universal coefficient theorem

E911366

theorem in algebraic topology

The universal coefficient theorem is a fundamental result in algebraic topology that relates the homology or cohomology groups of a space with coefficients in an arbitrary abelian group to those with integer coefficients via Ext and Tor functors.

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Statements (47)

Predicate	Object
instanceOf	theorem in algebraic topology ⓘ
appliesTo	CW complexes ⓘ chain complexes of abelian groups ⓘ singular cohomology ⓘ singular homology ⓘ topological spaces with reasonable finiteness conditions ⓘ
dependsOn	exactness properties of Hom functor ⓘ exactness properties of tensor product ⓘ
expresses	cohomology with coefficients as combination of Hom and Ext terms ⓘ homology with coefficients as combination of tensor and Tor terms ⓘ
field	algebraic topology ⓘ
generalizedTo	universal coefficient spectral sequence ⓘ
hasConsequence	computation of cohomology with finite coefficients ⓘ computation of homology with finite coefficients ⓘ reduction of coefficient computations to integer homology ⓘ
hasProperty	expresses universal behavior of coefficient change ⓘ functorial in the coefficient group ⓘ natural with respect to continuous maps ⓘ
hasVersion	universal coefficient theorem for cohomology NERFINISHED ⓘ universal coefficient theorem for generalized homology theories ⓘ universal coefficient theorem for homology ⓘ universal coefficient theorem for reduced homology ⓘ
involves	Hom functor ⓘ derived functors ⓘ short exact sequences ⓘ tensor product of abelian groups ⓘ
isRelatedTo	Ext groups ⓘ Hurewicz theorem NERFINISHED ⓘ Künneth theorem NERFINISHED ⓘ Tor groups ⓘ free resolutions ⓘ homological algebra ⓘ projective resolutions ⓘ
relates	cohomology groups with coefficients in an abelian group ⓘ cohomology groups with integer coefficients ⓘ homology groups with coefficients in an abelian group ⓘ homology groups with integer coefficients ⓘ
typicalDomain	abelian groups ⓘ modules over a principal ideal domain ⓘ
usedIn	K-theory NERFINISHED ⓘ classification of topological spaces up to homology ⓘ cohomology operations ⓘ computations in algebraic topology ⓘ spectral sequence calculations ⓘ stable homotopy theory ⓘ
usesFunctor	Ext functor ⓘ Tor functor NERFINISHED ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Characteristic Classes → hasSubject → universal coefficient theorem ⓘ

"Algebraic Topology" → developsConcept → universal coefficient theorem ⓘ

subject surface form: Algebraic Topology