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

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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.