Bibliographic record
Abstract
In this note, we describe a family of arguments that link the homotopy type of (a) the diffeomorphism group of the disc D^{n} , (b) the space of co-dimension one embedded spheres in S^{n} , and (c) the homotopy type of the space of co-dimension two trivial knots in S^{n} . We also describe some natural extensions to these arguments. We begin with Cerf’s “upgraded” proof of Smale’s theorem, showing that the diffeomorphism group of S^{2} has the homotopy type of the isometry group. This entails a cancelling-handle construction, related to recently studied “scanning” maps of spaces of embeddings \operatorname{Emb}(D^{n-1}, S^{1}\times D^{n-1}) \to \Omega^{j} \operatorname{Emb}(D^{n-1-j}, S^{1} \times D^{n-1}) . We further give a Bott-style variation on Cerf’s construction and a related embedding calculus framework for these constructions. We use these arguments to prove that the monoid of Schönflies spheres \pi_{0} \operatorname{Emb}(S^{n-1}, S^{n}) is a group with respect to the connected-sum operation for all n \geq 2 . This last result is perhaps only interesting when n=4 , as when n \neq 4 , it follows from the resolution of the various generalised Schönflies problems.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 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".