The motivic Mahowald invariant

TL;DR

This paper extends the Mahowald invariant to motivic stable homotopy theory over complex numbers, computing a motivic $C_2$-Tate construction and revealing exotic periodic phenomena in motivic homotopy groups.

Contribution

It introduces a motivic version of the Mahowald invariant, computes a motivic $C_2$-Tate construction, and relates it to $w_1$-periodic families, revealing new periodic phenomena.

Findings

Motivic Mahowald invariant of $\eta^i$ corresponds to $w_1$-periodic families.

Computed motivic $C_2$-Tate construction for various spectra.

Identified exotic periodic analogs of classical homotopy invariants.

Abstract

The classical Mahowald invariant is a method for producing nonzero classes in the stable homotopy groups of spheres from classes in lower stems. We study the Mahowald invariant in the setting of motivic stable homotopy theory over $Spec(\mathbb{C})$. We compute a motivic version of the $C_2$-Tate construction for various motivic spectra, and show that this construction produces "blueshift" in these cases. We use these computations to show that the Mahowald invariant of $\eta^i$, $i \geq 1$, is the first element in Adams filtration $i$ of the $w_1$-periodic families constructed by Andrews ~\cite{And14}. This provides an exotic periodic analog of Mahowald and Ravenel's computation ~\cite{MR93} that the classical Mahowald invariant of $2^i$, $i \geq 1$, is the first element in Adams filtration $i$ of the $v_1$-periodic families constructed by Adams ~\cite{Ada66}.