On realizations of the subalgebra a(1) of the r-motivic steenrod algebra

P. Bhattacharya, B. Guillou, A. Li

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2 Scopus citations

Abstract

In this paper, we show that the finite subalgebra A(1), generated by Sq1 and Sq2, of the R-motivic Steenrod algebra A can be given 128 different A-module structures. We also show that all of these A-modules can be realized as the cohomology of a 2-local finite R-motivic spectrum. The realization results are obtained using an ℝ-motivic analogue of the Toda realization theorem. We notice that each realization of A (1) can be expressed as a cofiber of an ℝ-motivic v1-self-map. The C2-equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the RO(C2)-graded Steenrod operations on a C2-equivariant space and the classical Steenrod operations on both its underlying space and its fixed-points. This technique is then used to identify the geometric fixed-point spectra of the C2-equivariant realizations of AC2 (1). We find another application of the ℝ-motivic Toda realization theorem: We produce an ℝ-motivic, and consequently a C2-equivariant, analogue of the Bhattacharya-Egger spectrum Ƶ, which could be of independent interest.

Original languageEnglish
Pages (from-to)700-732
Number of pages33
JournalTransactions of the American Mathematical Society Series B
Volume9
DOIs
StatePublished - 2022

Bibliographical note

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© 2022 by the author(s) under Creative Commons Attribution 3.0 License.

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

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