Space with Euler characteristic one: Difference between revisions

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==Definition==
==Definition==


Suppose <math>X</math> is a [[topological space]] that is a [[defining ingredient::space with finitely generated homology]], i.e., it has only finitely many nontrivial [[homology group]]s are all of them are finitely generated. We say that <math>X</math> is a '''space with Euler characteristic one''' if the [[defining ingredient::Euler characteristic]] of <math>X</math> equals <math>1</math>, i.e., <math>\chi(X) = 1</math>.
Suppose <math>X</math> is a [[topological space]] that is a [[defining ingredient::space with finitely generated homology]], i.e., it has only finitely many nontrivial [[homology group]]s and all of them are finitely generated. We say that <math>X</math> is a '''space with Euler characteristic one''' if the [[defining ingredient::Euler characteristic]] of <math>X</math> equals <math>1</math>, i.e., <math>\chi(X) = 1</math>.


==Relation with other properties==
==Relation with other properties==
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| [[Weaker than::weakly contractible space]] || weakly homotopy-equivalent to a point || (via acyclic) || (via acyclic) || {{intermediate notions short|space with Euler characteristic one|weakly contractible space}}
| [[Weaker than::weakly contractible space]] || weakly homotopy-equivalent to a point || (via acyclic) || (via acyclic) || {{intermediate notions short|space with Euler characteristic one|weakly contractible space}}
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| [[Weaker than::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]] || {{intermediate notions short|space with Euler characteristic one|acyclic space}}
| [[Weaker than::acyclic space]] || all the homology groups are zero, except for the zeroth homology group || (via rationally acyclic) || (via rationally acyclic) || {{intermediate notions short|space with Euler characteristic one|acyclic space}}
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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 || || ||
| [[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===
 
{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::space with finitely generated homology]] || finitely many nonzero homology groups, and they are all finitely generated || || || {{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]] || all homology groups are finitely generated || || || {{intermediate notions short|space with homology of finite type|space with Euler characteristic one}}
|}
|}

Latest revision as of 15:06, 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 finitely many nonzero homology groups, and they are all finitely generated |FULL LIST, MORE INFO
space with homology of finite type all homology groups are finitely generated |FULL LIST, MORE INFO
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