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, 
, G),
where (E, B, )
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 
i: –1(Ui) → Ui × F,
of the form i(p)
= ((p), i(p)),
with i a
homeomorphism; (Ui, i)
is called
a local
trivialization; (2) Transition functions, For all x 
Ui 
Uj,
the mapping i,x 
j,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, 
)
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)
(pg–1, gf).
* Property: The dimensions are related by dim(E) = dim(B)
+ dim(F).
* Property: As for all
quasifibrations, the sequence ...→ i(F) →i(E) →i(B) → i–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 [(x, f)]:= x,
and the local trivialization, using i–1: Ui × F →
–1(Ui),
is (x, f) [(x, f)].
- Remark: If we change gij(x) g'ij(x):=
i–1(x)
gij(x) 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, , 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)
=
(p)},
and ((a, p))
= (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: Obvious; 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, i),
the cartesian product is (E1
× E2, B1
×
B2, 1 ×
2),
where the fiber is given the tensor product linear structure; Example:
if M =
M1 ×
M2, TM = TM1 ×
TM2.
Other Related Concepts > see types of fiber bundles; principal fiber bundles.
And Physics > s.a. formulations
of quantum mechanics; [topology
in physics].
* Idea: Used
extensively in gauge theories, because it is the structure that allows one
to define "internal" symmetries.
@ References: Geroch notes; 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-in.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
5 jul 2008