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Homology of Linear Groups

Author: Kevin P Knudson
Publisher: Basel : Birkhäuser Basel : Imprint : Birkhäuser, 2001.
Series: Progress in Mathematics, 193.
Edition/Format:   eBook : Document : EnglishView all editions and formats
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.  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Kevin P Knudson
ISBN: 9783034883382 3034883382
OCLC Number: 851828530
Description: 1 online resource (xi, 192 pages).
Contents: 1. Topological Methods --
1.1. Finite Fields --
1.2. Quillen's Conjecture --
1.3. Étale 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. Étale Cohomology --
C.1. Étale Morphisms and Henselian Rings --
C.2. Étale Cohomology --
C.3. Simplicial Schemes.
Series Title: Progress in Mathematics, 193.
Responsibility: by Kevin P. Knudson.

Abstract:

Daniel Quillen's definition of the higher algebraic K-groups of a ring emphasized the importance of computing the homology of groups of matrices. It presents the stability theorems and  Read more...

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"A book for graduates and researchers in K-theory, cohomology, algebraic geometry and topology. The theme is the development of the computing of the homology of the groups of matrices from Daniel Read more...

 
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