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

本文从变换群论的角度研究了Borsuk-Ulam型定理。经典的Borsuk-Ulam定理说，对于任何从n球到n欧氏空间的连续映射f，存在一个点x使得f(x)=f（-x）。将这一结果重述如下：从n球到（n-1）球不存在具有对映作用的c_2等变映射。从这个角度看，Borsuk-Ulam型定理的研究就是等变或等变映射的存在性问题。首先阐明了等变映射与上同调指标的关系，利用它们得到了等变映射的Borsuk-Ulam型定理；特别地，对于具有初等阿贝尔p群的自由作用的Stiefel流形，Borsuk-Ulam型定理成立。此外，我们还得到了等变映射的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到表示球的等变映射，则Borsuk-Ulam型不等式成立。这一不等式有望在各个领域得到应用。接着，我们研究了Borsuk-Ulam型定理的逆性，并从等变阻塞理论中证明了该逆性在某些情况下成立，包括经典情况。这对Borsuk-Ulam型定理的研究有了新的认识，并提出了等变同伦类和多次的分类问题等新问题。因此，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.