Unstable one-point compactification map

Construction 10.2.18 (One-point compactification of vector spaces). By Proposition 10.2.11, the rank \(n\) component of \(\Vect _{\R }(\pt )^{\simeq }\) is isomorphic to \(\bB O(n)\). We denote the restriction of one-point compactification to this component by \[ J_n^{\mathrm {un}}\colon \bB O(n) \longrightarrow \An _*. \] It is the functor classified by the standard pointed action of \(O(n)\) on \(S^{\R ^n}\). Consequently, the restriction of \(J\) along the rank \(n\) component \(\bB O(n)\to \BOP \) is \(\Sigma ^\infty J_n^{\mathrm {un}}\). Its restriction along the stable map \[ \bB O(n)\longrightarrow \{0\}\times \bB O\subseteq \BOP , \qquad V\longmapsto V-\R ^n, \] is \(\Sigma ^\infty J_n^{\mathrm {un}}[-n]\).