Torsion Eilenberg–Mac Lane spaces are rationally acyclic

A finitely generated subgroup F of T is finite: if its generators have orders d_1,...,d_r, the product of those cyclic groups surjects onto F. For finite F, the covering J(F)→B_wF has |F| sheets. For each singular simplex, sum all its lifts to obtain a chain map τ. Existence and uniqueness of based lifts apply because a simplex is simply connected; restriction to a face bijects its lift set with that face's lift set. Thus ∂τ=τ∂ and p_#τ=|F| id, including degree zero. These are exactly the lift-sum identities proved in the published transfer supplier. On rational homology, τ_* is injective because p_τ_=|F| id. The map J(F)→* is a weak equivalence by the weak-join $K(G,1)$ lemma; the published weak-equivalence/homology lemma and the rationalization lemma give zero positive rational homology of J(F). Therefore B_wF has zero positive rational homology. A rational cellular cycle in B_wT has finite support. Row 3 places all its cells in B_wF for one finitely generated, hence finite, subgroup F. The cellular differential agrees with that in B_wF, and the inclusion of cellular chain groups is injective, so the same chain is a cycle in B_wF. It bounds there by the preceding paragraph, hence bounds in B_wT. Cellular/singular comparison proves positive rational acyclicity. Marked CW uniqueness identifies B_wT with every chosen K(T,1), and homotopy invariance transfers the result.

Take the actual path fibration ΩK(T,n)→PK(T,n)→K(T,n), with contractible total, from the mapping-path supplier. Its exact sequence shows its loop fiber is path-connected and has T as its only positive homotopy group, in degree n−1. Apply the relative version of CW approximation with a prescribed vertex mapping to the constant loop, obtaining a based weak equivalence L→ΩK(T,n). L is a marked CW K(T,n−1). Its rational homology is that of the loop fiber by the published weak-equivalence/homology lemma and the rationalization lemma, and is acyclic by the induction hypothesis. The base K(T,n) is simply connected. In the rational Serre sequence only the row b=0 remains, with E^2_{a,0}=H_a(K(T,n);Q). No differential can enter that row, and every outgoing target is zero. Strong convergence and contractibility of the total force H_a(K(T,n);Q)=0 for a>0. Path-connectedness gives H_0=Q. This is a finite induction for each specified n, and requires no homotopy equivalence from a CW complex to the strict loop fiber.