Transformation Group Theory and Critical Point Theory

点击翻译按钮获取中文摘要

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.