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monodromy (Definition)

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
    $\displaystyle \forall e\in F \quad e\cdot\gamma _1=e\cdot\gamma _2.$
  2. The map
    $\displaystyle 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$:
    $\displaystyle \operatorname{Stab}(x) = p_{*}\left(\pi_1(E,e)\right)\,.$
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
    $\displaystyle \forall e\in F,\quad e\cdot 1=e\,.$
    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
    $\displaystyle e\cdot (\gamma _1\gamma _2)=(e\cdot \gamma _1)\cdot \gamma _2\,.$
  3. This is a tautology: $ \gamma $ fixes $ e$ if and only if its lifting starting at $ e$ is a loop.
$ \qedsymbol$
Definition 2   The action described in the above theorem is called the monodromy action and the corresponding homomorphism
$\displaystyle \rho \colon\thinspace \pi \to {\rm Sym}(F) $
is called the monodromy of $ p$.



"monodromy" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: monodromy group

Also defines:  monodromy, monodromy action, monodromy homomorphism
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Cross-references: homomorphism, tautology, concatenation, action, well defined, discrete, path, lifts, lifting theorem, properties, homotopy, induced, image, stabilizer, right action, map, homotopic, endpoint, lifting, point, loop, fundamental group, basepoint, fiber, covering map, locally connected, connected
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This is version 5 of monodromy, born on 2003-02-08, modified 2004-02-18.
Object id is 4000, canonical name is Monodromy.
Accessed 6952 times total.

Classification:
AMS MSC: 55R05 (Algebraic topology :: Fiber spaces and bundles :: Fiber spaces)
Pending Errata and Addenda
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