论文标题
有限型类型的弦拓扑
String topology of finite groups of Lie type
论文作者
论文摘要
我们表明，任何有限的谎言类型的Mod $ \ ell $共同体，与$ \ ell $不同的特征性$ p $不同，该模块在Mod $ \ ell $共同体上的结构是通过环和模块结构的相应compact lie Group $ g $的自由回路$ bg $的自由回路空间，该结构是通过环形和模块的结构来构建的。如果有限的谎言类型的同源性中的某个基本类是非平凡的，那么该模块结构变得不含等级，并在配备了杯子产品的两个共同体学环之间提供结构化的同构，直至过滤。 我们在一系列情况下验证了基本类的非平地，包括所有简单地连接到$ q $元素的未介绍的古典组，其中$ q $是1 mod $ \ ell $的$ q $。我们还展示了如何通过将$ \ ell $ -compact固定点组替换为$ q $ mod $ \ ell $的订单，而无需更改有限组，从而消除了扭曲条件并摆脱了一致条件。通过这种修改，我们没有知道基本阶层是微不足道的例子，这提出了对Tezuka的开放问题的一般结构答案的可能性，他们猜测两个共同体圈之间存在同构。
We show that the mod $\ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $\ell$ admits the structure of a module over the mod $\ell$ cohomology of the free loop space of the classifying space $BG$ of the corresponding compact Lie group $G$, via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain fundamental class in the homology of the finite group of Lie type is non-trivial, then this module structure becomes free of rank one, and provides a structured isomorphism between the two cohomology rings equipped with the cup product, up to a filtration. We verify the nontriviality of the fundamental class in a range of cases, including all simply connected untwisted classical groups over the field of $q$ elements, with $q$ congruent to 1 mod $\ell$. We also show how to deal with twistings and get rid of the congruence condition by replacing $BG$ by a certain $\ell$-compact fixed point group depending on the order of $q$ mod $\ell$, without changing the finite group. With this modification, we know of no examples where the fundamental class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings.
