Triple

T11425542
Position Surface form Disambiguated ID Type / Status
Subject Introduction to Algebraic K-Theory E270737 entity
Predicate topic P261 FINISHED
Object Whitehead groups
Whitehead groups are algebraic K-theory invariants associated with groups that measure the failure of certain projective modules or h-cobordisms to be trivial, playing a central role in high-dimensional topology and geometric group theory.
E924762 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: Whitehead groups | Statement: [Introduction to Algebraic K-Theory, topic, Whitehead groups]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: Whitehead groups
Context triple: [Introduction to Algebraic K-Theory, topic, Whitehead groups]
  • A. 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.
  • B. 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.
  • 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. Picard group
    The Picard group is an algebraic invariant of a variety or scheme that classifies line bundles (or divisor classes) up to isomorphism, playing a central role in algebraic geometry.
  • E. Weil group
    The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
  • 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: Whitehead groups
Triple: [Introduction to Algebraic K-Theory, topic, Whitehead groups]
Generated description
Whitehead groups are algebraic K-theory invariants associated with groups that measure the failure of certain projective modules or h-cobordisms to be trivial, playing a central role in high-dimensional topology and geometric group theory.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: Whitehead groups
Target entity description: Whitehead groups are algebraic K-theory invariants associated with groups that measure the failure of certain projective modules or h-cobordisms to be trivial, playing a central role in high-dimensional topology and geometric group theory.
  • A. 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.
  • B. 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.
  • 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. Picard group
    The Picard group is an algebraic invariant of a variety or scheme that classifies line bundles (or divisor classes) up to isomorphism, playing a central role in algebraic geometry.
  • E. Weil group
    The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
  • 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_69d6aadeef688190874bcecd88b3dd9b completed April 8, 2026, 7:22 p.m.
NER Named-entity recognition batch_69d806be9c2c819084da13101cbb6c81 completed April 9, 2026, 8:06 p.m.
NED1 Entity disambiguation (via context triple) batch_69e5b8b553808190bf8b40d9b03e12b7 completed April 20, 2026, 5:25 a.m.
NEDg Description generation batch_69e5c28e2dd481909b45a43b5825f393 completed April 20, 2026, 6:07 a.m.
NED2 Entity disambiguation (via description) batch_69e5c4722c348190a4c49edb1f6df240 completed April 20, 2026, 6:15 a.m.
Created at: April 8, 2026, 9:35 p.m.