Localization of André–Quillen–Goodwillie towers, and the periodic homology of infinite loopspaces

Abstract

Let K ( n ) be the n th Morava K -theory at a prime p, and let T ( n ) be the telescope of a v n -self map of a finite complex of type n. In this paper we study the K ( n ) * -homology of Ω ∞ X , the 0th space of a spectrum X, and many related matters. We give a sampling of our results. Let P X be the free commutative S-algebra generated by X: it is weakly equivalent to the wedge of all the extended powers of X. We construct a natural map s n ( X ) : L T ( n ) P ( X ) → L T ( n ) Σ ∞ ( Ω ∞ X ) + of commutative algebras over the localized sphere spectrum L T ( n ) S . The induced map of commutative, cocommutative K ( n ) * -Hopf algebras s n ( X ) * : K ( n ) * ( P X ) → K ( n ) * ( Ω ∞ X ) , satisfies the following properties. It is always monic. It is an isomorphism if X is n-connected, π n + 1 ( X ) is torsion, and T ( i ) * ( X ) = 0 for 1 ⩽ i ⩽ n - 1 . It is an isomorphism only if K ( i ) * ( X ) = 0 for 1 ⩽ i ⩽ n - 1 . It is universal. The domain of s n ( X ) * preserves K ( n ) * -isomorphisms, and if F is any functor preserving K ( n ) * -isomorphisms, then any natural transformation F ( X ) → K ( n ) * ( Ω ∞ X ) factors uniquely through s n ( X ) * . The construction of our natural transformation uses the telescopic functors constructed and studied previously by Bousfield and the author, and thus depends heavily on the Nilpotence Theorem of Devanitz, Hopkins, and Smith. Our proof that s n ( X ) * is always monic uses Topological André-Quillen Homology and Goodwillie Calculus in nonconnective settings.