Why does a map from simply connected space to U(1) factors through R?

[eok20]

I'm having trouble seeing why the following is true: let M be a simply connected manifold and s a smooth map from M to U(1). Then why does it follow that s = e^(iu) for some smooth function u from M to R?

Thanks!

 

[Tinyboss]

Because R is the universal cover of U(1), and e^ix is a covering map.

 

[eok20]

Because R is the universal cover of U(1), and e^ix is a covering map.

Ahh...right, every map from a simply connected space lifts to the universal cover.

Thanks!