Algebraic Geometry of Topological Spaces I

ALGEBRAI C GEOMETR Y OF TOPOLOGICAL SP A CES I GUILLERMO COR TI ˜ NAS AND ANDREAS THOM Abstract. W e use tec hniques f rom b oth real and co mplex algebr aic geometry to s tudy K - theoretic and rela ted in v arian ts of the algebra C ( X ) of cont inuous complex-v alued f unctions on a compact Ha usdorﬀ to po lo gical space X . F or example, we prove a par ametrized version of a theor em of Joseph Gub eladze; we show t hat if M is a countable, ab elian, cancella- tive, torsion- free, semino r mal mono id, a nd X is co ntractible, then every ﬁnitely g e nerated pro jectiv e mo dule ov er C ( X )[ M ] is free. The particular case M = N n 0 gives a parametrized version of the ce lebrated theorem prov ed independently by Daniel Quillen and Andrei Suslin that ﬁnitely generated pro jectiv e mo dules ov er a p olynomia l ring ov er a ﬁeld are free. The conjecture of Jonathan Rosenber g which predicts the ho mo topy in v ar ia nce of the neg ative algebraic K -theory o f C ( X ) follows from the par ticular c a se M = Z n . W e als o give alge- braic conditions for a functor from commutativ e alg e br as to a belia n gr oups to b e homotopy inv ar iant on C ∗ -algebra s, and for a homolog y theory of commutativ e a lgebras to v anish on C ∗ -algebra s. These criteria hav e numerous applications. F or example, the v a nishing cri- terion applied to nil- K -theory implies that commutativ e C ∗ -algebra s a re K -regular . As another application, we show that the familiar formulas of Hochschild-Kostant -Rosenber g and Lo day-Quillen for the algebraic Ho chsc hild and cyc lic homo lo gy o f the co o rdinate ring of a smoo th alg ebraic v ariety remain v alid for the algebra ic Ho chsc hild and cyclic homolo gy of C ( X ). Applications to the co njectur e s of Be ˘ ılinson-Soul´ e and F arrell-J ones a re also given. Contents 1. In tro duction. 2 2. Split-exactness, homology theories, a nd excision. 8 2.1. Set-v alued split-exact functors on the catego r y of compact Hausdorﬀ spaces. 8 2.2. Algebraic K -theory . 10 2.3. Homology theories and excision. 11 2.4. Milnor squares and excision. 13 3. Real algebraic geometry and split exact functors. 13 3.1. General results ab out semi-algebraic sets. 14 3.2. The theorem on split exact functors and prop er maps. 16 4. Large semi-algebraic groups and the compact ﬁbration theorem. 17 4.1. Large semi-algebraic structures. 17 4.2. Construction of quotien ts of large semi-algebraic gro ups. 20 Key wor ds and phr ases. algebraic K- theory , Serre’s Co njecture, pro j ective mo dules, rings of cont inuous functions, algebra ic approximation. Corti ˜ nas’ r esearch was partly supp orted by gr ants PICT 2006-0 0 836, UBA CyT-X057 , and MTM20 0 7- 64704 . Thom’s resea rch was pa rtly suppo rted b y the DF G (GK Grupp en u nd Ge ometrie G¨ ottingen). 1 2 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM 4.3. Large semi-algebraic sets as compactly generated spaces. 21 4.4. The compact ﬁbration theorem for quotien ts of large semi-algebraic g roups. 22 5. Algebraic compactness, b ounded sequences and alg ebraic appro ximation. 23 5.1. Algebraic compactness. 23 5.2. Bounded sequence s, algebraic compactness and K 0 -trivialit y . 24 5.3. Algebraic appro ximation and b ounded sequences. 25 5.4. The algebraic compactness theorem. 26 6. Applications: pro jectiv e modules, low er K -theory , and bundle theory . 27 6.1. P arametrized Gub eladze’s theorem and Ro sen b erg’s conjecture. 27 6.2. Application to bundle theory: lo cal t rivialit y . 29 7. Homotop y in v ariance. 29 7.1. F rom compact p olyhedra to compact spaces: a result of Calder-Siegel. 29 7.2. Second pro of of Rosen b erg’s conjecture. 31 7.3. The homotop y inv ariance theorem. 32 7.4. A v anishing theorem for homology theories. 34 8. Applications of the homotopy in v ariance a nd v anishing homolog y theorems. 34 8.1. K -regularit y for comm utativ e C ∗ -algebras. 34 8.2. Ho c hsc hild and cyclic homolo gy of commutativ e C ∗ -algebras. 35 8.3. The F arrell-Jones Isomorphism Conjecture. 37 8.4. Adams op erations and the decomp osition of r ational K - theory . 38 References 39 1. Intr oduction. In his foundational paper [ 39 ], Jean-Pierre Serre ask ed whether all ﬁnitely generated pro jectiv e mo dules o v er the p olynomial ring k [ t 1 , . . . , t n ] o v er a ﬁeld k are free. This ques- tion, whic h became know n as Serre’s conjecture, remained op en for ab out tw en t y y ears. An aﬃrmativ e answ er w as giv en indep enden tly b y Daniel Quillen [ 35 ] a nd Andrei Suslin [ 42 ]. Ric hard G. Swan observ ed in [ 45 ] that t he Quillen-Suslin theorem implies tha t all ﬁnitely generated pro jectiv e modules ov er the Laurent p olynomial ring k [ t 1 , t − 1 1 . . . , t n , t − 1 n ] are free. This w as later generalized b y Joseph Gub eladze [ 20 ], [ 19 ], who pro v ed, among other things, that if M is an ab elian, cance llativ e, torsion-free, seminormal m onoid, the n ev ery ﬁnitely generated pro jectiv e mo dule o v er k [ M ] is free. Quillen-Suslin’s theorem and Swan’s theorem are the sp ecial cases M = N n 0 and M = Z n of Gub eladze’s result. On the ot her hand, it is classical that if X is a con tractible compact Ha usdorﬀ space, then all ﬁnitely generated pro jectiv e mo dules ov er the algebra C ( X ) of complex-v alued con tin uous functions on X – whic h b y another theorem of Sw an, a re the same thing as lo cally trivial complex v ector bundles on X – are free. In this pap er w e pro v e (see 6.1.3 ): Theorem 1.1. L et X b e a c o n tr actible c omp act sp ac e, and M a c ountable, c anc el la tive, torsion-fr e e, semino rmal, ab elian monoid . Then every ﬁnitely gener ate d pr o j e ctive mo dule over C ( X )[ M ] is fr e e. ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 3 Moreo v er w e show (Theorem 6.2.1 ) that bundles of ﬁnitely generated free C [ M ]-mo dules o v er a not necessarily con tractible, compact Hausdorﬀ X whic h are direct summands of trivial bundles, are lo cally trivial. The case M = N n 0 of Theorem 1.1 gives a parametrized v ersion of Quillen-Suslin’s theorem. The case M = Z n is connected with a conjecture of Jonathan Rosen b erg [ 38 ] whic h predicts that the negativ e alg ebraic K - theory groups of C ( X ) are homotop y inv ariant for compact Hausdorﬀ X . Indeed, if R is an y ring , then the negativ e alg ebraic K - t heory group K − n ( R ) is deﬁned as a certain canonical direct summand of K 0 ( R [ Z n ]); the theorem ab ov e thus implies that K − n ( C ( X )) = 0 if X is contractible. Using this and excision, w e deriv e the follow ing result (see Theorem 6.1.5 ). Theorem 1.2. L et Comp b e the c ate g o ry of c omp act Hausdorﬀ sp a c es and let n > 0 . Then the functor Comp → Ab , X 7→ K − n ( C ( X )) is homotopy invariant. A partial result in the direction of Theorem 1.2 w as obtained by Eric F riedlander and Mark E. W alk er in [ 15 ]. They pro v ed that K − n ( C (∆ p )) = 0 for p ≥ 0, n > 0. In Section 7 .2 , w e give a second pro of of Theorem 1.2 whic h uses the F riedlander-W alker result. Elab orating on their tec hniques, and combining them with our own methods, w e obta in the followin g general criterion for homotop y in v ariance (see Theorem 7.3.1 ). Theorem 1.3. L et F b e a functor on the c ate g ory Comm / C o f c ommutative C -alg e br as with values in the c ate gory Ab of ab elian gr oups. Assume that the fol lo wing thr e e c ond i tion s ar e satisﬁe d. (i) F is split-exact on C ∗ -algebr as. (ii) F van i s h es on c o or din a te rings of sm o oth aﬃn e varieties. (iii) F c ommutes with ﬁltering c olimits. Then the functor Comp → Ab , X 7→ F ( C ( X )) , is homotopy invariant and F ( C ( X )) = 0 for X c ontr actible. Observ e that K − n satisﬁes all the h yp othesis of the theorem ab ov e ( n > 0 ) . This giv es a third pro of of Theorem 1.2 . W e also use Theorem 1.3 t o pro v e the follo wing v anishing theorem fo r homology theories (see 7.4.1 ). In this pap er a homology theory on a category C of alg ebras is simply a functor E : C → Spt to the category of sp ectra whic h preserv es ﬁnite pro ducts up to homotop y . Theorem 1.4. L et E : Comm / C → Spt b e a homolo g y the ory of c omm utative C -algebr as and n 0 ∈ Z . Assume that the fol lo wing thr e e c ondi tion s ar e satisﬁe d. (i) E satisﬁes excision on c omm utative C ∗ -algebr as. (ii) E n c ommutes with ﬁltering c olimits for n ≥ n 0 . (iii) E n ( O ( V ) ) = 0 f o r e ach sm o oth aﬃn e algebr aic varie ty V for n ≥ n 0 . Then E n ( A ) = 0 for every c ommutative C ∗ -algebr a A for n ≥ n 0 . 4 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM Recall that a ring R is called K -regular if cok er ( K n ( R ) → K n ( R [ t 1 , . . . , t p ])) = 0 ( p ≥ 1 , n ∈ Z ) . As an application of Theorem 1.4 to the homology theory F p ( A ) = ho coﬁb er( K ( A ⊗ C O ( V )) → K ( A ⊗ C O ( V )[ t 1 , . . . , t p ]) where V is a smoo th algebraic v ariety , w e obtain the follo wing (Theorem 8.1.1 ). Theorem 1.5. L e t V b e a s m o oth aﬃne algebr aic variety over C , R = O ( V ) , and A a c ommutative C ∗ -algebr a. Then A ⊗ C R is K -r e g ular. The case R = C of the previous result w as disco vered by Jonathan Rosen b erg, see Remark 8.1.2 . W e also giv e an application of Theorem 1.4 whic h concerns the algebraic Ho c hsc hild a nd cyclic homolo gy of C ( X ). W e use the theorem in com bina t io n with the celebrated results o f Gerhard Ho ch sc hild, Bertram Kostan t and Alex Rosen b erg ([ 24 ]) and of D a niel Quillen and Jean-Louis Lo da y ([ 30 ]) on the Ho c hsc hild and cyclic ho molo gy of smoo t h aﬃne algebraic v arieties and the sp ectral sequence of Christian Kassel and Arne Sletsjøe ([ 27 ]), to pro v e the follow ing (see Theorem 8.2.6 fo r a full statemen t of our result and for the appropriate deﬁnitions). Theorem 1.6. L et k ⊂ C b e a subﬁeld. Write H H ∗ ( /k ) , H C ∗ ( /k ) , Ω ∗ /k , d and H ∗ dR ( /k ) for algeb r aic Ho chschild a nd cyclic homolo gy, alg e br aic K¨ ahler diﬀ e r ential forms, exterior diﬀer entiation, and alg e b r aic de Rham c ohomolo gy, al l taken r elative to the ﬁeld k . L et X b e a c omp act Hausdorﬀ sp ac e. Then H H n ( C ( X ) /k ) = Ω n C ( X ) /k H C n ( C ( X ) /k ) = Ω n C ( X ) /k /d Ω n − 1 C ( X ) /k ⊕ M 2 ≤ 2 p ≤ n H n − 2 p dR ( C ( X ) /k ) ( n ∈ Z ) W e also a pply Theorem 1.4 to the K -t heoretic isomorphism conjecture of F arrell-Jones and to the Beilison-Soul´ e conjecture. The K -theoretic isomorphism conjecture f o r the group Γ with co eﬃcien ts in a r ing R asserts that a certain assem bly map A Γ ( R ) : H Γ ( E V C (Γ) , K ( R )) → K ( R [Γ]) is a n equiv alence. Applying Theorem 1.4 t o the coﬁb er of the assem bly map, w e obtain that if A Γ ( O ( V ) ) is an equiv alence fo r eac h smo oth aﬃne algebraic v ariety V ov er C , then A Γ ( A ) is an equiv alence for any comm utativ e C ∗ -algebra A . The (rational) Be ˘ ılinson-Soul´ e conjecture concerns the decomp osition o f t he rational K -theory of a comm uta tiv e ring in to the sum of eigenspaces of the Adams op erations K n ( R ) ⊗ Q = ⊕ i ≥ 0 K ( i ) n ( R ) The conjecture asserts that if R is regular no etherian, then K ( i ) n ( R ) = 0 for n ≥ max { 1 , 2 i } It is w ell-kno wn that the v alidity of t he conjecture for R = C w ould imply that it also holds for R = O ( V ) whenev er V is a smo oth algebraic v ariety o ve r C . W e use Theorem 1.4 to ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 5 sho w that the v alidity of the conjecture for C would further imply that it holds for ev ery comm utativ e C ∗ -algebra. Next w e giv e an idea of the pro o fs of our main results , Theorem 1.1 and Theorem 1.3 . The basic idea of the pro of of Th eorem 1.1 go es bac k to Rosen b erg’s article [ 36 ] and ultimately to the usual pro of of the fact that lo cally trivial bundles ov er a contractible compact Hausdorﬀ space are trivial. It consists of translating the question of the freedom of pro jectiv e mo dules into a lifting pr o blem: (1.7) GL( C [ M ]) π   X e / / 3 3 P n ( C [ M ]) GL( C [ M ] ) GL [1 ,n ] ( C [ M ]) × GL [ n +1 , ∞ ) ( C [ M ]) ι o o Here we think