Spherical fibration of a vector bundle

Example 10.2.4 (Spherical fibration of vector bundle). Given a rank \(n\) vector bundle \(p\colon E \to X\), we may apply the previous example to the fiberwise one-point compactification \(S^E \to X\) from Construction 10.1.1. We denote the resulting spherical fibration by \[ \xi _E\colon \Pi _{\infty }(X) \to \An _*. \] Informally, \(\xi _E\) is the functor which sends a point \(x \in X\) to \(\Pi _{\infty }(S^{E_x})\), where \(S^{E_x}\) is the one-point compactification of the fiber \(E_x\) of \(p\) over \(x\).