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.’