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Little discs operads, graph complexes and Grothendieck-Teichmüller groups. (English) Zbl 1476.55030

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 405-441 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: This chapter aims to survey applications of the little discs operads which were motivated by the works of Kontsevich and Tamarkin on the deformation-quantization of Poisson manifolds and by the Goodwillie-Weiss embedding calculus in topology. Kontsevich used an explicit definition of such a comparison map in his first proof of the existence of deformation-quantizations. The theory of \(E_2\)-operads actually occurs in a second generation of proofs of this theorem. The existence of associators can be used to get insights into the structure of the rational Grothendieck-Teichmüller group. The chapter discusses the connection between the Grothendieck-Teichmüller group and the group of homotopy automorphisms of \(E_2\)-operads. The Grothendieck-Teichmüller group is defined as a group of automorphisms of the parenthesized braid operad. The hairy graph complex \(HGC_{mn}\) explicitly consists of formal series of connected graphs with internal vertices.
For the entire collection see [Zbl 1468.55001].

MSC:

55P48 Loop space machines and operads in algebraic topology
55P62 Rational homotopy theory
57R40 Embeddings in differential topology
20F36 Braid groups; Artin groups
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology

Citations:

Zbl 1468.55001
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