On the $\Gamma$-cohomology of rings of numerical polynomials

Abstract

We investigate $\Gamma$-cohomology of some commutative cooperation algebras $E_{\ast }E$ associated with certain periodic cohomology theories. For KU and $E(1)$, the Adams summand at a prime $p$, and for KO we show that $\Gamma$-cohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique $E_{\infty }$ structures. As a consequence we obtain an $E_{\infty }$ structure for the connective Adams summand. For the Johnson–Wilson spectrum $E(n)$ with $n\ge 1$ we establish the existence of a unique $E_{\infty }$ structure for its $I_n$-adic completion.