The finite Thom detector is an integral homology isomorphism below 2r−1

Set D=2r−1. The finite-type lemma and the away-from-two Thom theorem prove that for every field F of characteristic different from two, both reduced cohomology groups H̃^i(T_r;F) and H̃^i(P_r;F) vanish for 0<i<2r. In degree zero both spaces are connected, so the ordinary H⁰ map is also an isomorphism. The mod-two comparison above proves the F₂ cohomology isomorphism through every degree k≤D. Therefore f_r^* is an isomorphism in ordinary cohomology for F=Q and every F_p, in all degrees 0≤i≤D.

Let C_f be the integral singular-chain mapping cone of f_r. The integral finite-generation theorem and the finite-product cone lemma prove H_i(T_r;Z) and H_i(P_r;Z) are finitely generated, so its cone long exact sequence makes H_i(C_f) finitely generated in every degree. The cone is free degreewise as an abelian chain complex. For each field F, the degreewise-split cone cochain sequence gives H^{i−1}(P_r;F)→H^{i−1}(T_r;F)→H^i(Hom(C_f,F)) →H^i(P_r;F)→H^i(T_r;F). The adjacent cohomology isomorphisms force H^i(Hom(C_f,F))=0 for 0≤i≤D (with negative groups zero at i=0). The cohomological UCT over Z surjects this zero group onto Hom(H_i(C_f),F). Thus Hom(H_i(C_f),Q)=0 and Hom(H_i(C_f),F_p)=0 for every prime p. A finitely generated abelian group with all these Hom groups zero is zero: Q detects any nonzero free summand, and F_p detects any nonzero p-primary cyclic summand. Therefore H_i(C_f)=0 for i≤D. The integral cone exact sequence yields the stated isomorphism for i<D and surjection at D.

f_{r*} is an isomorphism for i<D=2r−1, f_{r*} is surjective for i=D=2r−1. This is exactly the endpoint needed below. It gives no injectivity assertion at degree 2r−1 and uses no comparison at or beyond 2r. The proof is the application of the finite-generation cohomological-UCT comparison; the finite-product cone computation independently records the same strict conclusion.