Space with Euler characteristic one: Difference between revisions
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| [[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. Equivalently, all integral homology groups except the zeroth one are torsion || [[rationally acyclic and finitely generated homology implies Euler characteristic one]] || [[Euler characteristic one not implies rationally acyclic]] || | | [[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. Equivalently, all integral homology groups except the zeroth one are torsion || [[rationally acyclic and finitely generated homology implies Euler characteristic one]] || [[Euler characteristic one not implies rationally acyclic]] || | ||
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===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::space with finitely generated homology]] || || || || {{intermediate notions short|space with finitely generated homology|space with Euler characteristic one}} | |||
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| [[Stronger than::space with homology of finite type]] || || || || {{intermediate notions short|space with homology of finite type|space with Euler characteristic one}} | |||
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Revision as of 15:05, 21 June 2016
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 is a topological space that is a space with finitely generated homology, i.e., it has only finitely many nontrivial homology groups and all of them are finitely generated. We say that is a space with Euler characteristic one if the Euler characteristic of equals , i.e., .
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 | (via rationally acyclic) | (via rationally 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. Equivalently, all integral homology groups except the zeroth one are torsion | rationally acyclic and finitely generated homology implies Euler characteristic one | Euler characteristic one not implies rationally acyclic |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| space with finitely generated homology | |FULL LIST, MORE INFO | |||
| space with homology of finite type | |FULL LIST, MORE INFO |