|
|
|
|
|
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 $e\cdot\Gg$ .
Theorem 1 With notation as above we have:
- If $\Gg_1$ and $\Gg_2$ are homotopic relative $\partial I$ then $$\forall e\in F \quad e\cdot\Gg_1=e\cdot\Gg_2.$$
- The map $$ F\times\pi\to F,\quad (e,\Gg)\mapsto e\cdot\Gg$$ defines a right action of $\pi$ on $F$ .
- 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.
- 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:
- $H(\bullet,0)=\Gg_1$ ,
- $H(\bullet,1)=\Gg_2$ ,
- $H(0,t)=H(1,t)=*,\quad \forall t\in I$ .
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 $\tilde \Gg_1(1)=\tilde \Gg_2(1)$ .
- 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 $$e\cdot (\Gg_1\Gg_2)=(e\cdot \Gg_1)\cdot \Gg_2\,.$$
- This is a tautology: $\Gg$ 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 $$\Gr\co \Gp\to {\rm Sym}(F) $$ is called the monodromy of $p$ .
|
"monodromy" is owned by mathcam. [ full author list (2) | owner history (1) ]
|
|
(view preamble | get metadata)
See Also: monodromy group
| Also defines: |
monodromy, monodromy action, monodromy homomorphism |
|
|
Cross-references: homomorphism, theorem, 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
There are 8 references to this entry.
This is version 5 of monodromy, born on 2003-02-08, modified 2004-02-18.
Object id is 4000, canonical name is Monodromy.
Accessed 9527 times total.
Classification:
| AMS MSC: | 55R05 (Algebraic topology :: Fiber spaces and bundles :: Fiber spaces) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
