Root invariants in the Adams spectral sequence

Abstract

Abstract. Let E be a ring spectrum for which the E-Adams spectral sequence converges. We define a variant of Mahowald's root invariant called the 'filtered root invariant' which takes values in the E 1 term of the E-Adams spectral sequence. The main theorems of this paper are concerned with when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the E-Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime 2. We use the filtered root invariants to compute some low-dimensional root invariants of v 1 -periodic elements at the prime 3. We also compute the root invariants of some infinite v 1 -periodic families of elements at the prime 3.

The motivic Mahowald invariant

“…Subsequent work of Sadofsky [33] showed that for p > 3 one has β k ∈ M cl (α k ). Computations of Behrens at p = 3 [5] and later at p = 2 [6] provided further evidence that the classical Mahowald invariant of a v n -periodic class in the stable stems is v n+1 -periodic. In this paper, we will produce motivic analogs of Mahowald and Ravenel's computations of the Mahowald invariant of 2 i for i ≥ 1.…”

Some root invariants at the prime 2

“…• For p = 3 we have R(α i ) = β h i for i ≤ 6, and we have R(α i ) = β i for for i ≡ 0, 1, 5 (mod 9) (see Behrens [2]).…”

Intersection forms of spin 4-manifolds and the pin(2)-equivariant Mahowald invariant

“…Then, apply the forgetful functor to the nonequivariant world. Bruner and Greenlees [BG95] proved that this construction produces the classical Mahowald invariant 𝑀(𝛼) of 𝛼, which has been studied extensively by Mahowald, Ravenel, and Behrens [MR93,Beh06,Beh07].…”