A streamlined proof of the convergence of the Taylor tower for embeddings in $\mathbb R^n$

TL;DR

This paper simplifies the proof of the convergence of the Taylor tower for embedding spaces into ^n, making the results more accessible and providing improvements to existing arguments.

Contribution

It offers a streamlined proof of convergence for the Taylor tower in the case of embeddings into ^n, with minimal machinery and enhanced clarity.

Findings

Convergence of the Taylor tower for embeddings into ^n is established.

The proof is simplified and made more accessible.

Several improvements and restatements of existing arguments are provided.

Abstract

Manifold calculus of functors has in recent years been successfully used in the study of the topology of various spaces of embeddings of one manifold in another. Given a space of embeddings, the theory produces a Taylor tower whose purpose is to approximate this space in a suitable sense. Central to the story are deep theorems about the convergence of this tower. We provide an exposition of the convergence results in the special case of embeddings into $\mathbb R^n$, which has been the case of primary interest in applications. We try to use as little machinery as possible and give several improvements and restatements of existing arguments used in the proofs of the main results.