Introduction to Fiber Bundles

Treating spaces as fiber bundles allows us to tame twisted beasts.

If we glue lines onto every point $b$ in a circle (or a circle to every point of a line), we get a cylinder. In other words, a cylinder is the product space $S^1 \times [0,1]$.

If we glue lines onto every point of a circle, progressively twisting each individual line, we get a Mobius strip.

A fiber bundle with fiber $F$ consists of: 2 topological spaces, and a projection map which projects the total space onto its base space.

In our example, $E$:= Mobius strip, $B$:= base circle, where the fiber $F$:= $[0,1]$.

If you flip the arrow around, $\pi^{-1}$, the inverse image of the projection map, maps every $b$ in the basespace to its corresponding fiber $\pi^{-1}(b)$ in the total space.

Similarly, the $\pi^{-1}(N)$ maps every point in the neighborhood $N$ of $b$ to their corresponding fibers $\pi^{-1}(N)$ in the total space.

We can locally treat the Mobius strip as a plane, in the same way that we can locally treat a cylinder as a plane.

This property allows us to vastly simplify calculations; it allows us to locally treat twisted spaces like their non-twisted counterparts. More formally:

For every $b \in B$ there is a neighborhood $N$ of $b$ s.t. the following diagram commutes.

Formally:

As an aside: How can we formally construct a twisted space?

A Mobius strip := $[0,1] \times [0,1] /\sim$, where the equivalence relation is $(0,t) \sim (1, 1-t)$.

Basically, this equivalence relation gives us gluing instructions.

We must twist the plane an odd number of times s.t. $(0,t)$ are the same as $(1, 1-t)$.