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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

##### MSC:
 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
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