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. The coverings are also a special particular form of a topological bundle (with discrete fibers).

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 $-x\in\mathbb S^{n-1}$, thus $F$ is a discrete two-point set $\mathbb Z_2=\{-1,1\}$. 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.
4. More generally, let $\pi:M^m\to N^n$ be a differentiable map between two smooth (connected) compact manifolds of dimensions $m\ge n$, which has the maximal rank (equal to $n$) everywhere. One can show then, using the implicit function theorem and partition of unity, that $\pi$ is a topological bundle with a fiber $F$ which itself is a smooth compact manifold.

Cocycle of a bundle

On a nonvoid overlapping $U_{\alpha\beta}=U_\alpha\cap U_\beta$ of two different trivializing charts $U_\alpha$ and $U_\beta$ two homeomorphisms $H_\alpha,H_\beta: \pi^{-1}(U_{\alpha\beta})\to F\times U_{\alpha\beta}$ are defined. Since both $H_\alpha$ and $H_\beta$ conjugate $\pi$ with the Cartesian projection on $U_{\alpha\beta}$, they map each fiber $F_b=\pi^{-1}(b)$ into the same space $F\times\{b\}$. The composition $H_\alpha\circ H_\beta^{-1}$ takes then the form $$ H_\alpha\circ H_\beta^{-1}:(v,b)\mapsto (H_{\alpha\beta}(x,v),x),\qquad H_{\alpha\beta}(\cdot,x)\in\operatorname{Homeo}(F) $$ with the homeomorphisms $H_{\alpha\beta}(\cdot, x)$ continuously depending on $x\in U_{\alpha\beta}$. The collection of these "homeomorphism-valued" functions defined in the intersections $U_{\alpha\beta}$ is called the cocycle associated with a given trivialization of the bundle $\pi$ (or simply the cocycle of the bundle. They homeomorphisms $\{H_{\alpha\beta}\}$ satisfy the following identities, obvious from their construction: $$ H_{\alpha\beta}\circ H_{\beta\alpha}=\operatorname{id},\qquad H_{\alpha\beta}\circ H_{\beta\gamma}\circ H_{\gamma\alpha}=\operatorname{id}, $$ the second being true on every nonvoid triple intersection $U_{\alpha\beta\gamma}=U_\alpha\cap U_\beta\cap U_\gamma$.

Bundles with additional structures

The general construction of bundle easily allows various additional structures, both on the base space and (more importantly) on the fibers. By far the most important special case is that of vector bundles.

To define a vector bundle, one has in addition to the principal definition assume the following:

1. The fiber $F$ is a vector space, and
2. The trivializing homeomorphisms must respect the linear structure of the fibers.

The second assumption means that rather than being arbitrary homeomorphisms, the maps $\{H_{\alpha\beta}\}$ forming the bundle cocycle, must be linear invertible of each "standard fiber" $F\times \{b\}$; if the fiber is identified with the canonical $n$-space $\Bbbk^n$ (over $\Bbbk=\R$ or $\Bbbk=\C$), then the cocycle will consist of invertible continuous matrix-functios $M_{\alpha\beta}:U_{\alpha\beta}\to\operatorname{GL}(n,\Bbbk)$, so that $H_{\alpha\beta}(v,x)=(M_{\alpha\beta}(x)\cdot v, x)$, $v\in\Bbbk^n$. The cocycle identities become then the identites relating the values of these matrix-valued functions, $$ M_{\alpha\beta}(x)\cdot M_{\beta\alpha}(x)\equiv E,\qquad M_{\alpha\beta}(x)\cdot M_{\beta\gamma}(x)\cdot M_{\gamma\alpha}(x)\equiv E, $$ where $E$ is the $n\times n$-identical matrix.

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