of a pro jectiv e mo dule o f constan t r ank n o v er C ( X ) as a map e to the set of all rank n idemp oten t matrices, whic h by Gub eladze’s theorem is the same a s the set P n ( C [ M ]) of those matrices whic h are conjugate to the diagonal matrix 1 n ⊕ 0 ∞ . Th us g 7→ g (1 n ⊕ 0 ∞ ) g − 1 deﬁnes a surjectiv e map G L( C [ M ]) → P n ( C [ M ]) whic h identiﬁes the latter set with the quotien t o f G L( C [ M ]) b y the stabilizer of 1 n ⊕ 0 ∞ , whic h is precisely the subgroup GL [1 ,n ] ( C [ M ]) × GL [ n +1 , ∞ ) ( C [ M ]). F or t his setup to mak e sense we need to equip eac h set in v olv ed in ( 1.7 ) with a top ology in suc h a w a y that all maps in the diag ram are contin uous. Moreov er for the lifting problem to hav e a solution, it will suﬃc e to show that ι is a homeomorphism and that π is a c omp a ct ﬁbr ation , i.e. that it restricts to a ﬁbration ov er eac h compact subset of the base. In Section 4 w e show that an y coun table dimensional R -algebra R is equipp ed with a canonical compactly generated top ology whic h mak es it in to a top olog ical algebra. A subset F ⊂ R is closed in this top o logy if and only if F ∩ B ⊂ B is closed for ev ery compact semi-algebraic subset B of ev ery ﬁnite dimensional subspace of R . In particular this applies to M ∞ R . The subset P n ( R ) ⊂ M ∞ ( R ) carries the induced top olog y , and the map e of ( 1.7 ) is con tin uous fo r this to p ology . The gr oup GL( R ) also carries a top olo gy , g enerated b y the compact semi-algebraic su bsets GL n ( R ) B . Here B ⊂ M n R is an y compact semi-algebraic subset as b efore, and GL n ( R ) B consists of those n × n inv ertible matrices g suc h that b ot h g and g − 1 b elong to B . The subgroup GL [1 ,n ] ( R ) × GL [ n +1 , ∞ ) ( R ) ⊂ GL( R ) turns out to b e closed, and w e sho w in Section 4.2 – with the aid of Gregory Brumﬁel’s theorem on quotients of semi-algebraic sets (see 3.1.4 ) – that for a top o logical group G of t his kind, the quotien t G/H by a closed subgroup H is a g ain compactly generated b y the images of the compact semi-algebraic subsets deﬁning the top ology of G , and these images are aga in compact, se mi-algebraic subsets. Moreo v er the restriction of the pro jection π : G → G/H ov er eac h compact semi-algebraic subset S ⊂ G is semi-algebraic. W e also sho w (Theorem 4.4.3 ) that π is a compact ﬁbration. This b oils do wn to sho wing that if S ⊂ G is compact semi-algebraic, and T = f ( S ), then w e can ﬁnd a n op en co v ering o f T suc h that π has a section o v er eac h op en set in the co v ering. Next w e observ e that if U is a n y space, then the gro up map( U, G ) acts on the set map( U, G/H ), and a map U → G/H lifts to U → G if and only if its class in the quotien t 6 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM F ( U ) = map( U, G/H ) / map( U, G ) is the class of the trivial elemen t: the constan t map u 7→ H ( u ∈ U ) . F or example t he class of the comp osite of π with the inclusion S ⊂ G is the trivial elemen t of F ( S ). Hence if p = π | S : S → T , then F ( p ) sends the inclusion T ⊂ G/H to the tr ivial elemen t of F ( S ). In Section 2 w e introduce a notio n of (we ak) split exactness for con tra v arian t functors of top ological spaces with v alues in pointed sets; for example the functor F in t r o duced ab o v e is split exact (Lemma 2.1.3 ). The k ey tec hnical to ol f or pro ving that π is a ﬁbrat io n is the follow ing (see 3.2.1 ); its pro of uses the go o d to p ological prop erties of semi-algebraic sets and maps, esp ecially Hardt’s triviality theorem 3.1.10 . Theorem 1.8. L et T b e a c omp act semi-algebr aic subset of R k . L et S b e a semi-algebr aic set and let f : S → T b e a pr op er c ontinuous semi-algebr aic surje c tion . Then, ther e exists a semi-algebr aic triangulation of T such that for every we akly split-exact c ontr avariant functor F fr om the c ate gory Pol of c omp act p olyhe dr a to the c a te gory Set + of p ointe d sets, and e v e ry simplex ∆ n in the triangulation, we have k er( F (∆ n ) → F ( f − 1 (∆ n ))) = ∗ Here k er is t he k ernel in the category of p oin ted sets, i.e. the ﬁb er ov er the base p oin t. In our situation Theorem 1.8 applies to sho w that there is a triangula t io n of T ⊂ G/H suc h that the pro jection π ha s a section o v er eac h simplex in the triangulation. A standard argumen t no w shows tha t T has an op en cov ering (by op en stars of a sub division of t he previous triangulat io n) suc h tha t π ha s section ov er eac h op en set in the cov ering. Th us in diagram ( 1.7 ) w e ha v e that π is a compact ﬁbration and that e is contin uous. The map ι : GL( R ) / GL [1 ,n ] ( R ) × GL [ n +1 , ∞ ) ( R ) → P n ( R ) is contin uous for ev ery coun table dimensional R -algebra R ( see 5.1 ). W e show in Prop osition 5.2.2 that it is a homeomorphism whenev er the map (1.9) K 0 ( ℓ ∞ ( R )) → Y n ≥ 1 K 0 ( R ) is injectiv e. Here ℓ ∞ ( R ) is t he set o f all sequences N 7→ R whose image is con tained in one of the compact semi-algebraic subsets B ⊂ R whic h deﬁne the top olo gy of R ; it is isomorphic to ℓ ∞ ( R ) ⊗ R (Lemma 5.2.1 ). The algebraic compactness theorem ( 5.4.1 ) says that if R is a coun table dimensional C -a lgebra suc h that (1.10) K 0 ( O ( V )) ∼ − → K 0 ( O ( V ) ⊗ C R ) ( ∀ smo oth aﬃne V ) , then ( 1.9 ) is injectiv e. A theorem of Swan (see 6 .1 .2 ) implies that R = C [ M ] satisﬁes ( 1.10 ). Th us the map ι of diagram ( 1.7 ) is a homeomorphism. This conclude s the ske tc h of the pro of of Theorem 1.1 . The pro of o f the algebraic compactness theorem uses the f ollo wing theorem (see 5.3.1 ). Theorem 1.11. L et F and G b e functors fr om c ommutative C -algebr as to sets. Assume that b oth F an d G pr e s e rve ﬁltering c olim its. L et τ : F → G b e a natur al tr ansf o rmation. Assume that τ ( O ( V )) is inje ctive (r esp. surje ctive) for e ach s mo oth aﬃne algebr aic variety V ov e r C . Then τ ( ℓ ∞ ( C )) is inje c tive (r esp. surje ctive). ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 7 The pro of o f 1.11 uses a tec hnique whic h we call algebr aic appr oximation , whic h we no w explain. An y comm utativ e C -algebra is the colimit of its subalgebras of ﬁnite ty p e, whic h form a ﬁltered system. If the algebra con ta ins no nilp o ten t eleme n ts, then eac h of its subal- gebras o f ﬁnite t yp e is of the form O ( Y ) for aﬃne variety Y , b y whic h w e mean a reduced aﬃne sche me of ﬁnite t yp e ov er C . If C [ f 1 , . . . , f n ] ⊂ ℓ ∞ ( C ) is the subalgebra generated b y f 1 , . . . , f n , and C [ f 1 , . . . , f n ] ∼ = O ( Y ), then Y is isomorphic t o a closed sub v ariet y of C n , and f = ( f 1 , . . . , f n ) deﬁnes a map fro m N to a precompact subset of the space Y an of closed p oin ts of Y equipp ed the top ology inherited b y the euclidean top olo gy on C n . The space Y an is equipp ed with the structure of a (p ossibly singular) analytic v ariety , whence the subscript. Summing up, w e ha v e (1.12) ℓ ∞ ( C ) = colim N → Y an O ( Y ) where the colimit runs ov er all aﬃne v arieties Y and all maps with precompact image. The pro of of Theorem 1.11 consists of sho wing that in ( 1.12 ) w e can restrict to maps N → V with V smo oth. This uses Hiro nak a’s desingularization [ 23 ] to lift a map f : N → V with V aﬃne and singular, to a map f ′ : N → ˜ V with ˜ V smo oth and p ossibly non- aﬃne, and Jouanoulou’s device [ 25 ] to further lift f ′ to a ma p f ′′ : N → W with W smo oth and aﬃne. The idea of algebraic appro ximation app ears in the w ork of Jonathan Ro sen b erg [ 36 , 37 , 38 ], and later in the ar t icle of Eric F riedlander and Mark E. W alk er [ 15 ]. One source of inspiration is the w ork of Andrei Suslin [ 41 ]. In [ 41 ], Suslin studies a n inclusion of alge- braically closed ﬁelds L ⊂ K and analyzes K successfully in terms of its ﬁnitely generated L -subalgebras. Next we sk etc h the pro of of Theorem 1.3 . The ﬁrst step is to reduce to the po lyhedral case. F or this w e use Theorem 1.13 below, prov ed in 7.1.3 . Its pro of uses another algebraic appro ximation arg umen t together with a result of Allan Calder and Jerrold Siegel, whic h sa ys that the righ t K an extension to Comp o f a homotopy inv aria nt functor deﬁned on Po l is homotop y in v arian t on Comp . Theorem 1.13. L et F : Comm → Ab b e a functor. Assume that F satisﬁes e ac h of the fol lowing c onditions. (i) F c ommutes with ﬁlter e d c olimi ts. (iii) The functor Pol → Ab , D 7→ F ( C ( D )) is homotopy invariant. Then the functor Comp → Ab , X 7→ F ( C ( X )) is homotopy invariant. Next, Prop osition 2.1.5 sa ys that w e can restrict to sho wing that F v anishes on con- tractible p olyhedra. Since any con tractible p olyhedron is a retract of its cone, whic h is a starlik e p olyhedron, w e further reduce to sho wing that F v anishes on starlik e p olyhedra. Using excision, w e may restrict once more, to prov ing that F (∆ p ) = 0 for all p . F or this w e follo w the strategy used by F riedlander-W alk er in [ 15 ]. T o star t , we use algebraic approxi- mation again. W e write (1.14) C (∆ p ) = colim ∆ p → Y an O ( Y ) 8 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM where the colimit runs o v er all con tin uous maps fr o m ∆ p to a ﬃne algebraic v arieties, equipped with the euclidean top ology . Since F is assumed to v anish on O ( V ) for smo oth aﬃne V , it w ould suﬃce to sho w that an y map ∆ p → Y an factors as ∆ p → V an → Y an with V smo ot h and a ﬃne. Actually using excision again w e ma y restrict to sho wing this for each simplex in a suﬃcien tly ﬁne triangulation of ∆ p . As in the pro of of Theorem 1.1 , this is done using Hironak a’s desingularization, Joua no ulou’s device and Theorem 1.8 . The rest of this pap er is organized as follo ws. In Section 2 w e giv e the appro pria te deﬁnitions and ﬁrst prop erties of split exactness. W e also recall some facts ab out a lgebraic K - theory and cyclic homology , suc h as the k ey r esults of Andrei Suslin and Mariusz W o dz- ic ki on excision for algebraic K -theory and alg ebraic cyclic homolog y . In Section 3 , we recall some facts from r eal algebraic geometry , and prov e Theorem 1.8 ( 3.2.1 ). Large semi-algebraic groups and their asso ciated compactly generated top ological g roups ar e the sub ject of Sec- tion 4 . The ma in result of this se ction is the Fibration Theorem 4.4.3 whic h sa ys that the quotien t ma p of suc h a group by a closed subgroup is a compact ﬁbration. Section 5 is de- v oted to algebraic compactness , that is, to the problem of giving conditions o n a coun table dimensional algebra R so that the map ι : GL( R ) / GL [1 ,n ] ( R ) × GL [ n +1 , ∞ ) ( R ) → P n ( R ) b e a homeomorphism. The connection b et ween this problem and the algebra ℓ ∞ ( R ) of b o unded sequence s is established by Prop osition 5.2.2 . Theorem 1.11 is prov ed in 5.3.1 . Theorem 5.4.1 establishes that the map ( 1 .9 ) is injectiv e whenev er ( 1 .10 ) holds. Section 6 con tains the pro of s of Theorems 1.1 and 1.2 ( 6.1.3 a nd 6 .1.5 ). W e also sho w (Theorem 6.2 .1 ) that if M is a monoid as in Theorem 1.1 then an y bundle o f ﬁnitely generated free C [ M ] mo dules o v er a compact Hausdorﬀ space whic h is a direct summand of a tr ivial bundle is lo cally triv- ial. Section 7 deals with homotop y inv ariance. Theorems 1.13 , 1.3 and 1.4 ( 7.1.3 , 7.3.1 and 7.4.1 ) are pro v ed in this section, where also a second pro of of Rosen b erg’s conjecture, using a res ult of F riedlander and W alk er, is giv en (see 7.2 ). Section 8 is devoted t o applications of the homotopy inv aria nce and v anishing homology theorems, including Theorems 1.5 and 1.6 ( 8.1.1 and 8.2.6 ) and also to the a pplications to the conjectures of F arrell-Jones ( 8.3.2 , 8.3.6 ) and of Be ˘ ılinson-Soul´ e ( 8.4.4 ). 2. Split-exactness, homolo gy theories , and excision. 