Detecting Periodic Elements in Higher Topological Hochschild Homology
Detecting Periodic Elements in Higher Topological Hochschild Homology
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检测高级拓扑 Hochschild 同调中的周期元素
DOI：
10.2140/gt.2018.22.693
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发表时间：
2013
影响因子：
2
通讯作者：
Torleif Veen
中科院分区：
文献类型：
--
作者：
Torleif Veen
给定一个交换环谱$R$，设$\Lambda_XR$为由Brun， Carlson和Dundas构造的Loday函子。给定质数$p\geq 5$，我们计算$n\leq p$的$\pi_*(\Lambda_{S^n}H\mathbb{F}_p)$和$\pi_*(\Lambda_{T^n}H\mathbb{F}_p)$，并使用这些结果推断$(\Lambda_{T^{n}}H\mathbb{F}_p)^{hT^{n}}$的$n-1$ -th连接Morava $K$ -理论中的$v_{n-1}$是非零的，并且在同伦不动点谱序列中被显式元素检测到，我们将该类命名为Rognes类。
为了方便这些计算，我们引入了多重Hopf代数。$T^n$中的每个轴圆都会产生$\pi_*(\Lambda_{T^n}H\mathbb{F}_p)$上的Hopf代数结构，这些Hopf代数结构相互作用的方式是用多重Hopf代数结构编码的。该结构对$\pi_*(\Lambda_{T^n}H\mathbb{F}_p)$上可能的代数结构施加了一些限制，并且是上述计算中的重要工具。
Given a commutative ring spectrum $R$ let $\Lambda_XR$ be the Loday functor constructed by Brun, Carlson and Dundas. Given a prime $p\geq 5$ we calculate $\pi_*(\Lambda_{S^n}H\mathbb{F}_p)$ and $\pi_*(\Lambda_{T^n}H\mathbb{F}_p)$ for $n\leq p$, and use these results to deduce that $v_{n-1}$ in the $n-1$-th connective Morava $K$-theory of $(\Lambda_{T^{n}}H\mathbb{F}_p)^{hT^{n}}$ is non-zero and detected in the homotopy fixed point spectral sequence by an explicit element, which class we name the Rognes class.
To facilitate these calculations we introduce Multifold Hopf algebras. Each axis circle in $T^n$ gives rise to a Hopf algebra structure on $\pi_*(\Lambda_{T^n}H\mathbb{F}_p)$, and the way these Hopf Algebra structures interact is encoded with a Multifold Hopf algebra structure. This structure puts several restrictions on the possible algrebra structures on $\pi_*(\Lambda_{T^n}H\mathbb{F}_p)$ and is a vital tool in the calculations above.