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

The Lusternik–Schnirelmann π 1 -category, cat π 1 X , of a space X is the least integer k such that there is a covering of X by ( k + 1 ) open subsets, every loop in each of which is contractible in X. Let f : X → K ( π 1 ( X ) , 1 ) be a map inducing an isomorphism on π 1 . The π 1 -cohomological dimension, d π 1 ( X ) , of a space X is the largest integer k such that the homomorphism f ∗ : H k ( K ( π 1 ( X ) , 1 ) ; Λ ) → H k ( X ; Λ ) is non-trivial for some π 1 ( X ) -module Λ. The π 1 -cohomological dimension is a lower bound of the Lusternik–Schnirelmann π 1 -category. We determine the π 1 -cohomological dimension of all the non-simply connected compact simple Lie groups except for the projective orthogonal group PO ( 2 m ) with m ⩾ 5 . We also determine the Lusternik–Schnirelmann π 1 -category of SO ( m ) for 3 ⩽ m ⩽ 10 , Ss ( 2 r ) , PSp ( 2 r ) , PU ( p r ) and PO ( 8 ) .