The homotopy groups of a space are amongst the hardest invariants of a space to compute. In stable homotopy theory the perspective of chromatic homotopy theory, filtering the groups by \(v_h\)-periodicity, has had strong computational applications. Davis and Mahowald defined a similar filtration in the unstable case, allowing us to define \(v_h^{-1}\pi_*X\) for a space \(X\).
Bousfield and Kuhn defined the functor \(\phi:Top_*\to Spectra\), such that for any space \(\pi^S_* \phi X\simeq v_h^{-1}\pi_*X\). By work of Arone-Mahowald, and Arone-Dwyer the Goodwillie tower of \(\phi\), called the \(v_h\)-periodic Goodwillie tower, can be understood, though it often fails to converge. Behrens-Rezk gave a list of some spaces that are \(\phi\)-good, meaning that the Goodwillie tower converges when evaluate on these.
Using work of Brantner on Lie power operations allows us to access the \(E_1\)-page of the Morava-\(E\)-theory-based Goodwillie spectral sequence. Using this I will attempt to give a sketch of the computation of the spectral sequence arising from the \(v_h\)-periodic Goodwillie tower at height \(h=1\) for some \(\phi\)-good spaces.