Algebraic Topology · Problem 31

Homotopy Theory — Allen Hatcher

Mathematics — Topology · fiber-bundle-fiber-null-homotopic-split

Problem

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^3 \to S^7 \to S^4

and

S^7 \to S^{15} \to S^8

this yields isomorphisms

\pi_n(S^4) \approx \pi_n(S^7) \oplus \pi_{n-1}(S^3)

\pi_n(S^8) \approx \pi_n(S^{15}) \oplus \pi_{n-1}(S^7)

Thus

\pi_7(S^4)

and

\pi_{15}(S^8)

contain

\mathbb{Z}

summands.