Künneth formula

E860097

mathematical theorem result in algebraic topology result in homological algebra

The Künneth formula is a fundamental result in algebraic topology and homological algebra that expresses the (co)homology of a product space or object in terms of the (co)homology of its factors.

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Observed surface forms (1)

Surface form	Occurrences
Künneth theorem in generalized cohomology	1

Statements (50)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in algebraic topology ⓘ result in homological algebra ⓘ
appliesTo	cellular homology ⓘ derived functors ⓘ group cohomology ⓘ group homology ⓘ sheaf cohomology ⓘ simplicial homology ⓘ singular cohomology ⓘ singular homology ⓘ
describes	cohomology of product spaces ⓘ cohomology of tensor products of chain complexes ⓘ homology of product spaces ⓘ homology of tensor products of chain complexes ⓘ
field	algebraic topology ⓘ homological algebra ⓘ
generalizes	product formula for Betti numbers ⓘ
hasCondition	finiteness conditions on homology groups ⓘ flatness of coefficient module ⓘ torsion-free coefficients ⓘ
hasVariant	Künneth spectral sequence NERFINISHED ⓘ cohomological Künneth formula NERFINISHED ⓘ homological Künneth formula NERFINISHED ⓘ
implies	decomposition of cohomology of product into tensor and Ext terms ⓘ decomposition of homology of product into tensor and Tor terms ⓘ
namedAfter	Hermann Künneth NERFINISHED ⓘ
relatedTo	Eilenberg–Zilber theorem NERFINISHED ⓘ Künneth theorem NERFINISHED ⓘ universal coefficient theorem NERFINISHED ⓘ
relates	cohomology of X ⓘ cohomology of X × Y ⓘ cohomology of Y ⓘ homology of X ⓘ homology of X × Y ⓘ homology of Y ⓘ
typicalAssumption	coefficients in a field ⓘ coefficients in a principal ideal domain ⓘ spaces are CW complexes ⓘ
usedIn	algebraic geometry ⓘ computation of cohomology rings ⓘ computation of homology of product manifolds ⓘ stable homotopy theory ⓘ topological K-theory NERFINISHED ⓘ
usesConcept	Ext functor ⓘ Tor functor NERFINISHED ⓘ chain complex ⓘ exact sequence ⓘ short exact sequence of chain complexes ⓘ tensor product ⓘ

Referenced by (4)

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

Weil conjectures → usesConcept → Künneth formula ⓘ

Weil cohomology → hasAxiom → Künneth formula ⓘ

Atiyah–Hirzebruch spectral sequence → relatedConcept → Künneth formula ⓘ

this entity surface form: Künneth theorem in generalized cohomology

"Algebraic Topology" → developsConcept → Künneth formula ⓘ

subject surface form: Algebraic Topology