New York Journal of Mathematics
Volume 26 (2020), 92-115


Michael A. Hill and Mingcong Zeng

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

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Published: January 11, 2020.
Keywords: Equivariant homotopy, slice spectral sequence, homotopy fixed points.
Subject: 55Q91, 55P42, 55T99.

We introduce a computationally tractable way to describe the Z-homotopy fixed points of a Cn-spectrum E, 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.


The first author was supported by NSF Grant DMS-1811189.

Author information

Michael A. Hill:
Department of Mathematics
University of California at Los Angeles
Los Angeles, CA 90095, USA


Mingcong Zeng:
Department of Mathematics
Utrecht University
Utrecht, the Netherlands