The unoriented bordism prespectrum

Construction 10.1.10 (The prespectrum \(\MO ^{\pre }\)). Consider the canonical inclusion \[ \Gr _n(\R ^{\infty }) \hookrightarrow \Gr _{n+1}(\R \times \R ^{\infty }) \cong \Gr _{n+1}(\R ^{\infty }), \quad V \mapsto \R \oplus V. \] The pullback of \(\gamma _{n+1}\) along this inclusion is the product bundle \(\gamma _n \times \ul {\R }\), and in particular we obtain a pointed continuous map \[ \Sigma (\Th (\gamma _n)) \cong \Th (\gamma _n \times \ul {\R }) \to \Th (\gamma _{n+1}). \] By adjunction, this defines a map \(\sigma _n\colon \Th (\gamma _n) \to \Omega (\Th (\gamma _{n+1}))\), turning the sequence of pointed animae \((\Pi _{\infty }\Th (\gamma _n))\) into a prespectrum, which we will denote by \(\MO ^{\pre }\).