Triple

T11085968
Position Surface form Disambiguated ID Type / Status
Subject Lefschetz fixed-point theorem E262120 entity
Predicate hasVariant P455 FINISHED
Object equivariant Lefschetz fixed-point theorem
The equivariant Lefschetz fixed-point theorem is a generalization of the classical Lefschetz fixed-point theorem that computes fixed points of maps respecting a group action using equivariant cohomological data.
E258614 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: equivariant Lefschetz fixed-point theorem | Statement: [Lefschetz fixed-point theorem, hasVariant, equivariant Lefschetz fixed-point theorem]
NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07
Target entity: equivariant Lefschetz fixed-point theorem
Context triple: [Lefschetz fixed-point theorem, hasVariant, equivariant Lefschetz fixed-point theorem]
  • A. Lefschetz fixed-point theorem
    The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.
  • B. Atiyah–Bott fixed-point theorem
    The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
  • C. Lefschetz number
    The Lefschetz number is a topological invariant, computed from the traces of induced maps on homology, that predicts the existence and number of fixed points of a continuous self-map on a topological space.
  • D. Grothendieck–Lefschetz trace formula
    The Grothendieck–Lefschetz trace formula is a fundamental result in algebraic geometry that expresses the number of rational points of a variety over a finite field in terms of traces of Frobenius acting on its étale cohomology groups.
  • E. equivariant index theorem
    The equivariant index theorem is a generalization of the Atiyah–Singer index theorem that computes indices of elliptic operators while taking into account the action of a symmetry group.
  • 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: equivariant Lefschetz fixed-point theorem
Triple: [Lefschetz fixed-point theorem, hasVariant, equivariant Lefschetz fixed-point theorem]
Generated description
The equivariant Lefschetz fixed-point theorem is a generalization of the classical Lefschetz fixed-point theorem that computes fixed points of maps respecting a group action using equivariant cohomological data.
NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07
Target entity: equivariant Lefschetz fixed-point theorem
Target entity description: The equivariant Lefschetz fixed-point theorem is a generalization of the classical Lefschetz fixed-point theorem that computes fixed points of maps respecting a group action using equivariant cohomological data.
  • A. Lefschetz fixed-point theorem
    The Lefschetz fixed-point theorem is a fundamental result in algebraic topology that relates the number of fixed points of a continuous map on a topological space to traces of the induced maps on its homology groups.
  • B. Atiyah–Bott fixed-point theorem chosen
    The Atiyah–Bott fixed-point theorem is a fundamental result in equivariant cohomology that expresses global invariants, such as indices of elliptic operators, in terms of local data at the fixed points of a group action.
  • C. Lefschetz number
    The Lefschetz number is a topological invariant, computed from the traces of induced maps on homology, that predicts the existence and number of fixed points of a continuous self-map on a topological space.
  • D. Grothendieck–Lefschetz trace formula
    The Grothendieck–Lefschetz trace formula is a fundamental result in algebraic geometry that expresses the number of rational points of a variety over a finite field in terms of traces of Frobenius acting on its étale cohomology groups.
  • E. equivariant index theorem
    The equivariant index theorem is a generalization of the Atiyah–Singer index theorem that computes indices of elliptic operators while taking into account the action of a symmetry group.
  • F. None of above.

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_69d6aa9983c08190b0ef61603b69feac completed April 8, 2026, 7:20 p.m.
NER Named-entity recognition batch_69d799c2c7d4819087ac793153340178 completed April 9, 2026, 12:21 p.m.
NED1 Entity disambiguation (via context triple) batch_69e42d66ded88190877a20a10f012d6b completed April 19, 2026, 1:18 a.m.
NEDg Description generation batch_69e42e1daa3c8190b598adcf9bac00f3 completed April 19, 2026, 1:21 a.m.
NED2 Entity disambiguation (via description) batch_69e42f415b1081909f9eedcb3640cdc3 completed April 19, 2026, 1:26 a.m.
Created at: April 8, 2026, 9:27 p.m.