Topological Operads Research Articles

Dold spaces in homotopy theory

Numerably contractible spaces play an important role in the theory of homotopy pushouts and pullbacks. The corresponding results imply that a number of well known weak homotopy equivalences are genuine ones if numerably contractible spaces are involved. In this paper we give a first systematic investigation of numerably contractible spaces. We list the elementary properties of the category of these spaces. We then study simplicial objects in this category. In particular, we show that the topological realization functor preserves fibration sequences if the base is path-connected and numerably contractible in each dimension. Consequently, the loop space functor commutes with realization up to homotopy. We give simple conditions which assure that free algebras over a topological operad are numerably contractible.

The topological cyclic Deligne conjecture

Let O be a cyclic topological operad with multiplication. In the framework of the cosimplicial machinery by McClure and Smith, we prove that the totalization of the cosimplicial space associated to O has an action of an operad equivalent to the framed little 2‐discs operad. 18D50; 55P48

Monoidal Functors, Acyclic Models and Chain Operads

AbstractWe prove that for a topological operad P the operad of oriented cubical singular chains, , and the operad of simplicial singular chains, S*(P), are weakly equivalent. As a consequence, is formal if and only if is formal, thus linking together some formality results which are spread out in the literature. The proof is based on an acyclic models theorem for monoidal functors. We give different variants of the acyclic models theorem and apply the contravariant case to study the cohomology theories for simplicial sets defined by R-simplicial differential graded algebras.

Knots, operads, and double loop spaces

We show that the space of long knots in an Euclidean space of dimension larger than three is a double loop space, proving a conjecture by Sinha. We also construct a double loop space structure on framed long knots, and show that the map forgetting the framing is not a double loop map in odd dimension. However, there is always such a map in the reverse direction expressing the double loop space of framed long knots as a semidirect product. A similar compatible decomposition holds for the homotopy fiber of the inclusion of long knots into immersions. We also show via string topology that the space of closed knots in a sphere, suitably desuspended, admits an action of the little 2-discs operad in the category of spectra. A fundamental tool is the McClure-Smith cosimplicial machinery, that produces double loop spaces out of topological operads with multiplication.

Bar constructions for topological operads and the Goodwillie derivatives of the identity

We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.