2.1. Set-v alued split-exact functors on the category of compact H ausdorﬀ spaces. In this section w e consid er con tra v arian t functors from the category of compact Hausdorﬀ top ological spaces to the categor y Se t + of p ointed sets. Recall that if T is a p ointed set and f : S → T is a map, then k er f = { s ∈ S : f ( s ) = ∗} W e sa y that a functor F : Comp → Se t + is split exact if for eac h push-out square (2.1.1) X 12 ι 1 / / ι 2   X 1   X 2 / / X ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 9 of top ological spaces with ι 1 or ι 2 split injectiv e, the map F ( X ) → F ( X 1 ) × F ( X 12 ) F ( X 2 ) is a surjection with trivial k ernel. W e sa y that F is we akly split exact if the map ab ov e has trivial k ernel. In case the functor tak es v alues in ab elian groups, the notion of split exactness ab ov e is equiv alen t to the usual one. F or more details on split-exact functors ta king v a lues in the category Ab of ab elian groups, see Subsection 2.4 . In the next lemma and elsewhe re, if X and Y are top olog ical spaces, w e write map( X , Y ) = { f : X → Y con tin uous } for the set of contin uous maps from X to Y . Lemma 2.1.2. L et ( Y , y ) b e a p ointe d top olo gic al sp ac e. T he c o ntr avariant functor X 7→ map( X, Y ) fr om c omp act Hausdo rﬀ top olo gic al sp ac es to p oin te d sets is split-exact. Pr o of. Note that map( X, Y ) is natura lly p oin ted b y the constant map taking the v alue y ∈ Y . Let ( 2.1.1 ) b e a push-out of compact Hausdorﬀ top olo gical spaces and assume that ι 1 is a split-injection. It is suﬃcien t to sho w that the diagram map( X 12 , Y ) map( X 1 , Y ) o o map( X 2 , Y ) O O map( X , Y ) O O o o is a pull-bac k. But this is immediate from the univ ersal prop ert y of a push-out.  Lemma 2.1.3. L e t H ⊂ G b e an inclusion of top olo gic al gr oups. Then the p ointe d set map( X , G/H ) c arri e s a natur al left action of the gr oup map( X, G ) and the functor X 7→ map( X , G/H ) map( X , G ) is split exact. Pr o of. W e need to sho w that the map (2.1.4) map( X , G/H ) map( X , G ) → map( X 1 , G/H ) map( X 1 , G ) × map( X 12 ,G/H ) map( X 12 ,G ) map( X 2 , G/H ) map( X 2 , G ) is a surjection with trivial k ernel. Let f : X → G/H b e suc h that its pull-backs f i : X i → G/H admit con tin uous lifts ˆ f i : X i → G . Althoug h the pull- bac ks o f ˆ f 1 and ˆ f 2 to X 12 migh t not agree, w e can ﬁx this problem. L et σ b e a con tinuous splitting of the inclusion X 12 ֒ → X 1 . Deﬁne a map γ : X 1 → H, γ ( x ) = ( ˆ f 1 | X 12 ( σ ( x )) − 1 · ( ˆ f 2 | X 12 ( σ ( x )) 10 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM Note ˆ f 1 · γ is still a lift of f 1 and agrees with ˆ f 2 on X 12 ; hence they deﬁne a map ˆ f : X → G whic h lifts f . This prov es that ( 2.1.4 ) has tr ivial kerne l. Let now f 1 : X 1 → G/ H and f 2 : X 2 → G/H b e suc h that there exists a function θ : X 12 → G with θ ( x ) · f 1 ( x ) = f 2 ( x ) for a ll x ∈ X 12 . Using the splitting σ of the inclusion X 12 ֒ → X 1 again, we can extend θ to X 1 to obta in f ′ 1 ( x ) = θ ( σ ( x )) f 1 ( x ), for x ∈ X 1 . Note f ′ 1 is just ano t her represen tativ e o f the class o f f 1 . Since f ′ 1 and f 2 agree on X 12 , we conclude that there exists a con tin uous map f : X → G/H , whic h pulls bac k to f ′ 1 on X 1 and to f 2 on X 2 . This prov es that ( 2.1.4 ) is surjectiv e.  Prop osition 2.1.5. L et C b e either the c ate gory Comp of c omp act Hausdorﬀ sp ac es or the ful l sub c ate gory Pol of c o m p act p olyhe dr a. L et F : C → Ab b e a split-exa c t functor. Assume that F ( X ) = 0 for c ontr actible X ∈ C . Then F is homotopy inva riant. Pr o of. W e hav e to prov e that if X ∈ C and 1 X × 0 : X → X × [0 , 1] is the inclusion, then F (1 X × 0) : F ( X × [0 , 1]) → F ( X ) is a bijection. Since it is ob viously a split-surjection it remains to sho w that this map is injectiv e. Consider the pushout diagram X   1 X × 0 / / X × [0 , 1]   ⋆ / / cX By split-exactness , the map F ( cX ) → P := F ( ∗ ) × F ( X ) F ( X × [0 , 1]) is o nto. Since we are also assuming that F v anishes on con tractible spaces, w e further hav e F ( ∗ ) = F ( cX ) = 0, whence P = k er( F (1 X × 0)) = 0.  2.2. Algebraic K -theory . In the previous subsection w e considered contra v ariant functors on spaces; now w e turn our atten t io n to the dual picture o f co v arian t functors fr o m categories of alg ebras to p oin ted sets or ab elian groups. The most imp ortant example for us is alg ebraic K - theory . Before w e go on, w e w an t to quick ly recall some deﬁnitions and results. L et R b e a unita l ring. The ab elian group K 0 ( R ) is deﬁne d to b e the Grothendiec k group of the monoid o f isomorphism classes of ﬁnitely generated pro j ectiv e R - mo dules with direct sum as addition. W e deﬁne K n ( R ) = π n ( B GL( R ) + ) , ⋆ ) , ∀ n ≥ 1 , where X 7→ X + denotes Q uillen’s plus-construction [ 34 ]. Bass’ Nil K -gro ups of a ring are deﬁned as (2.2.1) N K n ( R ) = coke r ( K n ( R ) → K n ( R [ t ])) . The so-called fundamen tal theorem gives an isomorphism (2.2.2) K n ( R [ t, t − 1 ]) = K n ( R ) ⊕ K n − 1 ( R ) ⊕ N K n ( R ) ⊕ N K n ( R ) , ∀ n ≥ 1 whic h holds for all unital ring s R . One can use this to deﬁne K -groups and Nil- g roups in negativ e degrees. Indeed, if one puts K n − 1 ( R ) = cok er  K n ( R [ t ]) ⊕ K n ( R [ t − 1 ]) → K n ( R [ t, t − 1 ])  , ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 11 negativ e K -groups can b e deﬁned inductiv ely . There is a functorial sp ectrum K ( R ), suc h that (2.2.3) K n ( R ) = π n K ( R ) ( n ∈ Z ) . This sp ectrum can b e constructed in sev eral equiv alent w ays ( e.g. see [ 1 7 ], [ 3 2 ],[ 33 , § 5], [ 49 , § 6], [ 50 ]). F unctors from the category of algebras to sp ectra and their pro p erties will b e studied in more detail in the next subsection. A ring R is called K n - r e gular if the map K n ( R ) → K n ( R [ t 1 , . . . , t m ]) is an isomorphism for all m ; it is called K - r e gular if it is K n -regular for all n . It is well-kno wn that if R is a regular no etherian ring then R is K - regular and K n R = 0 for n < 0. In particular this applies when R is the co ordinate ring of a smo oth aﬃne algebraic v ariet y o v er a ﬁeld. W e think of the Lauren t p olynomial ring R [ t 1 , t − 1 1 , . . . , t n , t − 1 n ] as the gro