Geodesic
Equivariant $A$-Theory
Documenta mathematica, Tome 24 (2019), pp. 815-855
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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 XhH​→BH. We then use the framework of spectral Mackey functors to produce a second equivariant refinement AG​(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.
DOI : 10.4171/dm/694
Classification : 19D10, 55N91, 55P91, 55Q91
Mots-clés : equivariant, K-theory, transfers, A-theory, Mackey functor, G-spectrum, Waldhausen categories 
@article{10_4171_dm_694,
     author = {Cary Malkiewich and Mona Merling},
     title = {Equivariant $A${-Theory}},
     journal = {Documenta mathematica},
     pages = {815--855},
     year = {2019},
     volume = {24},
     doi = {10.4171/dm/694},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/694/}
}
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Cary Malkiewich; Mona Merling. Equivariant $A$-Theory. Documenta mathematica, Tome 24 (2019), pp. 815-855. doi: 10.4171/dm/694

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