Bulletin of the Manifold Atlas (2013) Lens spaces in dimension 3: a history*
Bibliographic record
Abstract
Abstract. In this article we present the early history of 3-dimensional lens spaces from their first appearance in Tietze’s paper (1908) to the late 1930’s including the problem of their classification. 01A55, 01A60 Lens spaces are a particular class of closed orientable 3-manifolds which played an important role in the history of manifolds; they were obtained by identifications on a 2-sphere bounding a 3-ball or by Heegard’s method using tori. The Heegard splitting is of genus one, that is the reason why lens spaces are rather simple 3-manifolds. The first mathematician who mentioned lens spaces- this name wasn’t introduced until 1931 (cf. below)- was W. Dyck. He did this in a talk delivered to the British Association for the Advancement of Science held in Montreal 1884 ([3, 110]). After describing the construction of 3-manifolds by identifying homeomorphic surfaces of handle bodies- today known as Heegard diagrams- in a rather general way, Dyck gave two examples. Take two solid tori and define the identifications of their surfaces by fixing the images of the meridians and the latitudinal curves of the first torus on
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.000 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.011 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".