Acyclic space: Difference between revisions
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===Weaker properties=== | ===Weaker properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Stronger than::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]]) || {{intermediate notions short|rationally acyclic space|acyclic space}} | |||
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| [[Stronger than::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]]|| {{intermediate notions short|space with finitely generated homology|acyclic space}} | |||
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| [[Stronger than::space with homology of finite type]] || all homology groups are finitely generated || (direct) || (via finitely generated homology) || {{intermediate notions short|space with homology of finite type|acyclic space}} | |||
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| [[Stronger than::space with free homology]] || all homology groups are free abelian groups || (direct) || a [[sphere]] has free homology but is not acyclic || {{intermediate notions short|space with free homology|acyclic space}} | |||
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| [[Stronger than::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 || {{intermediate notions short|space with perfect fundamental group|acyclic space}} | |||
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| [[Stronger than::space with Euler characteristic one]] || the [[Euler characteristic]] is well-defined and equals one || || || {{intermediate notions short|space with Euler characteristic one|acyclic space}} | |||
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==Metaproperties== | ==Metaproperties== | ||
Revision as of 14:57, 21 June 2016
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
| 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
A product of acyclic spaces is acyclic. The proof of this relies on the Kunneth formula.