monodromy

Let $(X,*)$ be a connected and locally connected based space and $p:E\to X$ a covering map. We will denote $p^{-1}(*)$, the fiber over the basepoint, by $F$, and the fundamentalgroup ${\pi }_1(X,*)$ by $\pi$. Given a loop$\gamma :I\to X$ with $\gamma (0)=\gamma (1)=*$ and a point $e\in F$ there exists aunique $\tilde{\gamma }:I\to E,$ with $\tilde{\gamma }(0)=e$ such that $p\circ \tilde{\gamma }=\gamma$,that is, a lifting of $\gamma$ starting at $e$. Clearly, the endpoint$\tilde{\gamma }(1)$ is also a point of the fiber, which we will denote by$e\cdot \gamma$.

Theorem 1.

With notation as above we have:

1. If ${\gamma }_1$ and ${\gamma }_2$ are homotopic relative $\partial I$ then

   $$\forall e\in F\mathit{\hspace{1em}}e\cdot {\gamma }_1=e\cdot {\gamma }_2.$$

2. The map

   $$F\times \pi \to F,(e,\gamma )\mapsto e\cdot \gamma$$

   defines a right action of $\pi$ on $F$.

3. The stabilizer of a point $e$ is the image of the fundamental group${\pi }_1(E,e)$ under the map induced by $p$:

   $$\mathrm{Stab}(x)=p_{*}\left({\pi }_1(E,e)\right).$$

Proof.

1. Let $e\in F$, ${\gamma }_1,{\gamma }_2:I\to X$ two loops homotopic relative$\partial I$ and ${\tilde{\gamma }}_1,{\tilde{\gamma }}_2:I\to E$ their liftingsstarting at $e$. Then there is a homotopy $H:I\times I\to X$ with thefollowing properties:

   • $H(\bullet ,0)={\gamma }_1$,

   • $H(\bullet ,1)={\gamma }_2$,

   • $H(0,t)=H(1,t)=*,\forall t\in I$.

   According to the lifting theorem $H$ lifts to a homotopy $\tilde{H}:I\times I\to E$ with $H(0,0)=e$. Notice that $\tilde{H}(\bullet ,0)={\tilde{\gamma }}_1$ (respectively $\tilde{H}(\bullet ,1)={\tilde{\gamma }}_2$)since they both are liftings of ${\gamma }_1$ (respectively ${\gamma }_2$) starting at$e$. Also notice that that $\tilde{H}(1,\bullet )$ is a path that lies entirely inthe fiber (since it lifts the constant path $*$). Since the fiber isdiscrete this means that $\tilde{H}(1,\bullet )$ is a constant path. In particular$\tilde{H}(1,0)=\tilde{H}(1,1)$ or equivalently ${\tilde{\gamma }}_1(1)={\tilde{\gamma }}_2(1)$.

2. By (1) the map is well defined. To prove that it is an action noticethat firstly the constant path $*$ lifts to constant paths and therefore

   $$\forall e\in F,e\cdot 1=e.$$

   Secondly the concatenation of two paths lifts to the concatenation of theirliftings (as is easily verified by projecting). In other words, the liftingof ${\gamma }_1{\gamma }_2$ that starts at $e$ is the concatenation of ${\tilde{\gamma }}_1$,the lifting of${\gamma }_1$ that starts at $e$, and ${\tilde{\gamma }}_2$ the lifting of ${\gamma }_2$ thatstarts in ${\gamma }_1(1)$. Therefore

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

3. This is a tautology: $\gamma$ fixes $e$ if and only if its liftingstarting at $e$ is a loop.

∎

Definition 2.

The action described in the above theorem is called the monodromyaction and the corresponding homomorphism

$$\rho :\pi \to \mathrm{Sym}(F)$$

is called the monodromy of $p$.