Space with finitely generated homology

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 spaces

View 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 finitely generated homology if it has only finitely many nonzero homology groups, and each of them is a finitely generated group. In other words, the Betti numbers are all finite and only finitely many of them are nonzero.

Relation with other properties

Stronger properties

• Compact manifold
• Finite CW-space

Weaker properties

• Space with homology of finite type
• Space with finitely generated homology groups