Geodesic
Equivariant $A$-Theory
Documenta mathematica, Tome 24 (2019), pp. 815-855.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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.
Classification : 19D10, 55N91, 55P91, 55Q91, 18D50
Keywords: equivariant, \(A\)-theory, \(K\)-theory, Mackey functor, transfers, \(G\)-spectrum, Waldhausen categories 
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     author = {Malkiewich, Cary and Merling, Mona},
     title = {Equivariant {\(A\)-Theory}},
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     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2019__24__a41/}
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Malkiewich, Cary; Merling, Mona. Equivariant \(A\)-Theory. Documenta mathematica, Tome 24 (2019), pp. 815-855. http://geodesic.mathdoc.fr/item/DOCMA_2019__24__a41/