Equivariant Weiss Calculus and Loop Spaces of Stiefel Manifolds

TL;DR

This paper develops an equivariant Weiss calculus approach to establish stable splittings of loop spaces of Stiefel manifolds, extending classical results and providing new equivariant decompositions.

Contribution

It introduces an equivariant Weiss calculus framework to prove stable splittings of loop spaces of Stiefel manifolds, including the real case, which was previously unknown.

Findings

Proves an equivariant stable splitting of $oldsymbol{oldsymbol{ ext{ extOmega}} U(V; W)}$.

Provides a stable splitting of path spaces in unitary and orthogonal groups.

Extends classical splittings to an equivariant setting for loop spaces.

Abstract

In the mid 1980s, Steve Mitchell and Bill Richter produced a filtration of the Stiefel manifolds $O(V; W)$ and $U(V; W)$ of orthogonal and unitary, respectively, maps $V \to V \oplus W$ stably split as a wedge sum of Thom spaces defined over Grassmanians. Additionally, they produced a similar filtration for loops on $SU(V)$, with a similar splitting. A few years later, Michael Crabb made explicit the equivariance of the Stiefel manifold splittings and conjectured that the splitting of the loop space was equivariant as well. However, it has long been unknown whether the loop space of the real Stiefel manifold (or even the special case of $\Omega SO_n$) has a similar splitting. Here, inspired by the work of Greg Arone that made use of Weiss' orthogonal calculus to generalize the results of Mitchell and Richter, we obtain an $\mathbb{Z}/2 \mathbb{Z}$-equivariant splitting theorem using an equivariant version of Weiss calculus. In particular, we show that $\Omega U(V; W)$ has an equivariant stable splitting when $dim(W) > 0$. By considering the (geometric) fixed points of this loop space, we also obtain, as a corollary, a stable splitting of the space $\Omega(U(V; W),O(V_{\mathbb{R}}; W_{\mathbb{R}}))$ of paths in $U(V; W)$ from $I$ to a point of $O(V_{\mathbb{R}}; W_{\mathbb{R}})$ as well. In particular, by setting $W = \mathbb{C}$, this gives us a stable splitting of $\Omega(SU_n / SO_n)$.