Space with Euler characteristic zero

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

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Definition

A topological space is said to have zero Euler characteristic if it has finitely generated homology, and its Euler characteristic is zero.

Relation with other properties

Stronger properties

• Compact connected Lie group (nontrivial): For full proof, refer: compact connected nontrivial Lie group implies zero Euler characteristic
• Odd-dimensional compact connected orientable manifold: For full proof, refer: Euler characteristic of odd-dimensional compact connected orientable manifold is zero

Weaker properties

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