Lusternik-Schnirelmann category of non-simply connected compact simple Lie groups

Let F → X → B be a fibre bundle with structure group G, where B is (d - 1)-connected and of finite dimension, d ≥ 1. We prove that the strong L-S category of X is less than or equal to m+ dim B/d, 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 n ≥ 1, which might be best possible, since we have cat (PU(p^r)) = 3(p^r - 1) for any prime p and r ≥ 1. Similarly, we obtain the L-S category of SO (n) for n ≤ 9 and PO(8). We remark that all the above Lie groups satisfy the Ganea conjecture on L-S category.