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Equivariant A-theory

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Malkiewich,  Cary
Max Planck Institute for Mathematics, Max Planck Society;

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Merling,  Mona
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Malkiewich, C., & Merling, M. (2019). Equivariant A-theory. Documenta Mathematica, 24, 815-855. doi:10.25537/dm.2019v24.815-855.


Cite as: https://hdl.handle.net/21.11116/0000-0004-99FE-2
Abstract
We give a new construction of the equivariant $K$-theory of group actions [\textit{C. Barwick}, "Spectral Mackey functors and equivariant algebraic -theory (I)", Adv. Math. 304, 646-727 (2017; Zbl 1348.18020) and \textit{C. Barwick} et al., "Spectral Mackey functors and equivariant algebraic -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.