Study of Hopf spaces and p-compact groups

假设空间是简单连接的并且位于固定的奇素数 p 处。我们假设每个Hopf空间的同调性没有p挠率，并且它的mod p上同调环是有限的。我们首先完成了秩小于或等于五的不可约Hopf空间的分类。然后，利用这个结果，我们确定了 Hopf 空间的不可约分解，满足生成元的最大维数和最小维数之差小于 8(p-1) 的条件。在研究这个问题的过程中，我们证明了一些低维不稳定球体同伦群是由 alpha 元素的长 Toda 括号给出的元素生成的。此外，我们扩展了关于四元数 Stiefel 流形和实 Stiefel 流形的拟 p 正则性的已知事实。

Spaces are assumed to be simply connected and localized at a fixed odd prime p. We assume that the homology of each Hopf space has no p torsion, and the mod p cohomology ring of it is finite. We first completed the classification of irreducible Hopf spaces with rank less than or equal to five. Then, by using this result, we determined the irreducible decomposition of Hopf spaces satisfying the condition that the difference between the maximum and the minimum dimensions of the generators is less that 8(p-1). In the course of studying this problem we showed that some low dimensional unstable homotopy groups of spheres are generated by elements given by long Toda bracket of alpha elements. Moreover we extended a known fact on quasi p regularity of quaternionic Stiefel manifolds and real Stiefel manifolds.