$\u211e_\u221e$ 操作子的稳定化与 $p$ 进稳定同伦论
Stabilizations of $\mathbb{E}_\infty$ Operads and $p$-Adic Stable Homotopy Theory
AI总结：
该研究构造了作为 $\u211e_\u221e$ 操作子稳定化版本的稳定操作子，建立其代数同伦与上同调运算理论，并证明谱上链赋予其结构后可作为 $p$ 进稳定同伦型的代数模型。
摘要：
我们研究微分分次操作子与 $p$ 进稳定同伦论。首先构造了一类新的微分分次操作子，将其称为稳定操作子，这类操作子在特定意义下是 $\u211e_\u221e$ 操作子的稳定化版本，例如我们构造了稳定 Barratt-Eccles 操作子。我们建立了这类稳定操作子上代数的同伦论，以及这类稳定操作子上代数的（上）同调运算理论。我们发现这类操作子具有若干有趣性质：非等变情况下，每个元数的同调几乎都是平凡的；但等变情况下，这些同调的和构成广义 Steenrod 代数的某种完备化，因此高度非平凡。我们还通过证明与这些操作子相关的单子在同伦协调（即 $\u221e$）意义下是加性的，佐证了“稳定”这一修饰词的合理性。随后我们将稳定操作子应用于 $p$ 进稳定同伦论。众所周知，空间上的上链是 $\u211e_\u221e$ 操作子上代数的例子。我们证明在稳定情形下，谱上的上链是我们构造的稳定操作子上代数的例子。此外，Mandell 的结果表明，赋予 $\u211e_\u221e$ 代数结构后，空间上链可给出 $p$ 进同伦型的代数模型。我们证明，赋予由我们的稳定操作子编码的代数结构后，谱上链可给出 $p$ 进稳定同伦型的代数模型。
英文摘要：
We study differential graded operads and $p$-adic stable homotopy theory. We first construct a new class of differential graded operads, which we call the stable operads. These operads are, in a particular sense, stabilizations of $\mathbb{E}_\infty$ operads. For example, we construct a stable Barratt-Eccles operad. We develop a homotopy theory of algebras over these stable operads and a theory of (co)homology operations for algebras over these stable operads. We note interesting properties of these operads, such as that, non-equivariantly, in each arity, they have (almost) trivial homology, whereas, equivariantly, these homologies sum to a certain completion of the generalized Steenrod algebra and so are highly non-trivial. We also justify the adjective "stable" by showing that, among other things, the monads associated to these operads are additive in the homotopy coherent, or $\infty$-, sense. We then provide an application of our stable operads to $p$-adic stable homotopy theory. It is well-known that cochains on spaces yield examples of algebras over $\mathbb{E}_\infty$ operads. We show that in the stable case, cochains on spectra yield examples of algebras over our stable operads. Moreover, a result of Mandell says that, endowed with the $\mathbb{E}_\infty$ algebraic structure, cochains on spaces provide algebraic models of $p$-adic homotopy types. We show that, endowed with the algebraic structure encoded by our stable operads, spectral cochains provide algebraic models for $p$-adic stable homotopy types.
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