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 fundamental group^{} ${\pi}_{1}(X,*)$ by $\pi $. Given a loop $\gamma :I\to X$ with $\gamma (0)=\gamma (1)=*$ and a point $e\in F$ there exists a unique $\stackrel{~}{\gamma}:I\to E,$ with $\stackrel{~}{\gamma}(0)=e$ such that $p\circ \stackrel{~}{\gamma}=\gamma $, that is, a lifting of $\gamma $ starting at $e$. Clearly, the endpoint $\stackrel{~}{\gamma}(1)$ 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 ${\stackrel{~}{\gamma}}_{1},{\stackrel{~}{\gamma}}_{2}:I\to E$ their liftings starting at $e$. Then there is a homotopy^{} $H:I\times I\to X$ with the following 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 $\stackrel{~}{H}:I\times I\to E$ with $H(0,0)=e$. Notice that $\stackrel{~}{H}(\bullet ,0)={\stackrel{~}{\gamma}}_{1}$ (respectively $\stackrel{~}{H}(\bullet ,1)={\stackrel{~}{\gamma}}_{2}$) since they both are liftings of ${\gamma}_{1}$ (respectively ${\gamma}_{2}$) starting at $e$. Also notice that that $\stackrel{~}{H}(1,\bullet )$ is a path that lies entirely in the fiber (since it lifts the constant path $*$). Since the fiber is discrete this means that $\stackrel{~}{H}(1,\bullet )$ is a constant path. In particular $\stackrel{~}{H}(1,0)=\stackrel{~}{H}(1,1)$ or equivalently ${\stackrel{~}{\gamma}}_{1}(1)={\stackrel{~}{\gamma}}_{2}(1)$.

–

2.
By (1) the map is well defined. To prove that it is an action notice that 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 their liftings (as is easily verified by projecting). In other words, the lifting of ${\gamma}_{1}{\gamma}_{2}$ that starts at $e$ is the concatenation of ${\stackrel{~}{\gamma}}_{1}$, the lifting of ${\gamma}_{1}$ that starts at $e$, and ${\stackrel{~}{\gamma}}_{2}$ the lifting of ${\gamma}_{2}$ that starts 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 lifting starting at $e$ is a loop.
∎
Definition 2.
The action described in the above theorem is called the monodromy action and the corresponding homomorphism^{}
$$\rho :\pi \to \mathrm{Sym}(F)$$ 
is called the monodromy of $p$.
Title  monodromy 

Canonical name  Monodromy 
Date of creation  20130322 13:26:20 
Last modified on  20130322 13:26:20 
Owner  mathcam (2727) 
Last modified by  mathcam (2727) 
Numerical id  8 
Author  mathcam (2727) 
Entry type  Definition 
Classification  msc 55R05 
Related topic  MonodromyGroup 
Defines  monodromy 
Defines  monodromy action 
Defines  monodromy homomorphism 