The wedge family of the cohomology of the \(\mathbb{C}\)-motivic Steenrod algebra
Periodicity in the motivic stable stems is more mysterious than in the classical stable stems. In both the motivic and classical settings, the \(E_2\)-term of the Adams spectral sequence provides an algebraic approximation to the stable stems. The main contribution of this article is the description of a \(\mathbb{C}\)-motivic analog of the Mahowald-Tangora wedge [\textit{M. Mahowald} and \textit{M. Tangora}, Trans. Am. Math. Soc. 132, 263--274 (1968; Zbl 0177.51401)], an infinite wedge-shaped pattern of elements in the \(E_2\)-term of the classical Adams \(E_2\)-term. The \(\mathbb{C}\)-motivic wedge is described in Theorem 3.8. The author produces the \(\mathbb{C}\)-motivic wedge by comparison to three related algebraic objects: the classical Adams \(E_2\)-term (i.e. the Mahowald-Tangora wedge), the Adams \(E_2\)-term for \(\mathbb{C}\)-motivic modular forms studied by \textit{D. C. Isaksen} [Homology Homotopy Appl. 11, No. 2, 251--274 (2009; Zbl 1193.55009)], and the \(h_1\)-localization of the \(\mathbb{C}\)-motivic Adams \(E_2\)-term studied by \textit{B. J. Guillou} and \textit{D. C. Isaksen} [J. Pure Appl. Algebra 219, No. 10, 4728--4756 (2015; Zbl 1327.14107)]. This gives a streamlined approach to proving the nontriviality of the \(\mathbb{C}\)-motivic wedge described in Theorem 3.8. The author points out that certain other, exceptional classes in the \(\mathbb{C}\)-motivic Adams \(E_2\)-term ought to appear in the wedge, but these classes cannot be detected by comparison to other existing computations. In Section 4, these exceptional classes are explored and two conjectures about their behavior (Conjectures 4.11 and 4.16) are formulated. These results are an important step towards understanding analogous phenomena over other base fields. The computations are clear and well-written. Figure 1 is especially illuminating.
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