Transformation Group Theory and Critical Point Theory

变换群理论和临界点理论

• 批准号: 09640113
• 负责人: KOMIYA Katsuhiro
• 金额: $ 1.73万
• 依托单位: Yamaguchi 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-09640113/
• 关键词: Borsuk-Ulam theorem; equivariant K-theory; representation ring; Euler class; equivariant maps; mapping degree; 同変写像; 同変K理論; トーラス作用

The Borsuk-Ulam theorem is useful and attractive in the study of Topology, Global Analysis and other areas in Mathematics. The theorem has a long history since it was published in 1933. For more than 60 years many researchers has been contributing to various kind of applications and generalizations of the theorem.The classical Borsuk-Ulam theorem is concerned with equivariant maps between spheres with the antipodal action of the cyclic group of order 2. In our study we generalize this to equivariant maps between unit spheres SU and SW of unitary representations U and W of a more general compact Lie group G.In 1997 we mainly concerned with the case in which G is a torus. Observing the algebraic structure of the equivariant K-ring of a representation sphere, we obtained a natural generalization of the classical Borsuk-Ulam theorem. Moreover, under some conditions on representations we showed U must be a subrepresentation of W if there exists an equivariant maps ftom SU to SW.In 1998 we obtained results on the degrees of equivariant maps between representation spheres of a compact Lie group C.The method heavily depends on algebraic observation of the Euler class of representation which is defined in the representation ring R(G) of G.

Borsuk-ulam定理在拓扑，全球分析和其他数学领域的研究中很有用且有吸引力。 The theorem has a long history since it was published in 1933. For more than 60 years many researchers has been contributing to various kind of applications and generalizations of the theorem.The classical Borsuk-Ulam theorem is concerned with equivariant maps between spheres with the antipodal action of the cyclic group of order 2. In our study we generalize this to equivariant maps between unit spheres SU and SW of unitary representations U and W 1997年，我们主要关注G是圆环的情况。观察表示球体的e象k形的代数结构，我们获得了经典的Borsuk-ulam定理的自然概括。此外，在某些条件下，我们证明您必须是w的子代理。

项目成果

Katsuhiro Komiya: "Equivariant maps between representation sphreres of a torus" Publ.RIMS, Kyoto Univ.34. 271-276 (1998)

Katsuhiro Komiya：“环面表示球面之间的等变映射”Publ.RIMS，京都大学 34。

Takao Kato: "Variety of special linear systems on k-sheeted coverings" Geom.Dedicata. 69. 53-65 (1998)

Takao Kato：“k 片状覆盖物上的各种特殊线性系统”Geom.Dedicata。

Katsuhiro Komiya: "Equivariant maps between representation spheres of a torus" Publ.RIMS,Kyoto Univ.34. 271-276 (1998)

Katsuhiro Komiya：“环面表示球体之间的等变映射”Publ.RIMS，Kyoto Univ.34。

Yoshihisa Sato: "3-dimensional homology handles and minimal second Betti numbers of 4-manifolds" Osaka J.of Math.35. 509-527 (1998)

Yoshihisa Sato：“3 维同调句柄和 4 流形的最小二阶 Betti 数”Osaka J.of Math.35。

Katsuhiro Komiya: "Equivariant maps between representation sphreres of a torus" Publ.RIMS,Kyoto Univ.34. 271-276 (1998)

Katsuhiro Komiya：“环面表示球体之间的等变映射”Publ.RIMS，Kyoto Univ.34。