up ring R [ Z n ] and use the fact that if the natural map K 0 ( R ) → K 0 ( R [ Z n ]) is an isomor phism for all n ∈ N , then all negative algebraic K -gr oups and a ll (iterated ) nil- K -groups in negativ e degrees v anish. This can b e pro v ed with an easy induction argumen t. R emark 2.2.4 . Iterating the nil-group construction, one obtains the following formula fo r the K - theory of the p olynomial ring in m -v ariables (2.2.5) K n ( R [ t 1 , . . . , t m ]) = m M p =0 N p K n ( R ) ⊗ p ^ Z m Here V p is the exterior p ow er and N p K n ( R ) denotes the iterated nil-group deﬁned using the analogue of formula ( 2.2.1 ). Th us a ring R is K n -regular if and only if N p K n ( R ) = 0 for all p ≥ 0. In [ 1 ] Hyman Bass raised the question of whether the condition that N K n ( R ) = 0 is already suﬃcien t f o r K n -regularity . This ques tion w as settled in the negat ive in [ 9 , Thm. 4.1], where an example of a comm utativ e a lg ebra R of ﬁnite type o v er Q was g iv en suc h that N K 0 ( R ) = 0 but N 2 K 0 ( R ) 6 = 0. On the other hand it was pro v ed [ 8 , Cor. 6.7 ] (see also [ 21 ]) that if R is of ﬁnite t yp e o v er a la r ge ﬁeld suc h as R or C , then N K n ( R ) = 0 do es imply that R is K n -regular. This is already suﬃcien t for our purp oses, since the r ing s this pap er is concerned with are alg ebras ov er R . F or completeness let us remark further that if R is an y ring suc h that N K n ( R ) = 0 for al l n then R is K -regula r, i.e. K n -regular for all n ∈ Z . As observ ed b y Jim Davis in [ 11 , Cor. 3] this f ollo ws from F rank Quinn’s t heorem that the F arrell-Jones conjecture is v alid for the group Z n (see also [ 8 , 4.2]). 2.3. Homology theories and excision. W e consider functors and homology t heories of asso ciativ e, not necessarily unital alg ebras ov er a ﬁxed ﬁeld k of characteristic zero. In what follo ws, C will denote either the category Ass /k of asso ciativ e k - algebras or the full sub category Co mm /k of comm utativ e algebras. A homol o gy the ory on C is a functor E : C → S pt to the catego ry of sp ectra whic h preserv es ﬁnite pro ducts up to ho mo t o p y . That is, E ( Q i ∈ I A i ) → Q i ∈ I E ( A i ) is a we ak equiv alence for ﬁnite I . If A ∈ C and n ∈ Z , w e write E n ( A ) = π n E ( A ) for the n -th stable homotopy group. Let E b e a homology theory and let (2.3.1) 0 → A → B → C → 0 12 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM b e an exact sequence (or extension) in C . W e say that E satisﬁes excisio n for ( 2.3.1 ), if E ( A ) → E ( B ) → E ( C ) is a homotop y ﬁbration. The algebra A is E -excisiv e if E satisﬁes excision on an y extension ( 2.3.1 ) with k ernel A . If A ⊂ C is a subcategory , and E satisﬁes excision for ev ery sequenc e ( 2.3.1 ) in A , then w e sa y that E satisﬁe s excis ion o n A . R emark 2.3.2 . If we ha v e a functor E whic h is o nly deﬁned on the sub category C 1 ⊂ C of unital alg ebras and unital homomorphisms, and whic h preserv es ﬁnite pro ducts up to homotop y , then w e can extend it to all of C b y setting E ( A ) = hoﬁb er( E ( A + k ) → E ( k )) Here A + k denotes the unitalization of A as a k -algebra. The restriction of the new functor E to unital algebras is not the same as the old one, but it is ho mo t o p y equiv alen t to it. Indeed, for A unital, w e hav e A + k ∼ = A ⊕ k as k - algebras. Since E preserv es ﬁnite pro ducts, this implies the claim. In this article, whenev er w e encoun ter a homolo gy theory deﬁned only on unital algebras, w e shall implicitly consider it extended to non-unital algebras b y the pro cedure just explained. Similarly , if F : C 1 → Ab is a functor to ab elian gr o ups whic h preserv es ﬁnite pro ducts, it extends to all of C b y F ( A ) = k er( F ( A + k ) → F ( k )) In particular this applies when F = E n is the ho mo lo gy functor associated to a homology theory as ab ov e. The main examples of homolo gy theories w e are intereste d in are K -t heory , Ho c hsc hild homology and the v arious v a r ian ts of cyclic homology . A milestone in understanding excision in K -theory is the follo wing result of Andrei Suslin and Mariusz W o dzic ki, see [ 43 , 44 ]. Theorem 2.3.3 (Suslin-W o dzic ki) . A Q -algebr a R is K -excisive if and only if for the Q - algebr a unitalization R + Q = R ⊕ Q we have T or R + Q n ( Q , R ) = 0 for al l n ≥ 0 . F or example it w as sho wn in [ 44 , Thm. C] that any ring satisfying a certain “triple factorization prop erty” is K -excisiv e; since an y C ∗ -algebra has this prop erty , ([ 44 , Prop. 10.2]) w e ha v e Theorem 2.3.4 (Suslin-W o dzic ki) . C ∗ -algebr as ar e K -excisi v e. Excision for Ho c hsc hild and cyclic homology of k -algebras, denoted resp ectiv ely H H ( /k ) and H C ( /k ), has b een studied in detail by W o dzic ki in [ 5 2 ]; as a pa rticular case of his results, w e cite the follo wing: Theorem 2.3.5 (W o dzic ki) . The fol low ing ar e e quivalent for a k -alge br a A . (1) A is H H ( /k ) -excisi v e. (2) A is H C ( /k ) -excisive. (3) T or A + k ∗ ( k , A ) = 0 . ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 13 Note that it follo ws f rom ( 2.3.3 ) and ( 2 .3 .5 ) that a k -algebra A is K -excisiv e if a nd only if it is H H ( / Q )-excisiv e. R emark 2.3.6 . W o dzic ki has pro v ed (see [ 53 , Theorems 1 and 4]) that if A is a C ∗ -algebra then A satisﬁes the conditions of Theorem 2.3.5 for an y subﬁeld k ⊂ C . 2.4. Milnor squares and excision. W e no w record some facts ab out Milnor squares of k -algebras and excision. Deﬁnition 2.4.1. A square of k -algebras (2.4.2) A / /   B f   C g / / D is said to b e a Milnor squar e if it is a pull-back square and either f or g is surjectiv e. It is said to b e split if either f or g has a section. Let F b e a functor from C to ab elian groups and let (2.4.3) 0 / / A / / B / / C u u / / 0 b e a split extension in C . W e sa y that F is sp l i t exact on ( 2.4.3 ) if 0 / / F ( A ) / / F ( B ) / / F ( C ) / / 0 is (split) exact. If A ⊂ C is a sub catego r y , and F is split exact on all split exact sequences con tained in A , then w e sa y that F is split exact on A . Lemma 2.4.4. L et E : C → Spt b e a homolo gy the o ry and ( 2.4.2 ) a Milnor squar e. Assume that ker( f ) is E -e x cisive. T hen E maps ( 2.4.2 ) to a homotopy c artesian squar e. Lemma 2.4.5. L et F : C → Ab b e a functor, A ⊂ C a sub c ate gory close d under kernels, an d ( 2.4.2 ) a split Milnor sq uar e in A . Assume that F is split exact on A . Then the se quenc e 0 → F ( A ) → F ( B ) ⊕ F ( C ) → F ( D ) → 0 is split exact. 