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., R_{g}:
*π*^{−1}(*x*) →
*π*^{−1}(*x*),
by R_{g}(*p*)
= *φ*_{i,x}^{−1}
\(\circ\) R_{g} \(\circ\)
*φ*_{i,x}(*p*), for all
*i* such that *x* ∈ *U*_{i}
(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
*g*_{ij} 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*(R_{g} *p*)
= R_{g} *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 *P*on *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
H^{2}(*M*, π_{1}(*X*)).

* __Over 4D, oriented, simply connected,
compact M__: The possible

*

@

**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 π_{1}(*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