Lie algebras and $v_n$-periodic spaces

Abstract

We consider a homotopy theory obtained from that of pointed spaces by inverting the maps inducing isomorphisms in vn-periodic homotopy groups. The case n = 0 corresponds to rational homotopy theory. In analogy with Quillen's results in the rational case, we prove that this vn-periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in T (n)-local spectra. We also compare it to the homotopy theory of commutative coalgebras in T (n)-local spectra, where it turns out there is only an equivalence up to a certain convergence issue of the Goodwillie tower of the identity. Contents 1. Introduction Acknowledgments 2. Main results 3. The v n -periodic homotopy theory of spaces 3.1. The ∞-category L f n 3.2. The ∞-category M f n 3.3. The stabilization of M f n 3.4. The Bousfield-Kuhn functor 4. Lie algebras in T (n)-local spectra 4.1. Operads and cooperads of T (n)-local spectra 4.2. The functor ΦΘ 4.3. The monad structure of ΦΘ 4.4. Some applications 4.5. A variation for K(n)-local homotopy theory 5. The Goodwillie tower of M f n 5.1. Stabilizations and Goodwillie towers 5.2. The Goodwillie tower of the Bousfield-Kuhn functor Appendix A. Dual Goodwillie calculus Appendix B. A nilpotence lemma for differentiation B.1. Statement of the lemma B.2. Some preliminaries on nilpotence B.3. Proof Appendix C. CoalgebrasReferences Bousfield-Kuhn functor and topological André-Quillen homology. I have benefitted much from reading their work, as well as from several stimulating talks by and conversations with Mark Behrens. Theorem 2.6 (the comparison with Lie algebras) offers a different (and sharper) perspective on v n -periodic homotopy theory, which builds on joint work with Rosona Eldred, Akhil Mathew, and Lennart Meier [19] carried out at the Hausdorff Research Institute for Mathematics. I wish to thank my collaborators for an inspiring semester and the Institute for its hospitality and excellent working conditions. Moreover, I thank Greg Arone and Lukas Brantner for useful conversations relating to this paper. A large intellectual debt is owed to

The $v_n$-periodic Goodwillie tower on wedges and cofibres

“…Our methods in fact provide a useful tool in the study of v h -periodic spaces through spectral Lie algebras. For example, Theorem 2.5 was originally used in [10] to establish a more direct relation between the Bousfield-Kuhn functor and spectral Lie algebras which does not suffer from the aforementioned convergence issues.…”

“…Remark 5.5. In Theorem 2.11 of [10] it is in fact shown that Φ v (M ℓ ) is simply the direct sum of all its homogeneous layers (and similarly for higher heights h and appropriate type h complexes in place of M ℓ ); by the arguments above this can indeed not be equivalent to the infinite product. However, the arguments of [10] require more technology than is necessary to include here.…”

“…Statement (C) is established in 5.9 of [7]. Statement (D) is Theorem 3.8 of [10]. In fact, all four of these statements are summarised in Section 3 of op.…”

“…Proof. The following proof is a condensed version of a more detailed analysis of Goodwillie calculus in v h -periodic homotopy theory appearing in [10]. Following Bousfield [6][7], we let V h+1 be a finite space of type h+1 which is also a suspension.…”

Ambidexterity in Chromatic Homotopy Theory

“…It was also recently used in [Heu18] to generalize Quillen's rational homotopy theory to higher chromatic heights. We shall now describe an application of Theorem A, i.e.…”