Embedding Calculus, Goodwillie Calculus and Link Invariants
Hyeonhee Jin
TL;DR
This work links Goodwillie–Weiss embedding calculus with Goodwillie’s functor calculus by constructing a functorial $T_{n}$-complement valued in the categorical $n$-excisive approximation $oldsymbol{P}_{n}oldsymbol{S}_{*}$. It develops a $T_{n}$-complement mechanism, extending complement data from genuine embeddings to $T_{n}$-embeddings and establishing a lax map between the embedding tower and the Goodwillie tower. A Stallings–type theorem for $T_{n}$-complements in $D^{d}$ shows that boundary data control the $n$-excisive approximations, enabling a stable, nilpotent-type analysis of embeddings. In the string-link case, the Artin representation factors through the $n$-th embedding tower, and Milnor invariants of length at most $n+1$ are detected at this stage, linking classical link invariants to modern homotopy-theoretic towers.
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.