The mod 2 homology of MO

Remark 10.4.12 (The mod \(2\) homology of \(\MO \)). The universal rank-zero virtual bundle on \(\bB O\) has a unique \(H\F _2\)-orientation. The Thom isomorphism gives \[ H\F _2 \otimes \MO \; \cong \; H\F _2[\bB O] \] of \(H\F _2\)-modules. This computes the mod \(2\) homology of \(\MO \) in terms of that of \(\bB O\), whose mod \(2\) cohomology is the polynomial algebra \(\F _2[w_1,w_2,\ldots ]\) on the Stiefel–Whitney classes. It is the first step of Thom’s calculation of \(\pi _*(\MO )\) recorded in Remark 10.3.8, but not the whole of it: splitting \(\MO \) into a wedge of shifts of \(H\F _2\), and identifying the multiplicative structure of the coefficient ring, requires further input.