Principal Fiber Bundles| Principal Fiber Bundles |
In General > s.a. fiber bundles [including triviality criteria].
$ Def: A fiber bundle
(E, M, π, G), where F ~ G,
and G acts on the fiber F by left translations.
* Dimension: dim(E) = dim(M) + dim(G).
* Right action of the group:
A group action (a realization) that preserves the fibers, i.e., Rg:
π−1(x) →
π−1(x),
by Rg(p)
= φi,x−1
\(\circ\) Rg \(\circ\)
φi,x(p), for all
i such that x ∈ Ui
(we can denote this by pg); The group acts transitively on each fiber.
@ References: Typed notes on fiber bundles by Geroch (*);
> s.a. Wikipedia page.
Associated Principal Fiber Bundle
* Idea: Given a fiber bundle
(E, M, π, G), one can construct a
principal fiber bundle P(E) using the same M and
gij as for E,
and G both as structure group and fiber, with the reconstruction method.
* Example: If E = T(M),
then P(E) = F(M), the frame bundle on M.
* Use: See triviality criteria.
Reduction
* Idea: A principal fiber bundle
(E, M, π, G) is reducible to (E',
M, π', G') if G' is a subgroup of G,
and E' a subspace of E such that the injection f
: E' → E be a bundle morphism commuting with the action
of G':
π f(p) = π'p for all p ∈ E', and f(Rg p) = Rg f(p) for all p ∈ E', g ∈ G' .
* Or: A reduction of E to
a subgroup G' is a submanifold Q' which meets each G-orbit
in exactly one G'-orbit (and a similar condition for tgt spaces).
* Use: The possibility of having
different structures on a manifold M can often be cast into the question
of whether the frame bundle can be reduced to some subgroup of GL(n,
\(\mathbb R\)) or GL(n, \(\mathbb C\)); > see e.g.
orientation and lorentzian structure.
* Remark: Reductions need not exist
nor be unique; E is reducible to {e} iff it is trivial.
Extension of the Group
* Remark: May not exist nor be unique.
* Example: A spinor bundle (spin structure)
is an extension of the bundle of space and time oriented tetrads.
Other Operations
* Product:
Given (P, M, G) and
(P', M', G'), the action of G
× G' on P × P' is defined by
(p, p') \(\mapsto\) (pg, p'g').
* Pullback Bundle:
* Disjoint Union:
@ References: Bunke & Schick RVMP(05)m.GT/04 [T-duality for U(1)-principal fiber bundles].
Classification
> s.a. characteristic classes.
* In principle: An answer to
the question of classification of principal fiber bundles can be given as follows;
Given a base space M and a group G, any G-principal
fiber bundle Pon M is the pullback f*ξ,
for some f: M → O(n)/(G ×
O(n−k)), of the (n−k−1)
universal principal fiber bundle with fiber G, for some n large enough;
Thus, the principal fiber bundles are classified by the homotopy classes of such
maps f; The calculations are difficult if not impossible, in general.
* In general: Classified by
H2(M, π1(X)).
* Over 4D, oriented, simply connected,
compact M: The possible G-bundles are classified by
homotopy classes of maps f : M → B(G), where
B(G) is the classifying space.
* Over M = Sn:
The classification is given by πn−1(G), and
the U(1)-bundles over X are classified by H2(X).
@ Over 2D CW-complexes: Kubyshin m.AT/99-proc.
Examples, Types and Generalizations
> see bundle [gerbes]; Frame Bundle.
* Examples: Any frame bundle; Any
Lie group G, with a closed subgroup H as fiber and base manifold
G/H; The Universal covering space of a topological space X,
with fiber π1(X) and base manifold X.
* Generalizations: Stratified manifolds
(> see gauge theories, geometrodynamics).
@ References: Rossi m.DG/04 [with grupoid structure];
Oriti et al CQG(05)gq/04 [simplicial base space]
Masson JPCS(08)-a0709 [SU(n) principal fiber bundle].
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send feedback and suggestions to bombelli at olemiss.edu – modified 6 feb 2016