Localization of André¿Quillen¿Goodwillie towers, and the periodic homology of infinite loopspaces
By N. Kuhn. 2003
50 citations recorded by the source corpus.
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 Andre-Quillen Homology and Goodwillie Calculus in nonconnective settings.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/074856f0b27ce8ee0982de72ae88da80372e3d0f
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Citations in the corpus
- Goodwillie towers and chromatic homotopy: an overview (2004, methodology): “Included in this section, as an application of my work in [30], is an outline of a new way to possibly find a counterexample to the Telescope Conjecture.”
- Homotopy completion and topological Quillen homology of structured ring spectra (2011, background): “As a special case, it recovers Kuhn’s [43] result for non-unital commutative ring spectra, and more generally, Lawson’s [45] original structured ring spectra result (which is based on [32]).”