Representation Theory And Higher Algebraic K-Theory

Description:
'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-Luck assembly maps.

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-Luck assembly map for BC conjecture and its identification with analytic assembly map Exercise Appendices A: Some computations B: Some open problems References Index