monodromyMonodromy2003-02-08 22:47:042004-02-18 15:36:44Definitionmonodromymonodromy actionmonodromy homomorphism\usepackage{amsmath}
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\def\co{\colon\thinspace}Let $(X,*)$ be a connected and locally connected based space and $p\co E \to X$ 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
$\Gg\co I \to X$ with $\Gg(0)=\Gg(1)=*$ and a point $e\in F$ there exists a
unique $\tilde\Gg\co I\to E,$ with $ \tilde\Gg(0)=e$ such that $p\circ\tilde\Gg=\Gg$,
that is, a lifting of $\Gg$ starting at $e$. Clearly, the endpoint
$\tilde\Gg(1)$ is also a point of the fiber, which we will denote by
$e\cdot\Gg$.
\begin{thm}
With notation as above we have:
\begin{enumerate}
\item If $\Gg_1$ and $\Gg_2$ are homotopic relative $\partial I$ then
$$\forall e\in F \quad e\cdot\Gg_1=e\cdot\Gg_2.$$
\item The map
$$ F\times\pi\to F,\quad (e,\Gg)\mapsto e\cdot\Gg$$
defines a right action of $\pi$ on $F$.
\item The stabilizer of a point $e$ is the image of the fundamental group
$\pi_1(E,e)$ under the map induced by $p$:
$$ \operatorname{Stab}(x) = p_{*}\left(\pi_1(E,e)\right)\,.$$
\end{enumerate}
\end{thm}
\begin{proof}\begin{enumerate}
\item Let $e\in F$, $\Gg_1,\Gg_2\co I\to X$ two loops homotopic relative
$\partial I$ and $\tilde\Gg_1,\tilde\Gg_2\co I\to E$ their liftings
starting at $e$. Then there is a homotopy $H\co I\times I \to X$ with the
following properties:
\begin{itemize}
\item $H(\bullet,0)=\Gg_1$,
\item $H(\bullet,1)=\Gg_2$,
\item $H(0,t)=H(1,t)=*,\quad \forall t\in I$.
\end{itemize}
According to the lifting theorem $H$ lifts to a homotopy $\tilde H\co
I\times I\to E$ with $H(0,0)=e$. Notice that $\tilde
H(\bullet,0)=\tilde\Gg_1$ (respectively $\tilde H(\bullet,1)=\tilde\Gg_2$)
since they both are liftings of $\Gg_1$ (respectively $\Gg_2$) 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 $\tilde \Gg_1(1)=\tilde
\Gg_2(1)$.
\item 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 $\Gg_1\Gg_2$ that starts at $e$ is the concatenation of $\tilde \Gg_1$,
the lifting of
$\Gg_1$ that starts at $e$, and $\tilde \Gg_2$ the lifting of $\Gg_2$ that
starts in $\Gg_1(1)$. Therefore
$$e\cdot (\Gg_1\Gg_2)=(e\cdot \Gg_1)\cdot \Gg_2\,.$$
\item This is a tautology: $\Gg$ fixes $e$ if and only if its lifting
starting at $e$ is a loop.
\end{enumerate}
\end{proof}
\begin{defn}
The action described in the above theorem is called the \emph{monodromy
action} and the corresponding homomorphism
$$\Gr\co \Gp\to {\rm Sym}(F) $$
is called the \emph{monodromy} of $p$.
\end{defn}