Types of Fiber Bundles

Differentiable Fibre Bundles
\$ Def: A fiber bundle (B, E, G, F, π), where B, E, G, F are differentiable manifolds, π is a differentiable mapping, the covering {Uj} of B is an admissible atlas, and the transition functions gjk
are differentiable.

Trivial Fiber Bundles
* Triviality Criteria:
- P(E) trivial iff P(E) admits a cross-section;
- E trivial iff the transition functions can be written as gij = λi(x) λj–1(x);
- P(E) trivial implies E trivial;
- B contractible implies E trivial;
- F contractible implies E has a cross-section;
- G contractible implies E trivial.
* Results: All SU(2) bundles over 3-manifolds are trivial.

Vector Bundles > s.a. Jet Bundles; tangent bundles.
* Idea: A topological space E, a continuous projection π: EB, and a vector space (over a field $$\mathbb K$$) structure on each fiber π–1(x), with local triviality, i.e., a fiber bundle with F = $$\mathbb K$$n and G = GL(n, $$\mathbb K$$).
@ References: in Milnor & Stasheff 74, ch 2–3.