PDF (1016 kB)

More documents

Recommendations

Info

In the periodic case, we consider Hamiltonians H1, H2 ∈ C ∞ (Ì×T ∗ M) such that H1(0, ·) = H2(0, ·) with all time derivatives. Assuming that H1, H2, and H1#H2 satisfy (H0) Λ , the pair-ofpants product on HF Λ (T ∗ M) will be induced by a chain map Υ Λ : F Λ h (H1, J1) ⊗ F Λ k (H2, J2) → F Λ h+k−n (H1#H2, J1#J2), where n is the dimension of M. Let H ∈ C∞ ([−1, 1] × T ∗M), respectively H ∈ C∞ (Ê/2�×T ∗M), be defined by H(t, x) = 1 2 H1#H2((t � H1(t + 1, x) if − 1 ≤ t ≤ 0, + 1)/2, x) = H2(t, x) if 0 ≤ t ≤ 1. Notice that x : [−1, 1] → T ∗M is an orbit of XH if and only if the curve t ↦→ x((t + 1)/2) is an orbit of XH1#H2. Given x1 ∈ PΩ (H1), x2 ∈ PΩ (H2), and y ∈ PΩ (H1#H2), consider the space of solutions of the Floer equation delbarJ,H(u) = 0 on the holomorphic triangle M Ω � Υ (x1, x2; y) := u ∈ C ∞ (Σ Ω Υ, T ∗ � � M) �∂J,H(u) = 0, u(z) ∈ T ∗ q0M ∀z ∈ ∂ΣΩΥ, lim s→−∞ u(s, t − 1) = x1(t), lim s→−∞ u(s, t) = x2(t), lim u(s, 2t − 1) = y(t), uniformly in t ∈ [0, 1] s→+∞ Similarly, for x1 ∈ PΛ (H1), x2 ∈ PΛ (H2), and y ∈ PΛ (H1#H2), we consider the space of solutions of the Floer equation on the pair-of-pants surface M Λ � Υ(x1, x2; y) := u ∈ C ∞ (Σ Λ Υ, T ∗ � � M) �∂J,H(u) = 0, lim u(s, t − 1) = x1(t), s→−∞ lim s→−∞ u(s, t) = x2(t), � lim u(s, 2t − 1) = y(t), uniformly in t ∈ [0, 1] . s→+∞ The following result is proved in section 5.10. 3.4. Proposition. For a generic choice of H1 and H2 as above, the sets M Ω Υ (x1, x2; y) and M Λ Υ (x1, x2; y) - if non-empty - are manifolds of dimension dimM Ω Υ (x1, x2; y) = µ Ω (x1) + µ Ω (x2) − µ Ω (y), dimM Λ Υ (x1, x2; y) = µ Λ (x1) + µ Λ (x2) − µ Λ (y) − n. These manifolds carry coherent orientations. The energy identity (38) implies that every map u in M Ω Υ (x1, x2; y) or in M Λ Υ (x1, x2; y) satisfies � � |∂su(s, t)| 2 ds dt =�H1(x1) +�H2(x2) −�H1#H2(y). (41) (Ê×]−1,1[)\]−∞,0]×{0}) As a consequence, we obtain the following compactness result, which is proved in section 6.1. 3.5. Proposition. Assume that the Hamiltonians H1 and H2 satisfy (H1), (H2). Then the spaces M Ω Υ (x1, x2; y) and M Λ Υ (x1, x2; y) are pre-compact in C ∞ loc . When µ Ω (y) = µ Ω (x1) + µ Ω (x2), M Ω Υ (x1, x2; y) is a finite set of oriented points, and we denote by nΩ Υ (x1, x2; y) the algebraic sum of the corresponding orientation signs. Similarly, when µ Λ (y) = µ Λ (x1) + µ Λ (x2) − n, M Λ Υ (x1, x2; y) is a finite set of oriented points, and we denote by nΛ Υ (x1, x2; y) the algebraic sum of the corresponding orientation signs. These integers are the coefficients of the homomorphisms Υ Ω : F Ω h (H1) ⊗ F Ω k (H2) → F Ω h+k(H1#H2), x1 ⊗ x2 ↦→ � n Ω Υ(x1, x2; y)y, y∈P Ω (H1#H2) µ Ω (y)=h+k Υ Λ : F Λ h (H1) ⊗ F Λ k (H2) → F Λ h+k−n (H1#H2), x1 ⊗ x2 ↦→ � 30 y∈P Λ (H1#H2) µ Λ (y)=h+k−n n Λ Υ (x1, x2; y)y. (40) � .
A standard gluing argument shows that the homomorphisms Υ Ω and Υ Λ are chain maps. Therefore, they define products Υ Ω : HF Ω h (T ∗ M) ⊗ HF Ω k (T ∗ M) → HF Ω h+k(T ∗ M), Υ Λ : HF Λ h (T ∗ M) ⊗ HF Λ k (T ∗ M) → HF Λ h+k−n (T ∗ M), in homology. By standard gluing arguments, it could be shown that these products have a unit element, are associative, and the second one is commutative. These facts will actually follow from the fact that these products correspond to the Pontrjagin and the loop products on H∗(Ω(M, q0)) and H∗(Λ(M)). 