The Mitchell-Richter filtration of loops on Stiefel manifolds stably splits (Q2701602)

When \(X\) is a connected space the iterated loop spaces \(\Omega^{n} \Sigma^{n}X\) have combinatorial models which are filtered by subspaces \(F_{k}\Omega^{n} \Sigma^{n}X\). The Snaith splittings of [the reviewer, J. Lond. Math. Soc., II. Ser. 7, 577-583 (1974; Zbl 0275.55019)] are natural stable homotopy decompositions of the form \(F_{m}\Omega^{n} \Sigma^{n}X \simeq \bigvee_{k \leq m} (F_{k}\Omega^{n} \Sigma^{n}X)/(F_{k-1}\Omega^{n} \Sigma^{n}X)\). These splittings and their generalizations have proved very useful and ubiquitous, because similar combinatorial models are abundant and they are usually expected stably to decompose in this manner. For example, there are stable homotopy decompositions of \(BO(2n), BU(n)\) and \(BSp(n)\) [the reviewer, Algebraic cobordism and K-theory; Mem. Am. Math. Soc. 221 (1979; Zbl 0413.55004)] and of Stiefel manifolds [\textit{H. R. Miller}, Topology 24, 411-419 (1985; Zbl 0581.55006)]. If \(V\), \(W\) are real (or complex) inner product spaces the space of linear isometric inclusions of \(V\) into \(V \oplus W\) is denoted by \(\text{Mor}(V,V \oplus W)\) and its loop space has an increasing Mitchell-Richter filtration by subspaces \(S^{n}(V,W)\). The author shows that there is a stable homotopy decomposition of the form \(\Omega \text{ Mor}(V,V \oplus W) \simeq \bigvee_{n=1}^{\infty} (S^{n}(V,W))/(S^{n-1}(V,W))\). When \(\dim(W) = 1\) this yields the Mitchell-Richter decomposition of \(\Omega SU(n)\). The proof is an adaptation, using the orthogonal calculus of [\textit{M. Weiss}, Trans. Am. Math. Soc. 347, No. 10, 3743-3796 (1995; Zbl 0866.55020)], of an unpublished proof due to Goodwillie using his calculus of functors, of the original Snaith splittings.