Acyclic space

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

This is a variation of contractibility. View other variations of contractibility

Definition

A topological space is said to be acyclic if the homology groups in all dimensions are the same as those of a point, for any homology theory. Equivalently, it suffices to say that the singular homology groups are the same as those for a point.

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, or equivalently, has a contracting homotopy	(via weakly contractible)	(via weakly contractible)	Weakly contractible space|FULL LIST, MORE INFO
weakly contractible space	acyclic space that is also simply connected	(obvious)		|FULL LIST, MORE INFO

Weaker properties

• Rationally acyclic space
• Space with finitely generated homology
• Space with homology of finite type
• Space with free homology
• Space with perfect fundamental group

Metaproperties

A product of acyclic spaces is acyclic. The proof of this relies on the Kunneth formula.