# Space with free homology

From Topospaces

This article defines a property of topological spaces that depends only on the homology of the topological space, viz it is completely determined by the homology groups. In particular, it is a homotopy-invariant property of topological spacesView all homology-dependent properties of topological spaces OR view all homotopy-invariant properties of topological spaces OR view all properties of topological spaces

## 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.