Fixed point theorems and cobordism

川久保构造的G-流形是G-S余边界的，但对于任意紧致李群G-长崎计算了LH群的结构，这是为了研究有限群的同伦表示的线性。作为应用，他确定了同伦表示总是线性的有限群的集合。村上构造了多变量Alexander多项式的图拉耶夫意义下的顶点型状态模型。利用该模型，得到了多元Alexander多项式的一组新公理。他还确定了量子群U_q(gl(n，c))的混合张量表示的中心化子代数的结构。这个代数可以看作是A型Iwahori-Hecke代数的推广。利用这个代数，他推广了空间图在S^3中嵌入的不变量--Yamada多项式。他还研究了用Kontsevich迭代积分表示的缠结范畴，并成功地给出了关于纽结、链环和缠结的Kontsevich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.