Given a fibration \(F \to E \xrightarrow{p} B\), define a natural action of \(B\) on the homotopy fiber \(F_p\) and use this to show that exactness at \(\langle X, F \rangle\) in the long exact sequence can be improved to the statement that two elements of \(\langle X, F \rangle\) have the same image in \(\langle X, E \rangle\) if they are in the same orbit of the induced action of \(\langle X, B \rangle\) on \(\langle X, F \rangle\).
Solution
Goal. Define the action of \(\pi_1(B)\) on the homotopy fiber \(F_p\) of a fibration, and use it to sharpen exactness of the long exact sequence at \(\langle X, F\rangle\).
1. The homotopy fiber and its \(\pi_1(B)\)-action.
1.1. The homotopy fiber over \(p \in B\) is \(F_p \definedas \theset{(e, \gamma) \suchthat e \in E,\ \gamma: I \to B,\ \gamma(0) = p(e),\ \gamma(1) = p}\).
this is the pullback of \(E \xrightarrow{p} B\) along the path fibration \(PB \to B\), \(\gamma \mapsto \gamma(1)\).
1.2. A loop \(\lambda \in \Omega B\) (based at \(p\)) acts by \(\lambda \cdot (e, \gamma) \definedas (e, \lambda \cdot \gamma)\), where \(\lambda \cdot \gamma\) is the concatenation of \(\gamma\) followed by \(\lambda\).
\(\lambda \cdot \gamma\) is a path from \(p(e)\) to \(p\) (since \(\lambda\) is a loop at \(p\)), so the result lies in \(F_p\); this is a continuous action of \(\Omega B\) on \(F_p\).
1.3. This descends to an action of \(\pi_1(B, p) = \pi_0(\Omega B)\) on \(F_p\), hence on \(\pi_n(F_p)\) and on \(\langle X, F_p\rangle\).
homotopic loops give homotopic self-maps of \(F_p\), so the action factors through \(\pi_1(B, p)\).
2. Sharpened exactness at \(\langle X, F\rangle\).
2.1. The map \(i: F_p \to E\), \((e, \gamma) \mapsto e\), fits into the fibration sequence \(F_p \xrightarrow{i} E \xrightarrow{p} B\), giving exactness \(\langle X, F_p\rangle \xrightarrow{i_*} \langle X, E\rangle \xrightarrow{p_*} \langle X, B\rangle\).
the long exact sequence of homotopy groups, applied to the fibration.
2.2. Claim: for \(f, g: X \to F_p\), one has \(i \circ f \simeq i \circ g\) iff \(f\) and \(g\) are in the same \(\pi_1(B)\)-orbit.
a homotopy \(H: X \times I \to E\) from \(i \circ f\) to \(i \circ g\) projects to \(p \circ H: X \times I \to B\), a homotopy of paths from \(p \circ f\) to \(p \circ g\); for each \(x \in X\) the loop \(p \circ H(x, -)\) (closed up by the constant paths at \(p\)) is an element of \(\pi_1(B, p)\), and this loop acts on \(f(x)\) to give \(g(x)\).
2.3. Hence two elements of \(\langle X, F\rangle\) have the same image in \(\langle X, E\rangle\) iff they are in the same orbit of the \(\pi_1(B)\)-action.
2.2 restates the kernel of \(i_*\) as the orbit relation.
3. Q.E.D.
2.3 is the sharpened exactness statement.