Quantitative null-cobordism
Bibliographic record
Abstract
For a given null-cobordant Riemannian \n\n \n n\n n\n \n\n-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on \n\n \n n\n n\n \n\n. In the appendix the bound is improved to one that is \n\n \n \n O\n (\n \n L\n \n 1\n +\n ε\n \n \n )\n \n O(L^{1+\\varepsilon })\n \n\n for every \n\n \n \n ε\n >\n 0\n \n \\varepsilon >0\n \n\n.\n\nThis construction relies on another of independent interest. Take \n\n \n X\n X\n \n\n and \n\n \n Y\n Y\n \n\n to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose \n\n \n Y\n Y\n \n\n is simply connected and rationally homotopy equivalent to a product of Eilenberg–MacLane spaces, for example, any simply connected Lie group. Then two homotopic \n\n \n L\n L\n \n\n-Lipschitz maps \n\n \n \n f\n ,\n g\n :\n X\n →\n Y\n \n f,g:X \\to Y\n \n\n are homotopic via a \n\n \n \n C\n L\n \n CL\n \n\n-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces \n\n \n Y\n Y\n \n\n.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.008 | 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 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".