Fixed point theorems and cobordism

对于任意紧李群，Kawakubo构造了g -s-协但不是g -同胚的g流形G.Nagasaki计算了为研究有限群的同伦表示的线性而引入的LH群的结构。作为应用，他确定了同伦表示总是线性的有限群的集合。对于多变量Alexander多项式，Murakami构造了Turaev意义上的顶点型状态模型。利用该模型，得到了多变量Alexander多项式的一组新的公理。他还确定了量子群U_q （gl (n, c)）的混合张量表示的中心化代数的结构。该代数可以看作是a型Iwahori Hecke代数的推广。利用该代数，他推广了S^3空间图嵌入的不变量Yamada多项式。他还利用Kontsevich的迭代积分研究了缠结范畴的表示，并成功地给出了Kontsevich关于结、环和缠结的积分的组合描述。

Kawakubo constructed G-manifolds which are G-s-cobordant, but are not G-homeomorphic for arbitrary compact Lie groups G.Nagasaki computed the structure of the LH groups which are introduced for the purpose of investigating the linearity of homotopy representations of finite groups. As an application, he determined the set of finite groups whose homotopy representations are always linear. Murakami constructed a vertex type state model in Turaev's sense for the multi-variable Alexander polynomial. By using this model, a new set of axioms for the multi-variable Alexander polynomial is obtained. He also determined the structure of the centralizer algebra of the mixed tensor representations of the quantum group U_q (gl (n, c) ). This algebra can be considered as a generalization of the Iwahori Hecke algebra of type A.Using this algebra, he generalized the Yamada polynomial, which is an invariant of embeddings of a spatial graph in S^3. He also investigated representations of the category of tangles using Kontsevich's iterated integral, and succeeded in giving a combinatorial description of Kontsevich's integrals of knots, links and tangles.