## Equivariant $A$-Theory

##### Doc. Math. 24, 815-855 (2019)
DOI: 10.25537/dm.2019v24.815-855

### Summary

We give a new construction of the equivariant $K$-theory of group actions [\textit{C. Barwick}, Spectral Mackey functors and equivariant algebraic $K$-theory (I)'', Adv. Math. 304, 646--727 (2017; Zbl 1348.18020) and \textit{C. Barwick} et al., Spectral Mackey functors and equivariant algebraic $K$-theory (II)'', Preprint (2015); \url{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.

### Mathematics Subject Classification

19D10, 19C99, 55N91, 55P91, 55Q91, 18D50

### Keywords/Phrases

equivariant, $A$-theory, $K$-theory, Mackey functor, transfers, $G$-spectrum, Waldhausen categories

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### Affiliation

Malkiewich, Cary
Department of Mathematics, University of Binghamton, Binghamton, New York, USA
Merling, Mona
Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania, USA