1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 5. MULTI-VALUED PERTURBATIONS 207 Example 5.1. Suppose that (M,w) is an 8-dimensional symplectic manifold which satisfie ...

208 D. SALAMON, FLOER HOMOLOGY The solutions of this differential inequality will be called (J, r)-holomorphic curves. To ensure ...

LECTURE 5. MULTI-VALUED PERTURBATIONS 209 • The vertical differential D1i(v): TvBk,p--> £i-l,p is a compact oper- ator for v ...

210 D. SALAMON, FLOER HOMOLOGY The tangent space of the ( 6-dimensional) orbit u · G = { u o rp rp E G} at u is given by Vertu = ...

LECTURE 5. MULTI-VALUED PERTURBATIONS 211 Figure 20. Local slices for the G-action on Map( S^2 , M) Corollary 5.5. Let u E Map(S ...

212 D. SALAMON, FLOER HOMOLOGY semigroup property of the class of perturbations considered above. The family of perturbations ta ...

LECTURE 5. MULTI-VALUED PERTURBATIONS 213 This space carries a natural rational label M(A; J, f) ~ Q : v r-t A(v) = A( v, 8 J ( ...

214 D. SALAMON, FLOER HOMOLOGY Remark 5.7. Let (M,>..) be a branched manifold with charts {(Mi, >..i,cpi)}iEJ· Then the se ...

LECTURE 5. MULTI-VALUED PERTURBATIONS 215 Suppose first that x E Mi - int(Mi )· Then there exists a sequence xv E M - Mi which c ...

216 1 £=-1 ... D. SALAMON, FLOER HOMOLOGY 112 Figure 21. A compact oriented branched I-manifold with BM C Mreg the outward unit ...

LECTURE 5. MULTI-VALUED PERTURBATIONS 217 closed sets with empty interior.^3 The set ex(M -Mreg) is the union of these images ov ...

218 D. SALAMON, FLOER HOMOLOGY The proof is by induction over the number of sets in the open cover M = LJj M(j). Fix any index j ...

LECTURE 5. MULTI-VALUED PERTURBATIONS Perturbations and marked points Consider the space B = B(A, k) = MapA(8^2 , M) x ((8^2 )k ...

220 D. SALAMON, FLOER HOMOLOGY carries an action of the group G(T) of automorphisms of the bubble tree. The elements of G(T) are ...

LECTURE 5. MULTI-VALUED PERTURBATIONS essentially the arguments in [31]. One considers the moduli space Mo,k,A(M,w, J,I') {(u,z1 ...

222 D. SALAMON, FLOER HOMOLOGY injective for each solution. The injective condition is redundant, by the "free" ax- iom. One can ...

LECTURE 5. MULTI-VALUED PERTURBATIONS 223 which is equivariant with respect to the JR-action on Z and E. Instead of (7) we shall ...

224 D. SALAMON, FLOER HOMOLOGY Exercise 5.13. Prove that every solution u E Z(x-,x+) of fh,H(u) = 1(u) satis- fies E(u) = aH(x-, ...

LECTURE 5. MULTI-VALUED PERTURBATIONS 225 for all x, z E P(H). As in the monotone case, this follows by examining the moduli spa ...

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