# Surjectivity of Lie group covering map

I am trying to solve Chapter's 3 exercise 11 in the "Lie Groups, Lie Algebras and Representations" Brian C. Hall's book. It reads as:

If $$\tilde{G}$$ is a universal cover of a connected Lie group $$G$$ with projection map $$\Phi$$, show that $$\Phi$$ maps $$\tilde{G}$$ onto $$G$$.

It looks like it has an easy and elegant proof but I am not arriving to it. Any help will be appreciated. Thanks!

• Isn’t surjectivity in the definition of covering map? Jul 10, 2019 at 7:36
• I think not. The definition in this book says that $\Phi$ from $\tilde{G}$ to $G$ is only a Lie group homomorphism (such that the associated Lie Algebra homomorphism is a Lie Algebra isomorphism). Jul 10, 2019 at 18:56

• Sorry for the delay, I have not seen the notification. It is indeed great! However, I am not sure how do you use the inverse theorem. Shall I use that since $\Phi$ is also a smooth map, then it maps any open set in $\tilde{G}$ to an open set in $G$? Jul 14, 2019 at 16:30