Algebraic models for equivariant rational homotopy theory were
developed by Triantafillou and Scull for finite group actions and
action, respectively. They showed that given a diagram of
rational cohomology algebras from the orbit category of a group
,
there is a unique minimal system of DGAs representing a unique
-rational
homotopy type that is weakly equivalent to it. However, there can be several
equivariant rational homotopy types with the same diagram of cohomology
algebras. Halperin, Stasheff, and others studied the problem of classifying
rational homotopy types up to cohomology in the nonequivariant case. In
this article, we consider this question in the equivariant case. For the case
, for
prime ,
under suitable conditions, we are able to determine the equivariant rational
homotopy types with isomorphic diagram of cohomology algebras in terms of
nonequivariant data. We give explicit examples to demonstrate how these theorems
can be applied to classify equivariant rational homotopy types with isomorphic
cohomology.
Keywords
systems of DGAs, minimal systems, equivariantly formal,
intrinsically formal, unstable equivariant rational
homotopy theory.
