Perverse sheaves and knot contact homology

Berest, Yuri; Eshmatov, Alimjon; Yeung, Wai-Kit

doi:10.1016/j.crma.2017.02.007

Homological algebra/Topology

Perverse sheaves and knot contact homology
[Faisceaux pervers et homologie de contact des nœuds]

Présenté par : the Editorial Board

Berest, Yuri ¹ ; Eshmatov, Alimjon ² ; Yeung, Wai-Kit ¹

¹ Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA
² Department of Mathematics and Statistics, University of Toledo, Toledo, OH 43606-3390, USA

Comptes Rendus. Mathématique, Tome 355 (2017) no. 4, pp. 378-399.

Résumé
Abstract

Dans cette Note, nous donnons une nouvelle construction algébrique de l'homologie de contact des nœuds, au sens de Ng [37]. Pour un entrelacs L dans $R^{3}$ , nous définissons une k-catégorie différentielle graduée ${\tilde{A}}_{L}$ ayant un nombre fini d'objets, dont la classe de quasi-équivalence est un invariant topologique de L. Lorsque L est un nœud, l'algèbre des endomorphismes d'un objet distingué de ${\tilde{A}}_{L}$ coïncide avec l'algèbre différentielle graduée, pleinement non commutative du nœud, définie par Ekholm, Etnyre, Ng et Sullivan dans [12]. Notre construction se base sur une action naturelle du groupe de tresses $B_{n}$ sur la catégorie des faisceaux pervers sur un disque de dimension deux avec singularités en n points marqués, étudiée par Gelfand, McPherson et Vilonen dans [19]. Comme application, nous montrons que la catégorie des représentations de dimension finie de la k-catégorie d'entrelacs ${\tilde{A}}_{L} = H_{0} ({\tilde{A}}_{L})$ , définie comme l'homologie de degré 0 de ${\tilde{A}}_{L}$ , est équivalente à la catégorie des faisceaux pervers sur $R^{3}$ qui sont singuliers le long de l'entrelacs L. Nous obtenons également plusieurs généralisations de la catégorie ${\tilde{A}}_{L}$ en étendant l'action du groupe de tresses de Gelfand–MacPherson–Vilonen.

In this paper, we give a new algebraic construction of knot contact homology in the sense of Ng [35]. For a link L in $R^{3}$ , we define a differential graded (DG) k-category ${\tilde{A}}_{L}$ 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 ${\tilde{A}}_{L}$ coincides with the fully noncommutative knot DGA as defined by Ekholm, Etnyre, Ng, and Sullivan in [13]. 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 [19]. As an application, we show that the category of finite-dimensional representations of the link k-category ${\tilde{A}}_{L} = H_{0} ({\tilde{A}}_{L})$ defined as the 0-th homology of ${\tilde{A}}_{L}$ is equivalent to the category of perverse sheaves on $R^{3}$ that are singular along the link L. We also obtain several generalizations of the category ${\tilde{A}}_{L}$ by extending the Gelfand–MacPherson–Vilonen braid group action. Detailed proofs of results announced in this paper will appear in [4].

Reçu le : 2016-11-16
Accepté le : 2017-02-22
Publié le : 2017-03-11

DOI : 10.1016/j.crma.2017.02.007

Affiliations des auteurs :

Berest, Yuri ¹ ; Eshmatov, Alimjon ² ; Yeung, Wai-Kit ¹

¹ Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, USA
² Department of Mathematics and Statistics, University of Toledo, Toledo, OH 43606-3390, USA

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     title = {Perverse sheaves and knot contact homology},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {378--399},
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Berest, Yuri; Eshmatov, Alimjon; Yeung, Wai-Kit. Perverse sheaves and knot contact homology. Comptes Rendus. Mathématique, Tome 355 (2017) no. 4, pp. 378-399. doi : 10.1016/j.crma.2017.02.007. http://www.numdam.org/articles/10.1016/j.crma.2017.02.007/

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