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Fiber bundle

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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.

Mobius band as fiber bundle.png

Exercise. (a) Describe the fiber bundle below. (b) What if the top and bottom are identified?

Mobius band standing up.jpg

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:

Hair brush w twist.png Hair brush wo twist.png

An important example of a discrete fiber bundle ${\bf Z} \xrightarrow{ i } {\bf R}^1 \xrightarrow{ \pi } {\bf S}^1$ is below:

Bundle R to S.jpg

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.