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Section 10.3 Fibre Bundles

Definition 10.11.

A map \(p \colon E \to B\) is a fibre bundle with fibre \(F\) if every \(b \in B\) has an open neighbourhood \(U \ni b\) together with a homeomorphism \(h \colon p^{-1}(U) \xrightarrow{\;\cong\;} U \times F\) making the diagram
commute. Such a pair \((U, h)\) is a local trivialisation. We sometimes write \(F \to E \to B\) to record the fibre.
Every fibre \(p^{-1}(b)\) is homeomorphic to \(\{b\} \times F \cong F\text{.}\) The product projection \(B \times F \to B\) is the trivial bundle.

Example 10.12.

Covering maps are fibre bundles with discrete fibre: if \(U\) is evenly covered, then \(p^{-1}(U) = \coprod_\alpha V_\alpha\) with each \(V_\alpha \cong U\text{,}\) so \(p^{-1}(U) \cong U \times p^{-1}(b)\text{.}\)

Example 10.13.

The Möbius band \(M = \bigl(I \times [-1, 1]\bigr)/(0, y) \sim (1, -y) \to S^1\) is a fibre bundle with fibre \([-1, 1]\text{.}\)
The long exact sequence applies to fibre bundles over paracompact bases by the following theorem.
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