Seminar

Global localization and equivariant Thom spectra

Higher algebraic structures in algebra, topology and geometry

15 March 14:15 - 16:00

Stefan Schwede - University of Bonn

The aim of this talk is twofold. Firstly, I want to explain a systematic formalism to construct and manipulate Thom spectra in global equivariant homotopy theory. The upshot is a colimit preserving symmetric monoidal global Thom spectrum functor from the infinity-category of global spaces over BOP to the infinity-category of global spectra. Here BOP is a particular globally-equivariant refinement of the space Z x BO, which simultaneously represents equivariant K-theory for all compact Lie groups.

Secondly, I want to use the formalism to derive certain universal properties of real and complex bordism in the world of highly structured globally-equivariant spectra. For this we recall that equivariantly, the key features of the complex bordism spectrum are embodied in two different objects: the spectrum mU is equivariantly connective and the natural target for the Thom-Pontryagin construction; the spectrum MU is equivariantly complex-oriented and features in the theory of equivariant formal group laws. These two features are incompatible, and the morphism mU --> MU is *not* an equivalence.

Both equivariant forms of complex bordism assemble into multiplicative global homotopy types. I will explain why the morphism mU --> MU is a localization, in the infinity-category of commutative global ring spectra, at the `inverse Thom classes'. The key principle underlying this result can be summarized by the slogan: `Thom spectra take group completion to localization at inverse Thom classes.'

 

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Organizers
Gregory Arone
Stockholm University
Tilman Bauer
KTH Royal Institute of Technology
Alexander Berglund
Stockholm University
Søren Galatius
University of Copenhagen
Jesper Grodal,
University of Copenhagen
Thomas Kragh
Uppsala University

Program
Contact

Alexander Berglund

alexb@math.su.se

Søren Galatius

galatius@math.ku.dk

Thomas Kragh

thomas.kragh@math.uu.se

Other
information

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