Operads of genus zero curves and the Grothendieck–Teichmüller group

Boavida de Brito, Pedro¹ ; Horel, Geoffroy² ; Robertson, Marcy³

¹ Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal
² LAGA, Institut Galilée, Université Paris 13, Villetaneuse, France
³ School of Mathematics and Statistics, The University of Melbourne, Melbourne, Victoria, Australia

Geometry & topology, Tome 23 (2019) no. 1, pp. 299-346.

Résumé

We show that the group of homotopy automorphisms of the profinite completion of the genus zero surface operad is isomorphic to the (profinite) Grothendieck–Teichmüller group. Using a result of Drummond-Cole, we deduce that the Grothendieck–Teichmüller group acts nontrivially on ${\overline{\mathcal{ℳ}}}_{0,\bullet +1}$, the operad of stable curves of genus zero. As a second application, we give an alternative proof that the framed little $2$–disks operad is formal.

Détail

DOI : 10.2140/gt.2019.23.299

Classification : 18D50, 14G32, 32G15, 55P48, 55U35
Keywords: infinity operads, Grothendieck–Teichmüller group, absolute Galois group, moduli space of curves

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