Mod \(p\) decompositions of mod \(p\) finite \(H\)-spaces (Q2717461)

Assume all spaces localized at an odd prime \(p\). Consider the bundle NEWLINE\[NEWLINEO(2n+2)/ O(2n+1)\to O(2n+3)/ O(2n+1)\to O(2n+3)/ O(2n+2)NEWLINE\]NEWLINE and recall that \(O(2n+2)/ O(2n+1)= S^{2n+1}\) and \(O(2n+3)/ O(2n+2)= S^{2n+2}\). Let \(B_n(p)\) denote the total space of the bundle induced by a map \(S^{2n+1+2(p-1)}\to S^{2n+2}\) representing the element \(\frac{1}{2} \alpha_1 (2n+2)\), where \(\alpha_1(t)\in \pi_{t+2p-3} (S^t)\cong \mathbb{Z}/p\) is the generator. The author proves that each simply connected \(H\)-space \(X\) such that \(H^*(X; \mathbb{Z}/p)\cong \Lambda (x_1,\dots, x_k)\) with \(\deg x_i= 2n_i+1\), \(n_1\leq n_2\leq\dots\leq n_k\), and \(n_k- n_1< 2(p-1)\) splits into a product of odd spheres and \(B_n(p)\)s. This result extends results by \textit{J. R. Harper} [Lect. Notes Math. 428, 44-51 (1974; Zbl 0302.55006)], \textit{C. Wilkerson} [ibid., 52-57 (1974; Zbl 0302.55007)], and \textit{J. McCleary} [ibid. 763, 70-87 (1979; Zbl 0417.55012)].