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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.016 | 0.001 |
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 teacher head, 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".