Eilenberg–Zilber theorem
E634847
The Eilenberg–Zilber theorem is a fundamental result in algebraic topology that establishes a chain homotopy equivalence between the singular chain complex of a product space and the tensor product of the singular chain complexes of the factors.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Eilenberg–Zilber theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7011115 — 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: Eilenberg–Zilber theorem Context triple: [Samuel Eilenberg, notableConcept, Eilenberg–Zilber theorem]
-
A.
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.
-
B.
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.
-
C.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
D.
Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
-
E.
Freyd–Mitchell embedding theorem
The Freyd–Mitchell embedding theorem is a fundamental result in category theory stating that every small abelian category can be faithfully represented as a full subcategory of a module category, thereby allowing the use of element-wise methods in abstract settings.
- 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: Eilenberg–Zilber theorem Target entity description: The Eilenberg–Zilber theorem is a fundamental result in algebraic topology that establishes a chain homotopy equivalence between the singular chain complex of a product space and the tensor product of the singular chain complexes of the factors.
-
A.
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.
-
B.
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.
-
C.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
D.
Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
-
E.
Freyd–Mitchell embedding theorem
The Freyd–Mitchell embedding theorem is a fundamental result in category theory stating that every small abelian category can be faithfully represented as a full subcategory of a module category, thereby allowing the use of element-wise methods in abstract settings.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appearsIn |
Eilenberg and Steenrod’s work on axiomatic homology theory
ⓘ
foundational texts on algebraic topology ⓘ |
| appliesTo |
simplicial chain complexes
ⓘ
singular chain complexes ⓘ |
| asserts | existence of a natural chain homotopy equivalence between C_*(X×Y) and C_*(X)⊗C_*(Y) ⓘ |
| concerns |
chain homotopy equivalence
ⓘ
product of topological spaces ⓘ singular chain complex ⓘ tensor product of chain complexes ⓘ |
| context |
homological algebra
ⓘ
singular homology ⓘ |
| domain | topological spaces ⓘ |
| ensures | C_*(X×Y) is chain homotopy equivalent to C_*(X)⊗C_*(Y) ⓘ |
| field | algebraic topology ⓘ |
| guarantees | induced isomorphism H_*(X×Y) ≅ H_*(C_*(X)⊗C_*(Y)) ⓘ |
| hasConsequence | Künneth theorem for singular homology NERFINISHED ⓘ |
| hasProofTechnique |
acyclic models
ⓘ
explicit combinatorial chain maps ⓘ |
| hasVersion | simplicial Eilenberg–Zilber theorem NERFINISHED ⓘ |
| holdsFor | pairs of topological spaces X and Y ⓘ |
| implies | homology of a product is isomorphic to homology of tensor product of chain complexes ⓘ |
| level | chain complexes rather than just homology groups ⓘ |
| namedAfter |
Joseph A. Zilber
NERFINISHED
ⓘ
Samuel Eilenberg NERFINISHED ⓘ |
| property | naturality with respect to continuous maps ⓘ |
| provides |
Alexander–Whitney map
NERFINISHED
ⓘ
Eilenberg–Zilber map NERFINISHED ⓘ |
| relatedTo |
Alexander–Whitney map
NERFINISHED
ⓘ
Eilenberg–MacLane spaces NERFINISHED ⓘ Künneth theorem NERFINISHED ⓘ tensor product of abelian groups ⓘ |
| relates |
singular chain complex of a product space
ⓘ
tensor product of singular chain complexes of the factors ⓘ |
| states | Eilenberg–Zilber map and Alexander–Whitney map are chain homotopy inverses NERFINISHED ⓘ |
| typeOf | chain-level Künneth-type result ⓘ |
| usedFor |
computing homology of product spaces
ⓘ
constructing cross product in homology ⓘ defining cup product in cohomology via dualization ⓘ |
| usedIn |
construction of spectral sequences for product spaces
ⓘ
monoidal structure on derived categories of chain complexes ⓘ |
| yearProvedApprox | 1940s ⓘ |
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: Eilenberg–Zilber theorem Description of subject: The Eilenberg–Zilber theorem is a fundamental result in algebraic topology that establishes a chain homotopy equivalence between the singular chain complex of a product space and the tensor product of the singular chain complexes of the factors.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.