Lagrangian Embeddings, Maslov Indexes and Integer Graded Symplectic Floer Cohomology
نویسنده
چکیده
We define an integer graded symplectic Floer cohomology and a spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopies. Such an integer graded Floer cohomology is an integral lifting of the usual Floer-Oh cohomology with ZΣ(L) grading. As one of applications of the spectral sequence, we offer an affirmative answer to an Audin’s question for oriented, embedded, monotone Lagrangian tori, i.e. Σ(L) = 2.
منابع مشابه
The Z–graded symplectic Floer cohomology of monotone Lagrangian sub–manifolds
We define an integer graded symplectic Floer cohomology and a Fintushel–Stern type spectral sequence which are new invariants for monotone Lagrangian sub–manifolds and exact isotopes. The Z–graded symplectic Floer cohomology is an integral lifting of the usual ZΣ(L) –graded Floer–Oh cohomology. We prove the Künneth formula for the spectral sequence and an ring structure on it. The ring structur...
متن کاملLink Homology Theories from Symplectic Geometry
For each positive integer n, Khovanov and Rozansky constructed an invariant of links in the form of a doubly-graded cohomology theory whose Euler characteristic is the sl(n) link polynomial. We use Lagrangian Floer cohomology on some suitable affine varieties to build a similar series of link invariants, and we conjecture them to be equal to those of Khovanov and Rozansky after a collapse of th...
متن کاملAnti-symplectic Involution and Maslov Indices
We carry out some first steps in setting up a theory for Lagrangian Floer theory, mimicking Seidel’s construction for Hamiltonian Floer homology [7], for the subgroup HamL(M,ω) of Ham(M,ω) which preserves the Lagrangian L. When the symplectic manifold M has anti-symplectic involution c and L is the fixed Lagrangian submanifold, we consider the subgroup Hamc(M,ω) which commute with c. In the lat...
متن کاملLocalization for Involutions in Floer Cohomology
We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M . Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two application...
متن کاملGraded Lagrangian Submanifolds
Floer theory assigns, in favourable circumstances, an abelian group HF (L0, L1) to a pair (L0, L1) of Lagrangian submanifolds of a symplectic manifold (M,ω). This group is a qualitative invariant, which remains unchanged under suitable deformations of L0 or L1. Following Floer [7] one can equip HF (L0, L1) with a canonical relative Z/N -grading, where 1 ≤ N ≤ ∞ is a number which depends on (M,ω...
متن کامل