The homotopy type of the space of symplectic balls in rational ruled 4–manifolds
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Bibliographic record
Abstract
be a 4-dimensional rational ruled symplectic manifold and denote by w M its Gromov width. Let Emb ! .B 4 .c/; M / be the space of symplectic embeddings of the standard ball of radius r , B 4 .c/ R 4 (parametrized by its capacity c WD r 2 ), into .M; !/. By the work of Lalonde and Pinsonnault [13], we know that there exists a critical capacity c crit 2 .0; w M such that, for all c 2 .0; c crit /, the embedding space Emb ! .B 4 .c/; M / is homotopy equivalent to the space of symplectic frames SFr.M /. We also know that the homotopy type of Emb ! .B 4 .c/; M / changes when c reaches c crit and that it remains constant for all c 2 OEc crit ; w M /. In this paper, we compute the rational homotopy type, the minimal model and the cohomology with rational coefficients of Emb ! .B 4 .c/; M / in the remaining case c 2 OEc crit ; w M /. In particular, we show that it does not have the homotopy type of a finite CW-complex. Some of the key points in the argument are the calculation of the rational homotopy type of the classifying space of the symplectomorphism group of the blow up of M , its comparison with the group corresponding to M and the proof that the space of compatible integrable complex structures on the blow up is weakly contractible.
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| 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) | 0.000 | 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 it