Commuting elements, simplicial spaces and filtrations of classifying spaces
Bibliographic record
Abstract
Abstract Let G denote a topological group. In this paper the descending central series of free groups are used to construct simplicial spaces of homomorphisms with geometric realizations B(q, G) that provide a filtration of the classifying space BG. In particular this setting gives rise to a single space constructed out of all the spaces of ordered commuting n–tuples of elements in G. Basic properties of these constructions are discussed, including the homotopy type and cohomology when the group G is either a finite group or a compact connected Lie group. For a finite group the construction gives rise to a covering space with monodromy related to a delicate result in group theory equivalent to the odd-order theorem of Feit–Thompson. The techniques here also yield a counting formula for the cardinality of Hom(π, G) where π is any descending central series quotient of a finitely generated free group. Another application is the determination of the structure of the spaces B(2, G) obtained from commuting n-tuples in G for finite groups such that the centralizer of every non–central element is abelian (known as transitively commutative groups), which played a key role in work by Suzuki on the structure of finite simple groups.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 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 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".