Space with Euler characteristic one

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

Definition

Suppose $X$ is a topological space that is a space with finitely generated homology, i.e., it has only finitely many nontrivial homology groups are all of them are finitely generated. We say that $X$ is a space with Euler characteristic one if the Euler characteristic of $X$ equals $1$, i.e., $\chi (X)=1$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
contractible space	homotopy-equivalent to a point	(via acyclic)	(via acyclic)	Acyclic space|FULL LIST, MORE INFO
weakly contractible space	weakly homotopy-equivalent to a point	(via acyclic)	(via acyclic)	Acyclic space|FULL LIST, MORE INFO
acyclic space	all the homology groups are zero, except for the zeroth homology group	acyclic implies Euler characteristic one	Euler characteristic one not implies acyclic	|FULL LIST, MORE INFO
Space with finitely generated homology that is also a rationally acyclic space	all the homology groups over the rationals, except the zeroth homology group, are zero