The motivic Mahowald invariant

Abstract

The 2-primary homotopy β-family, defined as the collection of Mahowald invariants of Mahowald invariants of 2 i , i ≥ 1, is an infinite collection of periodic elements in the stable homotopy groups of spheres. In this paper, we calculate tmf -based approximations to this family. Our calculations combine an analysis of the Atiyah-Hirzebruch spectral sequence for the Tate construction of tmf with trivial C 2 -action and Behrens' filtered Mahowald invariant machinery.

Real motivic and C2‐equivariant Mahowald invariants

“…The other map is 2 + ρη ∈ π k 0,0 (S 0,0 ), where 2 is the degree 2-self map of the sphere spectrum and ρ is a certain element in π k −1,−1 (S 0,0 ) arising from the element [−1] ∈ K M 1 (k) in Milnor K-theory. ‡ In [48], we defined the C-motivic Mahowald invariant M C (−) and computed M C (2 i ) and M C (η i ) for all i 1. We briefly recall its definition now.…”

“…The classes v 4i 1 τ η 2 and v 4i 1 τ η 3 , defined in Section 5, base-change to the classes with the same name [48] in π C * * (S 0,0 ).…”

“…The proof of the following is similar to the proofs of [48,. In particular, one can compute the Atiyah-Hirzebruch spectral sequence by replacing Betti realization Re B in those proofs by the base-change functor i * :…”

“…Remark 1.3. In the classical and C-motivic settings [41,48], the upper bound on M (2 4i+j ) is obtained using approximations to the Mahowald invariant based on connective real topological K-theory or its C-motivic analogs. A similar approach can be used to bound the stem of the R-motivic Mahowald invariant, but the requisite calculations are much more laborious than the comparison methods we use here.…”

“…• For Theorems A and B, the upper bounds are obtained using part (5) of Theorem E applied to the C-motivic calculations of [48]. The lower bounds are obtained using Atiyah-Hirzebruch calculations.…”