3.4 Floer homology for figure-8 loops The pair-of-pants product on cotangent bundles - unlike on an arbitrary symplectic manifold - has a natural factorization. Indeed, we will show that it factors through the Floer homology of figure-8 loops. The aim of this sections is to define this Floer homology. Let H1, H2 ∈ C ∞ ([0, 1] × T ∗ M) be two Hamiltonians satisfying (H1) and (H2). We are now interested in the set P Θ (H1 ⊕ H2) of pairs of orbits (x1, x2) of XH1 and XH2 such that π ◦ x1(0) = π ◦ x2(0) = π ◦ x1(1) = π ◦ x2(1), x1(1) − x1(0) + x2(1) − x2(0) = 0. This is the Hamiltonian version of the problem P Θ (L1 ⊕ L2), introduced in section 2.5. It is a non-local Lagrangian boundary value problem on the symplectic manifold (T ∗ M × T ∗ M, ω × ω), that is on the cotangent bundle of M 2 = M × M endowed with its standard symplectic structure. Indeed, consider the following n-dimensional submanifold of M 4 = M × M × M × M, and denote by N ∗ ∆ Θ M ∆ Θ M := {(q, q, q, q) | q ∈ M} , its conormal bundle, that is N ∗ ∆ Θ M := � ζ ∈ T ∗ M 4 � | ∆Θ | ζ| M T∆Θ = 0 M = � (ζ1, ζ2, ζ3, ζ4) ∈ (T ∗ M) 4 | π(ζ1) = π(ζ2) = π(ζ3) = π(ζ4), ζ1 + ζ2 + ζ3 + ζ4 = 0 � . We are looking at orbits x = (x1, x2) of the Hamiltonian vector field XH1⊕H2 on T ∗M 2 such that (x(0), −x(1)) belongs to N ∗∆Θ M . In other words, we are looking at the intersection of the graph of −φH1⊕H2 (1, ·) - a Lagrangian7 submanifold of T ∗M 2 × T ∗M2 = T ∗M4 - with the Lagrangian submanifold N ∗∆Θ M . The corresponding non-degeneracy condition is then the following: (H0) Θ every solution x = (x1, x2) ∈ PΘ (H1 ⊕ H2) is non-degenerate, meaning that the graph of the map −φH1⊕H2 (1, ·) is transverse to the submanifold N ∗∆Θ M at the point (x(0), −x(1)). Let x = (x1, x2) ∈ P Θ (H1 ⊕ H2). By means of a unitary trivialization [0, 1] ×Ê4n → x ∗ (TT ∗ M 2 ), mapping (0) ×Ê2n into the vertical subbundle T v T ∗ M 2 , we can transform the differential of φ H1⊕H2 (t, ·) at x(0) into a path Φ : [0, 1] → Sp(4n) into the symplectic group ofÊ4n , such that Φ(0) = I. Denoting by C ∈ L(Ê4n ,Ê4n ) the anti-symplectic involution C(q, p) = (q, −p), condition (H0) Θ states that the graph of CΦ(1) has intersection (0) with the Lagrangian subspace ofÊ4n ×Ê4n , N ∗ ∆ ΘÊn := � (z1, z1, z3, z4) ∈ (Ê2n ) 4 | πz1 = πz2 = πz3 = πz4, (I − π)(z1 + z2 + z3 + z4) = 0 � , π :Ê2n =Ên ×Ên →Ên denoting the projection onto the first factor. 7 Usually a non-local Lagrangian boundary value problem on the symplectic manifold (P, ω) is given by fixing a Lagrangian submanifold L of (P × P, ω ⊕ (−ω)) and considering its intersection with the graph of a Hamiltonian diffeomorphism of P, which is a Lagrangian submanifold of (P × P, ω ⊕ (−ω)). Here P is the cotangent bundle of a manifold, and so is P × P. Therefore we prefer to consider always the standard symplectic structure ω ⊕ ω on P × P, and not the flipped one, ω ⊕ (−ω). With this choice, φ is a symplectic diffeomorphism of P if and only if the graph of −φ is a Lagrangian submanifold of P × P. See also section 5.1 for the consequences of adopting this sign convention. 31
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 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 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
show all

PDF (1016 kB)

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?