Simple space

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

Contents

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

• Space with Abelian fundamental group
• Path-connected space

References

Textbook references

• A Concise Course in Algebraic Topology by J Peter May^{Full text PDFMore info}, Page 140 (formal definition)
• Algebraic Topology by Allen Hatcher^{Full text PDFMore info}, Page 342 (definition in paragraph): Hatcher uses the term Abelian space locally in the book
• Algebraic Topology by Edwin H. Spanier^{More info}, Page 384 (definition in paragraph)