Homology of Linear Groups - Concise Reference for K-Theory

In this review the book Homology of Linear Groups is recommended for graduate students and researchers who need a focused reference on the homological methods behind algebraic K-theory. The text collects foundational results from Quillen through later work by Suslin and van der Kallen, and the single biggest reason to buy is its concentration of proofs and stability theorems that are otherwise scattered across original papers. Readers will find a tightly organized account that emphasizes the calculation of group homology and its role in higher K-groups.

Key Features

• Historical development: Traces the theory from Quillen's calculation of the cohomology of GLn(Fq) to later contributions, helping readers follow the subject's progression.
• Stability theorems: Presents the stability results of Suslin and van der Kallen so readers can apply them directly in computations of homology of matrix groups.
• Low-dimensional results: Collects concrete low-dimensional calculations that are useful for hands-on work in K-theory and group cohomology.
• Rank one groups: Includes recent results for rank one groups, offering up-to-date perspectives not commonly found in a single volume.
• Friedlander-Milnor discussion: A dedicated chapter explains the Friedlander-Milnor conjecture and its implications for treating algebraic groups as discrete groups.

Who It's For

The book is aimed at graduate students, researchers and practitioners in K-theory, algebraic geometry, topology and group cohomology who require a concise reference that gathers important theorems and computations in one place. It is particularly helpful for those working on the homology of GLn, applications to algebraic K-groups, or following the development of stability techniques.

Those looking for elementary introductions or broad surveys of algebra may want a more introductory text, since this volume expects familiarity with group cohomology and algebraic groups and focuses on assembling and explaining specific advanced results.

Pros & Cons

Pros

• Well organized collection of foundational results from Quillen through later contributors for easy reference.
• Includes stability theorems and low-dimensional computations that facilitate practical work in K-theory.
• The Friedlander-Milnor chapter offers a clear presentation of a significant conjecture and its context.

Cons

• Not an introductory textbook; readers without background in group cohomology may find it terse.

Specifications

Title	Homology of Linear Groups (Progress in Mathematics)
Author	Kevin P. P. Knudson
Subject focus	Homology of matrix groups and algebraic K-theory
Includes	Stability theorems, low-dimensional results, rank one groups
Notable chapter	Friedlander-Milnor conjecture on discrete algebraic groups
Intended audience	Graduate students and researchers in mathematics

Our Verdict

This volume is a compact, authoritative reference for specialists who need theorems and proofs about the homology of linear groups collected in one place. It is good value for researchers and advanced students working on algebraic K-theory or group cohomology, but it is not a substitute for an introductory text on the subject.

Frequently Asked Questions

Does this book cover Quillen's work?
Yes, it traces the development beginning with Quillen's calculation of the cohomology of GLn(Fq) and explains its importance for higher algebraic K-groups.

Is this suitable for beginners?
No, the book assumes familiarity with group cohomology and algebraic groups and is aimed at graduate-level readers and researchers.

Does it include recent results?
Yes, it presents later results including stability theorems and recent work on rank one groups alongside classical material.