Higher cohomology operations and R-completion

TL;DR

This paper introduces higher cohomology operations that, combined with cohomology algebra, fully determine the R-completion of R-good spaces and classify maps between them up to R-equivalence.

Contribution

It defines new higher cohomology operations that characterize R-completion and R-equivalence of maps between R-good spaces.

Findings

Higher cohomology operations determine R-completion of spaces.

Operations classify when maps are R-equivalent.

Provides tools for understanding R-local homotopy types.

Abstract

Let $R=\mathbb{F}_p$ 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 $H^*(Y;R)$ 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 $H^*(Y;R)\to H^*(Z;R)$) are $R$-equivalent.