Spaces with finitely generated cohomology

The main result of the paper is the following: m. Suppose all the Čech cohomology groups of a space X are finitely generated and Hm(X;Z) is free for some m⩾2. There is a metrizable space Y∈Cm−1∩LC∞ and a map f :X→Y such that f∗ :Hk(Y;Z)→Hk(X;Z) is an isomorphism for all k⩾m. If dimX is finite or X is uniformly movable and Hk(X;Z) vanish for large k, then Y is a finite polyhedron.