Spaces of smooth embeddings and configuration categories

TL;DR

This paper uses functor calculus and operad theory to describe obstructions in deforming smooth immersions into embeddings, providing new insights into high-dimensional knot spaces.

Contribution

It offers an operadic framework for understanding obstructions in smooth embedding deformations, advancing the study of high-dimensional knot spaces.

Findings

Operadic description of obstructions to smooth embedding deformations

Improved results on high-dimensional long knots

Connections between functor calculus and operad theory

Abstract

In the homotopical study of spaces of smooth embeddings, the functor calculus method (Goodwillie-Klein-Weiss manifold calculus) has opened up important connections to operad theory. Using this and a few simplifying observations, we arrive at an operadic description of the obstructions to deforming smooth immersions into smooth embeddings. We give an application which in some respects improves on recent results of Arone-Turchin and Dwyer-Hess concerning high-dimensional variants of spaces of long knots.