Table of Contents

Introduction
REVIEW OF CLASSICAL ALGEBRAIC K-THEORY AND REPRESENTAION 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
K0 of symmetric monoidal categories
K0 of exact categories - definitions and examples
Exercises

Some Fundamental Results on K0 of Exact and Abelian Categories with Applications to Orders and Group Rings
Some fundamental results on K0 of exact and Abelian categories
Some finiteness results on K0 and G0 of orders and groupings
Class groups of Dedekind domains, orders, and group rings plus some applications
Decomposition of G0 (RG) (G Abelian group) and extensions to some non-Abelian groups
Exercises

K1, K2 of Orders and Group Rings
Definitions and basic properties
K1, SK1 of orders and group-rings; Whitehead torsion
The functor K2
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 Kn, Gn, SKn, SGn of orders and groupings
Ranks of Kn(?), Gn(?) of orders and group rings plus some consequences
Decomposition of Gn(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 Groupings
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

MACKEY FUNCTORS, EQUIVARIANT HIGHER ALGEBRAIC K-THEORY, AND EQUIVARIANT HOMOLOGY THEORIES
Exercises

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 KG0 -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