Equivariant Intrinsic Formality

Algebraic models for equivariant rational homotopy theory were developed by Triantafillou and Scull for finite group actions and $S^1$ action, respectively. They showed that given a diagram of rational cohomology algebras from the orbit category of a group $G$, there is a unique minimal system of DGAs and hence a unique equivariant rational homotopy type that is weakly equivalent to it. However, there can be several equivariant rational homotopy types with the same system of cohomology algebras. Halperin, Stasheff, and others studied the problem of classifying rational homotopy types up to cohomology in the non-equivariant case. In this article, we consider this question in the equivariant case. We prove that when $\mathbb{Z}_p$ under suitable conditions, the equivariant rational homotopy types with isomorphic cohomology can be reduced to the non-equivariant case.