Classical bordism spectra

Example 10.2.26. A continuous group homomorphism \(G \to O\) induces a map \(\bB G \to \bB O\), and we write \(MG\) for the resulting Thom spectrum. Taking \(B = \bB O\) with \(\phi \) the identity recovers \(\MO \). Further examples are \[ \MSO , \quad \MU , \quad \mathrm {MSU}, \quad \mathrm {MSp}, \quad \text {and}\quad \mathrm {MSpin}. \] Here \(\MU \) is the complex bordism spectrum. For \(\MO \) and \(\MU \), the defining maps to \(\BOP \) are restrictions of morphisms of commutative groups constructed above. Hence Construction 10.2.25 equips these Thom spectra with commutative ring structures. The other classical Thom spectra in the display also admit commutative ring structures, but their construction requires compatible infinite-loop refinements of the maps \(\bB G\to \bB O\), which we do not develop here.

The full group completions give the periodic unoriented and complex bordism spectra \[ \MOP :=\th \big (J\colon \BOP \to \Sp \big ) \qquadtext {and}\qquad \MUP :=\th \big (\BUP \longrightarrow \BOP \xrightarrow {J}\Sp \big ), \] respectively. Thus \(\MOP \) is obtained from the identity of \(\BOP \), while \(\MUP \) is obtained from the forgetful map \(\BUP \to \BOP \). Restricting these defining diagrams to their virtual-rank-zero components recovers \(\MO \) and \(\MU \). Translation by the classes of \(\R \) and \(\C \) permutes the components of \(\BOP \) and \(\BUP \) while shifting the associated stable spherical fibrations by \(1\) and \(2\), respectively. It follows that there are isomorphisms of spectra \[ \MOP \cong \MOP [1] \qquadtext {and}\qquad \MUP \cong \MUP [2], \] which explains the terminology.