Abstract. Let (M, ω) be a compact symplectic 2n-manifold, and L a compact embedded Lagrangian submanifold in M . Fukaya, Oh, Ohta and Ono [9] construct Lagrangian Floer cohomology for such M, L, yielding groups HF * (L, b; Λnov) for one Lagrangian or HF * `( L 1 , b 1 ), (L 2 , b 2 ); Λnov´for two, where b, b 1 , b 2 are choices of bounding cochains, and exist if and only if L, L 1 , L 2 have unobstructed Floer cohomology. These are independent of choices up to canonical isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M, ω), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich.The goal of this paper is to extend [9] to immersed Lagrangians L in M with immersion ι : L → M , with transverse self-intersections. In the embedded case, Floer cohomology HF * (L, b; Λnov) is a modified, 'quantized' version of singular homology H n− * (L; Λnov) over the Novikov ring Λnov. In our immersed case, HF * (L, b; Λnov) turns out to be a quantized version ofThe theory becomes simpler and more powerful for graded Lagrangians in Calabi-Yau manifolds, when we can work over a smaller Novikov ring ΛCY. The proofs involve associating a gapped filtered A∞ algebra over Λ 0 nov or Λ 0 CY to ι : L → M , which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called A N,0 algebras for N = 0, 1, 2, . . ..
We introduce Floer homology for transversely intersecting Lagrangian immersions L and L in a symplectic manifold (X, ω). By using this homology, if π 2 (X, L) = 0 and L is the image of L under a Hamiltonian isotopy, then the number of the intersection points of L and L is bounded below by the sum of the Z 2 -betti numbers of L (or rather, the manifold whose immersion is L) and a non-negative extra term coming from the self-intersections of L.
Let (M, ω) be a compact symplectic 2n-manifold, and L a compact embedded Lagrangian submanifold in M . Fukaya, Oh, Ohta and Ono [9] construct Lagrangian Floer cohomology for such M, L, yielding groups HF * (L, b; Λnov) for one Lagrangian or HF * `(L 1 , b 1 ), (L 2 , b 2 ); Λnov ´for two, where b, b 1 , b 2 are choices of bounding cochains, and exist if and only if L, L 1 , L 2 have unobstructed Floer cohomology. These are independent of choices up to canonical isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M, ω), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich. The goal of this paper is to extend [9] to immersed Lagrangians L in M with immersion ι : L → M , with transverse self-intersections. In the embedded case, Floer cohomology HF * (L, b; Λnov) is a modified, 'quantized' version of singular homology H n− * (L; Λnov) over the Novikov ring Λnov. In our immersed case, HF * (L, b; Λnov) turns out to be a quantized version of H n− * (L; Λnov)⊕ L (p − ,p + )∈R Λnov •(p − , p + ), where R = ˘(p − , p + ) : p − , p + ∈ L, p − = p + , ι(p − ) = ι(p + ) ¯is a set of two extra generators for each selfintersection point of L, and (p − , p + ) has degree η (p − ,p + ) ∈ Z, an index depending on how L intersects itself at ι(p − ) = ι(p + ).The theory becomes simpler and more powerful for graded Lagrangians in Calabi-Yau manifolds, when we can work over a smaller Novikov ring ΛCY. The proofs involve associating a gapped filtered A∞ algebra over Λ 0 nov or Λ 0 CY to ι : L → M , which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called A N,0 algebras for N = 0, 1, 2, . . ..
We introduce Morse homology for manifolds with boundary, and construct relative Morse homology sequences. Keywords: Morse homology; manifold with boundary. Mathematics Subject Classification 2000: 58E05, 55N35, 57R70 301 Commun. Contemp. Math. 2007.09:301-334. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SANTA CRUZ on 02/03/15. For personal use only. 302 M. Akaho∞)×∂M and d((a,T )·y) da | a=0 on ((0, T ) · y)| (−∞,0)×∂M together through partition of unity around (0, f ∂M (γ + )), a section w of φ 5 (x, y, T ) * T (R × ∂M ) by gluing d((0,−T +b)·x) db | b=0 on ((0, −T ) · x)| (0,∞)×∂M and d((0,T +b)·y) db | b=0 on ((0, T ) · y)| (−∞,0)×∂M together through partition of unity around (0, f ∂M (γ + )) and a section v of φ 5 (x, y, T ) * T (R × ∂M ) by gluing d((−b,−T +f ∂M (γ + )b)·x) db | b=0 on ((0, −T ) · x)| (0,∞)×∂M and d((b,T −f ∂M (γ + )b)·y) db | b=0 on ((0, T )·y)| (−∞,0)×∂M together through partition of unity around (0, f ∂M (γ + )), then Commun. Contemp. Math. 2007.09:301-334. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ SANTA CRUZ on 02/03/15. For personal use only.
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