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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Steenrod algebra module maps from $ H\sp *(B({\bf Z}/p)\sp n)$ to $ H\sp *(B({\bf Z}/p)\sp s)$


Authors: John C. Harris, Thomas J. Hunter and R. James Shank
Journal: Proc. Amer. Math. Soc. 112 (1991), 245-257
MSC: Primary 55S10; Secondary 55R35
MathSciNet review: 1047010
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Abstract: Let $ {H^{ \otimes n}}$ denote the $ \operatorname{mod}-p$ cohomology of the classifying space $ B{({\mathbf{Z}}/p)^n}$ as a module over the Steenrod algebra $ \mathcal{A}$. Adams, Gunawardena, and Miller have shown that the $ n \times s$ matrices with entries in $ {\mathbf{Z}}/p$ give a basis for the space of maps $ {\text{Ho}}{{\text{m}}_\mathcal{A}}({H^{ \otimes n}},{H^{ \otimes s}})$. For $ n$ and $ s$ relatively prime, we give a new basis for this space of maps using recent results of Campbell and Selick. The main advantage of this new basis is its compatibility with Campbell and Selick's direct sum decomposition of $ {H^{ \otimes n}}$ into $ ({p^n} - 1)$ $ \mathcal{A}$-modules.

Our applications are at the prime two. We describe the unique map from $ \bar H$ to $ D(n)$, the algebra of Dickson invariants in $ {H^{ \otimes n}}$, and we give the dimensions of the space of maps between the indecomposable summands of $ {H^{ \otimes 3}}$.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1047010-6
Article copyright: © Copyright 1991 American Mathematical Society