The universal MO-orientation

Example 10.4.9 (The universal \(\MO \)-orientation). The universal rank-zero virtual bundle on \(\bB O\) is canonically \(\MO \)-oriented. Indeed, for every \(W\in \bB O\), symmetric monoidality of \(J\) and the definition \(\MO =\colim _{V\in \bB O}J(V)\) give \[ J(W)\otimes \MO \cong \colim _{V\in \bB O}J(W+V) \cong \MO . \] The second isomorphism is induced by the equivalence \(W+(-)\colon \bB O\to \bB O\), and these isomorphisms vary coherently with \(W\). They therefore trivialize the associated local system of \(\MO \)-lines. Pulling this orientation back along the classifying map of an honest real vector bundle shows that every real vector bundle is canonically \(\MO \)-orientable.