Framed Discs Operads and Batalin-Vilkovisky Algebras

Abstract

The framed n-discs operad fD(n) 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 fD(n) is equivalent to the n-fold loop space on an SO(n)-space. Examples of fD(2)-spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of fD(n), 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.