Equivariant -Theory

  • Cary Malkiewich

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

    Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania, USA
Equivariant $A$-Theory cover
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We give a new construction of the equivariant -theory of group actions [C. Barwick, “Spectral Mackey functors and equivariant algebraic -theory (I)”, Adv. Math. 304, 646–727 (2017; Zbl 1348.18020) and C. Barwick et al., “Spectral Mackey functors and equivariant algebraic -theory (II)”, Preprint (2015); arXiv:1505.03098], producing an infinite loop -space for each Waldhausen category with -action, for a finite group . On the category of retractive spaces over a -space , this produces an equivariant lift of Waldhausen's functor , and we show that the -fixed points are the bivariant -theory of the fibration . We then use the framework of spectral Mackey functors to produce a second equivariant refinement whose fixed points have tom Dieck type splittings. We expect this second definition to be suitable for an equivariant generalization of the parametrized -cobordism theorem.

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

DOI 10.4171/DM/694