Galois Correspondences III

This time, we consider another kind of Galois correspondence: the one for covering spaces. This is actually somehow trivial.

Preliminaries

Let XX be a topological space. We first recall some notions in topology.

Definition: A covering space YY over XX is a bundle p ⁣:YXp\colon Y\to X which is locally trival and with discrete fibers. In plain words, the condition means for every point xXx\in X, there exists an open neighborhood UU such that the pullback p1(U)Up^{-1}(U)\to U is isomorphic to a projection U×FUU\times F\to U with FF a nonempty discrete set.

\begin{gather*} p^{-1}(U) \stackrel{\cong}{\longrightarrow} U\times F \\ p\searrow\quad\swarrow\mathrm{pr} \\ U \end{gather*}

Definition: The fundamental group π(X,x)\pi(X,x) of XX at a point xXx\in X is the group of homotopy classes of loops through xx and lie in XX.

Under the assumption that XX is connected, locally path-connected and semi-locally simply-connected, the group π(X,x)\pi(X,x) is independent of the choice of xx. In this case, we denote π(X)\pi(X) for short. However, we still need to fix a base point xx. We will keep this assumption throughout this post.

Galois correspondence for covering spaces

From covering spaces to π(X)\pi(X)-sets

Let p ⁣:YXp\colon Y\to X be a covering space. Then p1(x)p^{-1}(x) is a π(X)\pi(X)-set whose action is given as follows: the unique path-lifting lemma says that given a point y1p1(x)y_1\in p^{-1}(x), any loop ϕ\phi in XX start at xx can be uniquely lifted to a path in YY from y1y_1 to another point y2p1(x)y_2\in p^{-1}(x). The homotopic loops give the same point y2y_2, thus we define the result of the action of this homotopy class [ϕ][\phi] on y1y_1 to be y2y_2.

Let f ⁣:YYf\colon Y\to Y' be a morphism of covering spaces of XX. Then it induces a continuous π(X)\pi(X)-map from p1(x)p^{-1}(x) to p1(x)p'^{-1}(x). In this way, we get a functor from the category Cov/X\mathrm{Cov}/X of covering spaces of XX to the category of π(X)\pi(X)-sets:

Fx ⁣:Cov/Xπ(X)-Set.\mathcal{F}_x\colon\mathrm{Cov}/X\longrightarrow\pi(X)\text{-}\mathrm{Set}.

The universal covering space

Let X~\widetilde{X} be the topological space whose underlying set is the set of all homotopy classes of paths in XX starting at the base point xx and whose topology is the weakest one making the following map continuous:

p ⁣:X~X,[γ]γ(1).p\colon\widetilde{X}\to X,\quad [\gamma]\mapsto\gamma(1).

For convenience, we fix the bas point x~\tilde{x} of X~\widetilde{X} to be the homotopy class of the trivial loops through xx.

Then p ⁣:X~Xp\colon\widetilde{X}\to X is the universal covering space of XX in the sense that for any connected covering space YXY\to X, there exists a morphism of covering spaces of XX:

\begin{gather*} \widetilde{X} \stackrel{\exists!}{\longrightarrow} Y \\ \searrow\quad\swarrow \\ X \end{gather*}

and this morphism is unique if we require it to preserve the base points. When we choice yp1(x)y\in p^{-1}(x) to be the base point of YY, the morphism is given as follows: the unique path-lifting lemma allows us to lift every path γ\gamma start at xx in XX to a path start at yy in YY, we define the image of [γ][\gamma] to be the end of this path.

Note that the universal covering space naturally has a free π(X)\pi(X)-action given by composition of homotopy classes of paths. For any subgroup HH of π(X)\pi(X), the orbit space X~/H\widetilde{X}/H also gives a connected covering space of XX whose fibers are isomorphic to π(X)/H\pi(X)/H.

From π(X)\pi(X)-sets to covering spaces

Let SS be a π(X)\pi(X)-set. Then, we may write

S=π(X)s.S = \bigsqcup \pi(X)s.

We define

F(S) = \bigsqcup \widetilde{X}/\pi(X)_s, $$ and the bundle map $p\colon F(S)\to X$ is the one induced from the previous property of universal covering space. One can see the canonical $\pi(X)$-map

S\longrightarrow p^{-1}(x)

isbijective.Notethatthisinducesafunctor is bijective. Note that this induces a functor

F\colon \pi(X)\text{-}\mathrm{Set}\longrightarrow\mathrm{Cov}/X.

### The equivalence Given a covering space $p\colon Y\to X$, we have a $\pi(X)$-set $p^{-1}(x)$ where $x$ is a point of $X$. Assume $p^{-1}(x) = \bigsqcup \pi(X)y$, we have $$F(p^{-1}(x)) = \bigsqcup \widetilde{X}/\pi(X)_y.

As elements of π(X)y\pi(X)_y fix yp1(x)y\in p^{-1}(x), the unique morphism of covering spaces of XX from X~\widetilde{X} to YY mapping x~\tilde{x} to yy factors through X~/π(X)y\widetilde{X}/\pi(X)_y. In this way, we have a canonical morphism of covering spaces of XX:

F(p1(x))Y.F(p^{-1}(x))\longrightarrow Y.

It is not difficult to see that YY is connected if and only if π(X)\pi(X) acts transitive on p1(x)p^{-1}(x). Therefore, the orbit decomposition of p1(x)p^{-1}(x) corresponds to the connected components decomposition of YY. In this way, we see the moprhism $$F(p^{-1}(x))\longrightarrow Y$$ is an isomoprhism.

Now, we get the Galois correspondence for covering spaces.

Theorem: Let XX be a connected, locally path-connected and semi-locally simply-connected space. Then we have an equivalence of categories

Cov/Xπ(X)-Set.\mathrm{Cov}/X\cong\pi(X)\text{-}\mathrm{Set}.

Rethinking

It is easy to verify that the universal covering space X~\widetilde{X} represents the functor Fx\mathcal{F}_x. Moreover, we have $$\mathrm{Aut}_X(\widetilde{X}) \cong \pi(X),$$ and $$\mathrm{Aut}_X(\widetilde{X}) < \mathrm{Aut}(\mathcal{F}_x).$$

However, AutX(X~)\mathrm{Aut}_X(\widetilde{X}) is in general not isomorphic to Aut(Fx)\mathrm{Aut}(\mathcal{F}_x).

What’s more, by only assume XX is connected, the functor Fx\mathcal{F}_x still works and is independent of the choice of xx, while the universal covering X~\widetilde{X} may not exists.

Let’s stop this post here and leave further discussions next time.

Reference

One can take a loo at

  • Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002.