Jonginn Yun

The Point of the Definition

A fiber bundle is a space that looks like a product locally but may fail to be a product globally. The basic picture is a projection

$$\pi:E\to B,$$

where $B$ is the base space, $E$ is the total space, and the fiber over $b\in B$ is

$$E_b=\pi^{-1}(b).$$

The phrase “looks like a product locally” means that each point of the base has an open neighborhood $U\subset B$ for which there is a homeomorphism

$$\varphi_U:\pi^{-1}(U)\to U\times F$$

such that projection to $U$ agrees with $\pi$. In other words, over a small enough patch of the base, every point carries a copy of the same model fiber $F$.

Local Trivializations

The maps $\varphi_U$ are called local trivializations. They are not extra decoration; they are the data that tells us how the bundle is locally organized.

If $e\in \pi^{-1}(U)$ and $\varphi_U(e)=(b,f)$, then $b=\pi(e)$ and $f$ is the coordinate of $e$ inside the model fiber. A different trivialization over another open set $V$ may assign a different fiber coordinate to the same point.

This is why bundles are about gluing. The local pieces are simple, but their overlap rules can carry nontrivial global information.

Transition Functions

On an overlap $U\cap V$, two trivializations are related by

$$\varphi_V\circ\varphi_U^{-1}:(U\cap V)\times F\to (U\cap V)\times F.$$

Because both sides must project to the same base point, this map has the form

$$(b,f)\mapsto (b,g_{VU}(b)f),$$

where $g_{VU}(b)$ is a symmetry of the fiber. The functions $g_{VU}$ are the transition functions of the bundle.

They satisfy the cocycle condition on triple overlaps:

$$g_{WU}(b)=g_{WV}(b)g_{VU}(b).$$

This condition is exactly the statement that changing charts from $U$ to $V$ to $W$ agrees with changing directly from $U$ to $W$.

Sections

A section is a map

$$s:B\to E$$

such that $\pi\circ s=\operatorname{id}_B$. It chooses one point in every fiber.

In a local trivialization, a section becomes an ordinary function

$$s_U:U\to F.$$

On overlaps, these local representatives must transform by the transition functions:

$$s_V(b)=g_{VU}(b)s_U(b).$$

This is the first place where the bundle viewpoint becomes useful in physics. A field can often be read as a section, while a change of local trivialization looks like a gauge transformation.

Why Nontrivial Bundles Exist

If all transition functions can be removed by changing local trivializations, then the bundle is globally a product $B\times F$. If they cannot, the bundle has genuine global twisting.

The Mobius strip is the simplest geometric warning: locally it is an interval over a circle, but one full trip around the base reverses the fiber. The obstruction is not visible in any sufficiently small patch. It is visible only after comparing patches around a loop.