Coefficient rings of bordism spectra

Remark 10.3.8 (Coefficient rings). The Pontryagin–Thom theorem turns computations of bordism groups into computations of homotopy groups of spectra. For \(\MO \) and \(\MU \), Construction 10.2.25 constructs the multiplication on the Thom spectrum from Whitney sum. Under the Pontryagin–Thom isomorphism, this multiplication corresponds to cartesian product of bordism classes. We will not prove this geometric compatibility here.

Thom computed the unoriented coefficient ring as \[ \pi _*(\MO ) \cong \F _2[a_i \mid i \geq 1,\ i \neq 2^j-1], \qquad \abs {a_i}=i; \] see Thom (1954). The complex bordism spectrum has coefficient ring \[ \pi _*(\MU ) \cong \Z [x_1,x_2,\ldots ], \qquad \abs {x_i}=2i, \] a theorem of Milnor and Novikov; see Ravenel (1986). By Theorem 10.3.6 this is the ring of complex bordism classes. The spectrum \(\MU \) is central to chromatic homotopy theory, where its additional structure and the algebra of its cooperations organize stable homotopy according to height. At the other extreme, take \(B = \pt \) with the constant map to \(\bB O\), so that \(MB = \S \). By Example 10.3.2 the cycles are then stably framed manifolds, and the Pontryagin–Thom theorem becomes \[ \Omega _n^{\mathrm {fr}}:=\Omega _n^{\pt }(\pt ) \cong \pi _n(\S ), \] recovering Pontryagin’s construction from the beginning of Section 10.1. In degree one, a circle with its nonbounding stable framing represents the nonzero class. Under the displayed isomorphism, this class corresponds to the stable Hopf map \(\eta \in \pi _1(\S )\) of Example 4.4.26.