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4.5 The right-hand square is homotopy commutative In this section we prove that the chain maps ΦΛ L1#L2 ◦ MΓ and G ◦ ΦΘ are both homotopic to L1⊕L2 a third chain map, named KΘ . This fact implies that the right-hand square in the diagram (53) commutes up to chain homotopy. The chain map KΘ is defined by using the following spaces of solutions of the Floer equation on the half-cylinder for the Hamiltonian H1#H2: given γ ∈ PΘ (L1 ⊕ L2) and x ∈ PΛ (H1#H2), set M Θ � K (γ; x) := u ∈ C ∞ ([0, +∞[×Ì, T ∗ � � M) �∂J,H1#H2(u) = 0, π ◦ u(0, ·) ∈ Γ � W u (γ; −grad gΘËΘ L1⊕L2 ) � � , lim u(s, ·) = x uniformly in t . s→+∞ By Theorem 3.2 in [AS06b] (or by the arguments of section 5.10), the space M Θ K (γ; x) is a smooth manifold of dimension dimM Θ K(γ; x) = m Θ (γ) − µ Λ (x), for a generic choice of g Θ , H1, and H2. These manifolds carry coherent orientations. Compactness follows from the energy estimate � ]0,+∞[×Ì|∂su(s, t)| 2 dsdt ≤ËL1⊕L2(γ) −�H1#H2(x), implied by (49). Counting the elements of the zero-dimensional spaces, we define a homomorphism K Θ : Mj(ËΘ L1⊕L2 , g Θ ) → F Λ j (H1#H2, J), which is shown to be a chain map. It is easy to construct a chain homotopy P Γ K between ΦΛ L1#L2 ◦MΓ and K Θ by considering the space M Γ � K(γ; x) := (α, u) ∈]0, +∞[×C ∞ ([0, +∞[×Ì, T ∗ � � M) �∂J,H1#H2(u) = 0, φ Λ −α (π ◦ u(0, ·)) ∈ Γ�W u (γ; −grad gΘËΘ ) L1⊕L2 � � , lim u(s, ·) = x uniformly in t . s→+∞ where φ Λ s denotes the flow of −gradËΛ L1#L2 on Λ 1 (M). As before, we find that generically M Γ K (γ1, γ2; x) is a manifold of dimension dimM Γ K (γ; x) = mΛ (γ) − µ Θ (x) + 1. Compactness holds, so an algebraic count of the zero-dimensional spaces produces the homomorphism P Γ K : Mj(ËΘ L1⊕L2 , g Θ ) → F Λ j+1(H1#H2, J). A standard gluing argument shows that P Γ K is the required homotopy. Finally, the construction of the chain homotopy P K G between KΘ and G ◦ ΦΘ is based L1⊕L2 on the one-parameter family of Riemann surfaces ΣK G (α), α > 0, defined as the quotient of the disjoint union [0, +∞[×[−1, 0] ⊔ [0, +∞[×[0, 1] under the identifications (s, 0 − ) ∼ (s, 0 + ) and (s, −1) ∼ (s, 1) for s ≥ α. This object is a Riemann surface with boundary, the holomorphic structure at (α, 0) being given by the map {ζ ∈�|Re ζ ≥ 0, |ζ| < ǫ} → Σ K G (α), ζ ↦→ α + ζ2 , 50
and the holomorphic structure at (α, −1) ∼ (α, 1) being given by the map {ζ ∈�|Re ζ ≥ 0, |ζ| < ǫ} → Σ K � 2 α − i + ζ if Imζ ≥ 0, G(α), ζ ↦→ α + i + ζ2 if Imζ ≤ 0. Here ǫ is a positive number smaller than 1 and √ α. Given γ ∈ PΘ (L1 ⊕L2) and x ∈ PΛ (H1#H2), we consider the space M K G (γ, x) of pairs (α, u) where α is a positive number and u ∈ C∞ (ΣK G (α), T ∗M) solves the equation ∂J,H1#H2(u) = 0, satisfies the boundary conditions � π ◦ u(s, −1) = π ◦ u(s, 0 − ) = π ◦ u(s, 0 + ) = π ◦ u(s, 1), u(s, 0 − ) − u(s, −1) + u(s, 1) − u(s, 0 + ) = 0, and the asymptotic condition ∀s ∈ [0, α], (π ◦ u(0, · − 1), π ◦ u(0, ·)) ∈ W u (γ, −gradËΘ L1⊕L2 ), lim u(s, 2t − 1) = x(t), s→+∞ uniformly in t ∈ [0, 1]. The following result is proved in section 5.10. 4.8. Proposition. For a generic choice of J1, J2, and gΘ , M K G (γ, x) - if non-empty - is a smooth manifold of dimension dim M K G (γ, x) = mΘ (γ; L1 ⊕ L2) − µ Λ (x; H1#H2) + 1. The projection (α, u) ↦→ α is smooth on M K G (γ, x). These manifolds carry coherent orientations. The elements (α, u) of M K G (γ, x) satisfy the energy estimate �Ê×]−1,1[\{0}×[0,α] |∂su(s, t)| 2 ds dt ≤ËL1⊕L2(γ) −�H1#H2(x). This provides us with the compactness which is necessary to define the homomorphism P K G : Mj(ËΘ L1⊕L2 , g Θ ) −→ F Λ j+1(H1#H2, J), by the usual counting procedure applied to the spaces M K G . A standard gluing argument shows that P K G is a chain homotopy between KΘ and G ◦ ΦΘ L1⊕L2 . 4.6 Comparison between C, EV, I! and c, ev, i! In section 2.5 we have shown that the two upper squares in the diagram Hj(M) ∼ = c∗ �� Hj(Λ(M)) ev∗ �� HMc HjM(f, gM) �� HjM(ËΛ � L , g �� HC �� Λ ) �� HΦΛ HMev �� HjM(f, gM) L HjF Λ �� HEv �� (H, J) ∼ = 51 �� Hj(M) ∼=
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 42 and 43: 4.2. Proposition. For a generic cho
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 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
Page 92 and 93: Given periodic orbits x1 ∈ P(H1),
Page 94 and 95: Since u and ∂u are integrable ove
Page 96 and 97: with non-local boundary conditions
Page 98 and 99: to ũ0 uniformly on Σ, we may loca
Page 100 and 101: we see that associating ũ to u pro
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The energy of a solution u ∈ M K
Page 104 and 105:
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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