Perverse Sheaves and Knot Contact Homology
| Autor: | Alimjon Eshmatov, Wai-kit Yeung, Yuri Berest |
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| Rok vydání: | 2016 |
| Předmět: |
Pure mathematics
Endomorphism Braid group Homology (mathematics) Mathematics::Algebraic Topology 01 natural sciences Mathematics - Geometric Topology Mathematics::Algebraic Geometry Mathematics::Category Theory 0103 physical sciences FOS: Mathematics Algebraic Topology (math.AT) Mathematics - Algebraic Topology 0101 mathematics Invariant (mathematics) Representation Theory (math.RT) Mathematics::Representation Theory Mathematics 010308 nuclear & particles physics 010102 general mathematics Geometric Topology (math.GT) K-Theory and Homology (math.KT) General Medicine Algebraic construction Noncommutative geometry Mathematics - K-Theory and Homology Gravitational singularity Mathematics - Representation Theory Knot (mathematics) |
| DOI: | 10.48550/arxiv.1610.02438 |
| Popis: | In this paper, which is mostly a research announcement, we give a new algebraic construction of knot contact homology in the sense of L. Ng [Ng05a]. For a link $L$ in $ {\mathbb R}^3 $, we define a differential graded (DG) $k$-category $ \tilde{\mathscr A} $ with finitely many objects, whose quasi-equivalence class is a topological invariant of $ L $. In the case when $L$ is a knot, the endomorphism algebra of a distinguished object of $ \tilde{\mathscr A} $ coincides with the fully noncommutative knot DGA as defined by Ekholm, Etnyre, Ng and Sullivan in [EENS13a]. The input of our construction is a natural action of the braid group $B_n$ on the category of perverse sheaves on a two-dimensional disk with singularities at $n$ marked points, studied by Gelfand, MacPherson and Vilonen in [GMV96]. As an application, we show that the category of finite-dimensional representations of the link $k$-category $ \tilde{A} = H_0(\tilde{\mathscr A}) $ defined as the $0$th homology of our DG category $ \tilde{\mathscr A} $ is equivalent to the category of perverse sheaves on $ {\mathbb R}^3 $ which are singular along the link $ L $. We also obtain several generalizations of the category $ \tilde{\mathscr A} $ by extending the Gelfand-MacPherson-Vilonen braid action. Comment: 26 pages; 2 figures; some remarks and references are added |
| Databáze: | OpenAIRE |
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