The study of Borsuk-Ulam type theorems in transformation group theory

本文从变换群理论的角度研究了Borsuk-Ulam型定理。经典的Borsuk-Ulam定理说，对于任何从n-球面到n-欧几里德空间的连续映射f，存在一个点x使得f（x）=f（-x）。这个结果被重述如下：不存在从n-球面到具有对映作用的（n-1）-球面的C_2-等变映射。从这个观点出发，Borsuk-Ulam型定理的研究就是等变映射或等变映射存在性问题的研究.本文首先阐明了等变映射与上同调指标的关系，并利用它们得到了等变映射的Borsuk-Ulam型定理，特别是Borsuk-Ulam型定理对具有初等交换p-群自由作用的Stiefel流形成立.此外，我们还得到了一些关于等变映射的Borsuk-Ulam型定理，例如，如果存在从具有自由有限群作用的同调球面到表示球面的等变映射，则Borsuk-Ulam型不等式成立。我们接着研究了Borsuk-Ulam型定理的匡威，并从等变阻塞理论证明了在包括经典情形在内的某些情形下，该匡威定理也成立，这为Borsuk-Ulam型定理的研究提供了新的视角，并提供了一些新的问题，如同变同伦类和多重度的分类问题.最后对该研究的未来发展进行了展望。

In this study, we have studied Borsuk-Ulam type theorems from the viewpoint of transformation group theory. The classical Borsuk-Ulam theorem says that for any continuous map f from the n-sphere to the n-Euclidean space, there is a point x such that f(x)=f(-x). This result is restated as follows: There is no C_2-equivariant map from the n-sphere to the (n-1)-sphere with the antipodal action. From this point of view, the study of Borsuk-Ulam type theorems is that of the existence problem of equivariant or isovariant maps.We first clarified the relation of equivariant maps and the cohomological indices, and using them we obtained some Borsuk-Ulam type theorems for equivariant maps; in particular, the Borsuk-Ulam type theorem holds for Stiefel manifolds with free actions of an elementary abelian p-group. Moreover, we obtained some Borsuk-Ulam type theorems for isovariant maps; for example, if there exists an isovariant map from a homology sphere with free finite group action to a representation sphere, then a Borsuk-Ulam type inequality holds. This inequality is expected to give some applications to various fields.We next studied the converse of Borsuk-Ulam type theorems, and showed from equivariant obstruction theory that the converse holds for some cases including the classical case.This gives a new insight to the study of Borsuk-Ulam type theorems, and provides new problems such as the classification problem of isovariant homotopy classes and multidegrees. Thus the future development of this research is expected.