×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alperin, R.C; Dennis, R.K, K2 of quaternion algebras, J. algebra, 56, 262-273, (1979) · Zbl 0397.18011
[2] Baer, R, Linear algebra and projective geometry, (1952), Academic Press New York · Zbl 0049.38103
[3] Bass, H; Tate, J, The Milnor ring of a global field, (), 349-446 · Zbl 0299.12013
[4] Cheeger, J, Invariants of flat bundles, (), 3-6
[5] Dupont, J.L, Algebra of polytopes and homology of flag complexes, Osaka J. math., 19, 599-641, (1982) · Zbl 0499.51014
[6] Dupont, J.L, The dilogarithm as a characteristic class for flat bundles, Aarhus university preprint no. 29, (1985)
[7] Dupont, J.L; Sah, C.H, Scissors congruences, II, J. pure appl. algebra, 25, 159-195, (1982) · Zbl 0496.52004
[8] Garland, H, A finiteness theorem for K2 of a number field, Ann. of math., 94, 534-548, (1971) · Zbl 0247.12103
[9] Harris, B, Group cohomology classes with differential form coefficients, (), 278-282
[10] Helgason, S, Differential geometry, Lie groups, and symmetric spaces, (1978), Academic Press New York · Zbl 0451.53038
[11] Jessen, B, The algebra of polyhedra and the Dehn-sydler theorem, Math. scand., 22, 241-256, (1968) · Zbl 0183.49803
[12] MacLane, S, Homology, (1963), Springer-Verlag/Academic Press New York · Zbl 0133.26502
[13] Milnor, J.W, Introduction to algebraic K-theory, () · Zbl 0237.18005
[14] Milnor, J.W, On the homology of Lie groups made discrete, Comm. helv. math., 58, 72-85, (1983) · Zbl 0528.20033
[15] Parry, W; Sah, C.H, Third homology of SL(2,\(R\) made discrete, J. pure appl. algebra, 30, 181-209, (1983) · Zbl 0527.18006
[16] Sah, C.H, Hilbert’s third problem: scissors congruences, () · Zbl 0406.52004
[17] Sah, C.H, Scissors congruences. I. the Gauss-Bonnet map, Math. scand., 49, 181-210, (1981) · Zbl 0496.52003
[18] Sah, C.H, Homology of classical Lie groups made discrete. I. stability theorems and Schur multipliers, Comm. helv. math., 61, 308-347, (1986) · Zbl 0607.57025
[19] \scC. H. Sah, Homology of classical Lie groups made discrete. III. J. Pure Appl. Algebra, to appear · Zbl 0684.57020
[20] Sah, C.H; Wagoner, J.B, Second homology of Lie groups made discrete, Comm. algebra, 5, 611-642, (1977) · Zbl 0375.18006
[21] Spanier, E.H, Algebraic topology, (1966), Mc Graw-Hill New York · Zbl 0145.43303
[22] Steinberg, R, Lectures on Chevalley groups, (1967), Yale University New Haven, CT
[23] Suslin, A, Homology of GLn, characteristic classes and Milnor K-theory, (), 357-375
[24] Suslin, A, On the K-theory of algebraically closed fields, Invent. math., 73, 241-245, (1983) · Zbl 0514.18008
[25] Suslin, A, On the K-theory of local fields, J. pure appl. algebra, 34, 301-318, (1984) · Zbl 0548.12009
[26] ()
[27] Dieudonné, J, On the automorphisms of the classical groups, with a supplement by L.K. hua, Mem. amer. math. soc., 2, (1951)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.