The stable embedding tower and operadic structures on configuration spaces

Abstract

Given smooth manifolds $M$ and $N$, manifold calculus studies the space of embeddings $\operatorname{Emb}(M,N)$ via the "embedding tower", which is constructed using the homotopy theory of presheaves on $M$. The same theory allows us to study the stable homotopy type of $\operatorname{Emb}(M,N)$ via the "stable embedding tower". By analyzing cubes of framed configuration spaces, we prove that the layers of the stable embedding tower are tangential homotopy invariants of $N$. If $M$ is framed, the moduli space of disks $E_M$ is intimately connected to both the stable and unstable embedding towers through the $E_n$ operad. The action of $E_n$ on $E_M$ induces an action of the Poisson operad $\mathrm{pois}_n$ on the homology of configuration spaces $H_*(F(M,-))$. In order to study this action, we introduce the notion of Poincare-Koszul operads and modules and show that $E_n$ and $E_M$ are examples. As an application, we compute the induced action of the Lie operad on $H_*(F(M,-))$ and show it is a homotopy invariant of $M^+$.