Eilenberg–Steenrod axioms
E634843
The Eilenberg–Steenrod axioms are a foundational set of conditions that formally characterize homology theories in algebraic topology.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Eilenberg–Steenrod axioms canonical | 4 |
| Eilenberg–Steenrod axioms of homology | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7011080 — 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–Steenrod axioms Context triple: [Samuel Eilenberg, notableWork, Eilenberg–Steenrod axioms]
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A.
Atiyah–Segal axioms
The Atiyah–Segal axioms are a set of mathematical conditions that rigorously define topological quantum field theories as functorial assignments from geometric data to algebraic structures.
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B.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
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C.
"Algebraic Topology"
"Algebraic Topology" is a foundational mathematical text that develops topological concepts using algebraic methods such as homology and cohomology theories.
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D.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
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E.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
- 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–Steenrod axioms Target entity description: The Eilenberg–Steenrod axioms are a foundational set of conditions that formally characterize homology theories in algebraic topology.
-
A.
Atiyah–Segal axioms
The Atiyah–Segal axioms are a set of mathematical conditions that rigorously define topological quantum field theories as functorial assignments from geometric data to algebraic structures.
-
B.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
-
C.
"Algebraic Topology"
"Algebraic Topology" is a foundational mathematical text that develops topological concepts using algebraic methods such as homology and cohomology theories.
-
D.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
-
E.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
axiomatic system
ⓘ
foundational concept in algebraic topology ⓘ set of axioms ⓘ |
| appliesTo |
continuous maps
ⓘ
pairs of topological spaces ⓘ |
| assumes | abelian category structure on target ⓘ |
| characterizes |
ordinary homology theories
ⓘ
singular homology ⓘ |
| codomain |
graded abelian groups
ⓘ
graded modules ⓘ |
| contrastedWith | extraordinary cohomology theories ⓘ |
| defines | homology theory ⓘ |
| domain | topological spaces ⓘ |
| ensures |
Mayer–Vietoris sequence
ⓘ
homotopy invariance of homology ⓘ uniqueness of ordinary homology theories up to natural isomorphism ⓘ |
| field |
algebraic topology
ⓘ
homological algebra ⓘ |
| formalizes | properties of classical homology ⓘ |
| generalizedBy | Brown representability theorem ⓘ |
| implies |
homology of a point is concentrated in degree zero
ⓘ
homology of disjoint union is direct sum of homologies ⓘ |
| includes |
boundary homomorphism
ⓘ
long exact sequence of a pair ⓘ |
| influenced | development of modern algebraic topology ⓘ |
| introducedBy |
Norman Steenrod
ⓘ
Samuel Eilenberg ⓘ |
| language | category theory ⓘ |
| namedAfter |
Norman Steenrod
ⓘ
Samuel Eilenberg ⓘ |
| publication | Foundations of Algebraic Topology ⓘ |
| publicationYear | 1952 ⓘ |
| relatedTo |
cohomology theories
ⓘ
spectra in stable homotopy theory ⓘ |
| requires |
additivity axiom
ⓘ
dimension axiom ⓘ exactness axiom ⓘ excision axiom ⓘ functoriality ⓘ homotopy axiom ⓘ naturality of homology maps ⓘ |
| topicOf | many graduate textbooks in algebraic topology ⓘ |
| usedFor |
defining cellular homology
ⓘ
defining simplicial homology ⓘ defining singular homology ⓘ |
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
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Input
Subject: Eilenberg–Steenrod axioms Description of subject: The Eilenberg–Steenrod axioms are a foundational set of conditions that formally characterize homology theories in algebraic topology.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.