First rational Hurewicz after killing lower torsion homotopy

A based weak CW approximation L→Y induces isomorphisms on homotopy, integral homology and rational homology. Hurewicz naturality therefore reduces the claim to L. Choose its prescribed vertex as basepoint. It is simply connected. We describe a finite sequence of CW spaces Y_2=L, Y_3,...,Y_n with Y_j (j−1)-connected, and maps Y_{j+1}→Y_j inducing isomorphisms on π_i for i>j and on rational homology in every degree.

Suppose Y_j has been constructed for 2≤j<n. Its group T_j=π_j(Y_j) is the original π_j(L), because all earlier maps preserve higher homotopy; it is torsion. Integral Hurewicz gives H_j(Y_j;Z)≅T_j and H_{j−1}(Y_j;Z)=0. The cohomological UCT therefore identifies H^j(Y_j;T_j) with Hom(H_j(Y_j;Z),T_j). Take the class corresponding to the Hurewicz inverse. Represent it by a based map f_j:Y_j→K(T_j,j). The universal class evaluates as identity on T_j. Naturality of evaluation and integral Hurewicz shows (f_j)*:π_j(Y_j)→π_j(K(T_j,j)) is the identity under the chosen markings. Let F_j be its strict homotopy fiber in the actual mapping-path fibration. Its exact sequence shows F_j is j-connected and that F_j→Y_j is an isomorphism on π_i for i>j. Indeed K(T_j,j) has no higher groups, its degree-j map is an isomorphism, and both spaces have zero groups below j. The component segment shows F_j is path-connected. The base is simply connected and rationally acyclic by the torsion-acyclicity lemma. For any rational vector space V, the rationalization lemma makes H_a(K(T_j,j);V) zero for a>0 and equal to V for a=0. Therefore the Serre sequence of F_j→E{f_j}→K(T_j,j) has only column a=0. Its fiber edge is an isomorphism in every degree. Compose it with the deformation retraction E_{f_j}→Y_j: the projection F_j→Y_j is a rational homology isomorphism. Take a based weak CW approximation Y_{j+1}→F_j preserving a chosen fiber point. It transfers both homotopy and homology, and its composite into Y_j has exactly the properties promised. This finishes the construction. The case T_j=0 uses a CW K(0,j) and works with the same argument.

After the finite sequence j=2,...,n−1, Y_n is (n−1)-connected. The composite Y_n→L induces an isomorphism on π_n and on all rational homology. Integral Hurewicz on Y_n, the rationalization lemma, and naturality give the claimed isomorphism on L and then on Y; lower vanishing transfers as well. If n=2 the construction is empty and integral first Hurewicz on L suffices. Every comparison is induced by an actual continuous map. No use of a generalized Whitehead theorem modulo torsion has been concealed.