600 research outputs found

    Legendrian contact homology in R3\mathbb{R}^3

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    This is an introduction to Legendrian contact homology and the Chekanov-Eliashberg differential graded algebra, with a focus on the setting of Legendrian knots in R3\mathbb{R}^3.Comment: v3: 59 pages, 27 figures; introduction rewritten, sections 5 and 6 switched, many small revision

    Lagrangian Cobordisms via Generating Families: Constructions and Geography

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    Embedded Lagrangian cobordisms between Legendrian submanifolds are produced from isotopy, spinning, and handle attachment constructions that employ the technique of generating families. Moreover, any Legendrian with a generating family has an immersed Lagrangian filling with a compatible generating family. These constructions are applied in several directions, in particular to a non-classical geography question: any graded group satisfying a duality condition can be realized as the generating family homology of a connected Legendrian submanifold in R^{2n+1} or in the 1-jet space of any compact n-manifold with n at least 2.Comment: 34 pages, 11 figures. v2: corrected a referenc

    Tame stacks in positive characteristic

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    We introduce and study a class of algebraic stacks with finite inertia in positive and mixed characteristic, which we call tame algebraic stacks. They include tame Deligne-Mumford stacks, and are arguably better behaved than general Deligne-Mumford stacks. We also give a complete characterization of finite flat linearly reductive schemes over an arbitrary base. Our main result is that tame algebraic stacks are \'etale locally quotient by actions of linearly reductive finite group schemes. In a subsequent paper we will show that tame algebraic stacks admit a good theory of stable maps.Comment: 31 pages, 3 sections and 1 appendi

    Contact Structures on Plumbed 3-Manifolds

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    In this paper, we show that the Ozsv\'ath-Szab\'o contact invariant c+(ξ)∈HF+(−Y)c^+(\xi)\in HF^+(-Y) of a contact 3-manifold (Y,ξ)(Y,\xi) can be calculated combinatorially if YY is the boundary of a certain type of plumbing XX, and ξ\xi is induced by a Stein structure on XX. Our technique uses an algorithm of Ozsv\'ath and Szab\'o to determine the Heegaard-Floer homology of such 3-manifolds. We discuss two important applications of this technique in contact topology. First, we show that it simplifies the calculation of the Ozsv\'ath-Stipsicz-Szab\'o obstruction to admitting a planar open book. Then we define a numerical invariant of contact manifolds that respects a partial ordering induced by Stein cobordisms. We do a sample calculation showing that the invariant can get infinitely many distinct values.Comment: Added some examples and comment
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