# Real models for the framed little \$n\$-disks operad

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Speaker:
Thomas Willwacher
Affiliation:
ETH Zürich
Date:
Tue, 2017-06-20 11:00 - 12:00
Location:
MPIM Lecture Hall

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 \$\mathbb{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 \$\mathbb{R}\$ for \$n=2\$ due to work of Giansiracusa-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 \$\mathbb{R}\$ iff \$n\$ is even.

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