有限群的非交换de Rham上同调
Noncommutative de Rham cohomology of finite groups
摘要：
我们研究了8阶以下有限群G上各种微分演算的de Rham上同调。这些群包括置换群S_3、二面体群D_4和四元数群Q。庞加莱的对偶性在任何情况下都成立，在某些假设下（基本上是存在一个顶形式），我们发现它必须在一般情况下成立。简要回顾了有限G上的双协变（非交换）微分自洽。外导数、外积、度量、霍奇对偶、联络、挠率、曲率和双不变积分都可以用代数定义。找到了辫子算子的投影分解，并利用它构造了2-形式空间上的投影算子。利用辫子算子和度量定义了有限群的纽结不变量。
英文摘要：
We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S_3, the dihedral group D_4 and the quaternion group Q. Poincare' duality holds in every case, and under some assumptions (essentially the existence of a top form) we find that it must hold in general. A short review of the bicovariant (noncommutative) differential calculus on finite G is given for selfconsistency. Exterior derivative, exterior product, metric, Hodge dual, connections, torsion, curvature, and biinvariant integration can be defined algebraically. A projector decomposition of the braiding operator is found, and used in constructing the projector on the space of 2-forms. By means of the braiding operator and the metric a knot invariant is defined for any finite group.
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