We calculate the direct sum of the mod-two cohomology of all alternating groups, with both cup and transfer product structures, which in particular determines the additive structure and ring structure of the cohomology of individual groups. We show that there are no nilpotent elements in the cohomology rings of individual alternating groups. We calculate the action of the Steenrod algebra and discuss individual component rings. A range of techniques are developed, including an almost Hopf ring structure associated to the embeddings of products of alternating groups and Fox–Neuwirth resolutions, which are new techniques. We also extend understanding of the Gysin sequence relating the cohomology of alternating groups to that of symmetric groups and calculation of restriction to elementary abelian subgroups.

中文翻译：

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交替基团的模二同调环

我们计算所有交替基团的模二同调的直接和，同时具有杯子和转移产物的结构，这尤其决定了各个基团同调的加和结构和环结构。我们表明，在各个交替基团的同调环中没有幂幂元素。我们计算Steenrod代数的作用并讨论各个组成环。开发了一系列技术，包括与交替基团的产品嵌入和Fox-Neuwirth分辨率相关的几乎霍普夫环结构，它们是新技术。我们还扩展了对关于交替基团与对称基团的同调相关联的Gysin序列的理解，以及对基本阿贝尔亚群的限制的计算。