Framed Discs Operads and Batalin-Vilkovisky Algebras

Abstract

The framed n‐discs operad fDn is studied as semidirect product of SO(n) and the little n‐discs operad. Our equivariant recognition principle says that a grouplike space acted on by fDn is equivalent to the n‐fold loop space on an SO(n)‐space. Examples of fD2‐spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of fDn, which produces higher Batalin–Vilkovisky algebra structures for n even. We study quadratic duality for semidirect product operads and compute the double loop space homology of a manifold as BV‐algebra.