Embedding Calculus, Goodwillie Calculus and Link Invariants

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.