The stable embedding tower and operadic structures on configuration spaces

TL;DR

This paper explores the stable embedding tower for manifolds, revealing operadic and homotopy invariance properties of configuration spaces and their homology, and introduces Poincare-Koszul operads to analyze these structures.

Contribution

It introduces Poincare-Koszul operads and modules, linking operadic actions to configuration space invariants and establishing new homotopy invariants for manifolds.

Findings

Layers of the stable embedding tower are tangential homotopy invariants.

Operadic actions induce homology actions that are homotopy invariants.

The Lie operad acts on configuration space homology as a homotopy invariant.

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 $M^+$.