Constructing Topological manifolds using group actions. What can we say about the newly constructed charts?

Question

Let $X$ be a topological manifold. If $G$ is group with a free and properly discontinuous on $X$, then We can construct $X/G$ a new topological manifold, and we identify two elements if they are in the same orbit. What can we say about the number of the charts ($U$, $ \phi $). Are they the same number of charts as $X$. For example the sphere $\mathbb{S}^1$ can we viewed as a topological manifold using the action of translation by integers. In this case what are the charts of $\mathbb{S}^1$ ? (in this specific construction).

I am just a beginner in differetial geometry and the texbook I am reading doesn't mention anything about the charts of $\mathbb{S}^1$ in this specific construction.

Answer 1

Charts for $X/G$ may be derived from charts $(U,\phi)$ of $X$ by taking advantage of the following fact:

The quotient map $q : X \to X/G$ is a covering map. This means that $X/G$ has an open covering $\{V_i\}_{i \in I}$ such that each inverse image $q^{-1}(V_i)$ decomposes as a disjoint union of "slices" $$q^{-1}(V_i) = \bigsqcup_{j \in J} \tilde V_{i,j} $$ each slice being an open subset $\tilde V_{i,j} \subset X$ with the property that the restricted map $$q_{i,j} = q \bigm| \tilde V_{i,j} : \tilde V_{i,j} \to V_i $$ is a homeomorphism.

We can then combine the charts on $X$ with the slices to form charts on $X/G$ like this: for each chart $(U,\phi)$ on $X$ and for each slice $\tilde V_{i,j}$ one obtains chart on $X/G$ of the form $(W,\psi)$ where $$W_{i,j} = q_{i,j}(U \cap \tilde V_{i,j}) $$ $$\psi = \phi \circ q_{i,j}^{-1} : W_{i,j} \to \phi(U \cap \tilde V_{i,j}) $$

So for example we have $S^1 = \mathbb R / G$ where $G$ is the group of integer translations of $\mathbb R$. The covering map is $q(t)=e^{2\pi i t}$. As an example of a chart on $S^1$ let $W = q(0,1/2) = \{e^{2 \pi i t} \mid 0 \le t \le \frac{1}{2}\}$, and let $$\psi(e^{2 \pi i t}) = t \quad\text{for $0 \le t \le \frac{1}{2}$} $$ Using the principle branch of the natural logarithm function $\ln(z)$ defined on $\mathbb C$ minus the negative real half-line, one can also write this formula as $$\psi(z) = \frac{1}{2\pi} \ln(z) $$

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You ask specifically about the number of charts, but as you can see the actual count of charts obtained in this manner depends on a lot of variables: the given collection of charts on $X$; the group $G$ and its action; the covering $\{V_i\}$ of $X/G$. Perhaps from this it is clear that this number is not very significant. There's certainly not a very reasonable relation between the number of charts given on $X$ and the number of charts obtained on $X/G$.