TOPOLOGICAL AND BIVARIANT K-THEORY
CONTENTS THE ELEMENTARY ALGEBRA OF K-THEORY PROJECTIVE MODULES, IDEMPOTENTS, AND VECTOR BUNDLES GENERAL PROPERTIES SIMILARITY OF IDEMPOTENTS RELATIONSHIP TO VECTOR BUNDLES PASSAGE TO K-THEORY EULER CHARACTERISTICS OF ?NITE PROJECTIVE COMPLEXES DE?NITION OF K0 FOR NON-UNITAL RINGS EXACTNESS PROPERTIES OF K-THEORY HALF-EXACTNESS OF K0 INVERTIBLE ELEMENTS AND THE INDEX MAP NILPOTENT EXTENSIONS AND LOCAL RINGS FUNCTIONAL CALCULUS AND TOPOLOGICAL K-THEORY BORNOLOGICAL ANALYSIS SPACES OF CONTINUOUS MAPS BORNOLOGICAL TENSOR PRODUCTS LOCAL BANACH ALGEBRAS AND FUNCTIONAL CALCULUS HOMOTOPY INVARIANCE AND EXACT SEQUENCES FOR LOCAL BANACH ALGEBRAS HOMOTOPY INVARIANCE OF K-THEORY HIGHER K-THEORY THE PUPPE EXACT SEQUENCE THE MAYER–VIETORIS SEQUENCE PROJECTIONS AND IDEMPOTENTS IN C?-ALGEBRAS INVARIANCE OF K-THEORY FOR ISORADIAL SUBALGEBRAS ISORADIAL HOMOMORPHISMS NEARLY IDEMPOTENT ELEMENTS THE INVARIANCE RESULTS CONTINUITY AND STABILITY HOMOTOPY INVARIANCE OF STABILISED ALGEBRAIC K-THEORY INGREDIENTS IN THE PROOF SPLIT-EXACT FUNCTORS AND QUASI-HOMOMORPHISMS INNER AUTOMORPHISMS AND STABILITY A CONVENIENT STABILISATION HÖLDER CONTINUITY THE HOMOTOPY INVARIANCE RESULT A KEY LEMMA THE MAIN RESULTS WEAK VERSUS FULL STABILITY BOTT PERIODICITY TOEPLITZ ALGEBRAS THE PROOF OF BOTT PERIODICITY SOME K-THEORY COMPUTATIONS THE ATIYAH–HIRZEBRUCH SPECTRAL SEQUENCE THE K-THEORY OF CROSSED PRODUCTS CROSSED PRODUCTS FOR A SINGLE AUTOMORPHISM CROSSED TOEPLITZ ALGEBRAS THE PIMSNER–VOICULESCU EXACT SEQUENCE SOME CONSEQUENCES OF THE PIMSNER–VOICULESCU THEOREM A GLIMPSE OF THE BAUM–CONNES CONJECTURE TOEPLITZ CONES PROOF OF THE DECOMPOSITION THEOREM TOWARDS BIVARIANT K-THEORY: HOW TO CLASSIFY EXTENSIONS SOME TRICKS WITH SMOOTH HOMOTOPIES TENSOR ALGEBRAS AND CLASSIFYING MAPS FOR EXTENSIONS THE SUSPENSION-STABLE HOMOTOPY CATEGORY BEHAVIOUR FOR IN?NITE DIRECT SUMS AN ALTERNATIVE APPROACH EXACT TRIANGLES IN THE SUSPENSION-STABLE HOMOTOPY CATEGORY LONG EXACT SEQUENCES IN TRIANGULATED CATEGORIES LONG EXACT SEQUENCES IN THE SUSPENSION-STABLE HOMOTOPY CATEGORY THE UNIVERSAL PROPERTY OF THE SUSPENSION-STABLE HOMOTOPY CATEGORY BIVARIANT K-THEORY FOR BORNOLOGICAL ALGEBRAS SOME TRICKS WITH STABILISATIONS COMPARING STABILISATIONS A GENERAL CLASS OF STABILISATIONS SMOOTH STABILISATIONS EVERYWHERE DE?NITION AND BASIC PROPERTIES BOTT PERIODICITY AND RELATED RESULTS K-THEORY VERSUS BIVARIANT K-THEORY COMPARISON WITH OTHER TOPOLOGICAL K-THEORIES THE WEYL ALGEBRA A SURVEY OF BIVARIANT K-THEORIES K-THEORY WITH COE?CIENTS ALGEBRAIC DUAL K-THEORY HOMOTOPY-THEORETIC KK-THEORY BROWN–DOUGLAS–FILLMORE EXTENSION THEORY BIVARIANT K-THEORIES FOR C?