Prove the following: Theorem. Let G be a topological group with multiplication operation m: G times G rightarrow G and identity element e. Assume p: tildeG rightarrow G is a covering map. Given tildee with p(tildee)=e, there is a unique multiplication operation on barG that makes it into a topological group such that tildee is the identity element and p is a homomorphism. Proof Recall that, by our convention, G and barG are path connected and locally path connected. (a) Let I: G rightarrow G be the map I(g)=g⁻¹. Show there exist unique maps barm: tildeG times tildeG rightarrow tildeG and tildeI: tildeG rightarrow_- tildeG with tildem(tildee times tildee)=tildee and barI(bare)=bare such that p circ tildem=m circ(p times p) and p circ tildeI=I circ p. (c) Show the maps tildeG rightarrow tildeG given by barg rightarrow tildem(tildeg times tildeI(tildeg)) and tildeg rightarrow tildem(barI(tildeg) times tildeg) map tildeG to bare. (d) Show the maps barG times tildeG times barG rightarrow barG given by tildeg times tildeg^ prime times tildeg^ prime prime rightarrow vecm left(tildeg times barm left(tildeg^ prime times tildeg^ prime prime right) right) tildeg times tildeg^ prime times tildeg^ prime prime rightarrow barm left(tildem left(tildeg times tildeg^ prime right) times tildeg^ prime prime right) endarray are equal. (e) Complete the proof.
