Localization of Andre-Quillen-Goodwillie towers, and the periodic homology of infinite loopspaces

Let K(n) be the nth Morava K–theory at a prime p. This paper is a thorough study of questions like the following: to what extent does the K(n)–localization, or the K(n)–homology, of a spectrum X determine the K(n)–homology of its 0th space X_0? Our methods combine techniques from modern homotopical algebra with chromatic homotopy. In particular, we use the telescopic functors of Bousfield and the author (dependent on the Nilpotence Theorem of Devanitz, Hopkins, and Smith), as wel as Topological Andre–Quillen Homology and Goodwillie calculus in nonconnective settings.