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Moreover, �Dαn(βnvn)�0,p;Ê≤ �iβ ′ n vn�0,p;Ê+�βnDαnvn�0,p;Ê ≤ c2αn�vn� −1 0,p;[−αn ,0]∪[α −1 n ,2α −1 n ] + 3�Dαnvn�0,p;αn ≤ c2αn2�vn�1,p;αn + 3�Dαnvn�0,p;αn → 0 . Hence, we find a subsequence such that βnk vnk → vo ∈ ker∂ = {0} which means that �vnk �1,p;αn k → 0 in contradiction to �vn� = 1. Similarly, we see that the coherent orientation for the determinant of Dαn equals that of ∂ which is canonically 1. This completes the proof of the proposition. From the cobordisms M Ev(γ, x), i = 1, 2, 3, we now obtain the chain homotopy as claimed. Pi Finally, there remains to prove that the right-hand square in the diagram Hj(Λ(M)) i! �� Hj−n(Ω(M, q0)) ∼ = �� HjM(ËΛ L , g Λ ) HMi ! HΦ Λ L ∼ �� = �� Hj−nM(ËΩ L , gΩ HΦ ) �� Ω L Hj−nF Ω (H, J) �� HjF(H, J) commutes, the commutativity of the left-hand square having been established in section 2.5. Again, we work at the chain level, proving that the diagram Mj(ËΛ L , g Λ ) Mi ! �� Mj−n(ËΩ L , gΩ ) Φ Λ L Φ Ω L �� Fj(H, J) F Ω j−n � (H, J) is homotopy commutative. Indeed, we can show that both I! ◦ ΦΛ L and ΦΩL ◦ Mi! are homotopic to the same chain map K ! . The definition of K ! makes use of the following spaces: given γ ∈ PΛ (L) and x ∈ PΩ (H), set M ! � K (γ, x) := u ∈ C ∞ ([0, +∞[×[0, 1], T ∗ � � M) �∂J,H(u) = 0, π ◦ u(s, 0) = π ◦ u(s, 1) = q0 ∀s ≥ 0, π ◦ u(0, ·) ∈ W u (γ; −gradËΛ L ), � lim u(s, ·) = x uniformly in t . s→+∞ Again, details are left to the reader. 5 Linear theory 5.1 The Maslov index Let η0 be the Liouville one-form on T ∗Ên =Ên ×(Ên ) ∗ , that is the tautological one-form η0 = p dq: Its differential ω0 = dη0 = dp ∧ dq, η0(q, p)[(u, v)] = p[u], for q, u ∈Ên , p, v ∈ (Ên ) ∗ . ω0[(q1, p1), (q2, p1)] = p1[q2] − p2[q1], for q1, q2 ∈Ên , p1, p2 ∈ (Ên ) ∗ , is the standard symplectic form on T ∗Ên . 54 � I! HI!
The symplectic group, that is the group of linear automorphisms of T ∗Ên preserving ω0, is denoted by Sp(2n). Let L (n) be the Grassmannian of Lagrangian subspaces of T ∗Ên , that is the set of n-dimensional linear subspaces of T ∗Ên on which ω0 vanishes. The relative Maslov index assigns to every pair of Lagrangian paths λ1, λ2 : [a, b] → L (n) a half integer µ(λ1, λ2). We refer to [RS93] for the definition and for the properties of the relative Maslov index. Another useful invariant is the Hörmander index of four Lagrangian subspaces (see [Hör71], [Dui76], or [RS93]): 5.1. Definition. Let λ0, λ1, ν0, ν1 be four Lagrangian subspaces of T ∗Ên . Their Hörmander index is the half integer h(λ0, λ1; ν0, ν1) := µ(ν, λ1) − µ(ν, λ0), where ν : [0, 1] → L (n) is a Lagrangian path such that ν(0) = ν0 and ν(1) = ν1. Indeed, the quantity defined above does not depend on the choice of the Lagrangian path ν joining ν0 and ν1. If V is a linear subspace ofÊn , N ∗ V ⊂ T ∗Ên denotes its conormal space, that is N ∗ V := {(q, p) ∈Ên × (Ên ) ∗ | q ∈ V, V ⊂ kerp} = V × V ⊥ , where V ⊥ denotes the set of covectors in (Ên ) ∗ which vanish on V . Conormal spaces are Lagrangian subspaces of T ∗Ên . Let C : T ∗Ên → T ∗Ên be the linear involution C(q, p) := (q, −p) ∀(q, p) ∈ T ∗Ên . The involution C is anti-symplectic, meaning that ω0(Cξ, Cη) = −ω0(ξ, η) ∀ξ, η ∈ T ∗Ên . In particular, C maps Lagrangian subspaces into Lagrangian subspaces. Since the Maslov index is natural with respect to symplectic transformations and changes sign if we change the sign of the symplectic structure, we have the identity µ(Cλ, Cν) = −µ(λ, ν), (76) for every pair of Lagrangian paths λ, ν : [a, b] → L (n). Since conormal subspaces are C-invariant, we deduce that µ(N ∗ V, N ∗ W) = 0, (77) for every pair of paths V, W into the Grassmannian ofÊn . Let V0, V1, W0, W1 be four linear subspaces ofÊn , and let ν : [0, 1] → L (n) be a Lagrangian path such that ν(0) = N ∗ W0 and ν(1) = N ∗ W1. By (76), h(N ∗ V0, N ∗ V1; N ∗ W0, N ∗ W1) = µ(ν, N ∗ V1) − µ(ν, N ∗ V0) = −µ(Cν, N ∗ V1) + µ(Cν, N ∗ V0). But also the Lagrangian path Cν joins N ∗ W0 and N ∗ W1, so the latter quantity equals We deduce the following: −h(N ∗ V0, N ∗ V1; N ∗ W0, N ∗ W1). 5.2. Proposition. Let V0, V1, W0, W1 be four linear subspaces ofÊn . Then h(N ∗ V0, N ∗ V1; N ∗ W0, N ∗ W1) = 0. 55
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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 52 and 53: 4.5 The right-hand square is homoto
Page 54 and 55: commute. Our first aim in this sect
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
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6.9. Lemma. There exists a positive
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[ChS04] M. Chas and D. Sullivan, Cl
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