Triple

T11219624
Position Surface form Disambiguated ID Type / Status
Subject Characteristic Classes E265523 entity
Predicate hasSubject P450 FINISHED
Object Hurewicz homomorphism
The Hurewicz homomorphism is a fundamental map in algebraic topology that relates the homotopy groups of a space to its homology groups, often serving as a bridge between geometric and algebraic invariants.
E911367 NE FINISHED

How this triple was built (4 steps)

Every LLM step that produced this triple, in pipeline order — named-entity classification, the disambiguation choices (the exact options shown, with the pick highlighted), and the generated description. The batch + timestamp of each is in the Provenance table below.

NER Named-entity recognition gpt-5-mini
Instruction
Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).
Input
Phrase: Hurewicz homomorphism | Statement: [Characteristic Classes, hasSubject, Hurewicz homomorphism]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Hurewicz homomorphism
Context triple: [Characteristic Classes, hasSubject, Hurewicz homomorphism]
  • A. Hopf invariant
    The Hopf invariant is a topological integer-valued invariant that classifies certain continuous maps between spheres, playing a central role in homotopy theory and the study of higher-dimensional linking.
  • B. Whitehead product in homotopy theory
    The Whitehead product in homotopy theory is a bilinear operation on homotopy groups that captures how spheres can be nontrivially linked or composed within a topological space.
  • C. Whitehead product
    The Whitehead product is a fundamental operation in algebraic topology that combines homotopy classes of maps to produce higher-order homotopy information, playing a key role in the structure of homotopy groups of spheres and related spaces.
  • D. Eilenberg–MacLane spaces
    Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
  • E. Eilenberg–Zilber 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.
  • F. None of above. chosen
  • G. Unsure - the case is ambiguous/there is not enough information to decide.
NEDg Description generation gpt-5.1
Instruction
Generate a one-sentence description of the target entity. 
You are given a context triple in the form (subject, predicate, object), where the object is the target entity. 
# Instructions
Use the triple to infer relevant information about the entity. Describe the entity based on what is most defining, well-known. 
Avoid repeating the information from the triple, unless really essential.
# Response Format
Return only the sentence: "Description: [one-sentence description of the target entity]"
Input
Entity: Hurewicz homomorphism
Triple: [Characteristic Classes, hasSubject, Hurewicz homomorphism]
Generated description
The Hurewicz homomorphism is a fundamental map in algebraic topology that relates the homotopy groups of a space to its homology groups, often serving as a bridge between geometric and algebraic invariants.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Hurewicz homomorphism
Target entity description: The Hurewicz homomorphism is a fundamental map in algebraic topology that relates the homotopy groups of a space to its homology groups, often serving as a bridge between geometric and algebraic invariants.
  • A. Hopf invariant
    The Hopf invariant is a topological integer-valued invariant that classifies certain continuous maps between spheres, playing a central role in homotopy theory and the study of higher-dimensional linking.
  • B. Whitehead product in homotopy theory
    The Whitehead product in homotopy theory is a bilinear operation on homotopy groups that captures how spheres can be nontrivially linked or composed within a topological space.
  • C. Whitehead product
    The Whitehead product is a fundamental operation in algebraic topology that combines homotopy classes of maps to produce higher-order homotopy information, playing a key role in the structure of homotopy groups of spheres and related spaces.
  • D. Eilenberg–MacLane spaces
    Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
  • E. Eilenberg–Zilber 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.
  • F. None of above. chosen

Provenance (5 batches)

The batch behind each pipeline step, in order, with when it ran. Timestamps are batch-level — stages were processed in waves, so the object chain (NER → NED1 → NEDg → NED2) reads in order, but predicate / elicitation batches can sit in a different wave.

Step Stage Batch ID Status When
creating Elicitation batch_69d6aac59460819089b9848b27f57848 completed April 8, 2026, 7:21 p.m.
NER Named-entity recognition batch_69d7e8eb84c48190b4f3bede254afde2 completed April 9, 2026, 5:59 p.m.
NED1 Entity disambiguation (via context triple) batch_69e4976f38788190855aed6338d819b7 completed April 19, 2026, 8:50 a.m.
NEDg Description generation batch_69e49d37989881909c7e75ddfff06726 completed April 19, 2026, 9:15 a.m.
NED2 Entity disambiguation (via description) batch_69e49f41a1f8819087cc15527dc7ff63 completed April 19, 2026, 9:24 a.m.
Created at: April 8, 2026, 9:30 p.m.