Mod-$p$ homotopy decompositions of looped Stiefel manifolds
Homology, Homotopy and Applications
Let W n,k be the Stiefel manifold U (n)/U (n − k). For odd primes p and for k ≤ (p − 1)(p − 2), we give a homotopy decomposition of the based loop space ΩW n,k as a product of p − 1 factors, each of which is the based loops on a finite H-space. Similar decompositions are obtained for Sp(n)/Sp(n − k) and O(n)/O(n − k) and upper bounds on the homotopy exponents are obtained. Introduction Let V n,k = O(n)/O(n − k), W n,k = U (n)/U (n − k) and X n,k = Sp(n)/Sp(n − k) be the real, complex and
... complex and quaternionic Stiefel manifolds respectively. The topology of Stiefel manifolds is of long-standing interest, and many of their properties have been determined. James' book [J2] on the subject is an excellent exposition of what was done up to the late 1970's. In terms of homotopy decompositions, Miller [M] gave stable decompositions of W n,k and X n,k , which were later refined in different ways by Crabb [C] and Yang [Yan]. Unstably, a product decomposition of W n,k or X n,k is unlikely since, in general, Stiefel manifolds are not H-spaces, even when localized at an odd prime. Nevertheless, Hemmi [He] and Yamaguchi [Yam] have determined many cases when W n,k and X n,k are homotopy equivalent to a product of odd dimensional spheres when localized at an odd prime. It is more reasonable to ask for a product decomposition of the loop spaces ΩW n,k and ΩX n,k . Mimura, Nishida and Toda's [MNT2] work on mod-p homotopy decompositions of simple, compact Lie groups may lead to mod-p decompositions of ΩW n,k and ΩX n,k . However, the factors would only be opaquely identified as the homotopy fibres of maps between various factors of the Lie groups. Recently, using a different approach, Beben [B] and Grbić and Zhao [GZ] gave p-local loop space decompositions of ΩW n,k for n ≤ (p − 1)(p − 2) and ΩX n,k for n ≤ (p − 1)(p − 2)/2, where the factors are better identified as the loops on finite H-spaces. In this paper we greatly improve on Beben's and Grbić and Zhao's results. We show that if k ≤ (p − 1)(p − 2) then for any n there is p-local loop space decomposition of ΩW n,k as a product of loop spaces on finite H-spaces, and if k ≤ (p − 1)(p − 2)/2 then for any n there is a p-local loop space decomposition of ΩX n,k as a product of loop spaces on finite H-spaces. To state our results explicitly, we introduce some notation. From now on, assume that all spaces and maps have been localized at an odd prime p, and take homology with mod-p coefficients. Recall 2010 Mathematics Subject Classification. Primary 55P35, Secondary 55Q52.