REPRESENTATION THEORY AND HIGHER ALGEBRAIC K-THEORY
CONTENTS I REVIEW OF CLASSICAL ALGEBRAIC K-THEORY AND REP- RESENTATION THEORY 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 – DE?NITIONS AND EXAMPLES EXERCISES SOME FUNDAMENTAL RESULTS ON K0 OF EXACT AND ABELIAN CATE- GORIES – WITH APPLICATIONS TO ORDERS AND GROUPRINGS SOME FUNDAMENTAL RESULTS ON K0 OF EXACT AND ABELIAN CATE- GORIES (2 1)A DEVISSAGE THEOREM AND EXAMPLE (2 1)B RESOLUTION THEOREM AND EXAMPLES (2 1)C K0 AND LOCALIZATION IN ABELIAN CATEGORIES PLUS EXAMPLES SOME ?NITENESS RESULTS ON K0 AND G0 OF ORDERS AND GROUPRINGS CLASS GROUPS OF DEDEKIND DOMAINS, ORDERS, AND GROUPRINGS PLUS SOME APPLICATIONS (2 3)A CLASS GROUPS OF DEDEKIND DOMAINS (2 3)B CLASS GROUPS OF ORDERS AND GROUPRINGS (2 3)C APPLICATIONS – WALL ?NITENESS OBSTRUCTION DECOMPOSITION OF G0(RG) (G ABELIAN GROUP) AND EXTENSIONS TO SOME NON-ABELIAN GROUPS (2 4)A DECOMPOSITION OF G0(RG), G ABELIAN (2 4)B CONNECTIONS TO THE GROUP “SSF” (2 4)C EXTENSIONS TO SOME NON-ABELIAN GROUPS (DIHEDRAL AND QUATERNION GROUPS) K1, K2 OF ORDERS AND GROUPRINGS DE?NITIONS AND BASIC PROPERTIES (3 1)A K1 OF A RING (3 1)B K1 OF LOCAL RINGS AND SKEW ?ELDS (3 1)C MENNICKE SYMBOLS K1, SK1 OF ORDERS AND GROUPRINGS; WHITEHEAD TORSION (3 2)A K1, SK1 OF ORDERS AND GROUPRINGS (3 2)B APPLICATIONS – WHITEHEAD TORSION AND S-COBORDISM THEOREM (3 3)A K2 OF RINGS AND ?ELDS (3 3)B K2 OF DIVISION ALGEBRAS AND MAXIMAL ORDERS (3 3)C K2 AND PSEUDO-ISOTROPY 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 (4 4)A LF, NF FUNCTORS AND THE FUNCTORS K?N (4 4)B MAYER - VIETORIS SEQUENCE (4 4)C EXACT SEQUENCE ASSOCIATED TO AN IDEAL (4 4)D LOCALIZATION SEQUENCE (4 4)E K?N(A) := K0(SNA) (4 4)F K?N(A), A AN ADDITIVE CATEGORY LOWER K-THEORY OF GROUPRINGS OF VIRTUALLY IN?NITE CYCLIC GROUPS (4 5)A FARRELL - JONES ISOMORPHISM CONJECTURE (4 5)B A PRELIMINARY RESULT (4 5)C LOWER K-THEORY FOR V = G ?? T (4 5)E SOME APPLICATIONS II HIGHER ALGEBRAIC K-THEORY AND INTEGRAL REPRE- SENTATIONS HIGHER ALGEBRAIC K-THEORY – DE?NITIONS, CONSTRUCTIONS, AND RELEVANT EXAMPLES THE PLUS CONSTRUCTION AND HIGHER K-THEORY OF RINGS (5 1)A THE PLUS CONSTRUCTION CLASSIFYING SPACES AND HIGHER K-THEORY OF EXACT CATEGORIES – CONSTRUCTIONS AND EXAMPLES (5 2 )A SIMPLICIAL OBJECTS AND CLASSIFYING SPACES (5 2)B HIGHER K-THEORY OF EXACT CATEGORIES – DE?NITIONS AND EXAM- PLES (5 2)C K-GROUPS AS HOMOTOPY GROUPS OF SPECTRA HIGHER