Fiberwise one-point compactification

Construction 10.1.1 (Fiberwise one-point compactification). Let \(p\colon E \to X\) be a real vector bundle of rank \(n\). We define its fiberwise one-point compactification \(S^E\) as the topological space obtained from \(E\) by forming the one-point compactification fiberwise, replacing \(\R ^n\) with \(S^n = \R ^n \cup \{\infty \}\) in each fiber. More precisely, given a local trivialization \(\{\phi _i\colon U_i \times \R ^n \to p^{-1}(U_i)\}_{i \in I}\) of \(E\), we define \(S^E\) by gluing together the spaces \(U_i \times S^n\) along the transition functions: for \(x \in U_i \cap U_j\), the map \(\phi _j^{-1} \circ \phi _i\colon \{x\} \times \R ^n \to \{x\} \times \R ^n\) extends uniquely to a homeomorphism \(\{x\} \times S^n \to \{x\} \times S^n\) fixing the point at infinity. One can show \(S^E\) is independent of the choice of local trivializations of \(E \to X\).

The space \(S^E\) comes equipped with a map \(S^E \to X\). The ‘points at infinity’ assemble into a continuous section \(s_{\infty }\colon X \to S^E\).