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Lie algebras and \(v_n\)-periodic spaces. (English) Zbl 1481.55007

Through this important work, the author places rational homotopy theory and the analogous \(v_n\)-periodic homotopy theory for spaces in the context of higher category theory and higher algebra, develops and applies tools in this framework to obtain desirable (spectral) algebraic models for (unstable) \(v_n\)-periodic homotopy theory, along with a conceptual understanding of the Bousfield-Kuhn functor and its Goodwillie tower, among other things.
More precisely, for each \(n\geq1\), upon defining a (locally small) full \(\infty\)-subcategory \(\mathcal{S}_{v_n}\) of pointed spaces by inverting the \(v_n\)-periodic equivalence, the author establishes as a main result (Theorem 2.6) an equivalence between \(\mathcal{S}_{v_n}\) and the \(\infty\)-category of Lie algebras in \(T(n)\)-local spectra, where the Lie algebra structure arises from the adjunction between \[\Theta\!:\mathrm{Sp}_{T(n)} \to \mathcal{S}_{v_n}\text{ and }\Phi\!: \mathcal{S}_{v_n} \to \mathrm{Sp}_{T(n)}\] with \(\Phi\) equivalent to the restriction of the Bousfield-Kuhn functor to this subcategory, so that \(\Phi\) is also equivalent to the forgetful functor of the Lie algebra structure. For example, the classical Whitehead bracket on the homotopy groups of a pointed space corresponds to this Lie algebra structure in a precise way. A \(K(n)\)-local analogue of the main theorem follows as a formal consequence (but not the other way around).
On the other hand, for a Koszul-dual statement, a second stabilize-destabilize adjunction between \[\Sigma^\infty_{T(n)}\!:\mathcal{S}_{v_n} \to\mathrm{Sp}_{T(n)}\text{ and }\Omega^\infty_{T(n)}\!: \mathrm{Sp}_{T(n)}\to \mathcal{S}_{v_n}\] leads to a comparison between \(\mathcal{S}_{v_n}\) and the \(\infty\)-category of commutative ind-coalgebras in \(T(n)\)-local spectra, which is shown to be an equivalence only for levelwise truncations of the Goodwillie approximation of \(\mathcal{S}_{v_n}\) (along the tower of the identity). In this way, the author identifies the Goodwille tower of the Bousfield-Kuhn functor \(\Phi\) levelwise as primitives for this coalgebra structure. Due to the issue with convergence of the Goodwillie tower of the identity, not all spaces enjoy the full equivalence without truncation, and interesting examples and non-examples are studied by the author here, in joint work elsewhere, as well as by others, with computational consequences, particularly for spheres and \(n\leq2\).
The reviewer recommends the survey [M. Behrens and C. Rezk, Springer Proc. Math. Stat. 309, 275–323 (2020; Zbl 1456.55007)] for an accessible and generous introduction to this series of exciting developments in unstable \(v_n\)-periodic homotopy theory by the author here, by Behrens and Rezk, and by G. Arone and M. Ching, among others, through a variety of approaches, as vast generalizations of Quillen’s and others’ pioneering work in rational homotopy theory, as well as a set of seventeen lecture notes from the Harvard Thursday seminars on unstable chromatic homotopy theory organized by J. Lurie and M. J. Hopkins. It is also worth mentioning that algebra and coalgebra objects of \(\infty\)-categories, such as those appearing in this work, play a pivotal role in spectral algebraic geometry. Further computations with \(v_n\)-periodic homotopy theory will surely elucidate the advances made here.

MSC:

55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55P62 Rational homotopy theory
55Q51 \(v_n\)-periodicity

Citations:

Zbl 1456.55007
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References:

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