K-THEORY OF SYMMETRIC MONOIDAL CATEGORIES – DE?NITIONS AND EXAMPLES HIGHER K-THEORY OF WALDHAUSEN CATEGORIES – DE?NITIONS AND EXAMPLES SOME FUNDAMENTAL RESULTS AND EXACT SEQUENCES IN HIGHER K- THEORY SOME FUNDAMENTAL THEOREMS (6 1)A RESOLUTION THEOREM (6 1)B ADDITIVITY THEOREM (FOR EXACT AND WALDHAUSEN CATEGORIES) (6 1)C DEVISSAGE LOCALIZATION (6 2)A LOCALIZATION SEQUENCE PLUS EXAMPLES (6 2)B FUNDAMENTAL THEOREM FOR HIGHER K-THEORY SOME EXACT SEQUENCES IN THE K-THEORY OF WALDHAUSEN CATE- GORIES EXACT SEQUENCE ASSOCIATED TO AN IDEAL; EXCISION; AND MAYER - VIETORIS SEQUENCES SOME RESULTS ON HIGHER K-THEORY OF ORDERS, GROUPRINGS, AND MODULES OVER ‘EI’ CATEGORIES SOME ?NITENESS RESULTS ON KN, GN, SKN, SGN OF ORDERS AND GROUPINGS (7 1)A HIGHER K-THEORY OF MAXIMAL ORDERS (7 1)B KN, GN, SKN, SGN OF ARBITRARY ORDERS RANKS OF KN(?), GN(?) OF ORDERS AND GROUPRINGS PLUS SOME CONSEQUENCES (7 2)A RANKS OF KN AND GN OF ORDERS ? (7 2)B K2N(?), G2N(?) ARE ?NITE FOR ALL N ?1 AND FOR ALL R-ORDERS ? DECOMPOSITION OF GN(RG) N ?0, G ?NITE ABELIAN GROUP; EXTENSIONS TO SOME NON-ABELIAN GROUPS, E G , QUATERNION AND DIHEDRAL GROUPS (7 3)A LENSTRA FUNCTOR AND THE DECOMPOSITION (7 3)B GN(RH), H DIHEDRAL GROUP OR NON-ABELIAN GROUP OF ORDER PQ (7 3)C GN(RH), H THE GENERALIZED QUATERNION GROUP OF ORDER 4 2 (7 3)D GN(RH), (H A NILPOTENT GROUP) PLUS A CONJECTURE OF HAM- BLETON, TAYLOR, AND WILLIAMS HIGHER DIMENSIONAL CLASS GROUPS OF ORDERS AND GROUPRINGS (7 4)A GENERALITIES ON HIGHER CLASS GROUPS (7 4)B TORSION IN ODD DIMENSIONAL HIGHER CLASS GROUPS (7 4)C TORSION IN EVEN-DIMENSIONAL HIGHER CLASS GROUPS C?2R(?) OF ORDERS HIGHER K-THEORY OF GROUPRINGS OF VIRTUALLY IN?NITE CYCLIC GROUPS (7 5)A SOME PRELIMINARY RESULTS (7 5)B K-THEORY FOR THE ?RST TYPE OF VIRTUALLY IN?NITE CYCLIC GROUPS (7 5)C NIL-GROUPS FOR THE SECOND TYPE OF VIRTUALLY IN?NITE CYCLIC GROUPS HIGHER K-THEORY OF MODULES OVER ‘EI’ CATEGORIES (7 6)A GENERALITIES ON MODULES OVER ‘EI’ CATEGORIES C (7 6)B KN(RC), SKN(RC) (7 6)C GN(RC), SGN(RC) (7 6)D CARTAN MAP KN(RC) ?GN(RC) (7 6)E PAIRINGS AND MODULE STRUCTURES HIGHER K-THEORY OF P(A)G; A MAXIMAL ORDERS IN DIVISION ALGEBRAS; G ?NITE GROUP (7 7)A A TRANSFER MAP IN HIGHER K-THEORY NON-COMMUTATIVE ANALOGUE OF A RESULT OF R G SWAN (7 7)B HIGHER K-THEORY OF P(A)G, A A MAXIMAL ORDER IN A P-ADIC DIVISION ALGEBRA (7 7)C HIGHER K-THEORY OF P(A)G, A A MAXIMAL ORDER IN DIVISION ALGEBRAS OVER NUMBER ?ELDS MOD-M