Finite Thom classifying detector map

Assume AC. Let $\mathcal{A}$ be the mod-two square algebra acting on $M={\widehat{H}}^{\ast }(TO;{\mathbb{F}}_2)$, let ${\mathcal{A}}^{+}={⨁}_{j>0}{\mathcal{A}}^j$ be its positive-degree ideal, and put $Q=M\mathrm{/}{\mathcal{A}}^{+}M$ as in the freeness theorem. Fix once and for all a homogeneous basis $B={\coprod }_{d\ge 0}{B}_d$ of $Q$ and a degree-preserving section $s:Q\to M$ of the quotient map, using AC; these choices are shared by every rank. For $b\in B_d$ set $m_b=s(b)$ and $d_b=d$. For each $r\ge 2$, set $B(r)={\bigcup }_{0\le d<r}{B}_d$. The freeness theorem makes all $m_b$, for $b\in B$, a homogeneous free $\mathcal{A}$-basis of $M$. Via the eventual-constancy isomorphism $M^d\cong {\tilde{H}}^{r+d}(T_r;{\mathbb{F}}_2)$, let ${\bar{m}}_{b,r}$ be the rank-r coordinate of the same fixed $m_b$, and choose a based representative $f_{r,b}:T_r\to K({\mathbb{F}}_2,r+d_b)$ of its representing class. Set $P_r={\prod }_{b\in B(r)}K({\mathbb{F}}_2,r+d_b)$ and define the candidate based detector $f_r:T_r\to P_r$ by its coordinate maps. The same basis and lifts are used at every rank, so the coordinates are compatible with prespectrum stabilization. The finite existence and continuity argument is recorded in The finite Thom classifying detector map exists and is continuous ↗.

The construction fixes one global homogeneous ${\mathbb{F}}_2$-basis $B$ of $Q$ and one degree-preserving section $s$ of the quotient map, used at every rank; a given detector sees only the finite initial segment $B(r)$. The freeness theorem identifies the lifts $m_b=s(b)$ as a free $\mathcal{A}$-basis of $M$, and the eventual-constancy lemma identifies each stable class with its rank-$r$ coordinate whenever $r>d_b$. The finiteness of $B(r)$, the existence of the based representing maps $f_{r,b}$, and the continuity of $f_r$ are proved in The finite Thom classifying detector map exists and is continuous ↗. The same basis and lifts are used in every rank, so no new choice enters the suspension-compatibility computation.