Higher cohomology operations and $R$–completion

Abstract

Let $R$ be either ${\mathbb{F}}_p$ or a field of characteristic $0$. For each $R$–good topological space $Y$, we define a collection of higher cohomology operations which, together with the cohomology algebra $H^{\ast }\left(Y;R\right)$, suffice to determine $Y$ up to $R$–completion. We also provide a similar collection of higher cohomology operations which determine when two maps $f_0,f_1:Z\to Y$ between $R$–good spaces (inducing the same algebraic homomorphism $H^{\ast }\left(Y;R\right)\to H^{\ast }\left(Z;R\right)$) are $R$–equivalent.