3. Real algeb raic geometr y and split exact functors. In this section, w e recall sev eral res ults from real algebraic geometry and pro ve a the- orem on the b eha vior of w eakly split exact functors with resp ect to prop er semi-algebraic surjections (see Theorem 3.2.1 ). Recall that a semi-algebraic set is a priori a subset of R n whic h is describ ed as the solution set of a ﬁnite n um b er of p o lynomial equalities and inequal- ities. A map b etw een semi-algebraic sets is semi-algebraic if its graph is a semi-algebraic set. F or general bac kground on semi-algebraic sets, consult [ 2 ]. 14 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM 3.1. General results ab out semi-algebraic sets. Let us start with recalling the f o llo wing t w o prop ositions. Prop osition 3.1.1 (see Prop osition 3 . 1 in [ 2 ]) . The closur e of a semi-algebr a ic set is se m i- algebr aic. Prop osition 3.1.2 (see Prop o sition 2 . 83 in [ 2 ]) . L et S an d T b e semi-algebr aic sets, S ′ ⊂ S and T ′ ⊂ T se mi-algebr aic subsets and f : S → T a semi - a lgebr aic map . Then f ( S ′ ) and f − 1 ( T ′ ) ar e semi-a l g ebr aic. Note that a semi-algebraic map do es not need to b e contin uous. Moreo v er, within the class of contin uous maps, there are surjectiv e maps f : S → T , for whic h the quotien t top ology induced by S do es not a gree with the top o lo gy on T . An easy example is the pro jection map from { (0 , 0 ) } ∪ { ( t, t − 1 ) | t > 0 } t o [0 , ∞ ). This motiv at es the following deﬁnition. Deﬁnition 3.1.3. Let S, T b e semi-algebraic sets. A con tinu ous semi-algebraic surjection f : S → T is said to b e top olo gic al , if for ev ery semi-algebraic map g : T → Q the comp osition g ◦ f is con tin uous if and only if g is con tin uous. Gregory Brumﬁel pro v ed the followin g result, whic h says that (under certain conditions) semi-algebraic equiv alence relations lead t o g o o d quotien ts. Theorem 3.1.4 (see Theorem 1.4 in [ 5 ]) . L et S b e a semi-algebr aic set and let R ⊂ S × S b e a close d semi-algebr aic e quiva lenc e r elation. If π 1 : R → S i s pr o p er, then ther e exists a semi-algebr aic set T and a top olo gic al semi-algebr aic surje ction f : S → T such that R = { ( s 1 , s 2 ) ∈ S × S | f ( s 1 ) = f ( s 2 ) } . R emark 3.1.5 . Note that the prop erness a ssumption in the previous theorem is automatically fulﬁlled if S is compact. This is the case w e are in terested in. Corollary 3.1.6. L et S, S ′ and T b e c om p act sem i-algebr aic s ets a n d f : T → S and f ′ : T → S ′ b e c on tinuous semi-algeb r aic m a p s. Then, the top olo gic al push-out S ∪ T S ′ c arries a c anonic al semi-algebr aic structur e such that the natur al maps σ : S → S ∪ T S ′ and σ ′ : S ′ → S ∪ T S ′ ar e semi-algebr ai c . F or semi-algebraic sets, there is an in trinsic not io n of connectedness, whic h is giv en b y the follo wing deﬁnition. Deﬁnition 3.1.7. A semi-algebraic set S ⊂ R k is said to b e sem i-algebr aic al ly c onne cte d if it is not a non-trivial union of semi-algebraic subsets whic h are b o t h op en and closed in S . One o f the ﬁrst r esults o n connectedness of semi-algebraic sets is the follow ing theorem. Theorem 3.1.8 (see Theorem 5.20 in [ 2 ]) . Every semi-algebr aic set S is the disjoint union of a ﬁni te numb er of semi-a l g ebr aic al ly c onne cte d semi-alge br aic s e ts which ar e b oth op en and cl o se d i n S . ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 15 Next w e come to asp ects of semi-algebraic sets and con tinuous semi-algebraic maps whic h diﬀer dra stically from the exp ected results for general con tin uous maps. In fact, there is a far- reac hing generalization of Ehresmann’s theorem ab out lo cal triviality of submersions. Let us consider the follow ing deﬁnition. Deﬁnition 3.1.9. Let S and T b e t w o se mi-algebraic sets and f : S → T b e a con tin uous semi-algebraic function. W e sa y that f is a semi-algebr aic al ly trivial ﬁbr ation if there exist a semi-algebraic set F a nd a semi-algebraic homeomorphism θ : T × F → S suc h that f ◦ θ is the pro jection onto T . A seminal theorem is Ha r dt ’s triviality result, whic h sa ys that aw a y from a subset of T of smaller dimension, ev ery map f : S → T lo oks lik e a semi-algebraically tr ivial ﬁbration. Theorem 3.1.10 ([ 2 ] or § 4 in [ 22 ]) . L et S and T b e two semi-algebr aic sets and f : S → T a c ontinuous semi-algeb r aic function. Then ther e exists a close d semi-algebr aic subset V ⊂ T with dim V < dim T , such that f is a semi-a l g ebr aic al ly trivial ﬁbr ation over every semi- algebr aic c onne cte d c omp o nent of T \ V . W e shall also need the follo wing result ab out semi-algebraic triangulat io ns. Theorem 3.1.11 (Thm. 5 . 4 1 in [ 2 ]) . L et S ⊂ R k b e a c om p act semi- a lgebr aic set, and let S 1 , . . . , S q b e s e mi-algebr aic subsets. Ther e exists a simplicial c omplex K and a semi-a lgebr aic home omorp hism h : | K | → S such that e a c h S j is the union of images of op en simp l i c es of K . R emark 3.1.12 . In the preceding theorem, the case where t he subsets S j are closed is o f special in terest. Indeed, if the sub sets S j are closed, the theorem implies that the triangulation of S induces triangulations of S j for eac h j ∈ { 1 , . . . , q } . The follo wing prop osition is an application of Theorems 3.1.10 and 3 .1 .11 Prop osition 3.1.13. L et T ⊂ R m b e a c omp act semi-algebr aic subset, S a semi-al g ebr aic set and f : S → T a c o ntinuous semi-algebr a i c map. The n ther e exist a s e m i-algebr aic triangulation of T and a ﬁnite se quenc e of cl o se d s ub c omplexes ∅ = V r +1 ⊂ V r ⊂ V r − 1 ⊂ · · · ⊂ V 1 ⊂ V 0 = T such that the fol lowing c onditions ar e satisﬁe d (i) F or e a c h k = 0 , . . . , r , we have dim V k +1 < dim V k , and the map f | f − 1 ( V k \ V k +1 ) : f − 1 ( V k \ V k +1 ) → V k \ V k +1 is a semi- algebr aic a l ly trivial ﬁbr ation over every semi-algebr a ic c on n e cte d c om p o- nent. (ii) Each simplex in the triangulation lies in som e V k and has at most one fac e of c o di- mension one which interse cts V k +1 . 