Homology of Linear Groups: 193 - Brossura

Knudson, Kevin P.

Brossura

ISBN 10: 3034895232 ISBN 13: 9783034895231

Casa editrice: Birkhäuser, 2012

Riassunto

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.

Su questo libro

Recensione

"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

Contenuti

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.

Dettagli dell'edizione

• Editore: Birkhäuser
• Data di pubblicazione: 2012
• Lingua: Inglese
• ISBN 10: 3034895232
• ISBN 13: 9783034895231
• Rilegatura: Copertina flessibile
• Numero di pagine: 203
• Contatto del produttore: Springer Nature Customer Service Center GmbH; ProductSafety@springernature.com; ; Europaplatz 3; Heidelberg; 69115; Germania