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Fiber bundle
Fiber bundles over space, $B$ called the base, are locally $F × U$, where $F$ is the fiber and $U$ is an open subset of $B$. It is realized via the projection $$\pi_N:N \to B$$ of the bundle.
The Mobius band below is shown as a fiber bundle over the circle with $F$ the interval.
Exercise. (a) Describe the fiber bundle below. (b) What if the top and bottom are identified?
Notation: $$F \xrightarrow{ i } N \xrightarrow{ \pi } B,$$ where $$fiber \xrightarrow{inclusion} bundle \xrightarrow{projection} base.$$ To describe a fiber bundle means then to specify these five participants.
A "trivial" fiber bundle over $B$ is $N=F × B$, globally.
- In 2d, compare the Mobius band and the cylinder.
- In 3d, compare these hairbrushes -- with and without a twist:
An important example of a discrete fiber bundle ${\bf Z} \xrightarrow{ i } {\bf R}^1 \xrightarrow{ \pi } {\bf S}^1$ is below:
The fiber consists of infinitely many isolated points.
Exercise. Find the explicit formula for the projection.
Exercise. Describe a fiber bundle with fiber that consists of $n$ isolated points.
Tangent bundles are a particular case of fiber bundles.
Exercise. Elaborate.