Thomas Willwacher: Rational homotopy theory of the little disks operads

Abstract:
The little n-disks operads are classical objects in topology, introduced by Boardman-Vogt and May in the 1970's in their study of iterated loop spaces. They have since seen a wealth of applications in algebra and topology, and received much attention recently due to their appearance in the manifold calculus of Goodwillie and Weiss, and relatedly in the factorization (or topological chiral) homology by Lurie, Francis, Beilinson-Drinfeld and others.

I report on joint work with Victor Turchin and Benoit Fresse, in which we (mostly) settle the rational homotopy theory of the little n-disks operads, by showing that they are intrinsically formal for n>=3, and by computing the rational homotopy type of the function spaces between these objects. As an application we obtain complete rational invariants of long knots in codimension >=3.