# GDO

## Seminar

### Homotopy automorphisms of operads in topological spaces (mini-course)

Seminar Room 2, Newton Institute Gatehouse

#### Abstract

The Grothendieck-Teichmüller group can be defined algebraically, as the automorphism group of an operad in groupoids, the operad of parenthesized braids. I will explain that this operad represents the fundamental groupoid of the little 2-discs operad. The homotopy automorphisms considered in my lecture series represent a topological counterpart of the automorphisms of an algebraic operad. In this lecture, I will explain the precise definition of this notion, and of a rational version of the notion of a homotopy automorphism, where we neglect torsion phenomena.The result which I aim to establish precisely asserts that, in the case of the little 2-discs operad, the group of rational homotopy automorphisms reduce to homotopy automorphisms that can be detected by their action on fundamental groupoids.

General reference:B. Fresse, "Homotopy of operads and Grothendieck-Teichmüller Groups". Book project. First volume available on the web-page "http://math.univ-lille1.fr/%7Efresse/OperadGT-December2012Preprint.pdf"

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