Homotopy exponents of Stiefel manifolds in the stable range
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.
scientific article; zbMATH DE number 6646694
scientific article