Space with Euler characteristic zero: Difference between revisions
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{{ | {{homology-dependent topospace property}} | ||
==Definition== | ==Definition== | ||
A [[topological space]] is said to have ''zero'' Euler characteristic if it has [[space with finitely generated homology|finitely generated homology]], and its Euler characteristic is zero. | A [[topological space]] is said to have ''zero'' Euler characteristic if it has [[defining ingredient::space with finitely generated homology|finitely generated homology]], and its [[defining ingredient::Euler characteristic]] is zero. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
* [[Compact connected Lie group]] (nontrivial) | * [[Compact connected Lie group]] (nontrivial): {{proofat|[[compact connected nontrivial Lie group implies zero Euler characteristic]]}} | ||
* Odd-dimensional [[compact connected orientable manifold]]: {{proofat|[[Euler characteristic of odd-dimensional compact connected orientable manifold is zero]]}} | |||
===Weaker properties=== | ===Weaker properties=== | ||
* [[Space with finitely generated homology]] | * [[Space with finitely generated homology]] | ||
* [[Space with finite | * [[Space with homology of finite type]] | ||
Latest revision as of 15:03, 21 June 2016
This article defines a property of topological spaces that depends only on the homology of the topological space, viz it is completely determined by the homology groups. In particular, it is a homotopy-invariant property of topological spaces
View all homology-dependent properties of topological spaces OR view all homotopy-invariant properties of topological spaces OR view all properties of topological spaces
Definition
A topological space is said to have zero Euler characteristic if it has finitely generated homology, and its Euler characteristic is zero.
Relation with other properties
Stronger properties
- Compact connected Lie group (nontrivial): For full proof, refer: compact connected nontrivial Lie group implies zero Euler characteristic
- Odd-dimensional compact connected orientable manifold: For full proof, refer: Euler characteristic of odd-dimensional compact connected orientable manifold is zero