Simply connected space: Difference between revisions
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===Stronger properties=== | ===Stronger properties=== | ||
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! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions | |||
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| [[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}} | |||
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| [[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}} | |||
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| [[Weaker than::Multiply connected space]] || first few homotopy groups trivial || || || | |||
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===Weaker properties=== | ===Weaker properties=== | ||
{| class="wikitable" border="1" | |||
! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions | |||
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| [[Stronger than::Semilocally simply connected space]] || || || || | |||
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| [[Stronger than::Simple space]] || || || || {{intermediate notions short|simple space|simply connected space}} | |||
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| [[Stronger than::Space with abelian fundamental group]] || || || || | |||
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| [[Stronger than::Space with perfect fundamental group]] || || || || | |||
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==Metaproperties== | ==Metaproperties== | ||
Latest revision as of 15:42, 25 October 2009
This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spaces
View other homotopy-invariant properties of topological spaces OR view all properties of topological spaces
This property of topological spaces is defined as the property of the following associated group: fundamental group having the following group property: trivial group
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
Fill this in later
Relation with other properties
Stronger properties
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
|---|---|---|---|---|
| Contractible space | homotopy-equivalent to a point | contractible implies simply connected | simply connected not implies contractible | Weakly contractible space|FULL LIST, MORE INFO |
| Weakly contractible space | all homotopy groups trivial | weakly contractible implies simply connected | simply connected not implies weakly contractible | |FULL LIST, MORE INFO |
| Multiply connected space | first few homotopy groups trivial |
Weaker properties
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
|---|---|---|---|---|
| Semilocally simply connected space | ||||
| Simple space | |FULL LIST, MORE INFO | |||
| Space with abelian fundamental group | ||||
| Space with perfect fundamental group |
Metaproperties
Products
This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products
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-hereditariness
This property of topological spaces is hereditary on retracts, viz if a space has the property, so does any retract of it
View all retract-hereditary properties of topological spaces
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.