2,963 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
Fukaya categories and deformations
This is an informal (and mostly conjectural) discussion of some aspects of
Fukaya categories. We start by looking at exact symplectic manifolds which are
obtained from a closed Calabi-Yau by removing a hyperplane section. We look at
the possible geometric significance of Hochschild cohomology in this situation,
and how one can try to get from the Fukaya category of the exact manifold to
that of the closed Calabi-Yau. Also included is a brief discussion of the role
of Lefschetz pencils, and a bit of general deformation theory. To appear in the
Proceedings of the Beijing ICM
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