Progress in Mathematics

By: Kevin P. Knudson

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Hardcover | 1 December 2000

At a Glance

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208 Pages

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24.13 x 16.51 x 1.91

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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.

Industry Reviews

"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 Quillen's definitions of the higher algebraic K-groups. Stability theorems, low-dimensional results and the Friedlander-Milnor conjecture are discussed in this monograph."

-Aslib Book Guide

"This marks the first time that many of these results have been collected in a single volume..."

-Mathematical Reviews

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You Can Find This Book In

Non-Fiction Mathematics Groups & Group Theory Algebra Medicine Medicine in General Probability & Statistics Geometry Algebraic Geometry Topology Algebraic Topology

Preface
Topological Methods	p. 1
Finite Fields	p. 1
Quillen's Conjecture	p. 12
Etale homotopy theory	p. 14
Analytical Methods	p. 19
Unstable Calculations	p. 21
Congruence Subgroups	p. 23
Stability	p. 33
van der Kallen's Theorem	p. 34
Stability for rings with many units	p. 38
Local rings and Milnor K-theory	p. 46
Auxiliary stability results	p. 56
Stability via Homotopy	p. 59
The Rank Conjecture	p. 61
Low-dimensional Results	p. 65
Scissors Congruence	p. 65
The Bloch Group	p. 70
Extensions and Generalizations	p. 82
Invariants of hyperbolic manifolds	p. 87
Rank One Groups	p. 91
SL[subscript 2](Z[1/p])	p. 91
The Bruhat-Tits Tree	p. 95
SL[subscript 2](k[t])	p. 96
SL[subscript 2](k[t,t[superscript -1]])	p. 97
Curves of Higher Genus	p. 99
Groups of Higher Rank	p. 105
The Friedlander-Milnor Conjecture	p. 117
Lie Groups	p. 117
Groups over Algebraically Closed Fields	p. 121
Rigidity	p. 132
Stable Results	p. 139
H[subscript 1], H[subscript 2], and H[subscript 3]	p. 144
Homology of Discrete Groups	p. 149
Classifying Spaces and K-theory	p. 165
Etale Cohomology	p. 173
Bibliography	p. 183
Index	p. 191
Table of Contents provided by Blackwell. All Rights Reserved.

Progress in Mathematics

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