It is well known that very special -spaces and grouplike E 1 -spaces both model connective spectra. Both these models have equivariant analogues in the case when the group acting is finite. Shimakawa defined the category of equivariant -spaces and showed that special equivariant -spaces determine positive equivariant spectra. Costenoble and Waner [7] showed that grouplike equivariant E 1 -spaces determine connective equivariant spectra.We show that with suitable model category structures the category of equivariant -spaces is Quillen equivalent to the category of equivariant E 1 -spaces. We define the units of equivariant ring spectra in terms of equivariant -spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum.

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Algebraic models for equivariant rational homotopy theory were developed by Triantafillou [Tri82] and Scull [Scu02], [Scu08] 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 G = Z p under suitable conditions, the equivariant rational homotopy types with isomorphic cohomology can be reduced to the non-equivariant case.

In this article, we show that there is no cofibration category on the category of finite graphs with ×-homotopy equivalences as the class of weak equivalences. Further, we show that there is no cofibration structure possible on the category of finite graphs if the weak equivalences are a subclass of graph maps between graphs with isomorphic stiff subgraphs.