Fiber Bundles| 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
x ∈ Ui
∩ Uj, the mapping
φi,x \(\circ\)
φj,x−1:
F → F belongs to G, and thus defines
a mapping gij:
Ui ∩
Uj → G,
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 × F →
π−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 : 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 (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].
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