Rational homotopy comparison with one endpoint surjection implies homology comparison

Let F→E_f→X be the actual mapping-path fibration. The fiber exact sequence gives F path-connected and simply connected, because π_1(W)=π_1(X)=π_2(X)=0. All groups in its relevant higher exact segments are abelian. Exactness of rationalization from the rationalization lemma gives, for 2≤j≤D, zero π_j(F)⊗Q: the adjacent map π_{j+1}(W)⊗Q→π_{j+1}(X)⊗Q is surjective, and π_j(W)⊗Q→π_j(X)⊗Q is injective. The endpoint j=D uses precisely the stipulated surjection.

Inductively apply the rational first-Hurewicz lemma to F for j=2,...,D. Its lower homotopy groups are torsion, so H_j(F;Q)=0 in that range; H_1(F;Q)=0 by simple connectivity. In the rational Serre sequence over X, every term with 0<b≤D is zero. For a≤D the bottom-row term E^2_{a,0}=H_a(X;Q) has no incoming differential and no nonzero outgoing differential: each outgoing target has fiber degree r−1≤a−1≤D−1. There are no other nonzero stable filtration terms of total degree at most D. The base edge p_*:H_i(E_f;Q)→H_i(X;Q) is therefore an isomorphism through D. The inclusion j_f:W→E_f is a homotopy equivalence and p_f j_f=f, so the asserted isomorphism is H_i(f;Q). This proves the limited comparison locally; it does not invoke a mod-torsion Whitehead theorem.