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.