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Homology of classical Lie groups made discrete. II: \(H_ 2\), \(H_ 3\), and relations with scissors congruences. (English) Zbl 0657.55022
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 \(K_ 2({\mathbb{H}})\cong K_ 2({\mathbb{C}})^+\).
Reviewer: M.R.Stein

55R40 Homology of classifying spaces and characteristic classes in algebraic topology
57T10 Homology and cohomology of Lie groups
22E60 Lie algebras of Lie groups
20J05 Homological methods in group theory
Full Text: DOI
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