Global stable splittings of Stiefel manifolds

TL;DR

This paper develops refined equivariant stable splittings for classical groups and their homogeneous spaces, incorporating Galois actions and extending global stable homotopy theory.

Contribution

It provides new global equivariant refinements of Miller's stable splittings for classical groups and introduces a generalized global stable homotopy theory with Galois actions.

Findings

Refined equivariant stable splittings for orthogonal, unitary, and symplectic groups.

Inclusion of Galois group actions in the stable splitting framework.

Extension of global stable homotopy theory to accommodate extrinsic group actions.

Abstract

We prove global equivariant refinements of Miller's stable splittings of the infinite orthogonal, unitary and symplectic groups, and more generally of the spaces $O/O(m)$, $U/U(m)$ and $Sp/Sp(m)$. As such, our results encode compatible equivariant stable splittings, for all compact Lie groups, of specific equivariant refinements of these spaces. In the unitary and symplectic case, we also take the actions of the Galois groups into account. To properly formulate these Galois-global statements, we introduce a generalization of global stable homotopy theory in the presence of an extrinsic action of an additional topological group.