The Goodwillie Filtration and the Genealogy of Unstable Elements

Previously, we constructed the transfinite Goodwillie spectral sequence (TGSS), whose signature is roughly E t,J(S ) = πtS ⇒ πtS. This spectral sequence computes unstable homotopy from stable data. Over the next couple of talks, we’ll discuss the differentials in this spectral sequence, but for today, let’s just focus on understanding the relationship between the E page and the E∞ page at a high level. Before jumping in, recall that elements in the E t,J term of the TGSS are denoted α[J ], where α ∈ πtS is a stable element and J is CU sequence with excess ≥ n. Let β ∈ πtS. Where does it come from in the E1 page? In other words,