Sphere bundles as spherical fibrations

Example 10.2.3 (Sphere bundle). Let \(X\) be a topological space and consider an \(n\)-dimensional sphere bundle \(Y \to X\) over \(X\), i.e. a fiber bundle whose generic fiber is \(S^n\) and whose structure group is \(\mathrm {Homeo}_*(S^n)\). Then the underlying map of animae \(\Pi _{\infty }(Y) \to \Pi _{\infty }(X)\) is a spherical fibration: by Proposition 2.3.19 and Corollary 2.4.15 the fibers of this map are the spheres \(S^n\).