Abstract

I will show that the little n-disks operad is SO(n)- and O(n)-equivariantly formal over the rationals. Equivalently, the oriented and unoriented framed little disks operads are rationally formal as infinity-operads. The approach to prove these results relies on a “purity implies formality” technique. Specifically, we consider the action of the Grothendieck–Teichmüller group on the rationalized little disks operad, constructed using the additivity theorem. This induces an action on the rationalized Borel construction, with special purity properties.  As a consequence of our method, we also establish rational formality for algebraic group actions on smooth complex projective varieties, using Galois action on étale cohomology as input.  This is joint work with Pedro Boavida de Brito and Geoffroy Horel.