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Register a new userHomotopy type of the unitary group of the uniform Roe algebra on $mathbb{Z}^n$
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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 $operatorname{U}_1(C^ast_u(|mathbb{Z}^n|))tooperatorname{U}_infty(C^ast_u(|mathbb{Z}^n|))$ is a homotopy equivalence. Moreover, when $n=1,2$, we determine the homotopy type of $operatorname{U}_1(C^ast_u(|mathbb{Z}^n|))$, which is the product of the unitary group $operatorname{U}_1(C^ast(|mathbb{Z}^n|))$ (having the homotopy type of $operatorname{U}_infty(mathbb{C})$ or $mathbb{Z}times Boperatorname{U}_infty(mathbb{C})$ depending on the parity of $n$) of the Roe algebra $C^ast(|mathbb{Z}^n|)$ and rational Eilenberg--MacLane spaces.
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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 the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable $mathbb{C}$-valued $mathbb{Z}$-sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators.
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 give an algebraic proof for the result of Eilenberg and Mac Lane that the second cohomology group of a simplicial group G can be computed as a quotient of a fibre product involving the first two homotopy groups and the first Postnikov invariant of G. Our main tool is the theory of crossed module extensions of groups.
A unitary fusion category is called $mathbb{Z}/2mathbb{Z}$-quadratic if it has a $mathbb{Z}/2mathbb{Z}$ group of invertible objects and one other orbit of simple objects under the action of this group. We give a complete classification of $mathbb{Z}/2mathbb{Z}$-quadratic unitary fusion categories. The main tools for this classification are skein theory, a generalization of Ostriks results on formal codegrees to analyze the induction of the group elements to the center, and a computation similar to Larsons rank-finiteness bound for $mathbb{Z}/3mathbb{Z}$-near group pseudounitary fusion categories. This last computation is contained in an appendix coauthored with attendees from the 2014 AMS MRC on Mathematics of Quantum Phases of Matter and Quantum Information.
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 allow us to calculate the coefficients of the geometric fixed point spectrum $Phi^{(mathbb{Z}/p)^n}Hmathbb{Z}/p$, and more generally, the $mathbb{Z}$-graded coefficients of the localization of $Hmathbb{Z}/p_{(mathbb{Z}/p)^n}$ by inverting any chosen set of embeddings $S^0rightarrow S^{alpha_i}$ where $alpha_i$ are non-trivial irreducible representations. We also calculate the $RO(G)^+$-graded coefficients of $Hmathbb{Z}/p_{(mathbb{Z}/p)^n}$, which means the cohomology of a point indexed by an actual (not virtual) representation. (This is the non-derived part, which has a nice algebraic description.)
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