Algebraic & Geometric Topology

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

Piotr Beben

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Let p be an odd prime, and fix integers m and n such that 0<m<n(p1)(p2). We give a p–local homotopy decomposition for the loop space of the complex Stiefel manifold Wn,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 Wn,m. Upper bounds for p–exponents in the stable range 2m<n and 0<m(p1)(p2) are computed as well.

Article information

Algebr. Geom. Topol., Volume 10, Number 2 (2010), 1089-1106.

Received: 10 February 2009
Revised: 7 January 2010
Accepted: 7 January 2010
First available in Project Euclid: 19 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55P15: Classification of homotopy type 55P35: Loop spaces 55Q05: Homotopy groups, general; sets of homotopy classes 57T20: Homotopy groups of topological groups and homogeneous spaces

Stiefel manifold homotopy decomposition homotopy exponent


Beben, Piotr. $p$–Primary homotopy decompositions of looped Stiefel manifolds and their exponents. Algebr. Geom. Topol. 10 (2010), no. 2, 1089--1106. doi:10.2140/agt.2010.10.1089.

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  • \bibmarginparWhat is the title of your thesis? Is it online? P Beben, PhD thesis, University of Aberdeen (2009)
  • F R Cohen, J C Moore, J A Neisendorfer, The double suspension and exponents of the homotopy groups of spheres, Ann. of Math. $(2)$ 110 (1979) 549–565
  • F R Cohen, J A Neisendorfer, A construction of $p$–local $H$-spaces, from: “Algebraic topology, Aarhus 1982 (Aarhus, 1982)”, (I Madsen, B Oliver, editors), Lecture Notes in Math. 1051, Springer, Berlin (1984) 351–359
  • M C Crabb, K Knapp, James numbers, Math. Ann. 282 (1988) 395–422
  • B Harris, On the homotopy groups of the classical groups, Ann. of Math. $(2)$ 74 (1961) 407–413
  • P G Kumpel, Jr, Lie groups and products of spheres, Proc. Amer. Math. Soc. 16 (1965) 1350–1356
  • M Mimura, G Nishida, H Toda, Localization of ${\rm CW}$–complexes and its applications, J. Math. Soc. Japan 23 (1971) 593–624
  • M Mimura, G Nishida, H Toda, ${\rm Mod}\ p$ decomposition of compact Lie groups, Publ. Res. Inst. Math. Sci. 13 (1977) 627–680
  • M Mimura, H Toda, Topology of Lie groups. I, II, Transl. Math. Monogr. 91, Amer. Math. Soc. (1991) Translated from the 1978 Japanese edition by the authors
  • J A Neisendorfer, $3$–primary exponents, Math. Proc. Cambridge Philos. Soc. 90 (1981) 63–83
  • S D Theriault, $2$–primary exponent bounds for Lie groups of low rank, Canad. Math. Bull. 47 (2004) 119–132
  • S D Theriault, The odd primary $H$–structure of low rank Lie groups and its application to exponents, Trans. Amer. Math. Soc. 359 (2007) 4511–4535