𝐶₂-equivariant and ℝ-motivic stable stems II

Abstract

We show that the stable homotopy groups of the C 2 C_2 -equivariant sphere spectrum and the R \mathbb {R} -motivic sphere spectrum are isomorphic in a range. This result supersedes previous work of Dugger and the third author.

Real motivic and C2‐equivariant Mahowald invariants

“…We first demonstrate Lemma 3.5 by computing the C 2 -equivariant Mahowald invariants of the classes η and ν. We will need the following result of Belmont-Guillou-Isaksen [11], which supersedes previous work of Dugger-Isaksen [18]:…”

“…We first demonstrate Lemma 3.5 by computing the ‐equivariant Mahowald invariants of the classes and . We will need the following result of Belmont–Guillou–Isaksen [11], which supersedes previous work of Dugger–Isaksen [18]: Theorem The map induced by equivariant Betti realization is an isomorphism if and , and it is injective if .…”

“…The ‐Bockstein spectral sequence computing the cohomology of was first considered by Dugger–Isaksen in [17], where they applied it to compute the first 4 Milnor–Witt stems of . More recently, Belmont and Isaksen [12] have computed the first 11 Milnor–Witt stems. In both cases, many ‐Bockstein differentials are deduced using the following theorem which we will need in the sequel.…”

“…Recall that the coweight or Milnor–Witt stem of an element is defined by . Dugger and Isaksen [17] calculated up to coweight 4 and Belmont–Isaksen [12] calculated up to coweight 11. Theorem A provides a new construction of infinite ‐periodic families in and Theorem B provides the first construction of infinite ‐periodic families in .…”

“…On the other hand, for h c 1 y to have the same as x, we must have c = s − a. Substituting the values of s and w above, we have c = 20k − − 4/7w = 20k − − 4/7(12k + δ) = 20k − − 48/7k − 4/7δ 13k − − δ 11kfor all k 1, where ( , δ) ∈ {(3,2),(6,4),(9,6),(18,11)}. In particular, the Adams filtration of h c 1 y is at least 11k > 4k + 4 f for all k 1.…”