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Computations of classical Mahowald invariants at prime 2

Kaixu Zhang, Dongming Zhang

TL;DR

The paper addresses the computation of the 2-primary Mahowald invariant $M(\alpha)$ for elements in the stable homotopy groups of spheres by connecting homotopy invariants with algebraic and $E$-filtered invariants via Behrens' method. It combines the mod $2$ Adams spectral sequence, the AHSS on stunted projective spectra $P^{\infty}_k$, James periodicity, and algebraic data from Bruner to compute invariants up to the 26-stem, with five elements left unresolved. The main contributions are (i) the explicit computation strategy linking $M_{alg}$ and $M_E^{[k]}$ to the actual homotopy invariant $M$, and (ii) a near-complete table of invariants up to the 26-stem with clearly identified exceptions. This work advances understanding of stable homotopy periodic phenomena and provides concrete data to guide future investigations of periodic families in the 2-primary context.

Abstract

We review the definition of Mahowald invariants and discuss the computational method of Mark Brehens introduced in [arXiv:math/0507182]. Then we examine the relationship between the algebraic Mahowald invariants and the \( E \)-filtered Mahowald invariants, and compute the Mahowald invariants for most elements up to the 26-stem.

Computations of classical Mahowald invariants at prime 2

TL;DR

The paper addresses the computation of the 2-primary Mahowald invariant for elements in the stable homotopy groups of spheres by connecting homotopy invariants with algebraic and -filtered invariants via Behrens' method. It combines the mod Adams spectral sequence, the AHSS on stunted projective spectra , James periodicity, and algebraic data from Bruner to compute invariants up to the 26-stem, with five elements left unresolved. The main contributions are (i) the explicit computation strategy linking and to the actual homotopy invariant , and (ii) a near-complete table of invariants up to the 26-stem with clearly identified exceptions. This work advances understanding of stable homotopy periodic phenomena and provides concrete data to guide future investigations of periodic families in the 2-primary context.

Abstract

We review the definition of Mahowald invariants and discuss the computational method of Mark Brehens introduced in [arXiv:math/0507182]. Then we examine the relationship between the algebraic Mahowald invariants and the -filtered Mahowald invariants, and compute the Mahowald invariants for most elements up to the 26-stem.

Paper Structure

This paper contains 5 sections, 17 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Atiyah-Hirzebruch spectral sequence (AHSS) and $P^\infty_k$
  4. Algebraic Mahowald Invariant and Filtered Mahowald Invariant
  5. Computations of homotopy Mahowald invariants

Key Result

Theorem 1.2

The Mahowald invariants are determined for all elements in the stable homotopy groups of spheres up to the 26-stem, with the exception of the five elements $\nu_4,\bar{\sigma},\{P^h_2\},4\bar{\kappa}$ and $4\nu\bar{\kappa}$.

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 2.1: James periodicity, MR177411
  • Proposition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Theorem 2.8
  • ...and 17 more