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 work of Isaksen and the authors through the $61$–stem.

The proof of this result uses the $R{P}^{\infty }$ technique, which was introduced by the authors to prove ${\pi }_{61}=0$. This paper advertises this technique through examples that have simpler proofs than in our previous work.