Generalized Mahowald Invariants

Doctoral Dissertation

Abstract

The classical Mahowald invariant is a method for producing nonzero classes in the stable homotopy groups of spheres from classes in lower stems. In this thesis, we study the Mahowald invariant in the settings of motivic stable homotopy theory over Spec(C) and Spec(R), as well as C2-equivariant stable homotopy theory. In the C2-equivariant setting, we prove an analog of Lin’s Theorem by adapting the motivic Singer construction developed by Gregersen to the C2-equivariant setting. We compute a motivic version of the Tate construction for various motivic spectra, and show that this construction produces “blueshift” in these cases. In the complex motivic case, we use these computations to show that the Mahowald invariant of η^i , i ≥ 1, is the first element in Adams filtration i of the w1-periodic families constructed by Andrews. This provides an exotic periodic analog of Mahowald and Ravenel’s computation that the classical Mahowald invariant of 2^i , i ≥ 1, is the first element in Adams filtration i of the v1-periodic families constructed by Adams. We obtain similar results in both the real motivic and C2-equivariant settings. Finally, we also study the behavior of the Mahowald invariant under various functors between the motivic, equivariant, and classical stable homotopy categories.

Attributes

Attribute NameValues
Author J.D. Quigley
Contributor Laurence Taylor, Committee Member
Contributor Mark Behrens, Research Director
Contributor Chris Schommer-Pries, Committee Member
Contributor Stephan Stolz, Committee Member
Degree Level Doctoral Dissertation
Degree Discipline Mathematics
Degree Name Doctor of Philosophy
Banner Code
  • PHD-MATH

Defense Date
  • 2019-03-26

Submission Date 2019-04-06
Record Visibility and Access Public
Content License
  • All rights reserved

Departments and Units
Catalog Record

Files

Please Note: You may encounter a delay before a download begins. Large or infrequently accessed files can take several minutes to retrieve from our archival storage system.