Generalized \(\mathbb{Z}\)-homotopy fixed points of \(C_n\) spectra with applications to norms of \(MU_{\mathbb{R}}\)
In this paper, the authors introduce a way to describe the \(\mathbb{Z}\)-homotopy fixed points of a \(C_n\)-spectrum. In more detail, let \(E\) be a \(C_n\)-spectrum. A choice of generator for the cyclic group \(C_n\) gives a surjection \(\mathbb{Z} \to C_n\), and it is therefore possible to form the \(\mathbb{Z}\)-homotopy fixed points of \(E\). The first main result of the authors is that, from this data, one can associate a cofree \(C_n\)-spectrum \(E^{hn\mathbb{Z}}\) such that for all subgroups \(C_d \subseteq C_n\) we have \[ (E^{hn\mathbb{Z}})^{C_d} \simeq E^{h(\frac{n}{d}\mathbb{Z})}. \] The \(C_n\)-spectrum \(E^{hn\mathbb{Z}}\) is the quotient of \(E\) by the Euler class associated to the faithful irreducible representation of \(C_n\) given by choosing a primitive \(n\)-th root of unity. Choosing different representations of \(C_n\) and quotienting by the associated Euler class, the authors produce other interesting spectra. The second part of the paper is dedicated to applications. For example, let \(E_{\mathbb{R}}(n)\) denote the real Johnson-Wilson theory studied by \textit{N. Kitchloo} and \textit{W. S. Wilson} [Geom. Topol. Monogr. 10, 237--244 (2007; Zbl 1117.55001)]. The authors identify the homotopy fixed points \(E_{\mathbb{R}}(n)^{h\mathbb{Z}}\) as certain quotients of \(EO(n) = E_{\mathbb{R}}(n)^{C_2}\). This should be thought of as a higher analog of the identification of self conjugate \(K\)-theory as \(KO/\eta^2\). They also study norms of the real bordism spectrum \(MU_{\mathbb{R}}\), made famous by the Hill-Hopkins-Ravenel solution of the Kervaire invariant one problem [\textit{M. A. Hill} et al., Ann. Math. (2) 184, No. 1, 1--262 (2016; Zbl 1366.55007)].
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