Free abelian topological groups and collapsing maps

We show that for every pair ( X , Y ) of ANR compacta, Y ⊂ X , the free abelian topological group applied to a collapsing map q : X → X / Y produces a locally trivial bundle A ( q ) : A ( X ) → A ( X / Y ) with the fiber A ( Y ) . As a result we obtain a short proof of the classical Dold–Thom theorem which states that H ˜ n ( X ) = π n ( A ( X ) ) for all complexes X.