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In the following two sections we will show that also the lower two squares commute. By Theorem 3.11, the composition of the two lower arrows is the pair-of-pants product. We conclude that the pair-of-pants product corresponds to the loop product. Again, the commutativity of the two lower squares will be seen at the chain level, by proving that the two squares below � � M(ËΛ L ,g1)⊗M(ËΛ 1 L ,g2) 2 k Φ Λ L 1 ⊗Φ Λ L 2 � F Λ (H1,J)⊗HF Λ (H2,J) commute up to chain homotopies. �� k M! E �� Mk−n(ËΘ L 1 ⊕L 2 ,g Θ ) �� Φ �� Θ L1⊕L2 F Θ k−n (H1⊕H2,J) 4.4 The left-hand square is homotopy commutative MΓ G �� Mk−n(S Λ L 1 #L 2 ,g Λ ) �� Φ �� Λ L1 #L2 F Λ k−n (H1#H2,J) In this section we show that the chain maps ΦΘ L1⊕L2 ◦ M! and E ◦ (ΦΛ L1 ⊗ ΦΛ ) are homotopic. We L2 start by constructing a one-parameter family of chain maps K Λ α : � M(ËΛ , g1) ⊗ M(ËΛ , g2) L1 L2 � ∗ −→ F Θ ∗−n (H1 ⊕ H2, J1 ⊕ J2), where α is a non-negative number. The definition of KΛ α is based on the solution spaces of the consisting of a half-cylinder with a slit. More precisely, Floer equation on the Riemann surface ΣK α when α is positive ΣK α is the quotient of [0, +∞[×[0, 1] modulo the identifications (s, 0) ∼ (s, 1) ∀s ∈ [0, α]. with the holomorphic coordinate at (α, 0) ∼ (α, 1) obtained from (43) by a translation by α. When α = 0, Σ K α = Σ K 0 is just the half-strip [0, +∞[×[0, 1]. Fix γ1 ∈ P Λ (L1), γ2 ∈ P Λ (L2), and x ∈ P Θ (H1 ⊕ H2). Let M K α (γ1, γ2; x) be the space of solutions u ∈ C ∞ (Σ K α , T ∗ M 2 ) of the equation satisfying the boundary conditions ∂H1⊕H2,J(u) = 0, (53) π ◦ u(0, ·) ∈ W u (γ1; −grad g1ËΛ L1 ) × W u (γ2; −grad g2ËΛ L2 ), (54) (u(s, 0), −u(s, 1)) ∈ N ∗ ∆ Θ M , ∀s ≥ α, (55) lim u(s, ·) = x. s→+∞ (56) Let us fix some α0 ≥ 0. The following result is proved in section 5.10: 4.5. Proposition. For a generic choice of g1, g2, H1, and H2, M K α0 (γ1, γ2; x) - if non-empty - is a smooth manifold of dimension dimM K α0 (γ1, γ2; x) = m Λ (γ1; L1) + m Λ (γ2; L2) − µ Θ (x) − n. These manifolds carry coherent orientations. Compactness is again a consequence of the energy estimate � |∂su(s, t)| 2 dsdt ≤ËL1(γ1) +ËL2(γ2) −�H1⊕H2(x), (57) ]0,+∞[×]0,1[ implied by (49). When m Λ (γ1; L1) + m Λ (γ2; L2) = k and µ Θ (x; H1 ⊕ H2) = k − n, the space M K α0 (γ1, γ2; x) is a compact zero-dimensional oriented manifold. The usual counting process defines the homomorphism K Λ α0 : � M(ËΛ L1 , g1) ⊗ M(ËΛ L2 , g2) � k → F Θ k−n (H1 ⊕ H2, J), 44
and a standard gluing argument shows that KΛ is a chain map. α0 Now assume α0 > 0. By standard compactness and gluing arguments, the family of solutions M K α for α varying in the interval [α0, +∞[ allows to define a chain homotopy between KΛ α0 and the composition E ◦ (ΦΛ L1 ⊗ ΦΛL2 ). Similarly, a compactness and cobordism argument on the Morse side shows that KΛ 0 is chain homotopic to the composition ΦΘ L1⊕L2 ◦ M!. See Figure 5. α Θ E ◦ (Φ Λ L1 ⊗ ΦΛ L2 ) ≃ KΛ α Θ ! ≃ KΛ 0 ≃ ΦΘ ◦ M! L1⊕L2 Figure 5: The homotopy through K Λ α and K Λ 0 . There remains to prove that KΛ α0 is homotopic to KΛ 0 . Constructing a homotopy between these chain maps by using the spaces of solutions M K α for α ∈ [0, α0] presents analytical difficulties: if we are given a solution u of the limiting problem M K 0 , the existence of a (unique) one-parameter family of solutions ”converging” to u is problematic, because we do not expect u to be C0 close to the one-parameter family of solutions, due to the jump in the boundary conditions. Therefore, we use a detour, starting from the following algebraic observation. If two chain maps ϕ, ψ : C → C ′ are homotopic, so are their tensor products ϕ ⊗ψ and ψ ⊗ϕ. The converse is obviously not true, as the example of ϕ = 0 and ψ non-contractible shows. However, it becomes true under suitable conditions on ϕ and ψ. Denote by (�, 0) the graded group which vanishes at every degree, except for degree zero where it coincides with�. We see (�, 0) as a chain complex with the trivial boundary operator. We have the following: 4.6. Lemma. Let (C, ∂) and (C ′ , ∂) be chain complexes, bounded from below. Let ϕ, ψ : C → C ′ be chain maps. Assume that there is an element ǫ ∈ C0 with ∂ǫ = 0 and a chain map δ : C ′ → (�, 0) such that δ(ϕ(ǫ)) = δ(ψ(ǫ)) = 1. If ϕ ⊗ ψ is homotopic to ψ ⊗ ϕ then ϕ is homotopic to ψ Proof. Let π be the chain map π : C ′ ⊗ C ′ → C ′ ⊗ (�, 0) ∼ = C ′ , π = id ⊗ δ. Let H : C ⊗ C → C ′ ⊗ C ′ be a chain homotopy between ϕ ⊗ ψ and ψ ⊗ ϕ, that is If we define the homomorphism h : C → C ′ by we have ϕ ⊗ ψ − ψ ⊗ ϕ = ∂H + H∂. h(a) := π ◦ H(a ⊗ ǫ), ∀a ∈ C, ∂h(a) + h∂a = ∂π(H(a ⊗ ǫ)) + π(H∂(a ⊗ ǫ)) = π(∂H(a ⊗ ǫ) + H∂(a ⊗ ǫ)) = π(ϕ(a) ⊗ ψ(ǫ) − ψ(a) ⊗ ϕ(ǫ)) = ϕ(a) ⊗ δ(ψ(ǫ)) − ψ(a) ⊗ δ(ϕ(ǫ)) = ϕ(a) − ψ(a). Hence h is the required chain homotopy. 45
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 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
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
Page 102 and 103:
The energy of a solution u ∈ M K
Page 104 and 105:
Since we have applied a conformal r
Page 106 and 107:
6.9. Lemma. There exists a positive
Page 108 and 109:
[ChS04] M. Chas and D. Sullivan, Cl
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