Hopf fibration

The complex Hopf fibration is the principal circle bundleIt is the unit-sphere bundle of the tautological complex line bundle.

Let $a,b\ge 2$ with $a+b=2n+1$. Collapse the lower cells of $S^a\times S^b$ to obtain $q:S^a\times S^b\to S^{2n+1}$ and compose with the Hopf fibration $p:S^{2n+1}\to {\mathbb{CP}}^n$. Both factor inclusions are killed by q, so every induced homotopy group map is zero; the induced reduced homology maps also vanish. If the composite were nullhomotopic, homotopy lifting property would deform q into the circle fibre. Every map from the simply connected product to that circle is nullhomotopic, contradicting the degree-one top homology map of q.

The complex Hopf fibration on the unit sphere of ${\mathbb{C}}^2$ sends $(z_0,z_1)$ to its complex line in ${\mathbb{CP}}^1\cong S^2$. Its fibres are circles. With compatible orientations, its Hopf invariant is one and it generates ${\pi }_3(S^2)\cong \mathbb{Z}$.