AND PRO?NITE HIGHER K-THEORY OF EXACT CATEGORIES, ORDERS, AND GROUPINGS MOD-M K-THEORY OF EXACT CATEGORIES, RINGS, AND ORDERS PRO?NITE K-THEORY OF EXACT CATEGORIES, RINGS AND ORDERS PRO?NITE K-THEORY OF P-ADIC ORDERS AND SEMI-SIMPLE ALGEBRAS CONTINUOUS K-THEORY OF P-ADIC ORDERS III 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 (9 4)A BURNSIDE RING OF A BASED CATEGORY (9 4)B UNIVERSALITY OF THE BURNSIDE FUNCTOR (9 4)C ARITHMETIC STRUCTURE OF ?(B), B A BASED CATEGORY (9 4)D ARITHMETIC STRUCTURE OF ?(G), G A ?NITE GROUP INDUCTION THEOREMS FOR MACKEY AND GREEN FUNCTORS DEFECT BASIS OF MACKEY AND GREEN FUNCTORS DEFECT BASIS FOR KG 0 -FUNCTORS 10 EQUIVARIANT HIGHER ALGEBRAIC K-THEORY TOGETHER WITH RELATIVE GENERALIZATIONS – FOR ?NITE GROUP ACTIONS 10 1 EQUIVARIANT HIGHER ALGEBRAIC K-THEORY 10 2 RELATIVE EQUIVARIANT HIGHER ALGEBRAIC K-THEORY 10 3 INTERPRETATION IN TERMS OF GROUP-RINGS 10 4 SOME APPLICATIONS 11 EQUIVARIANT HIGHER K-THEORY FOR PRO?NITE GROUP ACTIONS 11 1 EQUIVARIANT HIGHER K-THEORY – (ABSOLUTE AND RELATIVE) 11 2 COHOMOLOGY OF MACKEY FUNCTORS (FOR PRO?NITE GROUPS) 12 EQUIVARIANT HIGHER K-THEORY FOR COMPACT LIE GROUP ACTIONS 12 1 MACKEY AND GREEN FUNCTORS ON THE CATEGORY A(G) OF HOMOGE- NEOUS SPACES (12 1)A THE ABELIAN GROUP U(G, X), G A COMPACT LIE GROUP, X A G-SPACE; THE CATEGORY A(G) (12 1)B MACKEY AND GREEN FUNCTORS ON A(G) 12 2 AN EQUIVARIANT HIGHER K-THEORY FOR G-ACTIONS 12 3 INDUCTION THEORY FOR EQUIVARIANT HIGHER K-FUNCTORS (12 3)A REMARKS ON POSSIBLE GENERALIZATIONS EXERCISE 13 EQUIVARIANT HIGHER K-THEORY FOR WALDHAUSEN CATEGORIES 13 1 EQUIVARIANT WALDHAUSEN CATEGORIES 13 2 EQUIVARIANT HIGHER K-THEORY CONSTRUCTIONS FOR WALDHAUSEN CATEGORIES (13 2)A ABSOLUTE AND RELATIVE EQUIVARIANT THEORY (13 2)B EQUIVARIANT ADDITIVITY THEOREM (13 2)C EQUIVARIANT WALDHAUSEN ?BRATION SEQUENCE 13 3 APPLICATIONS TO COMPLICIAL BI-WALDHAUSEN CATEGORIES 13 4 APPLICATIONS TO HIGHER K-THEORY OF GROUPRINGS 14 EQUIVARIANT HOMOLOGY THEORIES AND HIGHER K-THEORY OF GROUPRINGS 14 1 CLASSIFYING SPACE FOR FAMILIES AND EQUIVARIANT HOMOLOGY THEORY (14 1)A CLASSIFYING SPACES FOR FAMILIES AND G-HOMOLOGY THEORY 14 2 ASSEMBLY MAPS AND ISOMORPHISM CONJECTURES 14 3 FARRELL - JONES CONJECTURE FOR ALGEBRAIC K-THEORY 14 4 BAUM - CONNES CONJECTURE (14 4)A GENERALITIES ON BAUM - CONNES CONJECTURE 14 5 DAVIS - L¨UCK ASSEMBLY MAP FOR BC CONJECTURE AND ITS IDENTI?- CATION WITH ANALYTIC ASSEMBLY MAP