The stable embedding tower and operadic structures on configuration spaces

Abstract

$\def\EmbMN{\operatorname{Emb}(M,N)}\def\EM{E_M}\def\En{E_n}$ Given smooth manifolds $M$ and $N$, manifold calculus studies the space of embeddings $\EmbMN$ 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 $\EmbMN$ 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 $\EM$ is intimately connected to both the stable and unstable embedding towers through the $\En$ operad. The action of $\En$ on $\EM$ induces an action of the Poisson operad poisn on the homology of configuration spaces $H_\ast (F(M,-))$. In order to study this action, we introduce the notion of Poincaré–Koszul operads and modules and show that $\En$ and $\EM$ are examples. As an application, we compute the induced action of the Lie operad on $H_\ast (F(M,-))$ and show it is a homotopy invariant of $M^+$.