Nishida Relations in Bordism and Homology (June 1995. To Appear, C.R.Math.Rep.Acad.Sci.Canada))
Bibliographic record
Abstract
This is the second of a series of Compte Rendus. In the rst [1] we have presented a theory of Dyer-Lashof operations in unoriented bordism. Here we shall discuss the (Nishida) relations between Dyer-Lashof and Landweber-Novikov operations. They are used to represent the algebra N of covering manifolds in terms of their homology characteristic numbers. The proofs are based on the properties of the covering space operations and the notions of D-ring and Q-ring introduced in [1]. 1. The Nishida relations in homology. In homology mod 2 the Nishida relations are commutation relations between Dyer-Lashof and Steenrod operations (see [4] for instance). We shall express the Nishida relations as a commutative square which involves the Milnor coaction and the Q-structure on the homology of any E1-space discussed in [1]. Recall that the Milnor Hopf algebra A is the dual of the Steenrod algebra; see [6] for instance. As a graded algebra it is A = Z2[0; 1; 2; : ::] = −10 Z2[0; 1; 2; : ::] with grade(i) = 2 i − 1; the diagonal : A ! A ⊗ A is the unique ring homomorphism such that () = ( ⊗ 1) (1 ⊗ ) where (x) = P ix2i. We are diverging from the usual convention that puts 0 = 1. The homology of any space has a natural (left) coaction : H(X)! A ⊗H(X) = H(X)[0; 1; 2; : ::] which restricts to the usual (left) coaction when we put 0 = 1 and to the \\grading" coaction (xn) = −n 0 xn for x 2 Hn(X) when we put 0 = 1 = 2 = . For example, for X = RP1 we have (b)(x) = b((−1)(x)) where b(x) = P i bix i and b0; b1; : : : is the canonical basis of HRP1 and (−1)(x) is the composition inverse of the power series (x). Proposition. If R is a Q-ring, then there is a Q-structure on A ⊗ R determined by Qt()(x(x + t)) = (x)(x+ t). In particular, there is a unique Q-structure on A such that Qt()(x(x+ t)) = (x)(x+ t). Remark: Since the denition forces Qt(0) = 0 X i
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.006 | 0.005 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.067 | 0.014 |
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".