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 xUi (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 pE',   and   f(Rg p) = Rg f(p)   for all pE', gG' .

* 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(nk)), of the (nk–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].