# Representation Theory and Higher Algebraic K-Theory, by Aderemi Kuku

Aderemi Kuku's monograph Representation Theory and Higher Algebraic K-Theory was published by Chapman & Hall in their Pure and Applied mathematics Series in 2007. The following information is taken from the publisher's leaflet advertising the book.

The definitive resource for algebraic $K$-theory

'Representation Theory and Higher Algebraic K-Theory' is the first book to present higher algebraic K-theory of orders and group rings as well as characterize higher algebraic K-theory as Mackey functors that lead to equivariant higher algebraic K-theory and their relative generalizations. Thus, this book makes computations of higher K-theory of group rings more accessible and provides novel techniques for the computations of higher K-theory of finite and some infinite groups.

Authored by a premier authority in the field, the book begins with a careful review of classical K-theory, including clear definitions, examples, and important classical results. Emphasizing the practical value of the usually abstract topological constructions, the author systematically discusses higher algebraic K-theory of exact, symmetric monoidal, and Waldhausen categories with applications to orders and group rings and proves numerous results. He also defines profinite higher K- and G-theory of exact categories, orders, and group rings. Providing new insights into classical results and opening avenues for further applications, the book then uses representation-theoretic techniques - especially induction theory - to examine equivariant higher algebraic K-theory, their relative generalizations, and equivariant homology theories for discrete group actions. The final chapter unifies Farrell and Baum-Connes isomorphism conjectures through Davis-Lück assembly maps.

Features
1. Presents higher algebraic K-theory of orders and group rings for the first time in book form
2. Explores connections between C_G and higher algebraic K-theory of C_G for suitable categories, such as exact, symmetric monoidal, and Waldhausen
3. Collects methods that have been known to work for computations of higher K-theory of noncommutative rings, such as orders and group rings
4. Describes all higher algebraic K-theory as Mackey functors that lead to equivariant higher algebraic K-theory and their relative generalizations for finite, profinite, and compact Lie group actions
5. Obtains results on higher K-theory of orders lambda, and hence group rings, for all n ≥ 0
6. Uses computations of higher K-theory of orders that automatically yield results on higher K-theory of RG(G finite) to produce results on higher K-theory of some infinite groups
7. Provides appendices with many known computations and open problems in classical and higher algebraic K-theory of orders, group rings, and related structures
Contents
Introduction
REVIEW OF CLASSICAL ALGEBRAIC $K$-THEORY AND REPRESENTATION THEORY
Notes on Notations

Category of Representations and Constructions of Grothendieck Groups and Rings

Category of representations and G-equivariant categories

Grothendieck group associated with a semi-group

$K_{0}$ of symmetric monoidal categories

$K_{0}$ of exact categories - definitions and examples

Exercises
Some Fundamental Results on $K_{0}$ of Exact and Abelian Categories with Applications to Orders and Group Rings
Some fundamental results on$K_{0}$ of exact and Abelian categories

Some finiteness results on $K_{0}$ and $G_{0}$ of orders and group rings

Class groups of Dedekind domains, orders, and group rings plus some applications

Decomposition of $G_{0}(RG)$ ($G$ Abelian group) and extensions to some non-Abelian groups

Exercises
$K1, K2$ of Orders and Group Rings
Definitions and basic properties

$K_{1}, SK_{1}$ of orders and group-rings; Whitehead torsion

The functor $K_{2}$

Exercises
Some Exact Sequences; Negative K-Theory
Mayer-Vietoris sequences

Localization sequences

Exact sequence associated to an ideal of a ring

Negative K-theory $K_{-n}, n$ positive integer

Lower $K$-theory of group rings of virtually infinite cyclic groups
HIGHER ALGEBRAIC $K$-THEORY AND INTEGRAL REPRESENTATIONS
Higher Algebraic $K$-Theory - Definitions, Constructions, and Relevant Examples

The plus construction and higher $K$-theory of rings

Classifying spaces and higher $K$-theory of exact categories - constructions and examples

Higher $K$-theory of symmetric monoidal categories - definitions and examples

Higher $K$-theory of Waldhausen categories - definitions and examples

Exercises
Some Fundamental Results and Exact Sequences in Higher $K$-Theory
Some fundamental theorems

Localization

Fundamental theorem of higher $K$-theory

Some exact sequences in the $K$-theory of Waldhausen categories

Exact sequence associated to an ideal, excision, and Mayer-Vietoris sequences

Exercises
Some Results on Higher $K$-Theory of Orders, Group Rings and Modules over "EI" Categories
Some finiteness results on $K_{n}, G_{n}, SK_{n}, SG_{n}$ of orders and group rings

Ranks of Kn(L), Gn(L) of orders and in rings plus some consequences

Decomposition of $G_{n}(RG) n ≥ 0, G$finite Abelian group

Extensions to some non-Abelian groups, e.g., quaternion and dihedral groups

Higher dimensional class groups of orders and group rings

Higher $K$-theory of group rings of virtually infinite cyclic groups

Higher $K$-theory of modules over "EI" - categories

Higher $K$-theory of $P(A)G, A$ maximal orders in division algebras, $G$ finite group

Exercises
Mod-m and Profinite Higher $K$-Theory of Exact Categories, Orders, and Group Rings
$Mod-m K$-theory of exact categories, rings and orders

Profinite $K$-theory of exact categories, rings and orders

Profinite $K$-theory of $p$-adic orders and semi-simple algebras

Continuous $K$-theory of $p$-adic orders

Exercises
MACKEY FUNCTORS, EQUIVARIANT HIGHER ALGEBRAIC $K$-THEORY, AND EQUIVARIANT HOMOLOGY THEORIES
Mackey, Green, and Burnside Functors

Mackey functors

Cohomology of Mackey functors

Green functors, modules, algebras, and induction theorems

Based category and the Burnside functor

Induction theorems for Mackey and Green functors

Defect basis of Mackey and Green functors

Defect basis for $(KG_{0})^{G}$ -functors

Exercises
Equivariant Higher Algebraic $K$-Theory Together with Relative Generalizations for Finite Group Actions
Equivariant higher algebraic $K$-theory

Relative equivariant higher algebraic $K$-theory

Interpretation in terms of group rings

Some applications

Exercises
Equivariant Higher $K$-Theory for Profinite Group Actions
Equivariant higher $K$-theory (absolute and relative)

Cohomology of Mackey functors (for profinite groups)

Exercises
Equivariant Higher $K$-Theory for Compact Lie Group Actions
Mackey and Green functors on the category $A(G)$ of homogeneous spaces

An equivariant higher $K$-theory for $G$-actions

Induction theory for equivariant higher K-functors

Exercise
Equivariant Higher $K$-Theory for Waldhausen Categories
Equivariant Waldhausen categories

Equivariant higher $K$-theory constructions for Waldhausen categories

Applications to complicial bi-Waldhausen categories

Applications to higher $K$-theory of group rings

Exercise
Equivariant Homology Theories and Higher $K$-Theory of Group Rings
Classifying space for families and equivariant homology theory

Assembly maps and isomorphism conjectures

Farrell-Jones conjecture for algebraic $K$-theory

Baum-Connes conjecture

Davis-Lück assembly map for BC conjecture and its identification with analytic assembly map

Exercise
Appendices
A: Some computations

B: Some open problems
References

Index

Last Updated May 2019