Stable classifying anima for real vector bundles

Definition 10.2.12. Write \(O:=\colim _n O(n)\) for the stable orthogonal group and \(\bB O\) for its classifying anima. We define \[ \BOP := \big (\Vect _{\R }(\pt )^{\simeq }\big )^{\grp } \qin \CGrp (\An ) \] to be the group completion of the commutative monoid of finite-dimensional real vector spaces. The telescope calculation of Corollary 9.4.3, with real vector spaces in place of complex ones, identifies its underlying anima as \[ \BOP \simeq \Z \times \bB O. \] The only additional point in applying Theorem 5.5.2 is that \(\pi _1(\bB O) \cong \pi _0(O) \cong C_2\) is abelian. The virtual-rank-zero component has the presentation \[ \bB O\cong \colim _n\bB O(n), \] where the transition maps add a trivial line. Under this presentation, the map from the \(n\)-th stage to the virtual-rank-zero component sends \(V\) to \(V-\R ^n\).