Acyclic space

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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) complement of a point in a homology sphere |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
rationally acyclic space homology groups over the rationals are the same as those of a point (direct) real projective space in even dimension (see homology of real projective space) |FULL LIST, MORE INFO
space with finitely generated homology only finitely many nonzero homology groups, and each is finitely generated (direct) any space with nontrivial homology, such as a sphere Space with Euler characteristic one|FULL LIST, MORE INFO
space with homology of finite type all homology groups are finitely generated (direct) (via finitely generated homology) Space with Euler characteristic one|FULL LIST, MORE INFO
space with free homology all homology groups are free abelian groups (direct) a sphere has free homology but is not acyclic |FULL LIST, MORE INFO
space with perfect fundamental group the fundamental group is perfect. This is equivalent to the first homology group being trivial (direct) a 2-sphere has perfect fundamental group but is not acyclic |FULL LIST, MORE INFO
space with Euler characteristic one the Euler characteristic is well-defined and equals one |FULL LIST, MORE INFO

Metaproperties

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