Unoriented bordism spectrum from the prespectrum

Proof. We first identify the levels and structure maps of the classical prespectrum with the unstable Thom construction. We then pass to the colimit presentation of the virtual-rank-zero component of \(\BOP \).

Step 1: Consider the functor \(\xi _{E_n} \colon \Pi _{\infty }(\Gr _n(\R ^{\infty })) \to \An _*\) associated to \(\gamma _n\). Applying Lemma 10.2.19 to the identity of \(\Gr _n(\R ^\infty )\) identifies \(\xi _{E_n}\) with \(J_n^{\mathrm {un}}\). In particular, Proposition 10.2.5 provides an isomorphism \[ \MO ^{\pre }_n = \Pi _{\infty }(\Th (\gamma _n)) \cong \Th (J_n^{\mathrm {un}}) \] of animae. The identification \(E_{n+1}\vert _{\Gr _n(\R ^{\infty })} \cong E_n \times \R \) corresponds to the monoidal identification \[ J_{n+1}^{\mathrm {un}}\vert _{\bB O(n)} \cong \Sigma J_n^{\mathrm {un}}. \] Thus the structure maps of \(\MO ^{\pre }\) correspond to the canonical comparison maps \[ \Sigma \Th (J_n^{\mathrm {un}}) \cong \Th \big (J_{n+1}^{\mathrm {un}}\vert _{\bB O(n)}\big ) \longrightarrow \Th (J_{n+1}^{\mathrm {un}}). \]

Step 2: Applying Theorem 21.2.11 to the presentation \(\bB O \simeq \colim _{n \in \N } \bB O(n)\) of Definition 10.2.12 gives \[ \MO \cong \colim _n \Sigma ^{\infty }\Th (J_n^{\mathrm {un}})[-n]. \] The structure maps in this colimit diagram are precisely those of \(\MO ^{\pre }\), so the right-hand side is \((\MO ^{\pre })^{\mathrm {sp}}\). □