Algebra/Topology Seminar by Thomas Willwacher

Abstract: The little n-disks operad D_n is a topological operad of rectilinear embeddings of a number of disjoint "little" disks into the unit disk. Its real homotopy type is known: Due to work of Kontsevich it is formal (over R), i.e., there is a zigzag of (homotopy) Hopf cooperads relating the cooperad of differential forms Omega(D_n) with the cohomology cooperad H(D_n). The framed little n-disks operad fD_n is a generalization of D_n in which one allows the little disks to be rotated. It is known to be formal over R for n=2 due to work of Longoni-Salvatore and Severa. We describe the real homotopy type of fD_n for higher n. Concretely, we show that Omega(fD_n) is quasi-isomorphic to H(fD_n) (only) for n even, and quasi-isomorphic to an explicitly described combinatorial Hopf cooperad for n odd. In particular fD_n (n>=2) is formal over R iff n is even.