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AuthorKnudson, Kevin P. author
TitleHomology of Linear Groups [electronic resource] / by Kevin P. Knudson
ImprintBasel : Birkhรคuser Basel : Imprint: Birkhรคuser, 2001
Connect tohttp://dx.doi.org/10.1007/978-3-0348-8338-2
Descript XI, 192 p. online resource

SUMMARY

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 of the cohomology of GLn (Fq). The stability theorems and low-dimensional results of A. Suslin, W. van der Kallen and others are presented as well as recent results for rank one groups. A chapter on the Friedlander-Milnor-conjecture concerning the homology of algebraic groups made discrete is also included. This marks the first time that these results have been collected in a single volume. The book should prove useful to graduate students and researchers in K-theory, group cohomology, algebraic geometry and topology


CONTENT

1. Topological Methods -- 1.1. Finite Fields -- 1.2. Quillenโ{128}{153}s Conjecture -- 1.3. ร{137}tale homotopy theory -- 1.4. Analytical Methods -- 1.5. Unstable Calculations -- 1.6. Congruence Subgroups -- Exercises -- 2. Stability -- 2.1. van der Kallenโ{128}{153}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. ร{137}tale Cohomology -- C.1. ร{137}tale Morphisms and Henselian Rings -- C.2. ร{137}tale Cohomology -- C.3. Simplicial Schemes


Mathematics Algebraic topology Mathematics Algebraic Topology



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