On the construction and classification of the finite geometry

论有限几何的构造与分类

• 批准号: 09640306
• 负责人: ODA Nobuyuki
• 金额: $ 1.15万
• 依托单位: Fukuoka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640306/
• 关键词: geometry; homotopy; algebra; cohomology; plane; Schur ring; 射影平面; 共線変換群; 擬正則; 積差集合

Geometrical constructions in homotopy sets were studied. We obtained results on the GAMMA-Whitehead product and the GAMMA-Hopf construction. We introduced the transformation between pairings and copairings and showed its applications. We obtained a formula for the smash product. We obtained a generalization of the Hardie-Jansen product and studied its properties. Dual results are also studied.For geometrical construction in operator algebras, Tomita-Takesaki theory was studied. We obtained results on unbounded C^*seminorms on *-algebra and standard weights which enable us to develop unbounded Tomita-Takesaki theory.We constructed explicit examples of surfaces in affine spaces of dimension three and four. We gave a necessary and sufficient condition on surfaces in a three-dimensional affine space to be metric when the surfaces have non-zero constant Gauss-Kronecker curvature.The cohomology of mapping class groups was studied. We obtained a relation among periodic automorphisms of closed surfaces and the eta-invariant of their mapping tori. We also obtained various vanishing theorems of mod 2 Morita-Mumford classes.The Schur ring of product type was characterized by the existence of a subgroup of a collineation group. The existence of a Schur ring of produt difference set type is characterized by a finite projective plane of order n with a collineation group of order n(n - 1).

研究了同伦集合中的几何构造。我们得到了关于<$MA-Whitehead积和<$MA-Hopf构造的结果。介绍了对偶与余对偶之间的变换及其应用。我们得到了一个smash乘积的公式。得到了Hardie-Jansen积的一个推广，并研究了它的性质.对于算子代数的几何构造，研究了Tomita-Takesaki理论.我们得到了关于 *-代数和标准权的无界C^* 范数的结果，这些结果使我们能够发展无界Tomita-Takesaki理论，我们构造了三维和四维仿射空间中曲面的显式例子。给出了三维仿射空间中具有非零常Gauss-Kronecker曲率的曲面为度量的充要条件，并研究了映射类群的上同调.得到了闭曲面的周期自同构与其映射环面的η-不变量之间的关系。我们还得到了mod 2 Morita-Mumford类的各种消失定理，并利用直射群的子群的存在性刻画了乘积型Schur环。通过n阶有限射影平面与n（n - 1）阶共线群刻画了乘积差集型Schur环的存在性.

项目成果

A.Inoue: "Standard systems for semifinite O-algebras" Proc.Amer.Math.Soc.128. 3303-3312 (1997)

A.Inoue：“半有限 O 代数的标准系统”Proc.Amer.Math.Soc.128。

A.Inoue: "Tomita-Takesaki Theory in Algebras of Unbounded Operators" Lecture Notes in Mathematics 1699, Springer. (1998)

A.Inoue：“无界算子代数中的 Tomita-Takesaki 理论”数学讲义 1699，Springer。

N.Oda and T.Shimizu: "Transformation between pairings" Fukuoka Univ.Sci.Reports. 28(2). 53-64 (1998)

N.Oda 和 T.Shimizu：“配对之间的转变”福冈大学科学报告。

T.Akita: "Euler characteristics of Coxeter groups, PL-triangulations of closed manifolds, and cohomology of subgroups of Artin groups" Journal of the London Mathematical Society. (to appear).

T.Akita：“Coxeter 群的欧拉特征、闭流形的 PL 三角剖分以及 Artin 群子群的上同调”《伦敦数学会杂志》。

井上淳, J.P.Antoine, 荻秀和: "Standard generalized vectors for partial O^*-algebras" Ann.Inst.H.Poincare. 67. 223-258 (1997)

Jun Inoue、J.P.Antoine、Hidekazu Ogi：“部分 O^*-代数的标准广义向量”Ann.Inst.H.Poincare 67. 223-258 (1997)