Some Results on Operads and Configuration Spaces

The little n-disk operad E_n is constructed geometrically as the collection of configuration spaces of n-disks. This is not an isolated example of an interaction between operads and geometry: geometric objects often are either acted upon by operads, form operads, or have associated moduli spaces which form operads. One major area of study in geometric topology is the configuration spaces of nonoverlapping points in M, and more generally, embedding spaces of one manifold into another. In this thesis, we study these spaces from the perspective of operads in spectra. The category of spectra is the "linearization'' of the category of spaces, and as such it has a rich algebraic structure which parallels that of the category of chain complexes. As a result, many classical techniques from from the homological algebra of chain complexes can be used. One such technique is Koszul duality for operads, modules, and algebras.

We initially study the stabilization of configuration space as an object in spectra, computing its Spanier-Whitehead dual and deducing the homotopy invariance of stabilized configuration space. For a framed manifold M of dimension n, the collection of all configuration spaces Conf(M,i) has a right module action of the E_n operad. We study the E_n operad action on the collection of configuration spaces and deduce results about Browder brackets in the homology of configuration spaces. We also deduce homotopy invariance results concerning the associated graded of filtrations of stabilized embedding spaces. Ultimately, we prove that the stabilized configuration spaces of a framed manifold are a fixed point of Koszul duality for right modules. In other words, the collection of stabilized configuration spaces of a framed manifold is Koszul self dual. This statement generalizes Atiyah duality to include the action of the E_n operad and has applications to spaces of embeddings and configuration spaces with labels in an E_n-algebra.