Some computations of the homology of real grassmannian manifolds
Bibliographic record
Abstract
When computing the homology of Grassmannian manifolds, the first step is usually to look at the Schubert cell decomposition, and the chain complex associated with it. In the complex case and the real unoriented case with Z₂ coefficients the additive structure is obtained immediately (i.e., generated by the homology classes represented by the Schubert cells) because the boundary map is trivial. In the real unoriented case (with Z₂ coefficients) and the real oriented case, finding the additive structure is more complicated since the boundary map is nontrivial. In this paper, this boundary map is computed by cell orientation comparisons, using graph coordinates where the cells are linear, to simplify the comparisons. The integral homology groups for some low dimensional oriented and unoriented Grassmannians are determined directly from the chain complex (with the boundary map as computed). The integral cohomology ring structure for complex Grassmannians has been completely determined mainly using Schubert cell intersections (what is known as Schubert Calculus).. In this paper, a method using Schubert cell intersections to describe the Z₂ cohomology ring structure of the real Grassmannians is sketched. The results are identical to those for the complex Grassmannians (with coefficients), but the notation used for the cohomology generators is not the usual one. It indicates that the products are to a certain degree independent of the Grassmannian.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.009 | 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".