Space with Euler characteristic one - Revision history

Vipul: /* Weaker properties */

2016-06-21T15:06:48Z

Weaker properties

← Older revision		Revision as of 15:06, 21 June 2016
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	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Stronger than::space with finitely generated homology]] \|\| \|\| \|\| \|\| {{intermediate notions short\|space with finitely generated homology\|space with Euler characteristic one}}		\| [[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}}
	\|-		\|-
	\| [[Stronger than::space with homology of finite type]] \|\| \|\| \|\| \|\| {{intermediate notions short\|space with homology of finite type\|space with Euler characteristic one}}		\| [[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}}
	\|}		\|}

Vipul: /* Relation with other properties */

2016-06-21T15:05:12Z

Relation with other properties

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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]] ||
 |}

Vipul: /* Relation with other properties */

2016-06-21T15:00:31Z

Relation with other properties

← Older revision		Revision as of 15:00, 21 June 2016
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	\| [[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}}		\| [[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}}
	\|-		\|-
	\| ~~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]] \|\|
	\|}		\|}

Vipul at 00:20, 28 April 2011

2011-04-28T00:20:17Z

← Older revision		Revision as of 00:20, 28 April 2011
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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}}
	\|-		\|-
	\| [[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}}
	\|-		\|-
	\| 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]] \|\|
	\|}		\|}

Vipul: Created page with "{{topospace property}} ==Definition== Suppose $X$ is a topological space that is a defining ingredient::space with finitely generated homology, i.e., it has ..."

2011-04-02T14:42:27Z

Created page with "{{topospace property}} ==Definition== Suppose <math>X</math> is a topological space that is a defining ingredient::space with finitely generated homology, i.e., it has ..."

New page

{{topospace property}}

==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>.

==Relation with other properties==

===Stronger properties===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::contractible space]] || homotopy-equivalent to a point || (via acyclic) || (via acyclic) || {{intermediate notions short|space with Euler characteristic one|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}}
|-
| [[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}}
|-
| 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 Euler characteristic one - Revision history

Vipul: /* Weaker properties */

Vipul: /* Relation with other properties */

Vipul: /* Relation with other properties */

Vipul at 00:20, 28 April 2011

Vipul: Created page with "{{topospace property}} ==Definition== Suppose X is a topological space that is a defining ingredient::space with finitely generated homology, i.e., it has ..."

Vipul: Created page with "{{topospace property}} ==Definition== Suppose $X$ is a topological space that is a defining ingredient::space with finitely generated homology, i.e., it has ..."