Bundles, cohomology and truncated symmetric polynomials
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Bibliographic record
Abstract
The cohomology of the classifying space BU(n) of the unitary group can be identified with the the ring of symmetric polynomials on n variables by restricting to the cohomology of BT , where T\subset U(n) is a maximal torus. In this paper we explore the situation where BT = (\ CP^{\infty})^n is replaced by a product of finite dimensional projective spaces (\ CP^d)^n , fitting into an associated bundle U(n)\times_T (\ S^{2d+1})^n\to (\ CP^d)^n\to BU(n). We establish a purely algebraic version of this problem by exhibiting an explicit system of generators for the ideal of truncated symmetric polynomials. We use this algebraic result to give a precise descriptions of the kernel of the homomorphism in cohomology induced by the natural map (\ CP^d)^n\to BU(n) . We also calculate the cohomology of the homotopy fiber of the natural map E S_n\times_{S_n}(\ CP^d)^n\to BU(n) .
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Full frame distilled prediction
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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) | 0.002 | 0.000 |
Machine scores (provisional)
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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