Splittings in rational equivariant homotopy theory

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• Magdalena Kedziorek (Radboud Universiteit Nijmegen)
• Monday 12 May 2025, 11:45-12:45
• Seminar Room 1, Newton Institute.

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EHTW03 - New horizons for equivariance in homotopy theory

When G is a finite group, rational genuine G-spectra split as a product of simpler categories: rational spectra with Weyl group actions. Algebraically, this is the statement that rational Mackey functors are semi-simple and the category of rational G-Mackey functors splits as a product of module categories with Weyl group actions.  Classically, the homotopy theory of G-spectra depended on a choice of a universe – this was modelled, for example, by the orthogonal G-spectra of Mandell and May. In modern language, Blumberg and Hill generalised the construction of the homotopy theory of G-spectra and show that one can choose an additive structure of it to be modelled by a homotopy type of an N_\infty operad O (or equivalently a transfer system associated to O). Thus homotopy groups of a G-spectrum with additive structure given by O have those additive transfers that are parametrised by O. We call such a structure a G-Mackey functor for O. The purpose of this talk is to describe splittings of rational G-Mackey functors for incomplete transfer systems and discuss how this extends to the topological setting. This is joint work with Anna Marie Bohmann, Dave Barnes and Mike Hill.

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

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