Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The $\mathbb{C}$-motivic Adams Spectral Sequence for $s\leq5$

Jordan Benson

Abstract

We determine the $τ^n$-torsion in the first 5-lines of the $E_2$ page of the $\mathbb{C}$-motivic Adams spectral sequence using the techniques of Burklund-Xu. In particular, every element in this range is either $τ^1$-torsion or $τ$-free. We also show that $τ^n$-torsion elements can appear only in Adams filtration at least $2n+2$ and give further evidence of a possible $3n$ bound.

The $\mathbb{C}$-motivic Adams Spectral Sequence for $s\leq5$

Abstract

We determine the -torsion in the first 5-lines of the page of the -motivic Adams spectral sequence using the techniques of Burklund-Xu. In particular, every element in this range is either -torsion or -free. We also show that -torsion elements can appear only in Adams filtration at least and give further evidence of a possible bound.

Paper Structure

This paper contains 15 sections, 23 theorems, 193 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Cartan-Eilenberg Spectral Sequence
  3. The Algebraic Atiyah-Hirzebruch Spectral Sequence
  4. Proof of Main Results
  5. $\tau^n$-Torsion on the 6-Line
  6. Ordinary $d_1$ Cycles
  7. Overlap $d_1$ Cycles
  8. Finding $d_2$ Cycles
  9. Narrowing Down Surviving Families
  10. The $q_0^4\cdot h_{j+3}$ Family
  11. Low-Dimensional Calculations
  12. $h_0$-Divisibility
  13. Periodicity
  14. Unrealized Periodicity
  15. Exotic Extensions

Key Result

Proposition 1

There is a ring isomorphism

Figures (3)

  • Figure 1: Stems 0 Through 13
  • Figure 2: Stems 14 Through 21
  • Figure 3: Stems 20 Through 29

Theorems & Definitions (44)

  • Proposition : DI
  • Theorem 1
  • Remark 2
  • Remark 3
  • Corollary 4
  • proof
  • Theorem 5
  • Conjecture 6
  • Theorem 7
  • Proposition 8: BX
  • ...and 34 more