Splittings of global Mackey functors and regularity of equivariant Euler classes

Abstract

We establish natural splittings for the values of global Mackey functors at orthogonal, unitary and symplectic groups. In particular, the restriction homomorphisms between the orthogonal, unitary and symplectic groups of adjacent dimensions are naturally split epimorphisms. The interest in the splitting comes from equivariant stable homotopy theory. The equivariant stable homotopy groups of every global spectrum form a global Mackey functor, so the splittings imply that certain long exact homotopy group sequences separate into short exact sequences. For the real and complex global Thom spectra $\mathbf{MO}$ and $\mathbf{MU}$, the splittings imply the regularity of various Euler classes related to the tautological representations of $O(n)$ and $U(n)$.

References in corpus (1)

• Global homotopy theory

Cited by in corpus (2)

• The completion and local cohomology theorems for complex cobordism for all compact Lie groups
• Chern classes in equivariant bordism