# Representation Theory and Higher Algebraic K-Theory

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IntroductionREVIEW 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 less

CRC Press; September 2006

471 pages; ISBN 9781420011128

Read online, or download in secure PDF format

Title: Representation Theory and Higher Algebraic K-Theory

Author: Aderemi Kuku

471 pages; ISBN 9781420011128

Read online, or download in secure PDF format

Title: Representation Theory and Higher Algebraic K-Theory

Author: Aderemi Kuku

Buy, download and read Representation Theory and Higher Algebraic K-Theory (eBook) by Aderemi Kuku today!

Subject categories

- Academic > Mathematics > Geometry. Trigonometry.Topology > Algebraic topology
- Academic > Mathematics > Geometry. Trigonometry.Topology > Homology theory
- Academic > Mathematics > Geometry. Trigonometry.Topology > K-theory
- Academic > Mathematics > Tables
- Mathematics > Algebra
- Mathematics > Geometry
- Mathematics > Number Theory

ISBNs

142001112X

9781420011128

9781584886037