#### Volume 19, issue 2 (2019)

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On $\mathrm{BP}\langle 2\rangle$–cooperations

### Dominic Leon Culver

Algebraic & Geometric Topology 19 (2019) 807–862
##### Abstract

We develop techniques to compute the cooperations algebra for the second truncated Brown–Peterson spectrum $BP〈2〉$. We prove that the cooperations algebra $BP{〈2〉}_{\ast }BP〈2〉$ decomposes as a direct sum of an ${\mathbb{F}}_{2}$–vector space concentrated in Adams filtration $0$ and an ${\mathbb{F}}_{2}\left[{v}_{0},{v}_{1},{v}_{2}\right]$–module which is concentrated in even degrees and is ${v}_{2}$–torsion-free. We also develop a recursive method which produces a basis for the ${v}_{2}$–torsion-free component.

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