\(\mathrm{tmf}\)-based Mahowald invariants
type: Item
instance of: scholarly article
title: \(\mathrm{tmf}\)-based Mahowald invariants
author: J. D. Quigley
published in: Algebraic \& Geometric Topology
publication date: 2022-11-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1911.07975
review text: The Greek letter elements were defined in order to organize the stable homotopy groups of spheres into periodic families of elements of longer an longer wavelength, corrsponding to higher and higher chromatic heights. One way to construct Greek letter elements at the prime \(2\) is to take iterated Mahowald invariants of the powers of \(2\) in degree zero in the stable homotopy groups of spheres. This has the advantage that these homotopy Greek letter elements are always nontrivial. For example, the \(2\)-primary homotopy beta family \(\beta_{i}^{h}\) is the Mahowald invariant of the Mahowald invariant of \(2^{i}\). Such Greek letter elements are well understood at height one, but at higher heights they are only known explicitly in a range at the prime \(2\). One way to approximate Mahowald invariants is to use generalized homology theories such as real K-theory and topological modular forms. This reduces a largely intractable problem, depending on computations of the entire stable homotopy groups of spheres, so a significantly more tractable problem depending on the coefficients of your generalied homology theory. For example, the paper under review presents clean and detailed computations of iterated Mahowald invariants using topological modular forms. This gives an approximation to the iterated Mahowald invariants for the sphere spectrum and in fact recovers iterated Mahowald invaraints for the sphere in some low dimensions. The paper is elegant and well written and it includes numerous figures that cleanly illustrate the computations.
reviewed by: Gabriel Angelini-Knoll
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55Q51
zbMATH DE Number: 7611768
zbMATH Keywords: Mahowald invariants
zbMATH Keywords: topological modular forms
zbMATH Keywords: Atiyah-Hirzebruch spectral sequence
zbMATH Keywords: Greek letter family elements
zbMATH Keywords: stable homotopy groups of spheres
MaRDI profile type: Publication
OpenAlex ID: W4306154943
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DOI: 10.2140/AGT.2022.22.1789
description: scientific article; zbMATH DE number 7611768