Higher topological cyclic homology and the Segal conjecture for tori
Higher topological cyclic homology and the Segal conjecture for tori
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更高的拓扑循环同调性和环面的 Segal 猜想
DOI：
10.1016/j.aim.2010.08.016
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发表时间：
2008
影响因子：
1.7
通讯作者：
B. Dundas
中科院分区：
文献类型：
--
作者：
G. Carlsson;Christopher L. Douglas;B. Dundas
我们研究更高的拓扑循环同调作为研究同伦理论中色现象的一种方法。更高的拓扑循环同调是根据基于 n 维环面的拓扑 Hochschild 同调版本的不动点构建的，我们将其视为 n 重迭代代数 K 理论的计算上易于处理的表亲。拓扑拓扑 Hochschild 同调的不动点通过限制和 Frobenius 算子相互关联。我们引入了另外两个定点算子家族：Verschiebung（以 n 环面的自同构为索引）和微分（以 n 向量为索引）。我们对限制、Frobenius、Verschiebung 和微分之间的关系进行了详细分析，产生了 Hesselholt 和 Madsen 描述的一维拓扑循环同调结构的更高模拟。我们计算了球谱的两个重要的更高拓扑循环同调性，即拓扑限制同调性和拓扑弗罗贝尼乌斯同调性。后者的计算使我们能够建立环面的西格尔猜想，也就是说完全计算环面分类空间的上同伦类型。
We investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory. Higher topological cyclic homology is constructed from the fixed points of a version of topological Hochschild homology based on the n-dimensional torus, and we propose it as a computationally tractable cousin of n-fold iterated algebraic K-theory. The fixed points of toral topological Hochschild homology are related to one another by restriction and Frobenius operators. We introduce two additional families of operators on fixed points, the Verschiebung, indexed on self-isogenies of the n-torus, and the differentials, indexed on n-vectors. We give a detailed analysis of the relations among the restriction, Frobenius, Verschiebung, and differentials, producing a higher analog of the structure Hesselholt and Madsen described for 1-dimensional topological cyclic homology. We calculate two important pieces of higher topological cyclic homology, namely topological restriction homology and topological Frobenius homology, for the sphere spectrum. The latter computation allows us to establish the Segal conjecture for the torus, which is to say to completely compute the cohomotopy type of the classifying space of the torus.