Space with free homology

Definition
A topological space is said to have free homology if all its homology groups are free Abelian groups.

Facts
If a topological space admits a CW-decomposition where cells of adjacent dimensions are never attached, then its cellular chain complex does not contain any two consecutive nonzero terms. This forces that all the boundary maps in the cellular chain complex are zero maps, and hence the cellular homology groups are the same as the cellular chain groups.

Examples are complex projective space, quaternionic projective space and the sphere (for dimension greater than 1). Note that the sphere of dimension 1 also has free homology, but it does contain 0-cells and 1-cells.