For a fiber bundle \(F \to E \to B\) such that the inclusion \(F \hookrightarrow E\) is homotopic to a constant map, show that the long exact sequence of homotopy groups breaks up into split short exact sequences giving isomorphisms \(\pi_n(B) \approx \pi_n(E) \oplus \pi_{n-1}(F)\). In particular, for the Hopf bundles \(S^1 \to S^3 \to S^2\), \(S^3 \to S^7 \to S^4\), and \(S^7 \to S^{15} \to S^8\) this yields isomorphisms \(\pi_n(S^2) \approx \pi_n(S^3) \oplus \pi_{n-1}(S^1)\), \(\pi_n(S^4) \approx \pi_n(S^7) \oplus \pi_{n-1}(S^3)\), and \(\pi_n(S^8) \approx \pi_n(S^{15}) \oplus \pi_{n-1}(S^7)\). Thus \(\pi_3(S^2)\), \(\pi_7(S^4)\), and \(\pi_{15}(S^8)\) contain \(\mathbb{Z}\) summands.
Solution
Let \[ F\xrightarrow{i}E\xrightarrow{p}B \] be the bundle, and suppose \(i\) is nullhomotopic. Then \(i_*\) is zero on every homotopy group. The long exact sequence therefore breaks into short exact sequences \[ 0\longrightarrow\pi_n(E) \xrightarrow{p_*}\pi_n(B) \xrightarrow{\partial}\pi_{n-1}(F) \longrightarrow0. \]
To split this sequence, represent \(\alpha\in\pi_{n-1}(F)\) by \[ a:S^{n-1}\to F. \] Since \(i\) is nullhomotopic, \(ia\) extends to a map \[ A:D^n\to E. \] Then \(pA\) sends \(\partial D^n\) to the basepoint of \(B\), so it defines a class \[ s(\alpha)=[pA]\in\pi_n(B). \] Different choices of representative and extension change \(pA\) by a based homotopy, so \(s\) is well-defined; juxtaposing extensions shows that \(s\) is a homomorphism. By the definition of the connecting homomorphism, lifting \(pA\) by \(A\) gives \[ \partial s(\alpha)=\alpha. \] Thus \(s\) is a section of \(\partial\), and \[ \boxed{\pi_n(B)\cong\pi_n(E)\oplus\pi_{n-1}(F).} \]
For the Hopf bundles the fiber inclusions are nullhomotopic because the total space sphere has no homotopy in the fiber dimension. Hence \[ \pi_n(S^2)\cong\pi_n(S^3)\oplus\pi_{n-1}(S^1), \] \[ \pi_n(S^4)\cong\pi_n(S^7)\oplus\pi_{n-1}(S^3), \] and \[ \pi_n(S^8)\cong\pi_n(S^{15})\oplus\pi_{n-1}(S^7). \] In particular the second summands give copies of \(\mathbb Z\) in \(\pi_3(S^2)\), \(\pi_7(S^4)\), and \(\pi_{15}(S^8)\).