Framed Discs Operads and Batalin–vilkovisky Algebras

The framed n-discs operad f Dn 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 f Dn is equivalent to the n-fold loop space on an SO(n)-space. Examples of f D2-spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of f Dn , 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.