Before I begin discussing the Hopf fibration of the 3-spbhere, one of the simplest yet deeply profound example of a non-trivial fiber bundle, I’d like to recall the definition of a fiber bundle.
Let represent the entire space, base space and the fiber respectively where
are connected. If
is a continuous surjection onto the base space, then the structure
is said to be a fiber bundle if for every
, there exists a neighborhood
of
such that there exists a homeomorphism
such that
.
What this basically means is that locally, the fiber bundle looks like the product but globally, it may have different topological properties.
A trivial fiber bundle is a fiber bundle which in which the total space is . In fact, any fiber bundle over a contractible CW Complex is trivial.
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