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We introduce a computationally tractable way to describe the $mathbb Z$-homotopy fixed points of a $C_{n}$-spectrum $E$, producing a genuine $C_{n}$ spectrum $E^{hnmathbb Z}$ whose fixed and homotopy fixed points agree and are the $mathbb Z$-homotopy fixed points of $E$. These form a piece of a contravariant functor from the divisor poset of $n$ to genuine $C_{n}$-spectra, and when $E$ is an $N_{infty}$-ring spectrum, this functor lifts to a functor of $N_{infty}$-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 $E^{hnmathbb Z}$, giving the homotopy groups of the $mathbb Z$-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 $mathbb Z$-homotopy fixed point case, giving us a family of new tools to simplify slice computations.
We study the homotopy type of the space of the unitary group $operatorname{U}_1(C^ast_u(|mathbb{Z}^n|))$ of the uniform Roe algebra $C^ast_u(|mathbb{Z}^n|)$ of $mathbb{Z}^n$. We show that the stabilizing map $operatorname{U}_1(C^ast_u(|mathbb{Z}^n|))
This note contains a generalization to $p>2$ of the authors previous calculations of the coefficients of $(mathbb{Z}/2)^n$-equivariant ordinary cohomology with coefficients in the constant $mathbb{Z}/2$-Mackey functor. The algberaic results by S.Kriz
We explain how to reconstruct the category of Artin-Tate $mathbb{R}$-motivic spectra as a deformation of the purely topological $C_2$-equivariant stable category. The special fiber of this deformation is algebraic, and equivalent to an appropriate ca
The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that $pi_0$ of the space is determined by t
If K is a discrete group and Z is a K-spectrum, then the homotopy fixed point spectrum Z^{hK} is Map_*(EK_+, Z)^K, the fixed points of a familiar expression. Similarly, if G is a profinite group and X is a discrete G-spectrum, then X^{hG} is often gi