On three classical results about compact groups (Q2842085)

The author presents very short proofs of three classical results on compact topological groups: (1) a compact connected \(n\)-dimensional topological group \(G\) is a Lie group if its Čech cohomology group \(H^n(G)=\mathbb Z\); (2) a closed connected subgroup \(H\) of a compact connected group \(G\) is normal if \(\dim G/H=1\); and (3) a closed subgroup \(H\) of a compact topological group \(G\) coincides with \(G\) if the quotient space \(G/H\) is contractible (more precisely, \(\mathbb Q\)-acyclic and \(\mathbb Z/2\mathbb Z\)-acyclic).