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arXiv 2401.04826math.AT
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$RO(G)$-分级Bredon上同调 of 欧几里得配置空间

$RO(G)$-graded Bredon cohomology of Euclidean configuration spaces

University of Oregon(俄勒冈大学);Grinnell College(格林内尔学院)

Daniel Dugger, Christy Hazel

AI总结:

本文研究了欧几里得配置空间的$RO(G)$-分级Bredon上同调,探讨了在不同条件下上同调的结构和环性质。

摘要:

设$G$为有限群,$V$为$G$-表示。我们研究$V$中有序配置空间的$RO(G)$-分级Bredon上同调,使用常积分系数。当$V$包含平凡子表示时,我们证明该上同调作为点的上同调模是自由的,并给出了环结构的生成元和关系描述。当$V$不包含平凡表示时,我们给出一个模块结构的计算,只要Bredon上同调中某消失条件成立。我们验证当$\dim(V)\geq 3$且$G$为$C_p$、$C_{p^2}$($p$为素数)或三个字母的对称群时,该消失条件成立。

英文摘要:

Let $G$ be a finite group and $V$ be a $G$-representation. We investigate the $RO(G)$-graded Bredon cohomology with constant integral coefficients of the space of ordered configurations in $V$. In the case that $V$ contains a trivial subrepresentation, we show the cohomology is free as a module over the cohomology of a point, and we give a generators-and-relations description of the ring structure. In the case that $V$ does not contain a trivial representation, we give a computation of the module structure that works as long as a certain vanishing condition holds in the Bredon cohomology of a point. We verify this vanishing condition holds in the case that $\dim(V)\geq 3$ and $G$ is any of $C_p$, $C_{p^2}$ ($p$ a prime), or the symmetric group on three letters.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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