Some extensions in the Adams spectral sequence and the 51-stem

Abstract

We show a few nontrivial extensions in the classical Adams spectral sequence. In particular, we compute that the 2-primary part of ${\pi }_{51}$ is $\mathbb{Z}/8\oplus \mathbb{Z}/8\oplus \mathbb{Z}/2$. This was the last unsolved 2-extension problem left by the recent works of Isaksen and the authors (\cite{Isa1}, \cite{IX}, \cite{WX1}) through the 61-stem. The proof of this result uses the $R{P}^{\infty }$ technique, which was introduced by the authors in \cite{WX1} to prove ${\pi }_{61}=0$. This paper advertises this method through examples that have simpler proofs than in \cite{WX1}.

Accepted version. arXiv admin note: text overlap with arXiv:1601.02184

References in corpus (4)

• Classical and motivic Adams charts
• The strong Kervaire invariant problem in dimension 62
• The algebraic Atiyah-Hirzebruch spectral sequence of real projective spectra
• Motivic Stable Homotopy and the Stable 51 and 52 Stems

Cited by in corpus (2)

• The special fiber of the motivic deformation of the stable homotopy category is algebraic
• The topological modular forms of $\mathbb{R}{P}^2$ and $\mathbb{R}{P}^2\wedge \mathbb{C}{P}^2$