Simple space: Difference between revisions
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* It is [[path-connected space|path-connected]] | * It is [[path-connected space|path-connected]] | ||
* The [[fundamental group]] is [[Abelian group|Abelian]] | * The [[fundamental group]] is [[Abelian group|Abelian]] | ||
* The fundamental group acts trivially on all the higher homotopy groups | * The fundamental group [[actions of the fundamental group|acts]] trivially on all the higher homotopy groups | ||
==Relation with other properties== | ==Relation with other properties== | ||
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* [[Space with Abelian fundamental group]] | * [[Space with Abelian fundamental group]] | ||
* [[Path-connected space]] | * [[Path-connected space]] | ||
==References== | |||
===Textbook references=== | |||
* {{booklink|Concise}}, Page 140 (formal definition) | |||
* {{booklink|Hatcher}}, Page 342 (definition in paragraph): Hatcher uses the term '''Abelian space''' locally in the book | |||
* {{booklink|Spanier}}, Page 384 (definition in paragraph) | |||
Latest revision as of 19:58, 11 May 2008
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
Definition
A topological space is termed simple if it satisfies the following three conditions:
- It is path-connected
- The fundamental group is Abelian
- The fundamental group acts trivially on all the higher homotopy groups
Relation with other properties
Stronger properties
- Simply connected space
- Aspherical space with Abelian fundamental group
Weaker properties
References
Textbook references
- A Concise Course in Algebraic Topology by J Peter MayFull text PDFMore info, Page 140 (formal definition)
- Algebraic Topology by Allen HatcherFull text PDFMore info, Page 342 (definition in paragraph): Hatcher uses the term Abelian space locally in the book
- Algebraic Topology by Edwin H. SpanierMore info, Page 384 (definition in paragraph)