Abstract
Let p be an odd prime, M be an unstable locally finite module over the Steenrod algebra, and let \({\Phi_{S}}\) be the localization out of the Euler class of the mod-p cohomology ring of the group Z/p. We prove that the Singer evaluation map \({d : \Phi_S^{GL_{1}} \otimes M \longrightarrow M }\) is dually related to a total operation \({\chi P : M \longrightarrow \Phi_S^{GL_{1}} \otimes M}\) . We determine the exotic \({\mathcal A (p)}\) -module structure on the target which makes χ P an \({\mathcal A (p)}\) -linear map, give a new proof of the Adem relations and find some new identities involving the Bockstein and the pth reduced powers.
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Brunetti, M., Ciampella, A. & Lomonaco, L.A. A total Steenrod operation as homomorphism of Steenrod algebra-modules. Ricerche mat. 61, 1–17 (2012). https://doi.org/10.1007/s11587-011-0111-3
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DOI: https://doi.org/10.1007/s11587-011-0111-3