2,915 research outputs found
Braids and symplectic four-manifolds with abelian fundamental group
We explain how a version of Floer homology can be used as an invariant of
symplectic manifolds with . As a concrete example, we look at
four-manifolds produced from braids by a surgery construction. The outcome
shows that the invariant is nontrivial; however, it is an open question whether
it is stronger than the known ones.Comment: 9 pages, LaTe
Picard-Lefschetz theory and dilating C^*-actions
We consider C^*-actions on Fukaya categories of exact symplectic manifolds.
Such actions can be constructed by dimensional induction, going from the fibre
of a Lefschetz fibration to its total space. We explore applications to the
topology of Lagrangian submanifolds, with an emphasis on ease of computation.Comment: v3: appendectomy performed; to appear in Journal of Topolog
More about vanishing cycles and mutation
The paper continues the discussion of symplectic aspects of Picard-Lefschetz
theory begun in "Vanishing cycles and mutation" (this archive). There we
explained how to associate to a suitable fibration over a two-dimensional disc
a triangulated category, the "derived directed Fukaya category" which describes
the structure of the vanishing cycles. The present second part serves two
purposes. Firstly, it contains various kinds of algebro-geometric examples,
including the "mirror manifold" of the projective plane. Secondly there is a
(largely conjectural) discussion of more advanced topics, such as (i)
Hochschild cohomology, (ii) relations between Picard-Lefschetz theory and Morse
theory, (iii) a proposed "dimensional reduction" algorithm for doing certain
Floer cohomology computations.Comment: 33 pages, LaTeX2e, 9 eps figure
Graded Lagrangian submanifolds
In the usual setup, the grading on Floer homology is relative: it is unique
only up to adding a constant. "Graded Lagrangian submanifolds" are Lagrangian
submanifolds with a bit of extra structure, which fixes the ambiguity in the
grading. The idea is originally due to Kontsevich. This paper contains an
exposition of the theory. Several applications are given, amongst them:
(1) topological restrictions on Lagrangian submanifolds of projective space,
(2) the existence of "symplectically knotted" Lagrangian spheres on a K3
surface, (3) a result about the symplectic monodromy of weighted homogeneous
hypersurface singularities.
Revised version: minor modifications, journal reference added.Comment: LaTex2e, 32 pages, one eps figur
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