Homology of linear groups
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Bibliographic Information
Homology of linear groups
（Progress in mathematics, v. 193）
Springer Basel AG, 2001
- : pbk
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Note
Originally published by Birkhäuser Verlag in 2001
Includes bibliographical references (p. [183]-189) and index
Description and Table of Contents
Description
Daniel Quillen's definition of the higher algebraic K-groups of a ring emphasized the importance of computing the homology of groups of matrices. This text traces the development of this theory from Quillen's fundamental calculation. It presents the stability theorems and low-dimensional results of A. Suslin, W. van der Kallen and others are presented. Coverage also examines the Friedlander-Milnor-conjecture concerning the homology of algebraic groups made discrete.
Table of Contents
1. Topological Methods.- 1.1. Finite Fields.- 1.2. Quillen's Conjecture.- 1.3. Etale homotopy theory.- 1.4. Analytical Methods.- 1.5. Unstable Calculations.- 1.6. Congruence Subgroups.- Exercises.- 2. Stability.- 2.1. van der Kallen's Theorem.- 2.2. Stability for rings with many units.- 2.3. Local rings and Milnor K-theory.- 2.4. Auxiliary stability results.- 2.5. Stability via Homotopy.- 2.6. The Rank Conjecture.- Exercises.- 3. Low-dimensional Results.- 3.1. Scissors Congruence.- 3.2. The Bloch Group.- 3.3. Extensions and Generalizations.- 3.4. Invariants of hyperbolic manifolds.- Exercises.- 4. Rank One Groups.- 4.1. SL2(?[1/p]).- 4.2. The Bruhat-Tits Tree.- 4.3. SL2(k[t]).- 4.4. SL2(k[t, t?1]).- 4.5. Curves of Higher Genus.- 4.6. Groups of Higher Rank.- Exercises.- 5. The Friedlander-Milnor Conjecture.- 5.1. Lie Groups.- 5.2. Groups over Algebraically Closed Fields.- 5.3. Rigidity.- 5.4. Stable Results.- 5.5. H1, H2, and H3.- Exercises.- Appendix A. Homology of Discrete Groups.- A.1. Basic Concepts.- A.2. Spectral Sequences.- B.1. Classifying Spaces.- Appendix C. Etale Cohomology.- C.1. Etale Morphisms and Henselian Rings.- C.2. Etale Cohomology.- C.3. Simplicial Schemes.
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