Project supported by the Japanese Government Scholarship;the Japan Society for the Promotion of Science Postdoctoral Fellowship for Foreign Researchers;the Focused Research Group Postdoctoral Fellowship;the Program of Visiting Scholars at Chern Institute of Mathematics;the National Natural Science Foundation of China (No. 10601053).
We give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain compact fiber bundles with non-trivial four dimensional fibers in the sense of cobordism, by virtue of the rigid...