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

Abstract

We identify the group of homomorphisms HomGF (F, RU Q ) in the abelian category GF of (fin)global functors to the rationalization of the unitary representation ring functor RU Q . The result is a certain inverse limit over the values of F at cyclic groups and we deduce that the higher Ext-groups Ext n GF (F, RU Q ) have to vanish for n 2. This leads to a rational splitting of the (fin)global equivariant K-theory spectrum KU 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.

Orbifolds, Orbispaces and Global Homotopy Theory

“…The rationalized K-theory is represented by the rationalized periodic global K-theory spectrum KU Q . By [42,Thm. 1.6], this spectrum decomposes up to Fin-global equivalence into a sum of shifted Eilenberg-MacLane spectra…”