The Möbius strip as the quintessential non-trivial fiber bundle over S¹ — one twist that changes everything
Controls
Non-trivial: Z₂ structure group
Fiber bundle: E → B with fiber F and structure group G.
Cylinder: S¹ × [-1,1] — trivial bundle, global product structure. A point traversing the base returns to the same fiber position.
Möbius strip: S¹ ×_Z₂ [-1,1] — non-trivial, twisted. The structure group Z₂ = {id, flip} acts on the fiber. After one full traversal, a point is flipped to -f.
Key consequence: The Möbius strip is non-orientable — it has only one side and one edge. A vector field on the base cannot be lifted consistently to a global section.