A filtration of unoriented cobordism (Q1928418)

The authors construct a filtration of \(BO\) recovering the generators of \(H_*(BO; \mathbb{F}_2)\) one at a time. Using the James construction, they define CW complexes \(B_{(n,i, j)},\) ordered lexicographically, such that \(BO \simeq \mathrm{hocolim} B_{(n,i, j)}\). Here \(H_*(B_{(n,i, j)}; \mathbb{F}_2)\) is a polynomial algebra on generators of degree \( \leq ((4n-2)2^{i-1}-1) \times 2^{j-1} - 1\) and the induced map \(H_*(B_{(n,i, j)}; \mathbb{F}_2) \to H_*(BO; \mathbb{F}_2)\) is injective. Taking the Thom spectra \(M_{(n, i, j)}\) of the maps \(B_{(n, i, j)} \to BO\) they obtain a corresponding filtration of the unoriented cobordism ring \(\pi_*(MO).\)