Existence and classification problems of equivariant maps preserving orbit structures

本文从Borsuk-Ulam定理和o-极小拓扑的角度研究了等变映射的存在性和分类问题。 G-空间之间的等变映射是保持它们的轨道结构的等变映射。 我们得到了以下结果：（1）发现了一类新的有限群族，对它们等变Borsuk-Ulam定理成立。 这就为等变映射的存在提供了一个必要条件。 (2)在分类问题中，我们考虑了从闭自由G-流形到表示空间球面的等变映射的等变同伦类，并在适当的情况下，得到了等变同伦类的多度分类.这个结果是经典的Hopf定理的推广。

In this research, we studied the existence and classification problems of isovariant maps from the viewpoint of the Borsuk-Ulam theorem and the o-minimal topology. The isovariant map between G-spaces is an equivariant map preserving their orbit structures. We obtain the following results: (1) We found new families of finite groups for which the isovariant Borsuk-Ulam theorem holds. This provides a necessary condition for the existence of isovariant maps. (2) In the classification problem, we consider isovariant homotopy classes of isovariant maps from a closed free G-manifolds to a sphere of a representation space, and in suitable situation, we obtain that the multidegree classifies isovariant homotopy classes. This result is a generalization of the classical Hopf theorem.