Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2004.05710

On a generalization of the topological Brauer group

Andrei V. Ershov
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the present paper we propose some generalization of the topological Brauer group that includes higher homotopical information and contains the classical one as a direct summand. Our approach is based on some kind of bundle-like objects called ``lax algebra bundles'' that occupy an intermediate position between ``Morita bundle gerbes'' and matrix algebra bundles. The main results of the paper include the descripion of the homotopy type of their classifying space. The obtained results can be applied to the twisted $K$-theory because the lax algebra bundles are geometric representatives of the ``higher'' twists of topological $K$-theory that have finite order. v.2: major changes, especially in the second half of the paper v.3: to clarify the presentation the significant part of the text has been rewritten v.4: major changes, completely different methods comparing with previous versions v.5: major changes and corrections v.6: section 3 added v.7: the definition of equivalence of LABs fixed v.8: section 3 has been rewritten v.9: remark 3.2 and some explanation in subsection 3.3 have been added v.10 in this version we omit the UHF algebra approach; otherwise, correction and clarifications have been made, in subsection 4.3 an outline of the theory of modules over LABs has been added v.11: some corrections and clarifications v.12: theorem 6.2 added, minor corrections v.13: some additions (the most important are section 7 and remark 3.5) v.14: subsections 6.2-6.4 added v.15: subsection 6.5 added v.16: The appendix has been completely rewritten v.17: Some issues with simplicial sets and simplicial spaces in the Appendix were corrected v.18: major correctness in subsection 7.5 v.19: major changes and corrections

Explore related subjects

math.KTmath.AT

Keep this discovery

BibTeXRIS

Andrei V. Ershov. 2020-04-12. On a generalization of the topological Brauer group. https://arxiv.org/abs/2004.05710

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Oka principle for \'etale Chow groups

The celebrated theorems of Shilov, Arens--Royden, and Forster give direct descriptions of the first three integral cohomology groups of the Gelfand spectrum of a commutative complex Banach algebra. In his 1974 ICM address, Taylor asked whether the higher cohomology groups admit descriptions in terms of the underlying ring. We give a solution to this question in even degrees: The \'etale (aka Lichtenbaum) Chow group in every codimension is canonically isomorphic to the corresponding even integral cohomology group of the Gelfand spectrum.

math.KT↗
arXiv2026

General linear and Steinberg groups over the Leavitt algebra $L_{\mathbb F_2}(1,2)$

Let $R=L_{\F_2}(1,2)$. We prove that $\GL_r(R)$ is integrally acyclic for every $r\geq1$ and that the canonical map $\St_r(R)\to\GL_r(R)$ is an isomorphism for every $r\geq3$. The homology calculation combines simultaneous extensions of ordered frames with scalar actions of the multiplicative groups of finite fields on their stabilizers. The presentation associated with the same frame complex defines a surjective section of the Steinberg map. An explicit finite presentation of $R^\times$ then follows from the theorem of Krsti\'c and McCool. We formulate separate criteria for acyclicity and for the Steinberg comparison over other rings.

math.KT↗
arXiv2026

The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups

For a uniformly locally finite coarse space $X$, the uniform Roe algebra $C_u^*(X)$ is the operator norm closure of the controlled operators on $\ell^2(X)$. The $K$-theory of uniform Roe algebras is known in asymptotic dimension zero, but it is not fully understood in higher dimensions. We compute $K_0(C_u^*(G,\mathcal E))$ and $K_1(C_u^*(G,\mathcal E))$ for every countable discrete abelian group $G$ and every finite-rank free abelian subgroup $H\leq G$, where $\mathcal E$ is the coarse structure generated by $H$. We use the Proietti--Yamashita spectral sequence to express the $K$-theory in terms of $H_*(H;\ell^\infty(G,\mathbb Z))$, which we then compute.

math.KT↗