Equivariant a-theory

We give a new construction of the equivariant K-theory of group actions (cf. Barwick et al.), producing an infinite loop Gspace for each Waldhausen category with G-action, for a finite group G. On the category R(X) of retractive spaces over a G-space X, this produces an equivariant lift of Waldhausen's functor A(X), and we show that the H-fixed points are the bivariant A-theory of the fibration X_hH → BH. We then use the framework of spectral Mackey functors to produce a second equivariant refinement A_G(X) whose fixed points have tom Dieck type splittings. We expect this second definition to be suitable for an equivariant generalization of the parametrized h-cobordism theorem.