The motivic Mahowald invariant

Quigley, J D

doi:10.2140/agt.2019.19.2485

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The motivic Mahowald invariant

J D Quigley

Algebraic & Geometric Topology 19 (2019) 2485–2534

DOI: 10.2140/agt.2019.19.2485

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 $Spec (ℂ)$ . 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 for $i \geq 1$ , the Mahowald invariant of $η^{i}$ 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 $i \geq 1$ , 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).

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Keywords

Mahowald invariant, root invariant, motivic $v_1$–periodicity, motivic $w_1$–periodicity, motivic Tate construction

Mathematical Subject Classification 2010

Primary: 55P42

References

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

Volume 19, issue 5 (2019)