Genuine equivariant E_\infty ring spectra, normed algebras, and lax limits

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• Tobias Lenz (Rheinische Friedrich-Wilhelms-Universität Bonn)
• Tuesday 06 May 2025, 14:00-15:00
• Seminar Room 2, Newton Institute.

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EHT - Equivariant homotopy theory in context

Multiplicative structures on cohomology theories have been exploited fruitfully throughout the history of algebraic topology, ranging from the basic and classical argument that the Hopf maps are stably non-trivial to the more recent (and very much non-basic) use of equivariant power operations in the solution of the Kervaire invariant one problem due to Hill—Hopkins—Ravenel. The latter crucially relies on a refined notion of `genuine commutative algebras’ in equivariant spectra, containing more information (in the form of `twisted power operations’) than just E_\infty algebras in the \infty-category of spectra. While originally these objects were defined using well-behaved pointset models of spectra, in more recent years an alternative \infty-categorical approach has been suggested by Bachmann-Hoyois and Nardin-Shah. In this talk I will report on joint work with Sil Linskens and Phil Pützstück, in which we construct an equivalence between the two suggested definitions of `genuine commutative G-ring spectra.’ I will further explain how our methods yield an \infty-categorical description of Schwede’s `ultra-commutative global ring spectra’ (again originally defined by a pointset model), and how this allows one to make precise the idea that an ultra-commutative global ring spectrum ought to be a compatible family of genuine commutative G-E_\infty ring spectra for all (finite) groups G.

This talk is part of the Isaac Newton Institute Seminar Series series.

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