Some hyperbolic three-manifolds that bound geometrically
Bibliographic record
Abstract
A closed connected hyperbolic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -manifold <italic>bounds geometrically</italic> if it is isometric to the geodesic boundary of a compact hyperbolic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(n+1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension. We construct here infinitely many explicit examples in dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 3"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n=3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> using right-angled dodecahedra and 120-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, for every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-slanted-equals 1"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo> ⩾ </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k\geqslant 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we build an orientable compact closed 3-manifold tessellated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="16 k"> <mml:semantics> <mml:mrow> <mml:mn>16</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">16k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> right-angled dodecahedra that bounds a 4-manifold tessellated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="32 k"> <mml:semantics> <mml:mrow> <mml:mn>32</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">32k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> right-angled 120-cells. A notable feature of this family is that the ratio between the volumes of the 4-manifolds and their boundary components is constant and, in particular, bounded.
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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 | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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".