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In the mid 1980s, Steve Mitchell and Bill Richter produced a filtration ofthe 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 Thomspaces defined over Grassmanians. Additionally, they produced a similarfiltration for loops on $SU(V)$, with a similar splitting. A few years later,Michael Crabb made explicit the equivariance of the Stiefel manifold splittingsand 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 Stiefelmanifold (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' orthogonalcalculus to generalize the results of Mitchell and Richter, we obtain an$\mathbb{Z}/2 \mathbb{Z}$-equivariant splitting theorem using an equivariantversion of Weiss calculus. In particular, we show that $\Omega U(V; W)$ has anequivariant stable splitting when $dim(W) > 0$. By considering the (geometric)fixed points of this loop space, we also obtain, as a corollary, a stablesplitting of the space $\Omega(U(V; W),O(V_{\mathbb{R}}; W_{\mathbb{R}}))$ ofpaths 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 stablesplitting of $\Omega(SU_n / SO_n)$.

Equivariant Weiss Calculus and Loop Spaces of Stiefel Manifolds