Rational extensions of the representation ring global functor and a splitting of global equivariant $K$-theory

Abstract

We identify the group of homomorphisms ${\mathrm{Hom}}_{\mathcal{GF}}(F,{\mathbf{RU}}_{\mathbb{Q}})$ in the category of ($\mathrm{fin}$)-global functors to the rationalization of the unitary representation ring functor and deduce that the higher $\mathrm{Ext}$-groups ${\mathrm{Ext}}_{\mathcal{GF}}^n(F,{\mathbf{RU}}_{\mathbb{Q}})$, $n\ge 2$ have to vanish. This leads to a rational splitting of the ($\mathrm{fin}$)-global equivariant $K$-theory spectrum into a sum of Eilenberg-MacLane spectra. Interpreted in terms of cohomology theories, it means that the equivariant Chern character is compatible with restrictions along all group homomorphisms.

10 pages

References in corpus (3)

• Global homotopy theory
• Equivariant properties of symmetric products
• Symmetric products and subgroup lattices

Cited by in corpus (1)

• Orbifolds, Orbispaces and Global Homotopy Theory