Embedding Calculus, Goodwillie Calculus and Link Invariants

TL;DR

This paper explores the relationship between Goodwillie-Weiss embedding calculus and functor calculus, constructing new functorial tools, establishing analogues of classical theorems, and applying these to detect Milnor invariants in string links.

Contribution

It introduces a functorial complement for T_n-embeddings, establishes an analogue of Stallings' theorem for T_n-embeddings, and applies these to detect Milnor invariants in string links.

Findings

Constructed a functorial complement for T_n-embeddings.

Established an analogue of Stallings' theorem for T_n-embeddings.

Showed the embedding tower of string links detects Milnor invariants.

Abstract

We study Goodwillie-Weiss embedding calculus through its relationship with Goodwillie's functor calculus. Specifically, building on a result of Tillmann and Weiss, we construct a functorial complement for \(T_{n}\)-embeddings that takes values in Heuts's categorical \(n\)-excisive approximation of pointed spaces. We also establish an analogue of Stallings' theorem for lower central series in the context of \(T_{n}\)-embeddings of \(P \times I\) into \(D^{d}\) for any compact manifold \(P\). As an application, we show that the embedding tower of string links detects Milnor invariants.