Splittings of global Mackey functors and regularity of equivariant Euler classes

TL;DR

This paper proves natural splittings for global Mackey functors at classical groups, leading to insights on the structure of equivariant stable homotopy groups and regularity of Euler classes in global Thom spectra.

Contribution

It establishes natural splittings for global Mackey functors at classical groups, revealing new structural properties relevant to equivariant stable homotopy theory.

Findings

Restriction homomorphisms are split epimorphisms.

Long exact sequences split into short exact sequences.

Euler classes in global Thom spectra are regular.

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 $U(n)$.