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

Let K(n)K(n) be the nnth Morava KK-theory at a prime p  , and let T(n)T(n) be the telescope of a vnvn-self map of a finite complex of type n  . In this paper we study the K(n)*K(n)*-homology of Ω∞XΩ∞X, the 0th space of a spectrum X, and many related matters.We give a sampling of our results.Let PXPX 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 mapsn(X):LT(n)P(X)→LT(n)Σ∞(Ω∞X)+sn(X):LT(n)P(X)→LT(n)Σ∞(Ω∞X)+of commutative algebras over the localized sphere spectrum LT(n)SLT(n)S. The induced map of commutative, cocommutative K(n)*K(n)*-Hopf algebrassn(X)*:K(n)*(PX)→K(n)*(Ω∞X),sn(X)*:K(n)*(PX)→K(n)*(Ω∞X),satisfies the following properties.It is always monic.It is an isomorphism if X is n  -connected, πn+1(X)πn+1(X) is torsion, and T(i)*(X)=0T(i)*(X)=0 for 1⩽i⩽n-11⩽i⩽n-1. It is an isomorphism only if K(i)*(X)=0K(i)*(X)=0 for 1⩽i⩽n-11⩽i⩽n-1.It is universal. The domain of sn(X)*sn(X)* preserves K(n)*K(n)*-isomorphisms, and if F   is any functor preserving K(n)*K(n)*-isomorphisms, then any natural transformation F(X)→K(n)*(Ω∞X)F(X)→K(n)*(Ω∞X) factors uniquely through sn(X)*sn(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 sn(X)*sn(X)* is always monic uses Topological André-Quillen Homology and Goodwillie Calculus in nonconnective settings.