Book on smooth covering spaces

Question

Hi I am looking for a book (I need to cite it.) about smooth manifolds and covering spaces. I need the prove that action $\pi_1(M)$ on universal cover is smooth and the space of orbits $\widetilde{M}/\pi_1(M)$ is diffeomorphic to M. All I can find are these results in topological category. Any suggestions?

Answer 1

I don't know of a specific reference, but I can outline a program for you to proceed to prove both these things (I am assuming all manifolds are connected in what follows)

1. Show that given any smooth manifold $M$, there is a unique smooth structure on its universal cover $\tilde{M}$, for which the covering map $\pi : \tilde{M} \rightarrow M$ is a local diffeomorphism (or a smooth covering map, its the same smooth structure in the end).

2. Use that action of the fundamental group on the universal cover of $M$ corresponds to a deck transformation $d : \tilde{M} \rightarrow \tilde{M}$. Then show that deck transformations are automatically smooth using the next point 3. (use 3. and the fact that a deck transformation is a lift of the covering map itself with respect to the same covering map)

3. Let $X$ be a smooth manifold with universal cover $\tilde{X}$. Then for any smooth map $f : Z \rightarrow X$, if $f$ has a lift with respect to the covering map $\pi : \tilde{X} \rightarrow X$, say $\tilde{f} : Z \rightarrow \tilde{X}$, then $\tilde{f}$ is smooth (use the $\pi$ is a local diffeomorphism and $\pi \circ \tilde{f} = f$ to prove this).

Finally, to show that $\tilde{M} / \pi_{1}(M)$ is diffeomorphic to $M$, you may trace through the proof of the quotient manifold theorem (see e.g. Theorem 9.19 of his Introduction to Smooth Manifolds), and apply it to the zero dimensional lie group of deck transformations. This gives a unique smooth structure on the quotient space, such that the quotient map is a smooth covering map (or you could just do this directly, check that there is a unique smooth structure on $\tilde{M} / \pi_{1}(M)$ such that the quotient map $\pi : \tilde{M} \rightarrow \tilde{M} / \pi_{1}(M)$ is a smooth covering map). Then check, using this smooth structure, that the map $\tilde{M} / \pi_{1}(M) \rightarrow M$ is a diffeomorphism.

Answer 2

If you really just need a citations, everything can be found in John M. Lee's Introduction to Smooth Manifolds, Second Edition. Here is the list you need:

• the smooth structure on the universal cover is given by Proposition 4.40 and Corollary 4.43.

• the fact that the fundamental group of a smooth manifold acts freely, smoothly and properly is a particular case of Proposition 21.12.

• assuming you already know that the quotient map is an homeomorphism, the fact that it is a diffeomorphism is a consequence of Theorem 21.13.