Acyclic space - Revision history

Vipul: /* Stronger properties */

2016-06-21T17:24:36Z

Stronger properties

← Older revision		Revision as of 17:24, 21 June 2016
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	\| [[Weaker than::contractible space]] \|\| homotopy-equivalent to a point, or equivalently, has a [[contracting homotopy]] \|\| (via weakly contractible) \|\| (via weakly contractible) \|\| {{intermediate notions short\|acyclic space\|contractible space}}		\| [[Weaker than::contractible space]] \|\| homotopy-equivalent to a point, or equivalently, has a [[contracting homotopy]] \|\| (via weakly contractible) \|\| (via weakly contractible) \|\| {{intermediate notions short\|acyclic space\|contractible space}}
	\|-		\|-
	\| [[Weaker than::weakly contractible space]] \|\| acyclic space that is also [[simply connected space\|simply connected]] \|\| (obvious) \|\| ~~any [[classifying space]] for~~ a ~~nontrivial action of~~ a ~~nontrivial~~ [[~~perfect group~~]] ~~on a weakly contractible space offers a counterexample~~ \|\| {{intermediate notions short\|acyclic space\|weakly contractible space}}		\| [[Weaker than::weakly contractible space]] \|\| acyclic space that is also [[simply connected space\|simply connected]] \|\| (obvious) \|\| complement of a point in a [[homology sphere]] \|\| {{intermediate notions short\|acyclic space\|weakly contractible space}}
	\|}		\|}

Vipul: /* Stronger properties */

2016-06-21T14:59:30Z

Stronger properties

← Older revision		Revision as of 14:59, 21 June 2016
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	\| [[Weaker than::contractible space]] \|\| homotopy-equivalent to a point, or equivalently, has a [[contracting homotopy]] \|\| (via weakly contractible) \|\| (via weakly contractible) \|\| {{intermediate notions short\|acyclic space\|contractible space}}		\| [[Weaker than::contractible space]] \|\| homotopy-equivalent to a point, or equivalently, has a [[contracting homotopy]] \|\| (via weakly contractible) \|\| (via weakly contractible) \|\| {{intermediate notions short\|acyclic space\|contractible space}}
	\|-		\|-
	\| [[Weaker than::weakly contractible space]] \|\| acyclic space that is also [[simply connected space\|simply connected]] \|\| (obvious) \|\| ~~the~~ [[classifying space]] of ~~any~~ [[perfect group]] offers a counterexample \|\| {{intermediate notions short\|acyclic space\|weakly contractible space}}		\| [[Weaker than::weakly contractible space]] \|\| acyclic space that is also [[simply connected space\|simply connected]] \|\| (obvious) \|\| any [[classifying space]] for a nontrivial action of a nontrivial [[perfect group]] on a weakly contractible space offers a counterexample \|\| {{intermediate notions short\|acyclic space\|weakly contractible space}}
	\|}		\|}

Vipul: /* Stronger properties */

2016-06-21T14:58:52Z

Stronger properties

← Older revision		Revision as of 14:58, 21 June 2016
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	\| [[Weaker than::contractible space]] \|\| homotopy-equivalent to a point, or equivalently, has a [[contracting homotopy]] \|\| (via weakly contractible) \|\| (via weakly contractible) \|\| {{intermediate notions short\|acyclic space\|contractible space}}		\| [[Weaker than::contractible space]] \|\| homotopy-equivalent to a point, or equivalently, has a [[contracting homotopy]] \|\| (via weakly contractible) \|\| (via weakly contractible) \|\| {{intermediate notions short\|acyclic space\|contractible space}}
	\|-		\|-
	\| [[Weaker than::weakly contractible space]] \|\| acyclic space that is also [[simply connected space\|simply connected]] \|\| (obvious) \|\| \|\| {{intermediate notions short\|acyclic space\|weakly contractible space}}		\| [[Weaker than::weakly contractible space]] \|\| acyclic space that is also [[simply connected space\|simply connected]] \|\| (obvious) \|\| the [[classifying space]] of any [[perfect group]] offers a counterexample \|\| {{intermediate notions short\|acyclic space\|weakly contractible space}}
	\|}		\|}

Vipul: /* Weaker properties */

2016-06-21T14:57:37Z

Weaker properties

← Older revision		Revision as of 14:57, 21 June 2016
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	===Weaker properties===		===Weaker properties===

	* [[~~Rationally~~ acyclic space]]		{\| class="sortable" border="1"
	* [[~~Space~~ with finitely generated homology]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[~~Space~~ with homology of finite type]]		\|-
	* [[~~Space~~ with free homology]]		\| [[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}}
	* [[~~Space~~ with perfect fundamental group]]		\|-
			\| [[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}}
			\|-
			\| [[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}}
			\|-
			\| [[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}}
			\|-
			\| [[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}}
			\|-
			\| [[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}}
			\|}

	==Metaproperties==		==Metaproperties==

Vipul: /* Relation with other properties */

2016-05-30T22:08:37Z

Relation with other properties

← Older revision		Revision as of 22:08, 30 May 2016
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	===Stronger properties===		===Stronger properties===

	* [[~~Contractible~~ space]]		{\| class="sortable" border="1"
	* [[~~Weakly~~ contractible space]]~~: In fact a weakly contractible~~ space is ~~precisely the same thing as a~~ [[simply connected space\|simply connected]] acyclic space		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::contractible space]] \|\| homotopy-equivalent to a point, or equivalently, has a [[contracting homotopy]] \|\| (via weakly contractible) \|\| (via weakly contractible) \|\| {{intermediate notions short\|acyclic space\|contractible space}}
			\|-
			\| [[Weaker than::weakly contractible space]] \|\| acyclic space that is also [[simply connected space\|simply connected]] \|\| (obvious) \|\| \|\| {{intermediate notions short\|acyclic space\|weakly contractible space}}
			\|}

	===Weaker properties===		===Weaker properties===

Vipul: 10 revisions

2008-05-11T19:31:32Z

10 revisions

← Older revision	Revision as of 19:31, 11 May 2008
(No difference)

Vipul: /* Weaker properties */

2007-12-03T14:33:29Z

Weaker properties

← Older revision		Revision as of 14:33, 3 December 2007
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	===Weaker properties===		===Weaker properties===

			* [[Rationally acyclic space]]
	* [[Space with finitely generated homology]]		* [[Space with finitely generated homology]]
	* [[Space with homology of finite type]]		* [[Space with homology of finite type]]

Vipul at 18:15, 2 December 2007

2007-12-02T18:15:53Z

← Older revision		Revision as of 18:15, 2 December 2007
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	* [[Space with homology of finite type]]		* [[Space with homology of finite type]]
	* [[Space with free homology]]		* [[Space with free homology]]
			* [[Space with perfect fundamental group]]

	==Metaproperties==		==Metaproperties==

Vipul: /* Stronger properties */

2007-12-01T23:58:51Z

Stronger properties

← Older revision		Revision as of 23:58, 1 December 2007
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	* [[Contractible space]]		* [[Contractible space]]
	* [[Weakly contractible space]]		* [[Weakly contractible space]]: In fact a weakly contractible space is precisely the same thing as a [[simply connected space\|simply connected]] acyclic space

	===Weaker properties===		===Weaker properties===

Vipul: /* Stronger properties */

2007-12-01T23:58:21Z

Stronger properties

← Older revision		Revision as of 23:58, 1 December 2007
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	* [[Contractible space]]		* [[Contractible space]]
			* [[Weakly contractible space]]

	===Weaker properties===		===Weaker properties===