Locally symmetric spaces and \(K\)-theory of number fields
type: Item
instance of: scholarly article
title: Locally symmetric spaces and \(K\)-theory of number fields
zbMATH Open document ID: 1271.57062
author: Thilo Kuessner
published in: Algebraic \& Geometric Topology
publication date: 2012-05-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/0904.0919
review text: Let \(G/K\) be a symmetric space of noncompact type and dimension \(2n-1\), and let \(\rho: G\to\text{GL}(N,{\mathbb C})\) be a representation such that \(\rho^*b_{2n-1}\neq0\) for the Borel class \(b_{2n-1}\in H^{2n-1}(\text{U}(N);{\mathbb R})\). Among other things, the author proves in the paper under review that there exists a constant \(c_\rho\neq0\) such that for every locally symmetric space \(M=\Gamma\setminus G/K\) which is orientable, has finite volume, and either is compact or \(\text{rk}(G/K)=1\), there exists \(\gamma(M)\in K_{2n-1}(\overline{\mathbb Q})\otimes{\mathbb Q}\) such that \(\langle b_{2n-1},\gamma(M)\rangle=c_\rho\text{vol}(M)\). He also determines the irreducible symmetric spaces \(G/K\) and the fundamental representations satisfying the above conditions.
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 11R70
zbMATH DE Number: 6035474
zbMATH Keywords: locally symmetric spaces
zbMATH Keywords: Borel class
zbMATH Keywords: algebraic \(K\)-theory
reviewed by: Daniel Beltiţă
MaRDI profile type: Publication
OpenAlex ID: W3100042307
arXiv ID: 0904.0919
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DOI: 10.2140/AGT.2012.12.155
description: scientific article; zbMATH DE number 6035474