Generalized Z-homotopy fixed points of Cn spectra with applications to norms of MUR

摘要

We introduce a computationally tractable way to describe the Z-homotopy fixed points of a Cn-spectrumE, producing a genuine Cn spectrum EhnZ whose fixed and homotopy fixed points agree and are the Z-homotopy fixed points of E. These form the bottom piece of a contravariant functor from the divisor poset of n to genuine Cn-spectra, and when E is an N∞-ring spectrum, this functor lifts to a functor of N∞-ring spectra.For spectra like the Real Johnson--Wilson theories or the norms of Real bordism, the slice spectral sequence provides a way to easily compute the RO(G)-graded homotopy groups of the spectrum EhnZ, giving the homotopy groups of the Z-homotopy fixed points. For the more general spectra in the contravariant functor, the slice spectral sequences interpolate between the one for the norm of Real bordism and the especially simple Z-homotopy fixed point case, giving us a family of new tools to simplify slice computations.