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(). We compute a motivic version of the C2–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 1, the Mahowald invariant of ηi is the first element in Adams filtration i of the w1–periodic families constructed by Andrews (2018). This provides an exotic periodic analog of the computation of Mahowald and Ravenel (1993) that for i 1, the classical Mahowald invariant of 2i, is the first element in Adams filtration i of the v1–periodic families constructed by Adams (1966).

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