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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology


Authors: David J. Pengelley and Frank Williams
Journal: Trans. Amer. Math. Soc. 352 (2000), 1453-1492
MSC (1991): Primary 55S99; Secondary 16W30, 16W50, 55S10, 55S12, 57T05
Published electronically: September 9, 1999
MathSciNet review: 1653375
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Abstract: The mod 2 Steenrod algebra $\mathcal{A}$ and Dyer-Lashof algebra $\mathcal{R} $ have both striking similarities and differences arising from their common origins in ``lower-indexed'' algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra $\mathcal{K}$, whose module actions are equivalent to, but quite different from, those of $ \mathcal{A}$ and $\mathcal{R}$. The exact relationships emerge as ``sheared algebra bijections'', which also illuminate the role of the cohomology of $ \mathcal{K}$. As a bialgebra, $\mathcal{K}^{*}$ has a particularly attractive and potentially useful structure, providing a bridge between those of $\mathcal{A^{*}}$ and $\mathcal{R^{*}}$, and suggesting possible applications to the Miller spectral sequence and the $\mathcal{A}$ structure of Dickson algebras.


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Additional Information

David J. Pengelley
Affiliation: Department of Mathematics, New Mexico State University, Las Cruces, New Mexico 88003
Email: davidp@nmsu.edu

Frank Williams
Affiliation: Department of Mathematics, New Mexico State University, Las Cruces, New Mexico 88003
Email: frank@nmsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02468-X
PII: S 0002-9947(99)02468-X
Keywords: Steenrod algebra, Dyer-Lashof algebra, bialgebras, sheared algebra map, Kudo-Araki-May algebra, Nishida relations
Received by editor(s): December 2, 1997
Published electronically: September 9, 1999
Article copyright: © Copyright 2000 American Mathematical Society