Transformation Group Theory and Critical Point Theory

Borsuk-Ulam定理在拓扑学、全局分析等数学领域的研究中具有重要的应用价值和吸引力。这一定理自1933年发表以来已经有很长的历史了。60多年来，许多研究者一直致力于这一定理的各种应用和推广。经典的Borsuk-Ulam定理涉及具有2阶循环群的对极作用的球面之间的等变映射。在我们的研究中，我们将其推广到单位球面SU和Sw之间的等变映射U和W之间的等变映射，U和W是更一般的紧李群G。1997年我们主要讨论了G是环面的情况。通过观察表示球上等变K-环的代数结构，得到了经典Borsuk-Ulam定理的一个自然推广。此外，在关于表示的某些条件下，我们证明了如果存在从SU到SW的等变映射，则U一定是W的次表示.1998年，我们得到了关于紧李群C的表示球面之间的等变映射度的结果.该方法严重依赖于对定义在G的表示环R(G)中的Euler表示类的代数观察.

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.