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4.2. Proposition. For a generic choice of gΘ , H1, H2, the space M Θ Φ (γ, x) - if non-empty - is a manifold of dimension These manifolds carry coherent orientations. dimM Θ Φ (γ, x) = m Θ (γ) − µ Θ (x). The inequality (49) implies the energy estimate which can be used to prove C∞ loc compactness of the space M Θ Φ (γ, x). When µΘ (x) = mΘ (γ), the space M Θ Φ (γ, x) consists of finitely many oriented points, which define an integer nΘ Φ (γ, x). These integers are the coefficients of a homomorphism Φ Θ L1⊕L2 : Mk(ËΘ , g L1⊕L2 Θ ) → Fk(H1 ⊕ H2, J), γ ↦→ � n Θ Φ (γ, x)x. x∈P Θ (H1⊕H2) µ Θ (x)=k This is a chain complex isomorphism. See [APS08] for the construction of this isomorphism for arbitrary non-local conormal boundary conditions. 4.2 The Ω ring isomorphism Let L1, L2 ∈ C ∞ ([0, 1] × TM) be two Lagrangians such that L1(1, ·) = L2(0, ·) with all the time derivatives, and such that L1 and L2 satisfy (L0) Ω , (L1), (L2), and (24). Assume also that the Lagrangian L1#L2 defined by (23) satisfies (L0) Ω . Let H1 and H2 be the Fenchel transforms of L1 and L2, so that H1#H2 is the Fenchel transform of L1#L2, and the three Hamiltonians H1, H2, and H1#H2 satisfy (H0) Ω , (H1), (H2). In section 2.6 we have shown how the Pontrjagin product can be expressed in a Morse theoretical way. In other words, we have constructed a homomorphism M# : Mh(ËL1, g1) ⊗ Mj(ËL2, g2) −→ Mh+j(ËL1#L2, g) such that the upper square in the following diagram commutes Hh(Ω(M, q0)) ⊗ Hj(Ω(M, q0)) ∼= �� HMh(ËΩ , g1) ⊗ HMj(ËΩ , g2) L1 L2 HΦ Ω L 1 ⊗HΦ Ω L 2 �� HF Ω h (H1, J) ⊗ HF Ω j (H2, J) # HM# HΥ Ω �� Hh+j(Ω(M, q0)) �� ∼= HMh+j(ËΩ L1#L2 , g) HΦ Ω L 1 #L 2 �� HF Ω h+j (H1#H2, J) The aim of this section is to show that also the lower square commutes. Actually, we will show more, namely that the diagram � M(ËΩ L1 , g1) ⊗ M(ËΩ L2 , g2) � Φ Ω L 1 ⊗Φ Ω L 2 � �� Ω F (H1, J) ⊗ F Ω (H2, J) � k k M# Υ Ω �� Mk(ËΩ L1#L2 , g) Φ Ω L 1 #L 2 �� F Ω k (H1#H2, J) is chain-homotopy commutative. Instead than constructing a direct homotopy between ΦΩ L1#L2 ◦ M# and ΥΩ ◦ ΦΩ L1 ⊗ ΦΩ , we shall prove that both chain maps are homotopic to a third one, that L2 we name KΩ , see Figure 4. The definition of KΩ is based on the following space of solutions of the Floer equation for the Hamiltonian H defined in (40): given γ1 ∈ PΩ (L1), γ2 ∈ PΩ (L2), and x ∈ PΩ (H1#H2), let M Ω K (γ1, γ2; x) be the space of solutions of the Floer equation ∂J,H(u) = 0, 40
α Υ Ω ◦ Φ Ω L1 ⊗ ΦΩ L2 with boundary conditions ≃ ≃ α ΦΩ ◦ M# L1#L2 Figure 4: The homotopy through K Ω . K Ω π ◦ u(s, −1) = π ◦ u(s, 1) = q0, ∀s ≥ 0, π ◦ u(0, · − 1) ∈ W u (γ1, −gradËΩ L1 ), π ◦ u(0, ·) ∈ W u (γ2, −gradËΩ L2 ), and the asymptotic behavior lim u(s, 2t − 1) = x(t), s→+∞ uniformly in t ∈ [0, 1]. Theorem 3.2 in [AS06b] (or the arguments of section 5.10) implies that for a generic choice of g1, g2, H1, and H2, M Ω K (γ1, γ2; x) - if non-empty - is a smooth manifold of dimension dimM Ω K (γ1, γ2; x) = m Ω (γ1; L1) + m Ω (γ2; L2) − µ Ω (x; H1#H2). These manifolds carry coherent orientations. The energy identity is now � |∂su(s, t)| 2 ds dt =�H1(x1) +�H2(x2) −�H1#H2(x), ]0,+∞[×]−1,1[ where x1(t) = u(0, t − 1) and x2(t) = u(0, t). Since π ◦ x1 is in the unstable manifold of γ1 and π ◦ x2 is in the unstable manifold of γ2, the inequality (49) implies that �H1(x1) ≤ËL1(γ1),�H2(x2) ≤ËL2(γ2), so the elements u of M Ω K (γ1, γ2; x) satisfy the energy estimate � |∂su(s, t)| 2 ds dt ≤ËL1(γ1) +ËL2(γ2) −�H1#H2(x). (52) ]0,+∞[×]−1,1[ When mΩ (γ1; L1) + mΩ (γ2; L2) = µ Ω (x; H1 ⊕ H2), the space M Ω K (γ1, γ2; x) is a compact zerodimensional oriented manifold. If nΩ K (γ1, γ2; x) is the algebraic sum of its points, we can define the homomorphism K Ω : � M(ËΩ , g1) ⊗ M(ËΩ , g2) L1 L2 � k → F Ω k (H1#H2, J), γ1 ⊗ γ2 ↦→ � n Ω K (γ1, γ2; x)x. 41 x∈P Ω (H1#H2) µ Ω (x)=k