-ALGEBRAS ADAPTING OUR MACHINERY ANOTHER VARIANT RELATED TO E-THEORY COMPARISON WITH KASPAROV’S DE?NITION SOME REMARKS ON THE KASPAROV PRODUCT EQUIVARIANT BIVARIANT K-THEORIES ALGEBRAS OF CONTINUOUS TRACE, TWISTED K-THEORY ALGEBRAS OF CONTINUOUS TRACE TWISTED K-THEORY 10 CROSSED PRODUCTS BY R AND CONNES’ THOM ISOMORPHISM 10 1 CROSSED PRODUCTS AND TAKAI DUALITY 10 2 CONNES’ THOM ISOMORPHISM THEOREM 10 2 1 CONNES’ ORIGINAL PROOF 10 2 2 ANOTHER PROOF 11 APPLICATIONS TO PHYSICS 11 1 K-THEORY IN PHYSICS 11 2 T-DUALITY 12 SOME CONNECTIONS WITH INDEX THEORY 12 1 PSEUDO-DI?ERENTIAL OPERATORS 12 1 1 DE?NITION OF PSEUDO-DI?ERENTIAL OPERATORS 12 1 2 INDEX PROBLEMS FROM PSEUDO-DI?ERENTIAL OPERATORS 12 1 3 THE DOLBEAULT OPERATOR 12 2 THE INDEX THEOREM OF BAUM, DOUGLAS, AND TAYLOR 12 2 1 TOEPLITZ OPERATORS 12 2 2 A FORMULA FOR THE BOUNDARY MAP 12 2 3 APPLICATION TO THE DOLBEAULT OPERATOR 12 3 THE INDEX THEOREMS OF KASPAROV AND ATIYAH–SINGER 12 3 1 THE THOM ISOMORPHISM AND THE DOLBEAULT OPERATOR 12 3 2 THE DOLBEAULT ELEMENT AND THE TOPOLOGICAL INDEX MAP CONTENTS 13 LOCALISATION OF TRIANGULATED CATEGORIES 13 1 EXAMPLES OF LOCALISATIONS 13 1 1 THE UNIVERSAL COE?CIENT THEOREM 13 1 2 THE BAUM–CONNES ASSEMBLY MAP VIA LOCALISATION 13 2 THE OCTAHEDRAL AXIOM CHAPTER THE ELEMENTARY ALGEBRA OF K-THEORY ORIGINALLY, K-THEORY WAS THE STUDY OF VECTOR BUNDLES ON TOPOLOGICAL SPACES BUT IT WAS SOON REALISED THAT THE NOTION OF VECTOR BUNDLE CAN BE FORMULATED MORE ALGE- BRAICALLY: SWAN’S THEOREM IDENTI?ES THE MONOID OF VECTOR BUNDLES OVER A COMPACT SPACE X WITH THE MONOID OF ?NITELY GENERATED PROJECTIVE MODULES OVER THE ALGEBRA C(X) OF CONTINUOUS FUNCTIONS ON X; WE TAKE REAL- OR COMPLEX-VALUED FUNCTIONS HERE TO GET REAL OR COMPLEX VECTOR BUNDLES, RESPECTIVELY THE ?NITELY GENERATED PROJECTIVE MODULES OVER A UNITAL RING R FORM A COMMU- TATIVE MONOID, WHICH WE DENOTE BY V(R) WE DESCRIBE V(R) MORE CONCRETELY USING EQUIVALENCE CLASSES OF IDEMPOTENT ELEMENTS IN MATRIX RINGS OVER R; THIS IS USED IN MANY PROOFS THE K-THEORY K0(R) OF R IS DE?NED AS THE GROTHENDIECK GROUP OF V(R) THIS GROUP IS USUALLY MUCH EASIER TO HANDLE THAN V(R) ITSELF BECAUSE WE MAY USE TOOLS FROM HOMOLOGICAL ALGEBRA THAT DO NOT WORK FOR MONOIDS WE EXTEND K0 TO NON-UNITAL RINGS BY A STANDARD TRICK WE NEED THIS IN ORDER TO FORMULATE THE EXACTNESS PROPERTIES OF K0, WHICH ARE CRUCIAL FOR MANY COMPUTATIONS WE WILL SEE THAT K0 IS HALF-EXACT AND SPLIT-EXACT WE ALSO STUDY THE FAILURE OF LEFT-EXACTNESS OF K0 THIS LEADS US TO THE INDEX MAP AND HINTS AT A DE?NITION OF K1: THIS GROUP SHOULD CLASSIFY INVERTIBLE ELEMENTS IN MATRIX RINGS UP TO AN APPROPRIATE EQUIVALENCE RELATION NOW K-THEORY SPLITS INTO TWO BRANCHES: THE ALGEBRAIC APPROACH USES COMMUTATORS OR, EQUIVALENTLY, EL- EMENTARY MATRICES TO GENERATE THE EQUIVALENCE RELATION, WHEREAS THE