Eilenberg–Zilber theorem

E634847

mathematical theorem

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.

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All labels observed (5)

Label	Occurrences
Alexander–Whitney map	2
Eilenberg–Zilber theorem canonical	2
Eilenberg–Zilber map	1
Eilenberg–Zilber map and Alexander–Whitney map are chain homotopy inverses	1
simplicial Eilenberg–Zilber theorem	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 ⓘ linked to: Künneth formula
hasProofTechnique	acyclic models ⓘ explicit combinatorial chain maps ⓘ
hasVersion	simplicial Eilenberg–Zilber theorem ⓘ linked to: Eilenberg–Zilber theorem
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 ⓘ Samuel Eilenberg ⓘ
property	naturality with respect to continuous maps ⓘ
provides	Alexander–Whitney map ⓘ linked to: Eilenberg–Zilber theorem Eilenberg–Zilber map ⓘ linked to: Eilenberg–Zilber theorem
relatedTo	Alexander–Whitney map ⓘ linked to: Eilenberg–Zilber theorem Eilenberg–MacLane spaces ⓘ Künneth theorem ⓘ linked to: Künneth formula 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 ⓘ linked to: Eilenberg–Zilber theorem
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