Künneth formula
E860097
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Künneth formula canonical | 3 |
| Künneth theorem in generalized cohomology | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10388804 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Künneth formula Context triple: [Weil conjectures, usesConcept, Künneth formula]
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A.
Lefschetz duality
Lefschetz duality is a generalization of Poincaré duality that relates the homology of a compact manifold with boundary to the cohomology of the manifold relative to its boundary.
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B.
Serre spectral sequence
The Serre spectral sequence is a fundamental tool in algebraic topology that relates the homology or cohomology of a fibration to that of its base and fiber, enabling complex computations in a systematic way.
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C.
Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
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D.
Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
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E.
Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Künneth formula Target entity description: 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.
-
A.
Lefschetz duality
Lefschetz duality is a generalization of Poincaré duality that relates the homology of a compact manifold with boundary to the cohomology of the manifold relative to its boundary.
-
B.
Serre spectral sequence
The Serre spectral sequence is a fundamental tool in algebraic topology that relates the homology or cohomology of a fibration to that of its base and fiber, enabling complex computations in a systematic way.
-
C.
Grothendieck–Riemann–Roch theorem
The Grothendieck–Riemann–Roch theorem is a fundamental result in algebraic geometry that generalizes the classical Riemann–Roch theorem by relating pushforwards in K-theory to pushforwards in cohomology via characteristic classes.
-
D.
Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
-
E.
Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Künneth formula Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Künneth theorem in generalized cohomology
subject surface form:
Algebraic Topology