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