Equivariant AA-Theory

  • Cary Malkiewich

    Department of Mathematics, University of Binghamton, Binghamton, New York, USA
  • Mona Merling

    Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania, USA
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We give a new construction of the equivariant KK-theory of group actions [C. Barwick, "Spectral Mackey functors and equivariant algebraic KK-theory (I)", Adv. Math. 304, 646--727 (2017; Zbl 1348.18020) and C. Barwick et al., "Spectral Mackey functors and equivariant algebraic KK-theory (II)", Preprint (2015); arXiv:1505.03098], producing an infinite loop GG-space for each Waldhausen category with GG-action, for a finite group GG. On the category R(X)R(X) of retractive spaces over a GG-space XX, this produces an equivariant lift of Waldhausen's functor A(X)A(X), and we show that the HH-fixed points are the bivariant AA-theory of the fibration XhHBHX_{hH}\to BH. We then use the framework of spectral Mackey functors to produce a second equivariant refinement AG(X)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 hh-cobordism theorem.

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Cary Malkiewich, Mona Merling, Equivariant AA-Theory. Doc. Math. 24 (2019), pp. 815–855

DOI 10.4171/DM/694