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+dimBd, 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(pr))=3(pr−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.