# Space with Euler characteristic zero

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

## Contents

## 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*