Diagrams up to cohomology

Abstract

We compute (under suitable assumptions) how many ways there are to take a diagram in the homotopy category of spaces and perturb it to get another diagram which looks the same up to cohomology. Sometimes there are no perturbations. This can shed light on the question of whether the pcompletion of the classifying space of a particular connected compact Lie group is determined up to homotopy by cohomological data.