A piece of paper is a collection of vertical strips. A piece of paper glued together at its ends remains as such, as does a piece of paper twisted and then glued together at its ends into a Mobius strip also remains as such.
Fiber bundles: an innocent re-representation of topological structures that has its own distinct generalizations.
Typically bundle is over a vector space. A real vector bundle \(\xi\) over \(B\) consists of
i) a topological space \(E = E(\xi)\) called the total space
ii) a continuous map \(\pi: E \to B\) called the projection map
iii) for each \(b \in B\), the structure of a vector space over the real numbers in the set \(\pi^{-1}(b)\).
iv) local triviality condition: for each \(b\) of \(B\) a neighborhood \(U \subset B\), an integer \(n \geq 0\), and a homeomorphism \(h : U \times \mathbb{R}^n \to \pi^{-1}(U)\) such that, for each \(b \in U\), the map \(x \mapsto h(b, x)\) defines an isomorphism between the vector space \(\mathbb{R}^n\) and the vector space \(\pi^{-1}(b)\).
A trivial bundle is one homeomorphic to \(B \times \mathbb{R}^n\): the piece of paper remains in its original form.
As with the corresponding trivial bundle, a fiber bundle has no natural association between fibers. A ‘twisted’ cylinder remains topologically a cylinder: a twisting of coordinates \((x, \phi) \mapsto (x, (\phi + \epsilon x)\mod 2 \pi)\) gives an equally valid homeomorphism. The logic applies to every bundle region \(\pi^{-1}(U)\).
The vector bundle is a fiber bundle with a vector space structure for the fibers. Other structures are possible; replace the vector space with the desired structure and required isomorphism in iv).
A principal bundle is a fiber bundle with groups as fibers, groups acting freely and transitively on fibers, the base space then the quotient of the total space by the action of the group.
A section \(\sigma: B \to E\) is a smooth right inverse of \(\pi\)
\[\pi(\sigma(x)) = x\]visualized as a plot in the Cartesian plane with \(E\) the plane, \(B\) the x-axis, and each fiber a vertical strip of the plane.
It can be demonstrated that a vector bundle is trivial if and only if there exist a collection of sections that form a basis for the corresponding fiber at every point.
Tangent bundle: in a sense the ‘original’ fiber bundle, motivated by the problem of defining smooth vector fields on manifolds. The problem: no relation between tangent spaces at distinct points on a manifold, so how to define smooth vector fields? The classical definition for smooth vector fields is in terms of the local coordinate chart about any given point. This is well-defined, but unsatisfactory from the point of view of being defined in terms of a chart (into \(\mathbb{R}^N\)) rather than as a map into each (distinct) tangent space. The tangent bundle solution: let \(TM = \bigcup_i T_{p_i}M\), the union of tangent spaces, each considered as a set. Define a vector field as a map \(v: M \to TM\), with each point mapping into its corresponding tangent space. \(TM\) can be turned into a manifold by defining the chart mapping a point and vector in the corresponding tangent space to its coordinate and vector coordinate representation: \((p, v) \mapsto (x_i, v_i)\). Then \(v\) is a smooth vector field on \(M\) if it is simply a smooth function on \(TM\).
Examples
Can demonstrate tangent space \(T\mathbb{S_1}\) of the 1-sphere \(\mathbb{S_1}\) is trivial, \(T\mathbb{S_2}\) is not.
The cylinder \(\mathbb{S_1} \times \mathbb{R}\) (with \(\mathbb{S_1}\) considered the base space and \(\mathbb{R}\) a fiber) is a trivial bundle, while the Mobius strip given as the quotient space \((\mathbb{S_1} \times \mathbb{R})/ \sim\) with the equivalence relation \((p, x) \sim (p + 2\pi, -x)\) is a non-trivial bundle. It can be demonstrated non-trivial by looking for sections that form a basis. Sections are of form \(\sigma(x) = (x, f(x))\) for \(f: \mathbb{S_1} \to \mathbb{R}\). For the section to be a basis f must be non-zero everywhere, and for the section to be smooth (continuous) \(f\) must invert at the boundaries, \(f(2\pi) = -f(0)\). But the intermediate value theorem then dictates that f is zero somewhere, so no such section exists and the Mobius band is non-trivial.