Fiber Bundles

In General > s.a. bundles; Trivialization.
* Idea: The formalization of the concept of local product of manifolds, and a generalization of covering space.
* History: Axiomatic definitions were tried around 1935–1940; & Ehresmann, Hurewicz, Hopf, Steenrod.
$Def 1: A quadruple (E, B, π, G), where (E, B, π) is a bundle and G is a topological group of homeomorphisms of the typical fiber F onto itself such that, for some open cover {Ui} of B (part of its manifold structure) we have (1) Local triviality, In every patch of the base space there is a homeomorphism φi: π−1(Ui) → Ui × F, of the form φi(p) = (π(p), φi(p)), with φi a homeomorphism; (Ui, φi) is called a local trivialization; (2) Transition functions, For all xUiUj, the mapping φi,x $$\circ$$ φj,x−1: FF belongs to G, and thus defines a mapping gij: UiUjG, called transition function; The latter is continuous, and gij(x) gjk(x) = gik(x).$ Def 2: More simply, it is a principal fiber bundle (P, G, ζ) and a manifold F with an action of G on F (not necessarily free); To construct the actual manifold E, divide P × F by the action of G defined by (p, f) $$\mapsto$$(pg−1, gf).
* Property: The dimensions are related by dim(E) = dim(B) + dim(F).
* Property: As for all quasifibrations, the sequence ...→ πi(F) → πi(E) → πi(B) → πi−1(F) → ... is exact.
@ References: Steenrod 51; Husemoller 75.

Constructions and Operations
* Reconstruction method: Given the base space B (with covering Ui), transition functionns gij(x), fiber F, and group G, the total space is E:= ∪i (Ui × F)/~, where (x, f) ~ (x', f ') if x = x' and gij(x) f = f '; The projection is π[(x, f)]:= x, and the local trivialization, using φi−1: Ui × Fπ−1(Ui), is (x, f) $$\mapsto$$[(x, f)].
- Remark: If we change gij(x) $$\mapsto$$g'ij(x):= λi−1(x) gij(x) λj(x), the new bundle is topologically the same as before, so we might want to consider equivalence classes of fiber bundles.
* Reduction: To reduce the group of transformations preserving the fiber structure, require the fibers to have more structure.
* Pullback bundle: Given (E, B, π, G), and thus Ui, gij, F, and a map f : AB, one can construct f*E on A, by straightforward pullback of the transition functions, same F and G, and reconstruction.
- Equivalently: f*E = {(a, p) ∈ A × E | f(a) = π(p)}, and π((a, p)) = π(p).
- Remark:E trivial implies f*E trivial; f, g: AB homotopic implies f*E and g*E equivalent.
* Restriction to a subset of the base space: The definition is the obvious one; The pullback of E under the base space inclusion map.
* Other constructions: Associated fiber bundle to a principal fiber bundle; Direct sum of tensor bundles; Disjoint union; Tensor product of vector bundles; Whitney sum; More generally, one can use continuous functors of several variables (in a category where morphisms are isomorphisms) to construct new bundles.
* Cartesian product: Given vector bundles (Ei, Bi, πi), the cartesian product is (E1 × E2, B1 × B2, π1 × π2), where the fiber is given the tensor product linear structure; Example: if M = M1 × M2, TM = TM1 × TM2.

Other Related Concepts > s.a. types of fiber bundles; principal fiber bundles.
* Natural bundle: A category of fibre bundles, considered together with a special class of morphisms.
* Gauge-natural bundle: A generalization of natural bundles, used in functorial approach to classical field theories.
* Remark: Every classical field theory can be regarded as taking place on some jet prolongation of some (vector or affine) gauge-natural bundle associated with some principal bundle over a given base manifold.
@ Natural bundles: Nijenhuis in(72); in Kolár et al 93; in Matteucci pr(02).
@ Gauge-natural bundles: Eck MAMS(81); in Kolár et al 93; in Matteucci pr(02).

And Physics > s.a. field theory; formulations of quantum mechanics; gauge theory; topology in physics.
* Idea: Used extensively in gauge theories, because it is the structure that allows one to define "internal" symmetries.
@ References: Geroch ln; Trautman RPMP(76) [rev]; Daniel & Viallet RMP(80); Nash & Sen 83, ch7; Morand 84; Coquereaux & Jadczyk 88; Iliev PS(03)qp/02 [and relativistic quantum mechanics]; Sen & Sewell JMP(02) [in quantum physics]; Collinucci & Wijns ht/06-ln; Sardanashvily a0908-ln [and jet manifolds and Lagrangian theory]; Weatherall a1411, Marsh a1607 [and gauge theory].