4. Classification of Covering

Definition 4.1. The universal cover of $B$ is a covering map $p:E\to B$ with $E$ simply connected.

The universal cover is unique (if exists) up to homeomorphism. This follows from the lifting criterion and the unique lifting property of covering maps. We left it as an exercise to readers.

Definition 4.2. A space is semi-locally simply connected if for any $x_0\in X$, there is a neighbourhood $U_0$ such that the image of the map $i_{\ast }:{\pi }_1(U_0,x_0)\to {\pi }_1(X,x_0)$ is trivial.

We recall the following theorem from point-set topology.

Theorem 4.3 (Existence of the universal cover). Assume $B$ is path connected and locally path connected. Then universal cover of $B$ exists if and only if $B$ is semi-locally simply connected.

Definition 4.4. We define the category $\text{Cov}(B)$ of coverings of $B$ where

•	an object is a covering map $p:E\to B$;
•	a morphism between two coverings $p_1:E_1\to B$ and $p_2:E_2\to B$ is a map $f:E_1\to E_2$ such that the following diagram is commutative

Definition 4.5. Let $B$ be connected. We define ${\text{Cov}}_0(B)\subset \text{Cov}(B)$ to be the subcategory whose objects consist of coverings of $B$ which are connected spaces.

Proposition 4.6. Let $B$ be connected and locally path connected. Then any morphism in ${\text{Cov}}_0(B)$ is a covering map.

Proof. Exercise.

$\square$

Definition 4.7. We define the category $G-\mathsf{Set}$, where

•	an object is a set $S$ with $G$-action and
•	morphisms are $G$-equivariant set maps, i.e. $f:S_1\to S_2$ such that $f\circ g=g\circ f$, for any $g\in G$.

Given a covering $p:E\to B$, $b\in B$, the transport functor shows$$p^{-1}(b)\in {\pi }_1(B,b)-\mathsf{Set}.$$

Lemma 4.8. Let $B$ be path connected. Then ${\pi }_1(B,b)$ acts transitively on $p^{-1}(b)$ if and only if $E$ is path connected.

Proof. The "if" part follows from Proposition 3.23. We prove the "only if" part.

Let us fix a point $e_0\in p^{-1}(b)$. Assume ${\pi }_1(B,b)$ acts transitively on $p^{-1}(b)$. This implies that any point in $p^{-1}(b)$ is connected to $e_0$ by a path. Given any point $e\in E$, let $\gamma$ be a path in $B$ that connects $p(e)$ to $b$. The transport functor $T_{[\gamma ]}$ gives a path connecting $e$ to some point in $p^{-1}(b)$. This further implies that $e$ is path connected to $e_0$. This proves the "only if" part.

[h]

Figure 1. Transitivity v.s. path connectedness

$\square$

Corollary 4.9. Let $B$ be path connected, $p:E\to B$ be a covering. Then there is a one-to-one correspondence between path connected components of $E$ and ${\pi }_1(B,b)$-orbits in $p^{-1}(b)$.

Proof. Exercise.

$\square$

Example 4.10. Let $p:\tilde{B}\to B$ be the universal covering. By Proposition 3.23, the fiber $p^{-1}(b)$ can be identified with ${\pi }_1(B,b)$ itself.

Proposition 4.11. Assume $B$ is path connected and locally path connected. Let $p_1,p_2\in \text{Cov}(B)$. Then$${\mathrm{Hom}}_{\text{Cov}(B)}(p_1,p_2)\cong {\mathrm{Hom}}_{{\pi }_1(B,b)-\mathsf{Set}}(p_1^{-1}(b),p_2^{-1}(b))$$for any $b\in B$.

Proof. Let $f\in {\mathrm{Hom}}_{\text{Cov}(B)}(p_1,p_2)$, i.e.By restricting $f$ to the fiber $p^{-1}(b)$, it induces a map$$f_b:p_1^{-1}(b)\to p_2^{-1}(b).$$By the same argument as in Proposition 3.30, we find $f_b$ is ${\pi }_1(B,b)$-equivariant. Thus we obtain a map$$\begin{array}{rl}\Phi :{\mathrm{Hom}}_{\text{Cov}(B)}(p_1,p_2)& \to {\mathrm{Hom}}_{{\pi }_1(B,b)-\mathsf{Set}}(p_1^{-1}(b),p_2^{-1}(b))\\ f& \to f_b\end{array}$$

The injectivity of $\Phi$ follows from the uniqueness of the lifting.

To prove surjectivity, we can assume $E_1$ is path connected, thus ${\pi }_1(B,b)$ acts transitively on $p_1^{-1}(b)$ (Corollary 4.9). Given $f_b:p_1^{-1}(b)\to p_2^{-1}(b)$, let us fix two points $e_i\in p_i^{-1}(b)$. The ${\pi }_1(B,b)$-equivariance of $f_b$ gives rise to the homomorphism$$\begin{array}{lll}& {\mathrm{Stab}}_{e_1}({\pi }_1(B,b))⟶& {\mathrm{Stab}}_{e_2}({\pi }_1(B,b))\\ & ={\pi }_1(E_1,e_1)& ={\pi }_1(E_2,e_2).\end{array}$$By Lifting Criterion (Theorem 3.24), we obtain a morphism $f:E_1\to E_2$ as required.

