Existence and classification problems of equivariant maps preserving orbit structures

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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.