Show that a principal fibration \(\Omega C \to E \xrightarrow{p} B\) is fiber homotopy equivalent to the product \(\Omega C \times B\) if it has a section, a map \(s: B \to E\) with \(ps = \mathbb{1}\).
Solution
1. A principal fibration \(\Omega C \to E \xrightarrow{p} B\) is a fibration whose fiber is the loop space \(\Omega C\), with a compatible action of \(\Omega C\) on \(E\).
definition of a principal fibration.
2. The section \(s: B \to E\) gives a basepoint in each fiber \(p^{-1}(b)\).
\(s(b) \in p^{-1}(b)\) since \(ps = 1\).
3. Define \(\Phi: \Omega C \times B \to E\) by \(\Phi(\omega, b) = \omega \cdot s(b)\) (the action of the loop \(\omega\) on the basepoint \(s(b)\)).
use the principal action of the fiber \(\Omega C\) on \(E\).
4. \(\Phi\) is a fiber-preserving map over \(B\).
\(p(\Phi(\omega, b)) = p(\omega \cdot s(b)) = p(s(b)) = b\) (the action preserves fibers).
5. \(\Phi\) restricts to a homotopy equivalence on each fiber.
on the fiber over \(b\), \(\Phi(-, b): \Omega C \to p^{-1}(b)\) is the orbit map \(\omega \mapsto \omega \cdot s(b)\), which is a homeomorphism (the action of \(\Omega C\) on a principal fiber is free and transitive).
6. Hence \(\Phi\) is a fiber homotopy equivalence.
a fiber-preserving map that is a homotopy equivalence on each fiber is a fiber homotopy equivalence (for fibrations over a CW base, or by the standard criterion).
7. Q.E.D.
6.