Volume 18, issue 7 (2018)

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Some extensions in the Adams spectral sequence and the $51$–stem

Guozhen Wang and Zhouli Xu

Algebraic & Geometric Topology 18 (2018) 3887–3906
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 $ℤ∕8\oplus ℤ∕8\oplus ℤ∕2$. 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.

Keywords
Adams spectral sequence, Atiyah–Hirzebruch spectral sequence
Primary: 55Q40