Malkiewich, Cary; Merling, Mona

Equivariant \(A\)-Theory

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


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


equivariant, \(A\)-theory, \(K\)-theory, Mackey functor, transfers, \(G\)-spectrum, Waldhausen categories


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