\( \mathbb{R} \)-motivic \(v_1\)-periodic homotopy (Q6579956)

In this paper, the authors seek to understand \(v_1\)-periodic \(\mathbb{R}\)-motivic stable homotopy groups. More precisely, they calculate the homotopy groups of a spectrum \(L\) that is a convenient motivic replacement for the classical connective image of \(j\) spectrum. The calculations use the effective slice spectral sequence, working with \(2\)-complete spectra.\N\NLet \(\mathrm{ko}\) be the very effective connective Hermitian \(K\)-theory spectrum, of which the effective slice filtration is understood [\textit{A. Ananyevskiy} et al., Math. Z. 294, No. 3--4, 1021--1034 (2020; Zbl 1453.14064)]. Using [\textit{T. Bachmann} and \textit{M. J. Hopkins}, ``\(\eta\)-periodic motivic stable homotopy theory over fields'', Preprint, \url{arXiv:2005.06778}], one has the Adams operation \(\psi^3\) (after inverting \(3\)); the local version of the spectrum \(L\) is defined as the fibre of \( (\psi^3 -1) : \mathrm{ko} [\frac{1}{3}] \rightarrow \mathrm{ko} [\frac{1}{3}] \) and the effective slice filtrations are defined at this level.\N\NPassage to \(2\)-completion gives the fibre sequence \N\[\NL \rightarrow \mathrm{ko} \stackrel{\psi^3 -1}{\rightarrow } \mathrm{ko}. \N\]\NThe above slice filtrations induce filtrations on the \(2\)-completed spectra, with respect to which the respective effective slice spectral sequences are defined. The authors trigrade these spectral sequences by \((s, f, w)\), respectively the topological dimension, the Adams-Novikov filtration, and the motivic weight (the Adams-Novikov filtration is twice the effective filtration minus the stem).\N\NThe authors first treat the analogous calculations in \(\mathbb{C}\)-motivic stable homotopy theory. They calculate the effective slice spectral sequence for \(\mathrm{ko}^{\mathbb{C}}\), then deduce the \(E_1\)-page for \(L^{\mathbb{C}}\) before analysing the effective slice spectral sequence for the \(\eta\)-periodic \(L^{\mathbb{C}}[\eta^{-1}]\). This feeds into the analysis of the effective slice spectral sequence for \(L^\mathbb{C}\); they conclude by an analysis of the hidden extensions in the \(E_\infty\)-page of the spectral sequence.\N\NThey then repeat this strategy for the \(\mathbb{R}\)-motivic case, using the \(\mathbb{C}\)-motivic information as input; this is more delicate. They first analyse the effective slice spectral sequence for \(\mathrm{ko}\) and for \(\mathrm{ko}[\eta^{-1}]\). They then pass to \(L\); the higher differentials in the effective slice spectral sequence for \(L [\eta^{-1}]\) are analysed by considering the map \(S[\eta^{-1}] \rightarrow L[\eta^{-1}]\) together with the known homotopy of \(S[\eta^{-1}]\). They conclude by treating the effective slice spectral sequence for \(L\) and hidden extensions, as before.\N\NResults are presented in tabular form and in charts, separating out the different coweights (coweight is the stem minus the motivic weight). These are essential for understanding the results.\N\NThe authors observe that it would be possible to proceed more directly, without using the effective slice spectral sequences, by using the fibre sequence defining \(L\) and the homotopy of \(\mathrm{ko}\). They argue that the effective filtration on the homotopy of \(L\) and the slice differentials provide additional information which may be useful in analysing higher structure such as Toda brackets.