#### Volume 19, issue 5 (2019)

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The motivic Mahowald invariant

### J D Quigley

Algebraic & Geometric Topology 19 (2019) 2485–2534
##### 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 $Spec\left(ℂ\right)$. 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 for $i\ge 1$, the Mahowald invariant of ${\eta }^{i}$ is the first element in Adams filtration $i$ of the ${w}_{1}$–periodic families constructed by Andrews (2018). This provides an exotic periodic analog of the computation of Mahowald and Ravenel (1993) that for $i\ge 1$, the classical Mahowald invariant of ${2}^{i}$, is the first element in Adams filtration $i$ of the ${v}_{1}$–periodic families constructed by Adams (1966).

##### Keywords
Mahowald invariant, root invariant, motivic $v_1$–periodicity, motivic $w_1$–periodicity, motivic Tate construction
Primary: 55P42
##### Publication
Received: 12 February 2018
Revised: 8 November 2018
Accepted: 20 November 2018
Published: 20 October 2019
##### Authors
 J D Quigley Department of Mathematics University of Notre Dame Notre Dame, IN United States Department of Mathematics Cornell University Ithaca, NY United States