Complex \(K\)-theory of \(B\text{SL}_ 3({\mathbb{Z}{}})\)
type: Item
instance of: scholarly article
title: Complex \(K\)-theory of \(B\text{SL}_ 3({\mathbb{Z}{}})\)
zbMATH Open document ID: 0770.55007
DOI: 10.1007/BF00961336
author: Nobuaki Yagita
author: Michishige Tezuka
published in: \(K\)-Theory
publication date: 1993-04-01T00:00:00Z
review text: The authors first study the Brown-Peterson (BP) cohomology of the classifying space of SL\(_ 3(\mathbb{Z})\); then, using the Conner-Floyd relation \[ K^*_{(p)}(-)\cong K^*\otimes_{\text{BP}^*}\text{BP}^*(-) \] they compute the complex \(K\)-theory of \(B\text{SL}_ 3(\mathbb{Z})\) and \(B\text{St}_ 3(\mathbb{Z})\), where St\(_ 3(\mathbb{Z})\) denotes the Steinberg group. Also, the authors prove that \(K^*(BG)\) is generated by Chern classes of complex representations, for \(G=\text{SL}_ 3(\mathbb{Z})\) and \(p=2\).
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55R35
zbMATH DE Number: 149767
zbMATH Keywords: Brown-Peterson cohomology
zbMATH Keywords: complex \(K\)-theory
zbMATH Keywords: classifying space
zbMATH Keywords: Chern classes
reviewed by: Cristian N. Costinescu
MaRDI profile type: Publication
cites work: Filtrations on the representation ring of a finite group
cites work: The cohomology of \(SL_3(Z)\)
cites work: Q3827226
cites work: Symmetric Product Spectra and Splittings of Classifying Spaces
cites work: Cohomological behaviour of the reduction modulo a prime of \(GL_ 3(Z)\)
cites work: Generalized group characters and complex oriented cohomology theories
Recommended article: \(K\)-theory of the full \(C^*\)-algebra of \(SL_{3}(\mathbb C)\)
Recommended article: Twisted equivariant \(K\)-theory and \(K\)-homology of \(\mathrm{SL}_3\mathbb Z\)
Recommended article: The Bernstein-Gelfand-Gelfand complex and Kasparov theory for \(SL(3,\mathbb C)\)
Recommended article: Controlled algebraic \(K\)-theory of integral group ring of \(SL(3,\mathbb{Z})\)
Recommended article: Bredon homology and equivariant \(K\)-homology of \(\mathrm{SL}(3,\mathbb Z)\)
Recommended article: The cohomology of \(\text{SL}(3,\mathbb{Z}[1/2])\)
Recommended article: Third homology of \({\mathrm{SL}}_2\) and the indecomposable \(K_3\)
Recommended article: Boundary and Eisenstein cohomology of \(\mathrm{SL}_3(\mathbb{Z})\)
Recommended article: Equivariant \(K\)-theory of central extensions and twisted equivariant \(K\)-theory: \(SL_{3}\mathbb{Z}\) and \(St_{3}{\mathbb{Z}}\)
Recommended article: The third homology of \(\mathrm{SL}_2(\mathbb{Q})\)
description: scientific article; zbMATH DE number 149767