AI 中文总结
本文从拓扑视角出发，结合组合拓扑技术与Ihara-Yokonuma计算，证明有限单群和有限Coxeter群的博戈莫洛夫乘子均消失，为搭建其在代数几何与配边群中作用的桥梁提供了小步进展。
AI 中文摘要
自20世纪80年代以来，有限群的博戈莫洛夫乘子被认为会阻碍复代数几何中的有理性，近年更发现它与定向及稳定酉二维G-等变配边群Ω₂^{SO,G}和Ω₂^{U,G}中的挠有关。本文为在这两个相距甚远的作用间搭建桥梁迈出小步，讨论两类特定有限群的博戈莫洛夫乘子消失问题：首先，受Ore猜想的低维拓扑解释启发，重新审视Kunyavskiĭ的结论——所有有限单群的博戈莫洛夫乘子均消失；其次，避开复双有理几何中的论证（如Chevalley-Shephard-Todd定理的难点方向），结合拼贴组合拓扑技术与Ihara-Yokonuma的初等计算，证明所有有限Coxeter群的博戈莫洛夫乘子均消失。
英文摘要
Since the 1980s, the Bogomolov multiplier of a finite group has been known to obstruct rationality in complex algebraic geometry, and more recently it is understood to be responsible for any torsion in the oriented and stable unitary 2-dimensional $G$-equivariant bordism groups $Ω_2^{SO,G}$ and $Ω_2^{U,G}$. In this note, as a small step toward building a bridge between these two far-flung roles, we discuss the vanishing of Bogomolov multipliers of two specific families of finite groups. First, we revisit Kunyavskiĭ's result that the Bogomolov multipliers of all finite simple groups vanish, taking inspiration from the low-dimensional topological interpretation of the Ore conjecture. Second, in lieu of arguments in complex birational geometry (such as the hard direction of the Chevalley-Shephard-Todd theorem), we combine cut-and-paste combinatorial-topological techniques with elementary calculations of Ihara-Yokonuma to show that all finite Coxeter groups have vanishing Bogomolov multiplier.
