Affiliation: | aFaculty of Mathematics, Kyushu University, Ropponmatsu Fukuoka 810-8560, Japan bDepartment of Mathematics, Faculty of Science, Okayama University, 3-1 Tsushima-naka, Okayama 700-8530, Japan cDepartment of Welfare Business, Kinki Welfare University, Fukusaki-cho, Hyogo 679-2217, Japan |
Abstract: | Let FX→B be a fibre bundle with structure group G, where B is (d−1)-connected and of finite dimension, d1. We prove that the strong L–S category of X is less than or equal to , if F has a cone decomposition of length m under a compatibility condition with the action of G on F. This gives a consistent prospect to determine the L–S category of non-simply connected Lie groups. For example, we obtain cat(PU(n))3(n−1) for all n1, which might be best possible, since we have cat(PU(pr))=3(pr−1) for any prime p and r1. Similarly, we obtain the L–S category of SO(n) for n9 and PO(8). We remark that all the above Lie groups satisfy the Ganea conjecture on L–S category. |