You are here
Homemonodromy
Primary tabs
monodromy
Let $(X,*)$ be a connected and locally connected based space and $p\colon\thinspace E\to X$ 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 $\gamma\colon\thinspace I\to X$ with $\gamma(0)=\gamma(1)=*$ and a point $e\in F$ there exists a unique $\tilde{\gamma}\colon\thinspace I\to E,$ 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
$\forall e\in F\quad e\cdot\gamma_{1}=e\cdot\gamma_{2}.$ 2. The map
$F\times\pi\to F,\quad(e,\gamma)\mapsto e\cdot\gamma$ 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 $p$:
$\operatorname{Stab}(x)=p_{{*}}\left(\pi_{1}(E,e)\right)\,.$
Proof.
1. Let $e\in F$, $\gamma_{1},\gamma_{2}\colon\thinspace I\to X$ two loops homotopic relative $\partial I$ and $\tilde{\gamma}_{1},\tilde{\gamma}_{2}\colon\thinspace I\to E$ their liftings starting at $e$. Then there is a homotopy $H\colon\thinspace I\times I\to X$ with the following properties:

$H(\bullet,0)=\gamma_{1}$,

$H(\bullet,1)=\gamma_{2}$,

$H(0,t)=H(1,t)=*,\quad\forall t\in I$.
According to the lifting theorem $H$ lifts to a homotopy $\tilde{H}\colon\thinspace I\times I\to E$ 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 in the fiber (since it lifts the constant path $*$). Since the fiber is discrete 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 notice that firstly the constant path $*$ lifts to constant paths and therefore
$\forall e\in F,\quad e\cdot 1=e\,.$ 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 $\tilde{\gamma}_{1}$, the lifting of $\gamma_{1}$ that starts at $e$, and $\tilde{\gamma}_{2}$ 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.
∎
Definition 2.
The action described in the above theorem is called the monodromy action and the corresponding homomorphism
$\rho\colon\thinspace\pi\to{\rm Sym}(F)$ 
is called the monodromy of $p$.
Mathematics Subject Classification
55R05 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
Recent Activity
new correction: Error in proof of Proposition 2 by alex2907
Jun 24
new question: A good question by Ron Castillo
Jun 23
new question: A trascendental number. by Ron Castillo
Jun 19
new question: Banach lattice valued Bochner integrals by math ias
Jun 13
new question: young tableau and young projectors by zmth
Jun 11
new question: binomial coefficients: is this a known relation? by pfb
Jun 6
new question: difference of a function and a finite sum by pfb