## Algebraic & Geometric Topology

### 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 $π 5 1$ is $ℤ ∕ 8 ⊕ ℤ ∕ 8 ⊕ ℤ ∕ 2$. This was the last unsolved $2$–extension problem left by the recent work of Isaksen and the authors through the $6 1$–stem.

The proof of this result uses the $R P ∞$ technique, which was introduced by the authors to prove $π 6 1 = 0$. This paper advertises this technique through examples that have simpler proofs than in our previous work.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 3887-3906.

Dates
Revised: 12 July 2018
Accepted: 3 August 2018
First available in Project Euclid: 18 December 2018

https://projecteuclid.org/euclid.agt/1545102057

Digital Object Identifier
doi:10.2140/agt.2018.18.3887

Mathematical Reviews number (MathSciNet)
MR3892234

Zentralblatt MATH identifier
07006380

Subjects
Primary: 55Q40: Homotopy groups of spheres

#### Citation

Wang, Guozhen; Xu, Zhouli. Some extensions in the Adams spectral sequence and the $51$–stem. Algebr. Geom. Topol. 18 (2018), no. 7, 3887--3906. doi:10.2140/agt.2018.18.3887. https://projecteuclid.org/euclid.agt/1545102057

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