Introduction
One reason homotopy groups are so much harder to compute than homology groups is that excision fails for them: for a CW pair \((X, A)\) the relative groups \(\pi_n(X, A)\) are in general not isomorphic to \(\pi_n(X/A)\text{,}\) so the long exact sequence of a pair (Theorem 6.14) is of little use unless we know \(\pi_n(X, A)\) by other means.
This lecture studies a class of maps \(p \colon E \to B\text{,}\) the fibrations, for which the relative groups are computable: if \(F\) is a fibre of \(p\text{,}\) then \(\pi_n(E, F) \cong \pi_n(B)\text{.}\) The long exact sequence of the pair \((E, F)\) then becomes a long exact sequence
\begin{equation*}
\cdots \to \pi_n(F) \to \pi_n(E) \to \pi_n(B) \to \pi_{n-1}(F) \to \cdots
\end{equation*}
relating three different spaces. Fibrations are defined by a homotopy lifting property, formally dual to the homotopy extension property of Chapter 7. Covering maps are the first examples; fibre bundles supply many more. As applications we compute the homotopy groups of \(\mathbb{C}P^\infty\text{,}\) and show that \(\pi_2(S^2) \cong \mathbb{Z}\) and \(\pi_3(S^2) \cong \pi_3(S^3)\text{.}\)
