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Chromatic structures in stable homotopy theory. (English) Zbl 1476.55026

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 163-220 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: This chapter explains how the solution of the Ravenel Conjectures by Ethan S. Devinatz, Michael J. Hopkins, D. C. Ravenel, and Jeffrey H. Smith leads to a canonical filtration in stable homotopy theory. It also explains that the chromatic filtration arises canonically from the global structure of the stable homotopy category. The chapter describes the geometric origins of the chromatic filtration through the relation with the stack of formal groups. As expressed in Waldhausen’s vision of brave new algebra, the category Sp of spectra should be thought of as a homotopical enrichment of the derived category \(\mathcal {D}_{\mathbb Z}\) of abelian groups. The chapter demonstrates that many geometric structures have homotopical manifestations in the chromatic picture. The final ingredient in the formulation of the asymptotic algebraicity of chromatic homotopy theory is the algebraic model itself.
For the entire collection see [Zbl 1468.55001].

MSC:

55P42 Stable homotopy theory, spectra
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology

Citations:

Zbl 1468.55001
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