action of fundamental group on cover

Overview

By defining a (well-behaved) action of the fundamental group on a cover we can use many of the tools of group theory to understand the topology of the spaces.

The main results of this idea are that we can learn about automorphisms that preserve the base space from group action automorphisms, or vice versa.

Definition

Let $p:E\to X$ be a cover and $x_0\in X$. For each $e\in p^{-1}(x_0)$ and $[\gamma ]\in {\pi }_1(X,x_0)$ let

$[\gamma ]\cdot e:=\tilde{\gamma }(1)$

where $\tilde{\gamma }$ is the unique lift of $\gamma$ with $\tilde{\gamma }(0)=e$.

This defines a group action of ${\pi }_1(X,x_0)$ on $p^{-1}(x_0)$.

Proof

First, as a sanity check we can make sure that the action makes sense

$p(\tilde{\gamma }(1))=\gamma (1)=x_0⟹[\gamma ]\cdot e\in p^{-1}(x_0)$

Next, we check the axioms for a group action.

1. Identity action

   $[C_{x_0}]\cdot e=C_e(1)=e$

2. Associative because the path lifts uniquely. Thus,

$([{\gamma }_2]\cdot [{\gamma }_1])\cdot e=[{\gamma }_2]\cdot ([{\gamma }_1]\cdot e)$

Transitive action

${\pi }_1(X,x_0)$ acts transitively on $p^{-1}(x_0)$.

Proof

todo

Equivariance of cover automorphisms

Fix $x_0\in X$. Then each $f\in {\text{Aut}}_X(E)$ satisfies

$f([\gamma ]\cdot e)=[\gamma ]\cdot f(e)\forall [\gamma ]\in {\pi }_1(X,x_0),e\in p^{-1}(x_0).$

Proof

todo

Correspondence between automorphisms

${\text{Aut}}_{X}E\cong {\text{Aut}}_{{\pi }_1(X,x_0)}(p^{-1}(x_0))$

Proof

todo We can look at the map

$$\begin{array}{rl}{\text{Aut}}_{X}E& ⟶{\text{Aut}}_{{\pi }_1(X,x_0)}(p^{-1}(x_0))\\ f& ⟼f{|}_{p^{-1}(x_0)}\end{array}$$

Just need to show that it is injective and surjective.

Corollaries

${\text{Aut}}_{X}E\cong N_{{\pi }_1(X,x_0)}(p_{\ast }{\pi }_1(E,e_0))/p_{\ast }{\pi }_1(E,e_0)$

2. If $p:E\to X$ is a regular cover,

${\text{Aut}}_X(E)\cong {\pi }_1(X,x_0)/p_{\ast }{\pi }_1(E,e_0)$

3. If $p:E\to X$ is a universal cover,

${\text{Aut}}_{X}E\cong {\pi }_1(X,x_0)$