Global stable splittings of Stiefel manifolds
Stefan Schwede
TL;DR
This work provides global equivariant refinements of Miller’s stable splittings for the infinite orthogonal, unitary, and symplectic groups by organizing the Stiefel manifolds into a Galois-global framework. The authors introduce, for each $m\ge0$, unstable filtrations ${\mathbf{L}/m}$ whose strata are described as global Thom spaces over Grassmannians ${\mathbf{Gr}}_k^{\mathbb{K}}$, with explicit equivariant data given by $\nu(k,m)\oplus\mathfrak{ad}(k)$, and prove a $G({\mathbb K})$-global stable splitting in the form $\Sigma^\infty_+ {\mathbf{L}/m} \cong \bigvee_{k\ge0} \Sigma^\infty ( {\mathbf{Gr}}_k^{\mathbb{K}})^{\nu(k,m)\oplus\mathfrak{ad}(k)}$ for ${\mathbb K}\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. The method extends Crabb–Miller’s topology of collapsing open cells to a Galois-global setting, yielding compatible splittings across all compact Lie groups and introducing a $C$-global theory to handle external symmetries (notably the Galois groups for $\mathbb{C}$ and $\mathbb{H}$). In the limit $m\to\infty$, the results produce nontrivial global objects ${\mathbf{O}/\infty}$, ${\mathbf{U}/\infty}$, ${\mathbf{Sp}/\infty}$ intimately connected to global classifying spaces and global stable homotopy theory. Overall, the paper advances global refinements of stable splittings for Stiefel-type spaces and integrates them into a broader $C$-global framework with potential applications in equivariant and orbifold cohomology theories.
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.