Homotopy nilpotency of some homogeneous spaces
Let \(\mathbb{K}P^n\) be the projective \(n\)-space for \(\mathbb{K} = \mathbb{R}, \mathbb{C}\), the field of reals or complex numbers and \(\mathbb{H}\), the skew \(\mathbb{R}\)-algebra of quaternions. In [J. Math. Mech. 16, 853--858 (1967; Zbl 0148.17104)], \textit{T. Ganea} developed the homotopy nilpotency of the loop spaces \(\Omega (\mathbb{K}P^n)\), and in [Osaka J. Math. 13, 145-156 (1976; Zbl 0329.55010)], \textit{V. P. Snaith} gave necessary and sufficient conditions for an \(H\)-space to have homotopy nilpotency equal to \(n\). In [Math. Z. 161, 169--183 (1978; Zbl 0367.55009)], \textit{W. Meier} studied relations between the homotopy nilpotency of a loop space and its localizations and profinite completions. Let \(G_{n,m}(\mathbb{K})\), \(F_{n;n_1,\ldots,n_k} (\mathbb{K})\) and \(V_{n,m}(\mathbb{K})\) be the Grassmann, flag and Stiefel manifolds in the \(n\)-dimensional \(\mathbb{K}\)-vector space, respectively. In this paper under review, the author investigates the homotopy nilpotency of the loop spaces on \(G_{n,m}(\mathbb{K})\), \(F_{n;n_1,\ldots,n_k} (\mathbb{K})\) and \(V_{n,m}(\mathbb{K})\), and homotopy nilpotency classes of \(p\)-localized loop spaces \(\Omega(G_{n,m}^+ (\mathbb{K})_{(p)})\) and \(\Omega(V_{n,m} (\mathbb{K})_{(p)})\) at a prime \(p\), where \(G_{n,m}^+ (\mathbb{K})\) is the oriented Grassmann manifold. Moreover, the author calculates the nilpotency of the Cohen group \([J(X), \Omega(Y)]\), where \(J(X)\) is the James reduced product of a space \(X\): For any space \(X\), (1) if \(n \geq 1\), then \(\operatorname{nil}[J(X), \Omega(Y)] < \infty\), where \(Y = SO(2n)/U(n)\), \(Sp(n)/U(n)\), \(SU(2n)/Sp(n)\), \(SO(4n)/Sp(n)\); (2) if \(m<n \leq \infty\), then \(\operatorname{nil}[J(X), \Omega(Y)] < \infty\), where \(Y\) is the (oriented) Grassmann and flag manifolds and their localizations with additional conditions; and (3) if \(1\leq m \leq n\), then \(\operatorname{nil}[J(X), \Omega(Y)] < \infty\), where \(Y = V_{n,m}(\mathbb{C})\), \(V_{n,m}(\mathbb{H})\) and \(V_{n,m}(\mathbb{R})_{(p)}\) for \(p \geq 3\). The author also develops the homotopy nilpotency of the loop spaces \(\Omega(\mathbb{O}P^2)\) on the Cayley plane \(\mathbb{O}P^2\), the homogeneous spaces of exceptional Lie groups, and their localizations. Reviewer's remark: The reviewer believes that in page 251, Corollary 2.4, ``(4) \([J(X), \Omega(SO(4n)/Sp(n))] < \infty\)'' should be ``(4) \(\operatorname{nil}[J(X), \Omega(SO(4n)/Sp(n))] < \infty\)''.
scientific article; zbMATH DE number 7457114
scientific article