Homotopy exponents of Stiefel manifolds in the stable range
type: Item
instance of: scholarly article
title: Homotopy exponents of Stiefel manifolds in the stable range
zbMATH Open document ID: 1351.55010
author: Linan Zhong
author: Wenhuai Shen
author: Hao Zhao
publication date: 2016-11-02T00:00:00Z
review text: In this paper the authors study the homotopy exponent problem. For a space \(X\), they define the \(p\)-primary homotopy exponent of a space \(X\), written \(\exp(X)\), as \(p^t\) if \(t\) is the minimal power of \(p\) which annihilates the \(p\)-torsion of \(\pi_\ast(X)\). We know from the literature that many researchers computed homotopy exponents for different spaces such as spheres \(S^{2n+1}\) or \(\mod p^r\) Moore spaces \(P^m\). Theriault gave upper bounds for the homotopy exponents of some classical Lie groups of low-rank and two exceptional groups such as \(E_7\) and \(E_8\). By method of homotopy decomposition and constructing spherical fibrations, Grbić-Zhao studied the homotopy exponents of some homogeneous spaces such as low-rank complex Stiefel manifolds and complex Grassmann manifolds, \(\mathrm{SU}(2n)/\mathrm{Sp}(n)\), \(E_6/F_4\) and \(F_4/G_2\). Especially, for the Stiefel manifold \(W_{m+k,k}=\mathrm{SU}(m+k)/\mathrm{SU}(m)\), Grbić gave an upper bound of its homotopy exponent when \(m+k\leq(p-1)2+1\), where \(p\) is an odd integer. Since the homology of \(W_{m+k,k}\) has rank \(k\), not related to \(m\), the authors try to generalize this result to the case \(k\leq(p-1)2+1\). In this paper they solve this problem in the stable range.
reviewed by: Cenap Özel
Mathematics Subject Classification ID: 55P45
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 57T20
zbMATH DE Number: 6646694
zbMATH Keywords: Stiefel manifold
zbMATH Keywords: homotopy exponent
zbMATH Keywords: \(H\)-space
MaRDI profile type: Publication
full work available at URL: s10114-016-5223-y
OpenAlex ID: W2508881476
cites work: The double suspension and exponents of the homotopy groups of spheres
cites work: Q3704619
cites work: Torsion free mod p H-spaces of low rank
cites work: On the sphere of origin of infinite families in the homotopy groups of spheres
cites work: Suspension splittings and James–Hopf invariants
cites work: HOMOTOPY EXPONENTS OF SOME HOMOGENEOUS SPACES
cites work: Localization of CW-complexes and its applications
cites work: Cohomology operations and the homotopy of compact Lie groups. I
cites work: 3-primary exponents
cites work: Q3490634
cites work: On natural coalgebra decompositions of tensor algebras and loop suspensions
cites work: The functor \(A^{\min}\) on \(p\)-local spaces
cites work: The 5-primary homotopy exponent of the exceptional Lie group \(E_8\)
cites work: The odd primary $H$-structure of low rank Lie groups and its application to exponents
cites work: An upper bound for the 3-primary homotopy exponent of the exceptional Lie group \(E_7\)
cites work: The functor \(A^{\text{min}}\) for \((p-1)\)-cell complexes and EHP sequences
DOI: 10.1007/S10114-016-5223-Y
published in: Acta Mathematica Sinica, English Series
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description: scientific article; zbMATH DE number 6646694