$\square$

Theorem 4.12. Assume $B$ is path connected, locally path connected and semi-locally simply connected. $b\in B$. Then there exists an equivalence of categories$$\text{Cov}(B)\simeq {\pi }_1(B,b)-\mathsf{Set}.$$

Proof. Let us denote ${\pi }_1={\pi }_1(B,b)$. Let $\tilde{p}:\tilde{B}\to B$ be a fixed universal cover of $B$ and $\tilde{b}\in {\pi }^{-1}(b)$ chosen.

We will define the following functorsLet $p:E\to B$ be a covering, we define$$F(p):=p^{-1}(b).$$Let $S\in {\pi }_1-\mathsf{Set}$, we define$$G(S):=\tilde{B}{\times }_{{\pi }_1}S=\tilde{B}\times S/\sim ,$$where $(e\cdot g,s)\sim (e,g\cdot s)$ for any $e\in \tilde{B},s\in S,g\in {\pi }_1$. Note that here $e\cdot g$ represents the (right) ${\pi }_1$-action on $\tilde{B}$. Then we have natural isomorphisms$$F\circ G\stackrel{\eta }{\simeq }1,G\circ F\stackrel{\tau }{\simeq }1.$$Here $\eta$ is the natural isomorphism$${\eta }_S\in {\mathrm{Hom}}_{{\pi }_1-\mathsf{Set}}(F\circ G(S),S),{\eta }_S(e,s)=g\cdot s\text{if}e=\tilde{b}\cdot g.$$$\tau$ is the natural isomorphism$${\tau }_p\in {\mathrm{Hom}}_{\text{Cov}(B)}(p^{\prime },p)\cong {\mathrm{Hom}}_{{\pi }_1-\mathsf{Set}}(p^{-1}(b),p^{-1}(b)),p^{\prime }=G\circ F(p):\tilde{B}{\times }_{{\pi }_1}{p}^{-1}(b)\to B,$$which is determined by the identity map in ${\mathrm{Hom}}_{{\pi }_1-\mathsf{Set}}(p^{-1}(b),p^{-1}(b))$.

$\square$

Definition 4.13. Let $B$ be path connected and $p:E\to B$ be a connected covering. A deck transformation (or covering transformation) of $p$ is a homeomorphism $f:E\to E$ such that $p\circ f=p$.Let $\mathrm{Aut}(p)$ denote the group of deck transformations.

Note that $\mathrm{Aut}(p)$ acts freely on $E$ by the Uniqueness of Lifting.

Proposition 4.14. Let $B$ be path connected and $p:E\to B$ be a connected covering. Then $\mathrm{Aut}(p)$ acts properly discontinuously on $E$.

Proof. Exercise.

$\square$

Corollary 4.15. Assume $B$ is path connected, locally path connected. Let $p:E\to B$ be a connected covering, $e\in E,b=p(e)\in B$, $G={\pi }_1(B,b),H={\pi }_1(E,e)$. Then$$\mathrm{Aut}(p)\cong N_G(H)/H$$where$$N_G(H):=\{r\in G\mid rH{r}^{-1}=H\}$$is the normalizer of $H$ in $G$.

Proof. By Proposition 3.23, Lemma 4.8 and Theorem 4.12$$\mathrm{Aut}(p)\cong {\mathrm{Hom}}_{G-\mathsf{Set}}(G/H,G/H)=N_G(H)/H.$$

$\square$

Definition 4.16. We define the orbit category $Orb(G)$:

•	objects consist of (left) coset $G/H$, where $H$ is a subgroup of $G$;
•	morphisms are $G$-equivariant maps: $G/H_1\to G/H_2$.

Note $G/H_1$ and $G/H_2$ are isomorphic in $Orb(G)$ if and only if $H_1$ and $H_2$ are conjugate subgroups of $G$.

If we restrict Theorem 4.12 to connected coverings, we find

Theorem 4.17. Assume $B$ is path connected, locally path connected and semi-locally simply connected. $b\in B$. Then there exists an equivalence of categories$${\text{Cov}}_0(B)\simeq Orb({\pi }_1(B,b)).$$

The universal cover $\tilde{B}\to B$ corresponds to the orbit ${\pi }_1(B,b)$. For the orbit ${\pi }_1(B,b)/H$, it corresponds to$$E=\tilde{B}/H\to B.$$This can be illustrated by the following correspondence

A more intrinsic formulation is as follows. Given a covering $p:E\to B$, we obtain a transport functor$$T_p:{\Pi }_1(B)\to \mathsf{Set}.$$Given a commutative diagramwe find a natural transformation$$\begin{array}{r}\tau :T_{p_1}⟹T_{p_2},\tau =\{f:p_1^{-1}(b)\to p_2^{-1}(b)|b\in B\}.\end{array}$$

The above structure can be summarized by a functor$$T:\text{Cov}(B)\to \mathsf{Fun}({\Pi }_1(B),\mathsf{Set}).$$

Theorem 4.18. Assume $B$ is path connected, locally path connected and semi-locally simply connected. Then $$T:\text{Cov}(B)\to \mathsf{Fun}({\Pi }_1(B),\mathsf{Set})$$is an equivalence of categories.