Let S2n+1{p} denote the homotopy fibre of the degree p self map of S2n+1. For primes p ≥ 5, work by Selick shows that S2n+1{p} admits a non-trivial loop space decomposition if and only if n = 1 or p. Indecomposability in all but these dimensions was obtained by showing that a non-trivial decomposition of ΩS2n+1{p} implies the existence of a p-primary Kervaire invariant one element of order p in $\pi _{2n(p-1)-2}^S$. We prove the converse of this last implication and observe that the homotopy decomposition problem for ΩS2n+1{p} is equivalent to the strong p-primary Kervaire invariant problem for all odd primes. For p = 3, we use the 3-primary Kervaire invariant element θ3 to give a new decomposition of ΩS55{3} analogous to Selick's decomposition of ΩS2p+1{p} and as an application prove two new cases of a long-standing conjecture stating that the fibre of the double suspension $S^{2n-1} \longrightarrow \Omega ^2S^{2n+1}$ is homotopy equivalent to the double loop space of Anick's space.

中文翻译：

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3 阶映射的纤维、Anick 空间和双悬浮

让小号2n+1{p} 表示度数的同伦纤维p自己的地图小号2n+1. 对于素数p≥ 5，Selick 的工作表明小号2n+1{p} 承认非平凡的循环空间分解当且仅当n= 1 或p. 通过证明 Ω 的非平凡分解获得了除这些维度之外的所有维度的不可分解性小号2n+1{p} 意味着存在一个p- 初级 Kervaire 不变量 1 阶元素p在$\pi _{2n(p-1)-2}^S$. 我们证明了最后一个含义的反面，并观察到 ​​Ω 的同伦分解问题小号2n+1{p} 等价于强p-所有奇素数的主要 Kervaire 不变问题。为了p= 3，我们使用 3-primary Kervaire 不变元素 θ3给出 Ω 的新分解小号55{3} 类似于 Selick 的 Ω 分解小号2p+1{p} 并作为应用证明了一个长期猜想的两个新案例，该猜想表明双悬浮纤维$S^{2n-1} \longrightarrow \Omega ^2S^{2n+1}$同伦等价于 Anick 空间的双环空间。