p–Primary homotopy decompositions of looped Stiefel manifolds and their exponents

Abstract

Let p be an odd prime, and fix integers m and n such that 0 < m < n Ä .p 1/.p 2/. We give a p -local homotopy decomposition for the loop space of the complex Stiefel manifold W n;m . Similar decompositions are given for the loop space of the real and symplectic Stiefel manifolds. As an application of these decompositions, we compute upper bounds for the p -exponent of W n;m . Upper bounds for p -exponents in the stable range 2m < n and 0 < m Ä .p

Modpdecompositions of the loop spaces of compact symmetric spaces

“…Remark 5.3. Using a different approach, a homotopy decomposition for Ω(SU(n)/SU(n − m)) is obtained in [1,14] which holds for n ≤ (p − 1)(p − 2). This range includes the quasi-p-regular cases and more.…”

“…using Cohen and Neisendorfer's construction of finite H-spaces [11] (see Theorem 2.2). Here, (2) is an H-fibration with a different H-structure from that in (1), but the maps M(q i ) are simple enough to allow us to identify their homotopy fibres. The paper is organized as follows.…”

“…It is natural to extend the decomposition approach to other spaces related to Lie groups. Some work has been done to determine homotopy decompositions of the loops on certain homogeneous spaces [1,14] and analyze the exponent implications. In this paper we consider the loops on symmetric spaces with an eye towards deducing exponent information.…”

“…Compact, irreducible, simply-connected Riemannian symmetric spaces were classified by Cartan [6,7] and an explicit list as homogeneous spaces was given in [22]. In an ad-hoc manner, the homotopy groups of symmetric spaces have been studied in several papers, for example [1,5,16,17,19,29,36,37]. We give a more systematic approach.…”

“…The paper is organized as follows. In Sections 2 through 4 we obtain the homotopy fibration (2) from (1), and prove properties about it. In Section 5 we identify the maps q i in a case-by-case analysis, and thereby obtain a homotopy decomposition for Ω(G/H).…”