Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Homotopy type of the unitary group of the uniform Roe algebra on
  $\mathbb{Z}^n$   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
$\operatorname{U}_1(C^\ast_u(|\mathbb{Z}^n|))\to\operatorname{U}_\infty(C^\ast_u(|\mathbb{Z}^n|))$
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 $\mathbb{Z}\times
B\operatorname{U}_\infty(\mathbb{C})$ depending on the parity of $n$) of the
Roe algebra $C^\ast(|\mathbb{Z}^n|)$ and rational Eilenberg--MacLane spaces.

27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th