Simply connected space - Revision history

Vipul: /* Relation with other properties */

2009-10-25T15:42:01Z

Relation with other properties

← Older revision		Revision as of 15:42, 25 October 2009
Line 20:		Line 20:
	===Stronger properties===		===Stronger properties===

	* [[Contractible space]]		{\| class="wikitable" border="1"
	* [[Weakly contractible space]]		! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
	* [[Multiply connected space]]		\|-
			\| [[Weaker than::Contractible space]] \|\| homotopy-equivalent to a point \|\| [[contractible implies simply connected]] \|\| [[simply connected not implies contractible]] \|\| {{intermediate notions short\|simply connected space\|contractible space}}
			\|-
			\| [[Weaker than::Weakly contractible space]] \|\| all homotopy groups trivial \|\| [[weakly contractible implies simply connected]] \|\| [[simply connected not implies weakly contractible]] \|\| {{intermediate notions short\|simply connected space\|weakly contractible space}}
			\|-
			\| [[Weaker than::Multiply connected space]] \|\| first few homotopy groups trivial \|\| \|\| \|\|
			\|}

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

	* [[Semilocally simply connected space]]		{\| class="wikitable" border="1"
	* [[Simple space]]		! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
	* [[Space with ~~Abelian~~ fundamental group]]		\|-
	* [[Space with perfect fundamental group]]		\| [[Stronger than::Semilocally simply connected space]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Stronger than::Simple space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|simple space\|simply connected space}}
			\|-
			\| [[Stronger than::Space with abelian fundamental group]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Stronger than::Space with perfect fundamental group]] \|\| \|\| \|\| \|\|
			\|}

	==Metaproperties==		==Metaproperties==

Vipul: 6 revisions

2008-05-11T19:58:17Z

6 revisions

← Older revision	Revision as of 19:58, 11 May 2008
(No difference)

Vipul at 21:15, 15 December 2007

2007-12-15T21:15:25Z

← Older revision		Revision as of 21:15, 15 December 2007
Line 37:		Line 37:
	An arbitrary product of simply connected spaces is simply connected. This follows from the fact that the fundamental group of a product of path-connected spaces, is the product of their fundamental groups.		An arbitrary product of simply connected spaces is simply connected. This follows from the fact that the fundamental group of a product of path-connected spaces, is the product of their fundamental groups.

	{{retract-~~closed~~}}		{{retract-hereditary}}

	A retract of a simply connected space is simply connected. This follows from the fact that the fundamental group of a retract is a group-theoretic retract of the fundamental group of the whole space.		A retract of a simply connected space is simply connected. This follows from the fact that the fundamental group of a retract is a group-theoretic retract of the fundamental group of the whole space.

Vipul at 21:14, 15 December 2007

2007-12-15T21:14:02Z

← Older revision		Revision as of 21:14, 15 December 2007
Line 30:		Line 30:
	* [[Space with Abelian fundamental group]]		* [[Space with Abelian fundamental group]]
	* [[Space with perfect fundamental group]]		* [[Space with perfect fundamental group]]

			==Metaproperties==

			{{DP-closed}}

			An arbitrary product of simply connected spaces is simply connected. This follows from the fact that the fundamental group of a product of path-connected spaces, is the product of their fundamental groups.

			{{retract-closed}}

			A retract of a simply connected space is simply connected. This follows from the fact that the fundamental group of a retract is a group-theoretic retract of the fundamental group of the whole space.

Vipul at 18:37, 11 December 2007

2007-12-11T18:37:20Z

← Older revision		Revision as of 18:37, 11 December 2007
Line 1:		Line 1:
	{{homotopy-invariant topospace property}}		{{homotopy-invariant topospace property}}

			{{group-determined topospace property\|fundamental group\|trivial group}}

	==Definition==		==Definition==

Vipul at 18:15, 2 December 2007

2007-12-02T18:15:05Z

← Older revision		Revision as of 18:15, 2 December 2007
Line 13:		Line 13:

	{{fillin}}		{{fillin}}

			==Relation with other properties==

			===Stronger properties===

			* [[Contractible space]]
			* [[Weakly contractible space]]
			* [[Multiply connected space]]

			===Weaker properties===

			* [[Semilocally simply connected space]]
			* [[Simple space]]
			* [[Space with Abelian fundamental group]]
			* [[Space with perfect fundamental group]]

Vipul at 23:22, 2 November 2007

2007-11-02T23:22:15Z

← Older revision		Revision as of 23:22, 2 November 2007
Line 7:		Line 7:
	A [[topological space]] is said to be '''simply connected''' if it satisfies the following equivalent conditions:		A [[topological space]] is said to be '''simply connected''' if it satisfies the following equivalent conditions:

	* It is path-connected, and any loop at any point is homotopic to the constant loop at that point		* It is [[path-connected space\|path-connected]], and any loop at any point is homotopic to the constant loop at that point
	* It is path-connected, and its [[fundamental group]] is trivial		* It is path-connected, and its [[fundamental group]] is trivial

Vipul at 19:21, 10 May 2007

2007-05-10T19:21:00Z

New page

{{homotopy-invariant topospace property}}

==Definition==

===Symbol-free definition===

A [[topological space]] is said to be '''simply connected''' if it satisfies the following equivalent conditions:

* It is path-connected, and any loop at any point is homotopic to the constant loop at that point
* It is path-connected, and its [[fundamental group]] is trivial

===Definition with symbols===

{{fillin}}