Homology of classical Lie groups made discrete. II: \(H_ 2\), \(H_ 3\), and relations with scissors congruences
type: Item
instance of: scholarly article
title: Homology of classical Lie groups made discrete. II: \(H_ 2\), \(H_ 3\), and relations with scissors congruences
zbMATH Open document ID: 0657.55022
DOI: 10.1016/0021-8693(88)90191-3
author: Johan L. Dupont
author: W. R. Parry
author: Chih-Han Sah
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: This paper is one of a series dealing with a subtle and interesting complex of ideas from differential geometry, group cohomology, and algebraic K-theory [for Part I cf. the third author, Comment. Math. Helv. 61, 308-347 (1986; Zbl 0607.57025)]. Many partial results and conjectures in the earlier works are generalized, summarized and clarified (for example, questions of 2-torsion left open in earlier works are settled). Here is a theorem which gives the flavor of this work: Theorem. Let G be a simple, connected, simply-connected nonabelian Lie group such that its Lie algebra is absolutely simple and not among 10 exceptional ones of type E and F (3 compact, 7 noncompact, and all are not \({\mathbb{R}}\)-split). Let \(\rho\) : \(G\to SL(n,{\mathbb{C}})\) denote any nontrivial Lie group homomorphism. Then \(\rho_*: H_ 2(G)\to H_ 2(SL(n,{\mathbb{C}}))\) is injective and the image is \(K_ 2({\mathbb{C}})^+\), where \(H_ 2(SL(n,{\mathbb{C}}))\cong K_ 2({\mathbb{C}})\) for \(n\geq 2\). In particular, \(H_ 2(SL(n,{\mathbb{H}}))\to H_ 2(SL(n+1,{\mathbb{H}}))\) is bijective for \(n\geq 1\) and \(K_ 2({\mathbb{H}})\cong K_ 2({\mathbb{C}})^+\).
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 57T10
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 4073975
zbMATH Keywords: second algebraic K-group of \({\mathbb{C}}\)
zbMATH Keywords: homology of SL(n,\({\mathbb{C}})\) made discrete
zbMATH Keywords: homology of Lie groups made discrete
zbMATH Keywords: simple, simply-connected nonabelian Lie group
Wikidata QID: Q115366418
reviewed by: Michael R. Stein
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90191-3
OpenAlex ID: W2091692392
cites work: \(K_2\) of quaternion algebras
cites work: Q5817043
cites work: Q4053733
cites work: Q4098379
cites work: Algebra of polytopes and homology of flag complexes
cites work: Scissors congruences. II
cites work: A finiteness theorem for K\(_2\) of a number field
cites work: Q4190817
cites work: Q3897665
cites work: The Algebra of Polyhedra and the Dehn-Sydler Theorem.
cites work: Q5343374
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: On the homology of Lie groups made discrete
cites work: Third homology of \(\mathrm{SL}(2,\mathbb R)\) made discrete
cites work: Q4192613
cites work: Scissors congruences, I. The Gauss-Bonnet map.
cites work: Homology of classical Lie groups made discrete. I: Stability theorems and Schur multipliers
cites work: Homology of classical Lie groups made discrete. III
cites work: Second homology of lie groups made discrete
cites work: Q5522742
cites work: Q3310033
cites work: On the K-theory of algebraically closed fields
cites work: On the K-theory of local fields
cites work: On the automorphisms of the classical groups. With a supplement by Loo-Keng Hua
Recommended article: On self-H-equivalences of homotopy associative H-spaces
Recommended article: Some Results in the Connective K-Theory of Lie Groups
Recommended article: Q4896555
Recommended article: On the twisted \(K\)-homology of simple Lie groups
Recommended article: On the Third-Degree Continuous Cohomology of Simple Lie Groups
Recommended article: Homology of the Kac-Moody groups. II
Recommended article: A Fourier-Mukai approach to the \(K\)-theory of compact Lie groups
Recommended article: On the Coefficient Groups of Equivariant K-Theory
Recommended article: Cartan involutions and normalizers of maximal tori
Recommended article: Q3979653
description: scientific article; zbMATH DE number 4073975