The finite Thom detector is a homotopy isomorphism through 2r−2

By the connectivity lemma, $T_r$ is a simply connected CW complex. The target P_r is a finite product of K(F₂,q) with q≥r≥2, hence path-connected and simply connected; we do not assume its ordinary product topology is a CW topology. Apply the arbitrary-target extension by relative CW approximation to replace $f_r$ by a CW extension pair, then the integral homology-to-homotopy comparison theorem with $N=2r-1$. The integral comparison supplies the exact homology hypotheses, so $f_r$ induces ${\pi }_i$ isomorphisms for $1\le i<N$, namely through $2r-2$. The same comparison also gives surjectivity at ${\pi }_{2r-1}$; only the isomorphisms through $2r-2$ are needed here, and no conclusion at ${\pi }_{2r}$ is supplied.

A factor K(F₂,r+d_b) has its only nonzero positive homotopy group F₂ in degree r+d_b. Coordinatewise homotopy groups of a finite product are the products of the factor groups, as proved in the finite-product homotopy computation. Hence at i=r+n the target is F₂^{B_n}. The induced coordinate on a sphere class [α:S^{r+n}→T_r] is ⟨m_{r,b}, α_*[S^{r+n}]{F₂}⟩ = ⟨α^*m{r,b},[S^{r+n}]_{F₂}⟩, by representability and the normalized fundamental class. The cutoff i≤2r−2 is equivalent to r≥n+2.