Coarse Alexander duality for pairs and applications
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Bibliographic record
Abstract
For a group G (of type F ) acting properly on a coarse Poincar duality space X, Kapovich and Kleiner introduced a coarse version of Alexander duality between G and its complement in X.More precisely, the cohomology of G with group ring coefficients is dual to a certain ech homology group of the family of increasing neighborhoods of a G-orbit in X.This duality applies more generally to coarse embeddings of certain contractible simplicial complexes into coarse PD.n/ spaces.In this paper we introduce a relative version of this ech homology that satisfies the Eilenberg-Steenrod exactness axiom, and we prove a relative version of coarse Alexander duality.As an application we provide a detailed proof of the following result, first stated by Kapovich and Kleiner.Given a 2-complex formed by gluing k halfplanes along their boundary lines and a coarse embedding into a contractible 3-manifold, the complement consists of k deep components that are arranged cyclically in a pattern called a Jordan cycle.We use the Jordan cycle as an invariant in proving the existence of a 3-manifold group that is virtually Kleinian but not itself Kleinian.
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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 | 0.000 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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