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.