Lusternik–Schnirelmann π1-category of non-simply connected simple Lie groups

The Lusternik–Schnirelmann π1-category, catπ1X, 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 be a map inducing an isomorphism on π1. The π1-cohomological dimension, dπ1(X), of a space X is the largest integer k such that the homomorphism is non-trivial for some π1(X)-module Λ. The π1-cohomological dimension is a lower bound of the Lusternik–Schnirelmann π1-category. We determine the π1-cohomological dimension of all the non-simply connected compact simple Lie groups except for the projective orthogonal group PO(2m) with m⩾5. We also determine the Lusternik–Schnirelmann π1-category of SO(m) for 3⩽m⩽10, Ss(r2), PSp(r2), PU(pr) and PO(8).