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The Adams differentials on the $e$-family

Runji Li, Yuxuan Li

Abstract

The New Doomsday Conjecture (Minami, Amer. J. Math., 1995) states that, for any nonzero $\mathrm{Sq}^0$-family, only finitely many terms in this family survive to the $E_\infty$-page. On the Adams $1$ and $2$-line, the conjecture, which corresponds to the Hopf invariant problem and the Kervaire invariant problem, were solved by Adams (Ann. of Math., 1960) and Hill-Hopkins-Ravenel (arXiv:0908.3724), respectively. On the Adams $3$-line, Burklund and Xu (arXiv:2302.11869) established a family of nontrivial differentials on the $h_j^3$ family, and in particular developed the Burklund-Xu Spectral Sequence, to study the non-triviality of its target on the Adams $E_2$-page. In this paper, we use the Burklund-Xu Spectral Sequence to establish the non-triviality of a product on the Adams $6$-line. Combining this with Bruner's formula by Bruner et al. (LNM 1176, 1986), we prove the New Doomsday Conjecture for the $e$-family on the Adams $4$-line.

The Adams differentials on the $e$-family

Abstract

The New Doomsday Conjecture (Minami, Amer. J. Math., 1995) states that, for any nonzero -family, only finitely many terms in this family survive to the -page. On the Adams and -line, the conjecture, which corresponds to the Hopf invariant problem and the Kervaire invariant problem, were solved by Adams (Ann. of Math., 1960) and Hill-Hopkins-Ravenel (arXiv:0908.3724), respectively. On the Adams -line, Burklund and Xu (arXiv:2302.11869) established a family of nontrivial differentials on the family, and in particular developed the Burklund-Xu Spectral Sequence, to study the non-triviality of its target on the Adams -page. In this paper, we use the Burklund-Xu Spectral Sequence to establish the non-triviality of a product on the Adams -line. Combining this with Bruner's formula by Bruner et al. (LNM 1176, 1986), we prove the New Doomsday Conjecture for the -family on the Adams -line.
Paper Structure (3 sections, 13 theorems, 34 equations, 4 figures)

This paper contains 3 sections, 13 theorems, 34 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Burklund-Xu Spectral Sequence
  3. The Cartan-Eilenberg Spectral Sequence

Key Result

Theorem 1.3

$d_2(h_j) = h_0 h_{j-1}^2$, for $j\ge 4$.

Figures (4)

  • Figure 1: The BXSS in degree $(s,k,t) = (2,3,2^{n+1}\cdot 21+1)$ for Lemma \ref{['CESS filtration 3 die']}, where $n\geq 4$. The vertical axis is the BX filtration and the horizontal axis is the $s$-degree. Red arrows indicate $d_1$-differentials, and blue arrows indicate $d_2$-differentials.
  • Figure 2: The BXSS in degree $(s,k,t) = (2,3,2^{3+1}\cdot 21+1)$ for Lemma \ref{['CESS filtration 3 die']}. The vertical axis is the BX-filtration and the horizontal axis is the $s$-degree. Red arrows indicate $d_1$-differentials, and blue arrows indicate $d_2$-differentials.
  • Figure :
  • Figure :

Theorems & Definitions (24)

  • Conjecture 1.1: New Doomsday Conjecture, Minami
  • Conjecture 1.2: Uniform Doomsday Conjecture, IWXsurveyICMBXhj3
  • Theorem 1.3: Adams, Adams
  • Theorem 1.4: Hill-Hopkins-Ravenel, HHR
  • Theorem 1.5: Bruner's formula, BrunerBMMS
  • Theorem 1.6: Isaksen-Wang-Xu IWX0to90 for $j=5$, Burklund-Xu BXhj3 for $j\ge 6$
  • Remark 1.7
  • Theorem 1.8: Main theorem
  • Corollary 1.9
  • Remark 1.10
  • ...and 14 more