Embeddings, Normal Invariants and Functor Calculus

TL;DR

This paper explores the homotopy-theoretic structure of Poincare embeddings using functor calculus, introducing a tower that connects immersions to unlinked embeddings and relating it to manifold calculus.

Contribution

It constructs a new tower interpolating between Poincare immersions and unlinked embeddings, utilizing Goodwillie calculus and addressing an open question in the field.

Findings

Describes a tower of spaces connecting immersions to unlinked embeddings.

Identifies layers of the tower via coefficient spectra in homotopy calculus.

Proposes a conjectural link between Poincare embedding tower and manifold calculus.

Abstract

This paper investigates the space of codimension zero embeddings of a Poincare duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincare immersions to a certain space of "unlinked" Poincare embeddings. The layers of this tower are described in terms of the coefficient spectra of the identity appearing in Goodwillie's homotopy functor calculus. We also answer a question posed to us by Sylvain Cappell. The appendix proposes a conjectural relationship between our tower and the manifold calculus tower for the smooth embedding space.