Homology of classical Lie groups made discrete. II: \(H_ 2\), \(H_ 3\), and relations with scissors congruences
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 \(K_ 2({\mathbb{H}})\cong K_ 2({\mathbb{C}})^+\).
second algebraic K-group of \({\mathbb{C}}\)
homology of SL(n,\({\mathbb{C}})\) made discrete
simple, simply-connected nonabelian Lie group
scientific article; zbMATH DE number 4073975
scientific article