On the mod-ℓ homology of the classifying spacefor commutativity
Bibliographic record
Abstract
We study the mod- [math] homotopy type of classifying spaces for commutativity, [math] , at a prime [math] . We show that the mod- [math] homology of [math] depends on the mod- [math] homotopy type of [math] when [math] is a compact connected Lie group, in the sense that a mod- [math] homology isomorphism [math] for such groups induces a mod- [math] homology isomorphism [math] . In order to prove this result, we study a presentation of [math] as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an idea of Adem and Gómez. We also study the relationship between the mod- [math] type of a Lie group [math] and the locally finite group [math] , where [math] is a Chevalley group. We see that the naïve analogue for [math] of the celebrated Friedlander–Mislin result cannot hold, but we show that it does hold after taking the homotopy quotient of a [math] action on [math] .
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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 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".