PlanetMath: lifting theorem

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lifting theorem

(Theorem)

Let $p\colon\thinspace E\to B$ be a covering map and $f\colon\thinspace X\to B$ be a (continuous) map where , and are path connected and locally path connected. Also let $x\in X$ and $e\in E$ be points such that . Then lifts to a map $\tilde f\colon\thinspace X\to E$ with $\tilde f(x)=e$ if and only if $\pi_1(f)$ maps $\pi_1(X,x)$ inside the image $\pi_1(p)\left(\pi_1(E,e)\right)$ , where $\pi_1$ denotes the fundamental group functor. Furthermore $\tilde f$ is unique (provided it exists of course).

The following diagrams might be useful: To check

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ &{(E,e)}\ar[d]^{p}\ {(X,x)}\ar[r]_{f}\ar@{-->}[ur]^{?\tilde f}&{(B,b)} } } \end{xy}$

one only needs to check

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ &&{\pi_1(E,e)}\ar[d]^{\pi_1(p)}\... ...1(X,x)\right)}\ar@{-->}[ur]^{?}\ar[r]_{\quad\subset}& {\pi_1(B,b)} } } \end{xy}$

Corollary 1 Every map from a simply connected space lifts. In particular:

a path $\gamma \colon\thinspace I\to B$ lifts,
a homotopy of paths $H\colon\thinspace I\times I\to B$ lifts, and
a map $\sigma \colon\thinspace S^n\to B$ , lifts if $n\geq 2$ .

Note that (3) is not true for because the circle is not simply connected. So although by (1) every closed path in lifts to a path in it does not necessarily lift to a closed path.

"lifting theorem" is owned by Dr_Absentius.

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Cross-references: closed path, circle, homotopy of paths, simply connected, functor, fundamental group, image, lifts, points, path connected, map, continuous, covering map
There are 3 references to this entry.

This is version 6 of lifting theorem, born on 2003-02-02, modified 2004-02-28.
Object id is 3963, canonical name is LiftingTheorem.
Accessed 2484 times total.

Classification:

AMS MSC:

55R05 (Algebraic topology :: Fiber spaces and bundles :: Fiber spaces)

Pending Errata and Addenda

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