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 {*U*_{i}}
of *B* (part of its manifold structure) we have (1) *Local
triviality*, In every patch of the base space there is a homeomorphism
*φ*_{i}:
π^{−1}(*U*_{i})
→ *U*_{i} × *F*,
of the form *φ*_{i}(*p*)
= (*π*(*p*), *φ*_{i}(*p*)),
with *φ*_{i} a
homeomorphism; (*U*_{i},
*φ*_{i}) is called
a local trivialization; (2) *Transition functions*, For all
*x* ∈ *U*_{i}
∩ *U*_{j}, the mapping
*φ*_{i,x} \(\circ\)
*φ*_{j,x}^{−1}:
*F* → *F* belongs to *G*, and thus defines
a mapping *g*_{ij}:
*U*_{i} ∩
*U*_{j} → *G*,
called transition function; The latter is continuous, and
*g*_{ij}(*x*)
*g*_{jk}(*x*)
= *g*_{ik}(*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 ...→ *π*_{i}(*F*)
→ *π*_{i}(*E*)
→ *π*_{i}(*B*)
→ *π*_{i−1}(*F*)
→ ... is exact.

@ __References__: Steenrod 51;
Husemoller 75.

**Constructions and Operations**

* __Reconstruction method__: Given the base
space *B* (with covering *U*_{i}),
transition functionns *g*_{ij}(*x*),
fiber *F*, and group *G*, the total space is *E*:=
∪_{i} (*U*_{i}
× *F*)/~, where (*x*, *f*) ~ (*x*', *f*
') if *x* = *x*' and *g*_{ij}(*x*)
*f* = *f* '; The projection is *π*[(*x*, *f*)]:=
*x*, and the local trivialization, using
*φ*_{i}^{−1}:
*U*_{i} × *F* →
*π*^{−1}(*U*_{i}),
is (*x*, *f*) \(\mapsto\)[(*x*, *f*)].

- __Remark__: If we
change *g*_{ij}(*x*)
\(\mapsto\)*g*'_{ij}(*x*):=
*λ*_{i}^{−1}(*x*)
*g*_{ij}(*x*)
*λ*_{j}(*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 *U*_{i},
*g*_{ij}, *F*, and
a map *f *: *A* → *B*, 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*: *A* →
*B* 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 (*E*_{i},
*B*_{i},
*π*_{i}), the cartesian
product is (*E*_{1} ×
*E*_{2}, *B*_{1}
× *B*_{2}, *π*_{1}
× *π*_{2}), where the fiber is
given the tensor product linear structure; __Example__: if *M*
= *M*_{1} × *M*_{2},
T*M* = T*M*_{1} ×
T*M*_{2}.

**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].

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send feedback and suggestions to bombelli at olemiss.edu – modified 11 jul 2016