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Register a new userC_2-equivariant and R-motivic stable stems, II
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We show that the $C_2$-equivariant and $mathbb{R}$-motivic stable homotopy groups are isomorphic in a range. This result supersedes previous work of Dugger and the third author.
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We compute some R-motivic stable homotopy groups. For $s - w leq 11$, we describe the motivic stable homotopy groups $pi_{s,w}$ of a completion of the R-motivic sphere spectrum. We apply the $rho$-Bockstein spectral sequence to obtain R-motivic Ext groups from the C-motivic Ext groups, which are well-understood in a large range. These Ext groups are the input to the R-motivic Adams spectral sequence. We fully analyze the Adams differentials in a range, and we also analyze hidden extensions by $rho$, 2, and $eta$. As a consequence of our computations, we recover Mahowald invariants of many low-dimensional classical stable homotopy elements.
We give a method for computing the C_2-equivariant homotopy groups of the Betti realization of a p-complete cellular motivic spectrum over R in terms of its motivic homotopy groups. More generally, we show that Betti realization presents the C_2-equivariant p-complete stable homotopy category as a localization of the p-complete cellular real motivic stable homotopy category.
We generalize the Mahowald invariant to the $mathbb{R}$-motivic and $C_2$-equivariant settings. For all $i>0$ with $i equiv 2,3 mod 4$, we show that the $mathbb{R}$-motivic Mahowald invariant of $(2+rho eta)^i in pi_{0,0}^{mathbb{R}}(S^{0,0})$ contains a lift of a certain element in Adams classical $v_1$-periodic families, and for all $i > 0$, we show that the $mathbb{R}$-motivic Mahowald invariant of $eta^i in pi_{i,i}^{mathbb{R}}(S^{0,0})$ contains a lift of a certain element in Andrews $mathbb{C}$-motivic $w_1$-periodic families. We prove analogous results about the $C_2$-equivariant Mahowald invariants of $(2+rho eta)^i in pi_{0,0}^{C_2}(S^{0,0})$ and $eta^i in pi_{i,i}^{C_2}(S^{0,0})$ by leveraging connections between the classical, motivic, and equivariant stable homotopy categories. The infinite families we construct are some of the first periodic families of their kind studied in the $mathbb{R}$-motivic and $C_2$-equivariant settings.
We compute the stable homotopy groups up to dimension 90, except for some carefully enumerated uncertainties.
We construct a $C_2$-equivariant spectral sequence for RO$(C_2)$-graded homotopy groups. The construction is by using the motivic effective slice filtration and the $C_2$-equivariant Betti realization. We apply the spectral sequence to compute the RO$(C_2)$-graded homotopy groups of the completed $C_2$-equivariant connective real $K$-theory spectrum. The computation reproves the $C_2$-equivariant Adams spectral sequence results by Guillou, Hill, Isaksen and Ravenel.
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