Equivariant $A$-Theory

Abstract

We give a new construction of the equivariant $K$-theory of group actions [C. Barwick, "Spectral Mackey functors and equivariant algebraic $K$-theory (I)", Adv. Math. 304, 646--727 (2017; Zbl 1348.18020) and C. Barwick et al., "Spectral Mackey functors and equivariant algebraic $K$-theory (II)", Preprint (2015); arXiv:1505.03098], producing an infinite loop $G$-space 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 $X_{hH}\to BH$. 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.