Parallel transport

A very flexible construction aimed to represent a family of similar objects (fibres or fibers, depending on the preferred spelling) which are parametrized by the index set which itself has an additional structure (topological space, smooth manifold etc.).

The most known examples are the tangent and cotangent bundles of a smooth manifold.

Formal definition of a topological bundle

Let $\pi:E\to B$ be a continuous map between topological spaces, called the total space and the base, and $F$ yet another topological space called fiber, such that the preimage $F_b=\pi^{-1}(b)\subset E$ of every point of the base is homeomorphic to $X$. The latter condition means that $E$ is the disjoint union of "fibers", $E=\bigsqcup_{b\in B} F_b$ homeomorphic to each other.

The map $\pi$ is called fibration of $E$ over $B$, if the above representation is locally trivial: any point of the base admits an open neighborhood $U$ such that the restriction of $\pi$ on the preimage $\pi^{-1}(U)$ is topologically equivalent to the Cartesian projection $\pi_2$ of the product $F\times U$ on the second component: $\pi_2(v,b)=b$. Formally this means that there exists a homeomorhism $H_U=H:\pi^{-1}(U)\to F\times U$ such that $\pi=\pi_2\circ H$.

Examples

1. The trivial bundle $E=F\times B$, $\pi=\pi_2: F\times B\to B$, $(v,b)\mapsto b$. In this case all trivializing homeomorphisms are globally defined on the entire total space (as the identity map).
2. Let $E=\R^n\smallsetminus\{0\}$ be the punctured Euclidean space, $B=\mathbb S^{n-1}$ the standard unit sphere and $\pi$ the radial projection $\pi(x)=\|x\|^{-1}\cdot x$. This is a topological bundle with the fiber $F=(0,+\infty)\simeq\R^1$.
3. Let $E=\mathbb S^{n-1}$ as above, $B=\R P^{n-1}$ the real projective space (all lines in $\R^n$ passing through the origin) and $\pi$ the map taking a point $x$ on the sphere into the line $\ell_x$ passing through $x$. The preimage $\pi^{-1}(\ell)$ consists of two antipodal points $x$ and $-x\in\mathbb S$, thus $F$ is a discrete two-point set $\{-1,1\}$. This is a topological bundle, which cannot be trivial: indeed, if it were, then the total space $\mathbb S^{n-1}$ would consist of two connected components, while it is connected.

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