Page 1: Å�ÜÈÐ�Ò�ÁÒ×Ø�ØÙ
Page 4 and 5: 5.8 Non-local boundary conditions .
Page 6 and 7: M into Λ(M) as the space of consta
Page 8 and 9: and converging to Hamiltonian orbit
Page 10 and 11: 1.2 The Chas-Sullivan loop product
Page 12 and 13: 1.1. Remark. The loop product was d
Page 14 and 15: 2.2 Functoriality Let (M1, g1) and
Page 16 and 17: The orbits of −grad(f1 ⊕f2) are
Page 18 and 19: this intersection is a finite set o
Page 20 and 21: (L0) Ω every solution γ ∈ P
Page 22 and 23: transverse to the stable manifold o
Page 24 and 25: 2.7. Proposition. The homomorphism
Page 26 and 27: A submanifold L ⊂ T ∗ M is call
Page 28 and 29: The L 2 -negative gradient equation
Page 30 and 31: The Riemann surface Σ Ω Υ s = 0
Page 32 and 33: In the periodic case, we consider H
Page 34 and 35: 3.6. Definition. The Maslov index
Page 36 and 37: The Riemann surface ΣG is obtained
Page 38 and 39: 3.12. Proposition. For a generic ch
Page 40 and 41: The following compactness statement
Page 44 and 45: A standard gluing argument shows th
Page 46 and 47: In the following two sections we wi
Page 48 and 49: We shall apply the above lemma to t
Page 50 and 51: Since the inclusion c : M ֒→ Λ
Page 52 and 53: 4.5 The right-hand square is homoto
Page 54 and 55: commute. Our first aim in this sect
Page 56 and 57: Moreover, �Dαn(βnvn)�0,p;Ê
Page 58 and 59: We identify the product T ∗Ên ×
Page 60 and 61: Up to multiplying u by U, we can re
Page 62 and 63: The symbol C∞ S ,c indicates boun
Page 64 and 65: 5.11. Theorem. (Cauchy-Riemann oper
Page 66 and 67: 5.15. Lemma. Let p > 1 and α ∈Ê
Page 68 and 69: Proof. By Proposition 5.14 the oper
Page 70 and 71: Then ind∂A = ind∂A1 + ind∂A1.
Page 72 and 73: (V0, . . . , Vk), by V ′ the (k
Page 74 and 75: is bounded and Fredholm of index in
Page 76 and 77: so the transversality hypotheses of
Page 78 and 79: 5.9 Coherent orientations As notice
Page 80 and 81: compact subsets of Σ Ω Υ , for
Page 82 and 83: where Ã(z) = CA(z)C ⊕ A(z), C de
Page 84 and 85: By the additivity property of the M
Page 86 and 87: The space M Υ GE . Let x1 ∈ P(H1
Page 88 and 89: The space M Θ Φ . Let γ ∈ PΘ
Page 90 and 91: The embedding Q ֒→ÊN induces an
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Given periodic orbits x1 ∈ P(H1),
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Since u and ∂u are integrable ove
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with non-local boundary conditions
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to ũ0 uniformly on Σ, we may loca
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we see that associating ũ to u pro
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The energy of a solution u ∈ M K
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Since we have applied a conformal r
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6.9. Lemma. There exists a positive
Page 108 and 109:
[ChS04] M. Chas and D. Sullivan, Cl
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