The Mobius band is a fiber bundle over the circle S¹ with fiber the interval [-1, 1]. What is its structure group?
AThe trivial group {id}
Bℤ₂ = {id, reflection} acting on [-1,1] by x ↦ -x
CSO(2), the rotation group
DGL(1, ℝ), the group of nonzero scalars
Cover S¹ by two overlapping arcs U₁, U₂. Over each arc, the bundle is a product U_i × [-1,1]. On one component of U₁ ∩ U₂, the transition function is the identity; on the other, it is the reflection x ↦ -x. The transition functions take values in ℤ₂ = {id, reflection}. The Mobius band is nontrivial (not a product) because this transition function is not the identity — the twist encoded by the ℤ₂ element is the essential feature.
Question 2 True / False
A fiber bundle π : E → B is called trivial if E is diffeomorphic to B × F (as a bundle). The tangent bundle of the sphere S² is trivial.
TTrue
FFalse
Answer: False
The tangent bundle TS² is nontrivial. By the hairy ball theorem, there is no nowhere-vanishing vector field on S² — but if TS² were trivial (isomorphic to S² × ℝ²), then the constant section (p, e₁) would give a nowhere-vanishing vector field. The nontriviality of TS² is detected by the Euler class (which equals χ(S²) = 2 ≠ 0). By contrast, the tangent bundle of S¹ is trivial (S¹ has a nowhere-vanishing vector field — the angular direction), and the tangent bundle of any Lie group is trivial.
Question 3 Short Answer
What role do transition functions play in defining a fiber bundle, and why are they valued in a group?
Think about your answer, then reveal below.
Model answer: Transition functions describe how to glue local trivializations together over overlaps. On U_α ∩ U_β, the two trivializations give two different identifications of the fiber with F, and the transition function g_αβ : U_α ∩ U_β → G ⊂ Aut(F) is the change-of-identification map. They must be group-valued because: (1) g_αα = id (trivially compatible with itself), (2) g_βα = g_αβ⁻¹ (changing identification order inverts the map), and (3) g_αβ g_βγ g_γα = id on triple overlaps (the cocycle condition ensures consistency). The transition functions, up to equivalence, classify the bundle.
This is exactly analogous to how transition maps of coordinate charts define a smooth structure on a manifold. The base space is the same in both cases; what differs is the structure of the fibers and the group acting on them. The cocycle condition is what ensures the local pieces patch together into a well-defined global object.