galois correspondences iii

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

Definition: A covering space $Y$ over $X$ is a bundle $pcolon Yto X$ which is locally trival and with discrete fibers. In plain words, the condition means for every point $xin X$, there exists an open neighborhood $U$ such that the pullback $p^{-1}(U)to U$ is isomorphic to a projection $Utimes Fto U$ with $F$ a nonempty discrete set.
$$
array{
p^{-1}(U) stackrel{cong}{longrightarrow} Utimes F \
psearrowquadswarrowmathrm{pr} \
U
}
$$

Definition: The fundamental group $pi(X,x)$ of $X$ at a point $xin X$ is the group of homotopy classes of loops through $x$ and lie in $X$.

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

Let $pcolon Yto X$ be a covering space. Then $p^{-1}(x)$ is a $pi(X)$-set whose action is given as follows: the unique path-lifting lemma says that given a point $y_1in p^{-1}(x)$, any loop $phi$ in $X$ start at $x$ can be uniquely lifted to a path in $Y$ from $y_1$ to another point $y_2in p^{-1}(x)$. The homotopic loops give the same point $y_2$, thus we define the result of the action of this homotopy class $[phi]$ on $y_1$ to be $y_2$.

Let $fcolon Yto Y’$ be a morphism of covering spaces of $X$. Then it induces a continuous $pi(X)$-map from $p^{-1}(x)$ to $p’^{-1}(x)$. In this way, we get a functor from the category $mathrm{Cov}/X$ of covering spaces of $X$ to the category of $pi(X)$-sets:
$$
mathcal{F}_xcolonmathrm{Cov}/Xlongrightarrowpi(X)text{-}mathrm{Set}.
$$

Let $widetilde{X}$ be the topological space whose underlying set is the set of all homotopy classes of paths in $X$ starting at the base point $x$ and whose topology is the weakest one making the following map continuous:
$$
pcolonwidetilde{X}to X,quad
[gamma]mapstogamma(1).
$$

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

Then $pcolonwidetilde{X}to X$ is the universal covering space of $X$ in the sense that for any connected covering space $Yto X$, there exists a morphism of covering spaces of $X$: $$
array{
widetilde{X} stackrel{exists!}{longrightarrow} Y \
searrowquadswarrow \
X
}
$$ and this morphism is unique if we require it to preserve the base points. When we choice $yin p^{-1}(x)$ to be the base point of $Y$, the morphism is given as follows: the unique path-lifting lemma allows us to lift every path $gamma$ start at $x$ in $X$ to a path start at $y$ in $Y$, we define the image of $[gamma]$ to be the end of this path.

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

Let $S$ be a $pi(X)$-set. Then, we may write
$$
S = bigsqcup pi(X)s.
$$

We define $$
F(S) = bigsqcup widetilde{X}/pi(X)_s,
$$ and the bundle map $pcolon F(S)to X$ is the one induced from the previous property of universal covering space. One can see the canonical $pi(X)$-map $$
Slongrightarrow p^{-1}(x)
$$ is bijective.

Note that this induces a functor
$$
Fcolon pi(X)text{-}mathrm{Set}longrightarrowmathrm{Cov}/X.
$$

Given a covering space $pcolon Yto X$, 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 $pi(X)_y$ fix $yin p^{-1}(x)$, the unique morphism of covering spaces of $X$ from $widetilde{X}$ to $Y$ mapping $tilde{x}$ to $y$ factors through $widetilde{X}/pi(X)_y$. In this way, we have a canonical morphism of covering spaces of $X$: $$F(p^{-1}(x))longrightarrow Y.$$

It is not difficult to see that $Y$ is connected if and only if $pi(X)$ acts transitive on $p^{-1}(x)$. Therefore, the orbit decomposition of $p^{-1}(x)$ corresponds to the connected components decomposition of $Y$. 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 $X$ be a connected, locally path-connected and semi-locally simply-connected space. Then we have an equivalence of categories $$
mathrm{Cov}/Xcongpi(X)text{-}mathrm{Set}.
$$

It is easy to verify that the universal covering space $widetilde{X}$ represents the functor $mathcal{F}_x$. Moreover, we have $$mathrm{Aut}_X(widetilde{X}) cong pi(X),$$ and $$mathrm{Aut}_X(widetilde{X}) < mathrm{Aut}(mathcal{F}_x).$$ However, $mathrm{Aut}_X(widetilde{X})$ is in general not isomorphic to $mathrm{Aut}(mathcal{F}_x)$.

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

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

One can take a loo at

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