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New infinite families in the stable homotopy groups of spheres

Prasit Bhattacharya, Irina Bobkova, J. D. Quigley

Abstract

We identify seven new $192$-periodic infinite families of elements in the $2$-primary stable homotopy groups of spheres. Although their Hurewicz image is trivial for topological modular forms, they remain nontrivial after $\mathrm{T}(2)$- as well as $\mathrm{K}(2)$-localization. We also obtain new information about $2$-torsion and $2$-divisibility of some of the previously known $192$-periodic infinite families in the stable stems.

New infinite families in the stable homotopy groups of spheres

Abstract

We identify seven new -periodic infinite families of elements in the -primary stable homotopy groups of spheres. Although their Hurewicz image is trivial for topological modular forms, they remain nontrivial after - as well as -localization. We also obtain new information about -torsion and -divisibility of some of the previously known -periodic infinite families in the stable stems.
Paper Structure (11 sections, 19 theorems, 31 equations, 1 figure, 1 table)

This paper contains 11 sections, 19 theorems, 31 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. Some ${\sf tmf}$-homology calculations
  4. $v_1$-periodic families
  5. From ${\sf tmf}_*\mathrm{Y}$ to ${\sf tmf}_*\mathrm{M}$
  6. From ${\sf tmf}_*\mathrm{M}$ to ${\sf tmf}_*$
  7. Summary Table
  8. New infinite families
  9. Infinite families in $2$-local stable stems
  10. Infinite families in $\mathrm{T}(2)$-local stable stems
  11. Infinite families in $\mathrm{K}(2)$-local stable stems

Key Result

Theorem 1

For each $m \in \{23, 47, 71, 74, 95, 119, 167 \}$ and $k \in \mathbb{N}$, there exists an element of order $2$ in dimension $m+192k$ of the stable stems whose image is trivial under the ${\sf tmf}$-Hurewicz homomorphism.

Figures (1)

  • Figure 1: The lightning flash pattern $\mathrm{L}$

Theorems & Definitions (49)

  • Theorem 1
  • Remark 1.1
  • Theorem 2: \ref{['MT:T2']} and \ref{['MT:K2']}
  • Corollary
  • Conjecture 1.2
  • Theorem 3
  • Remark 1.3
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 39 more