TOPOLOGICAL APPROACH USES HOMOTOPY WE WILL ONLY STUDY THE TOPOLOGICAL VERSION OF K1 THIS THEORY IS CONSIDERABLY EASIER TO COMPUTE THAN ITS ALGEBRAIC COUNTERPART, USING VAR- IOUS TOOLS FROM ALGEBRAIC TOPOLOGY SINCE THE DE?NITION REQUIRES A CERTAIN AMOUNT OF FUNCTIONAL ANALYSIS, WE ONLY TREAT K1 IN CHAPTER 2 WHEREAS HIGHER ALGEBRAIC K-THEORY IS EVEN MORE COMPLICATED THAN ALGEBRAIC K1, HIGHER TOPOLOGICAL K-THEORY GIVES NOTHING ESSENTIALLY NEW BY BOTT PERIODICITY PROJECTIVE MODULES, IDEMPOTENTS, AND VECTOR BUNDLES LET R BE A RING WITH UNIT LET MOD(R) BE THE CATEGORY OF LEFT R-MODULES WITH MODULE HOMOMORPHISMS AS MORPHISMS AS USUAL, WE REQUIRE THE UNIT ELEMENT OF R TO ACT IDENTICALLY ON MODULES DE?NITION 1 1 A LEFT R-MODULE M IS CALLED ?NITELY GENERATED IF THERE EXIST ?NITELY MANY ELEMENTS X1, , XN ?M SUCH THAT THE MAP RN ?M, (A1, , AN) ?A1X1 + + ANXN IS SURJECTIVE DE?NITION 1 2 A LEFT R-MODULE M IS CALLED PROJECTIVE IF ANY SURJECTIVE MODULE HOMOMORPHISM P: N ?M FOR ANY LEFT R-MODULE N SPLITS, THAT IS, THERE IS A MODULE HOMOMORPHISM S: M ?N SUCH THAT P ?S = IDM DE?NITION 1 3 LET V(R) BE THE SET OF ISOMORPHISM CLASSES OF ?NITELY GENERATED PROJECTIVE LEFT R-MODULES (WE WILL SEE BELOW THAT THIS IS A SET, NOT JUST A CLASS ) THE SET V(R) CONTAINS THE ZERO MODULE AND IS CLOSED UNDER DIRECT SUMS THUS THE DIRECT SUM OPERATION TURNS V(R) INTO A COMMUTATIVE MONOID EXAMPLE 1 4 THE CATEGORY MOD(Z) IS NOTHING BUT THE CATEGORY OF ABELIAN GROUPS THE CLASSI?CATION OF ?NITELY GENERATED ABELIAN GROUPS IMPLIES THAT ANY ?NITELY GENERATED PROJECTIVE Z-MODULE IS FREE THUS WE GET A MONOID ISOMORPHISM V(Z) ?= {[ZN] | N ?N} ?= (N, +) A SIMILAR ARGUMENT YIELDS V(R) ?= (N, +) IF R IS A ?ELD AND, MORE GENERALLY, IF R IS A PRINCIPAL IDEAL DOMAIN NOW WE DESCRIBE V(R) USING IDEMPOTENTS IN MATRIX RINGS OVER R DE?NITION 1 5 LET R BE A (POSSIBLY NON-UNITAL) RING WE LET MN(R) FOR N ?N BE THE RING OF N × N-MATRICES WITH ENTRIES IN R FOR M ?N, WE VIEW MM(R) AS A SUBRING OF MN(R) VIA X ? X LET M?(R) := N?N MN(R) EXERCISE 1 6 LET R?BE THE DIRECT SUM OF COUNTABLY MANY COPIES OF R LET MN(R) ACT ON RN BY MATRIX-VECTOR MULTIPLICATION ON THE RIGHT FOR N ?N ?{?} CHECK THAT THIS IDENTI?ES MN(R) FOR N ?N WITH THE RING HOMR(RN, RN) OF LEFT R-MODULE ENDOMORPHISMS OF RN CHECK THAT AN R-MODULE ENDOMORPHISM OF R?BELONGS TO M?(R) IF AND ONLY IF IT FACTORS THROUGH RN ?R?FOR SOME N ?N, IF AND ONLY IF IT FACTORS THROUGH SOME ?NITELY GENERATED R-MODULE DE?NITION 1 7 LET R BE A POSSIBLY NON-UNITAL RING AN ELEMENT E ?R IS CALLED IDEMPOTENT IF E2 = E WE LET IDEM R BE THE SET OF IDEMPOTENT ELEMENTS IN R WE CALL E1, E2 ?IDEM R EQUIVALENT AND WRITE E1 ?E2 IF THERE ARE V, W ?R WITH VW = E1 AND WV = E2 1 1 PROJECTIVE MODULES, IDEMPOTENTS, AND VECTOR BUNDLES