Yasuhiro HARA

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Transformation groups and topology
Keywords Group action, equivariant map, fixed point, coincidence point

The field of my study is topology and, especially, I study the theory of transformation groups. The Borsuk-Ulam theorem is one of famous theorems about transformation groups. This theorem is often taken up as an application in elementary lectures about the homology theory. The content of the theorem is as follows: for every continuous map from the n-dimensional sphere to the n-dimensional Euclidian space, there exists a point such that the map takes the same value at the point and at the antipodal point. A famous application of this theorem is the following. ''On the earth, there is a point such that the temperature and humidity at the point are the same as those at the antipodal point.'' We consider a free action of a group of order two on the n-dimensional sphere to prove the Borsuk-Ulam theorem. Then for any equivariant map (any continuous map which preserves the structure of the group action) from the sphere to itself, the degree of the map is odd. By using this fact, we obtain the Borsuk-Ulam theorem. In the case of the Borsuk-Ulam theorem, we consider spheres and free actions of a group of order two. Actually, when we consider other manifolds and actions of other groups, there are some restrictions of homotopy types of equivariant maps. I study such restrictions of homotopy types of equivariant maps by using the cohomology theory, and I study relationships between homotopy types of equivariant maps and topological invariants.