Homology of classical Lie groups made discrete. II: \(H_ 2\), \(H_ 3\), and relations with scissors congruences.

*(English)*Zbl 0657.55022This 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 \(K_ 2({\mathbb{H}})\cong K_ 2({\mathbb{C}})^+\).

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 |

##### Keywords:

second algebraic K-group of \({\mathbb{C}}\); homology of SL(n,\({\mathbb{C}})\) made discrete; homology of Lie groups made discrete; simple, simply-connected nonabelian Lie group
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\textit{J. L. Dupont} et al., J. Algebra 113, No. 1, 215--260 (1988; Zbl 0657.55022)

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##### 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) |

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