Let M be an n-manifold, and let A be a space with a partial sum behaving as an n-fold loop sum. We define the space C(M ; A) of configurations in M with summable labels in A via operad theory. Some examples are symmetric products, labelled configuration spaces, and spaces of rational curves. We show that C(I n , ∂I n ; A) is an n-fold classifying space of C(I n ; A), and for n = 1 it is homeomorphic to the classifying space by Stasheff. If M is compact, parallelizable, and A is path connected, then C(M ; A) is homotopic to the mapping space M ap(M, C(I n , ∂I n ; A)).

The framed n-discs operad f D 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 f D n is equivalent to the n-fold loop space on an SO(n)-space. Examples of f D 2 -spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of f D 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. IntroductionThe topology of iterated loop spaces was thoroughly investigated in the seventies. These spaces have a wealth of homology operations parametrized by the famous operads of little discs, denoted by D n in the text. The notion of operad was introduced in the first place for this purpose [2,19]. Such machinery allows, for example, the reconstruction of an iterated delooping if one has full knowledge of the operad action on an iterated loop space. Moreover any connected space acted on by the little discs is weakly homotopy equivalent to an iterated loop space. This fact is the celebrated recognition principle.Our main objective is to extend this theory by adding the operations rotating the discs. The operad generated by the little n-discs D n and the rotations in SO(n) is the framed n-discs operad f D n , first introduced in [10]. Our recognition principle for framed n-discs (Theorem 3.1) says that a connected (or grouplike) space acted on by the framed n-discs operad is weakly homotopic to an n-fold loop space on an SO(n)-space.Thus the looping and delooping functors induce a categorical equivalence between SO(n)-spaces and spaces acted on by the framed n-discs operad, under the correct connectivity assumptions. The main technique consists in presenting the framed little discs as a semidirect product of the little discs and the special orthogonal group.

Abstract. We show that the Chas-Sullivan loop product, a combination of the Pontrjagin product on the fiber and intersection product on the base, makes sense on the total space homology of any fiberwise monoid E over a closed oriented manifold M . More generally the Thom spectrum E −T M is a ring spectrum. Similarly a fiberwise module over E defines a module over E −T M . Fiberwise monoids include adjoint bundles of principal bundles, and the construction is natural with respect to maps of principal bundles. This naturality implies homotopy invariance of the algebra structure on H * (LM ) arising from the loop product. If M = BG is the infinite dimensional classifying space of a compact Lie group, then we get a well-defined pro-ring spectrum, which we define to be the string topology of BG. If E has a fiberwise action of the little n-cubes operad then E −T M is an En-ring spectrum. This gives homology operations combining Dyer-Lashof operations on the fiber and Steenrod operations on the base. We give several examples where the new operations give homological insight, borrowed from knot theory, complex geometry, gauge theory, and homotopy theory.

We present a counterexample to the conjecture on the homotopy invariance of configuration spaces. More precisely, we consider the lens spaces L-7,L-1 and L-7,L-2, and prove that their configuration spaces are not homotopy equivalent by showing that their universal coverings have different Massey products. (C) 2004 Elsevier Ltd. All rights reserved

The framed n-discs operad f D 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 f D n is equivalent to the n-fold loop space on an SO(n)-space. Examples of f D 2 -spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of f D 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. IntroductionThe topology of iterated loop spaces was thoroughly investigated in the seventies. These spaces have a wealth of homology operations parametrized by the famous operads of little discs, denoted by D n in the text. The notion of operad was introduced in the first place for this purpose [2,19]. Such machinery allows, for example, the reconstruction of an iterated delooping if one has full knowledge of the operad action on an iterated loop space. Moreover any connected space acted on by the little discs is weakly homotopy equivalent to an iterated loop space. This fact is the celebrated recognition principle.Our main objective is to extend this theory by adding the operations rotating the discs. The operad generated by the little n-discs D n and the rotations in SO(n) is the framed n-discs operad f D n , first introduced in [10]. Our recognition principle for framed n-discs (Theorem 3.1) says that a connected (or grouplike) space acted on by the framed n-discs operad is weakly homotopic to an n-fold loop space on an SO(n)-space.Thus the looping and delooping functors induce a categorical equivalence between SO(n)-spaces and spaces acted on by the framed n-discs operad, under the correct connectivity assumptions. The main technique consists in presenting the framed little discs as a semidirect product of the little discs and the special orthogonal group.