Action Functional (M; !). Holomorphic Strips J is !-compatible if The Floer Complex Applications !(v; Jv) > 0; 8v 2 TxM; 8x 2 M; and !(J·;J·) = !(·; ·): If J is !-compatible, then it induces a Riemannian metric on M: gJ (·; ·) := !(·;J·) The space J (M; !) of !-compatible almost complex structures on M is non-empty and contractible. Riemannian Metric Lagrangian Floer Homology Now, let (γ; [Γ]) 2 P](^γ). John Gabriel P. Pelias The tangent space to P](^γ) at (γ; [Γ]) is the space of vector Introduction fields ξ along γ with initial and final vectors in the tangent Action Functional Holomorphic spaces to L0 and L1, respectively, i.e. Strips The Floer ∗ Complex T(γ;[Γ])P](^γ) = fξ 2 Γ(γ (TM)) : ξ(t) 2 Tγ(t)M ; ξ(i) 2 Tγ(i)Lig Applications Choose a smooth family J = fJtg0≤t≤1 ⊂ J (M; !), which gives rise to a smooth family of Riemannian metrics fgtg0≤t≤1. We define a Riemannian metric on P](^γ) as follows: If ξ; η 2 T(γ;[Γ])P](^γ), put Z 1 hξ; ηi := gt(ξ(t); η(t))dt: 0 Gradient Lagrangian Floer Homology We are now able to make sense of the gradient flow. Let John Gabriel P. Pelias (γ; [Γ]) 2 P](^γ) and ξ 2 T(γ;[Γ])P](^γ). Let fγη(t)g1−0≤η≤1+0 be a Introduction smooth homotopy such that γ1 = γ and Action Functional Holomorphic @ Strips γη(t) = ξ(t); @η The Floer η=1 Complex Applications and let Γ : [0; 1 + 0] × [0; 1] ! M be a smooth homotopy for which Γ(0; t) =γ ^(t) Γ(1; t) = γ(t) Γ(η; t) = γη(t); 8η 2 [1 − 0; 1 + 0]: For each η, we have the homotopy Γη : [0; 1] × [0; 1] ! M from γ^ to γη defined by Γη(s; t) = Γ(ηs; t) Gradient Lagrangian Floer Homology John Gabriel P. Pelias @ dA(γ;[Γ])(ξ) = A(γη; [Γη]) Introduction @η η=1 Action Functional Z @ Holomorphic = − ! Strips @η η=1 Γη The Floer Complex Z 1 Z 1 @ @Γη @Γη Applications = − ! ; ds ^ dt @η η=1 0 0 @s @t Z 1 Z η @ @Γ @Γ = − ! ; d~s ^ dt @η η=1 0 0 @s~ @t Z 1 @γ = − ! ξ(t); dt 0 @t Z 1 @γ = ! ; ξ(t) dt 0 @t Gradient Lagrangian Floer Homology With the family of !-compatible almost complex structures Jt, we John Gabriel P. Pelias then have Introduction Action Functional Z 1 @γ Holomorphic Strips dA(γ;[Γ])(ξ) = ! Jt ;Jtξ(t) dt 0 @t The Floer 1 Complex Z @γ Applications = gt Jt ; ξ(t) dt 0 @t @γ = J ; ξ t @t That is, with respect to the Riemannian metric we defined, @γ grad A(γ; [Γ]) = J t @t Critical Points Lagrangian Floer Homology John Gabriel P. Pelias @γ grad A(γ; [Γ]) = Jt Introduction @t Action Functional Holomorphic Strips The Floer Since Jt is an automorphism of TM for each t, Complex @γ grad A(γ; [Γ]) = 0 if and only if ≡ 0, i.e. γ 2 P(L ;L ) is a Applications @t 0 1 constant trajectory. That is: The critical points of A are the intersection points 2 L0 \ L1. Gradient Flowlines Lagrangian Floer Homology John Gabriel P. @γ Pelias grad A(γ; [Γ]) = J t @t Introduction Action Functional Holomorphic If p; q 2 L0 \ L1, a flowline of − grad A from p to q is a smooth Strips R The Floer function u~ : ! P](^γ) satisfying Complex d~u Applications = − grad A; lim u~(s) = p; and lim u~(s) = q: ds s→−∞ s!+1 Equivalently, we have a strip u : R × [0; 1] ! M satisfying: @u @u + J = 0; @s t @t lim u(s; t) = p; s→−∞ lim u(s; t) = q: s!+1 J-Holomorphic Strips Lagrangian Floer Homology John Gabriel P. @u @u Pelias + J = 0; @s t @t Introduction lim u(s; t) = p; Action Functional s→−∞ Holomorphic lim u(s; t) = q: Strips s!+1 The Floer Complex Note that the first equation above is the Cauchy-Riemann Applications equation @J (u) = 0 for u, with respect to the family J = fJtg of almost complex structures. A map u satisfying @J (u) = 0 is called a J-holomorphic strip in (M; !). That is, a connecting gradient flowline is a J-holomorphic strip. Riemann Mapping Theorem ) R × [0; 1] is biholomorphic to D2 − {±1g, and hence a J-holomorphic strip can also be regarded as a closed Whitney disk with two points removed (corresponding to the two critical points p; q) Finite Energy Lagrangian Floer Homology John Gabriel P. Pelias Introduction If M is not compact, we need to impose an extra condition: The Action Functional J-holomorphic strips u : R × [0; 1] ! M must have finite Holomorphic energy, i.e. Strips The Floer 1 1 2 Complex Z Z Z @u E(u) := ! = dsdt < 1: Applications u 0 0 @s If M is compact, the limiting conditions lim u(s; t) = p; lim u(s; t) = q s→−∞ s!+1 turn out to be equivalent to the finite energy condition. Moduli Space of J-Holomorphic Strips Lagrangian Floer Homology John Gabriel P. Pelias Mc(p; q; J) denotes the space of all smooth J-holomorphic strips Introduction u : R × [0; 1] ! M for which lims→−∞ u(s; t) = p, Action Functional lims!+1 u(s; t) = q, u(s; ·) 2 P(L0;L1), and E(u) < 1. Holomorphic Strips M(p; q; J) denotes the quotient of M(p; q; J) by the action of The Floer c Complex R by reparametrization, i.e.
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