We give a new proof, independent of Lin's theorem, of the Segal conjecture for the cyclic group of order two. The key input is a calculation, as a Hopf algebroid, of the Real topological Hochschild homology of ${\mathbb{F}}_2$. This determines the ${\mathrm{E}}_2$-page of the descent spectral sequence for the map $\mathrm{N}{\mathbb{F}}_2\to {\mathbb{F}}_2$, where $\mathrm{N}{\mathbb{F}}_2$ is the $C_2$-equivariant Hill–Hopkins–Ravenel norm of ${\mathbb{F}}_2$. The ${\mathrm{E}}_2$-page represents a new upper bound on the $RO(C_2)$-graded homotopy of $\mathrm{N}{\mathbb{F}}_2$, from which the Segal conjecture is an immediate corollary.

中文翻译：

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实拓扑霍克希尔德同调和西格尔猜想

我们对二阶循环群的 Segal 猜想给出了独立于 Lin 定理的新证明。关键输入是作为 Hopf 代数计算的实拓扑 Hochschild 同调${\mathbb{F}}_2$. 这决定了${\mathrm{乙}}_2$- 地图的下降谱序列的页面 $\mathrm{N}{\mathbb{F}}_2\to {\mathbb{F}}_2$， 在哪里 $\mathrm{N}{\mathbb{F}}_2$ 是个 $C_2$的等变 Hill–Hopkins–Ravenel 范数 ${\mathbb{F}}_2$. 这${\mathrm{乙}}_2$-page 表示新的上限 $\mathrm{电阻}哦(C_2)$- 分级同伦 $\mathrm{N}{\mathbb{F}}_2$，由此西格尔猜想是直接推论。