16 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM Pr o of. W e set n = dim T . By Theorem 3.1.10 , there exists a closed semi-algebraic subset V 1 ⊂ T with dim V 1 < n , suc h that f is a semi-algebraic trivial ﬁbration ov er ev ery semi- algebraic connected comp onen t of T \ V 1 . Consider no w f | f − 1 ( V 1 ) : f − 1 ( V 1 ) → V 1 and pro ceed as b efore to ﬁnd V 2 ⊂ V 1 . By induction, w e ﬁnd a c hain ∅ ⊂ V r ⊂ V r − 1 ⊂ · · · ⊂ V 1 ⊂ V 0 = T suc h that V k ⊂ V k − 1 is a closed semi-algebraic subset and f | f − 1 ( V k − 1 \ V k ) : f − 1 ( V k − 1 \ V k ) → V k − 1 \ V k is a semi-algebraically trivial ﬁbration o v er ev ery semi-algebraic connected componen t, for all k ∈ { 1 , . . . , r } . Using Theorem 3.1.11 , w e ma y now ch o ose a semi-algebraic triangulation of T suc h that the subsets V k are sub-complexes. T aking a barycen tric sub division, each simplex lies in V k for some k ∈ { 0 , . . . , r } and has at most one face of co dimension one whic h in tersects the set V k +1 .  3.2. The theorem on split exact functors and prop er maps. The following result is our main tec hnical result. It is k ey to the pro ofs of Theorems 4.4.3 and Theorem 7.3.1 . Theorem 3.2.1. L et T b e a c om p act semi-algebr aic subset of R k . L et S b e a s emi-algebr aic set and let f : S → T b e a pr op er c ontinuous semi-algebr aic surje c tion . Then, ther e exists a semi-algebr aic triangulation of T such that for every we akly split-exact c ontr avariant functor F : Pol → Set + and ev e ry sim plex ∆ n in the triangulation, we have k er( F (∆ n ) → F ( f − 1 (∆ n ))) = ∗ Pr o of. Cho ose a triangulatio n of T and a seque nce of sub complexes V k ⊂ T as in Prop osition 3.1.13 . W e shall sho w that k er( F (∆ n ) → F ( f − 1 (∆ n ))) = ∗ for eac h simplex in the c hosen triangulation. The pro of is b y induction on the dimension of the simplex. The statemen t is clear for zero-dimensional simplices since f is surjectiv e. Let ∆ n b e an n -dimensional simplex in t he tr ia ngulation. By assumption, f is a semi-algebraically trivial ﬁbration o v er ∆ n \ ∆ n − 1 for some fa ce ∆ n − 1 ⊂ ∆ n . Hence, there exis ts a sem i-algebraic set K and a semi-algebraic homeomorphism θ : (∆ n \ ∆ n − 1 ) × K → f − 1 (∆ n \ ∆ n − 1 ) o v er ∆ n \ ∆ n − 1 . Consider the inclusion f − 1 (∆ n − 1 ) ⊂ f − 1 (∆ n ). Since f − 1 (∆ n − 1 ) is an absolute neigh b orho o d retract, there exists a compact neigh b orho o d N of f − 1 (∆ n − 1 ) in f − 1 (∆ n ) whic h retracts o nto f − 1 (∆ n − 1 ). W e claim that the set f ( N ) con tains some standard neigh b o rho o d A of ∆ n − 1 . Indeed, assume t ha t f ( N ) do es not con tain standard neighborho o ds. Then t here exists a sequence in the complemen t of f ( N ) con v erging to ∆ n − 1 . Lifting this sequence one can choose a conv ergen t sequence in the complemen t o f N conv erging to f − 1 (∆ n − 1 ). This con tradicts the fact that N is a neighborho o d and hence, there exists a standard compact neigh b orho o d A of ∆ n − 1 in ∆ n suc h that f − 1 ( A ) ⊂ N . Since (∆ n \ ∆ n − 1 ) × K ∼ = f − 1 (∆ n \ ∆ n − 1 ), an y retraction o f ∆ n on to A yields a retraction o f f − 1 (∆ n ) on to f − 1 ( A ). W e ha v e that f − 1 ( A ) ⊂ N , and th us we can conclude that f − 1 (∆ n − 1 ) is a retract of f − 1 (∆ n ). ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 17 By Corollary 3.1.6 , the top olog ical push-out f − 1 (∆ n − 1 ) / /   f − 1 (∆ n )   ∆ n − 1 / / Z carries a semi-algebraic structure. Moreo v er, by weak split- exactness (3.2.2) k er( F ( Z ) → F (∆ n − 1 ) × F ( f − 1 (∆ n − 1 )) F ( f − 1 (∆ n ))) = ∗ . Note that f − 1 (∆ n \ ∆ n − 1 ) ⊂ Z b y deﬁnition of Z . W e claim that the natura l map σ : Z → ∆ n is a semi-algebraic split surjection. Indeed, iden tify f − 1 (∆ n \ ∆ n − 1 ) = (∆ n \ ∆ n − 1 ) × K , pick k ∈ K , and consider G k = { ( d, ( d, k ) ) ∈ ∆ n × Z | d ∈ ∆ n \ ∆ n − 1 } . Then G k is semi-algebraic by Prop osition 3.1.1 , a nd therefore deﬁnes a con tin uous semi- algebraic map ρ k : ∆ n → Z whic h splits σ . Th us F ( σ ) : F (∆ n ) → F ( Z ) is injectiv e, whence (3.2.3) k er( F (∆ n ) → F (∆ n − 1 ) × F ( f − 1 (∆ n − 1 )) F ( f − 1 (∆ n ))) = ∗ , b y ( 3.2.2 ). But the k ernel of F (∆ n − 1 ) → F ( f − 1 (∆ n − 1 )) is trivial b y induction, so the same m ust b e true of F (∆ n ) → F ( f − 1 (∆ n )), b y ( 3.2.3 ).  4. Large semi-alge braic groups and the comp act fibra tion theorem. 4.1. Large semi-algebraic struct ures. Recall a partially ordered set Λ is ﬁlter e d if for an y t w o λ, γ ∈ Λ there exists a µ suc h that µ ≥ λ and µ ≥ γ . W e shall say t hat a ﬁltered p oset Λ is ar chime dian if there exists a monotone map φ : N → Λ from the ordered set of natural n um b ers whic h is coﬁnal, i.e. it is suc h that for ev ery λ ∈ Λ there exists an n ∈ N suc h that φ ( n ) ≥ λ . If X is a set, we write P ( X ) f o r t he partially ordered set of all subsets of X , ordered b y inclusion. A lar ge semi-algebr aic structur e on a set X consists of (i) An arc himedian ﬁltered partially ordered set Λ. (ii) A monotone map X : Λ → P ( X ), λ 7→ X λ , suc h t hat X = S λ X λ . (iii) A compact semi-algebraic structure on each X λ suc h that if λ ≤ µ then the inclusion X λ ⊂ X µ is semi-algebraic and con tin uous. W e think o f large a semi-algebraic structure on X as an exhaustiv e ﬁltratio n { X λ } by compact semi-algebraic sets. A structure { X γ : γ ∈ Γ } is ﬁner than a structure { X λ : λ ∈ Λ } if f o r ev ery γ ∈ Γ there exists a λ ∈ Λ so that X γ ⊂ X λ and the inclusion is con tin uous and semi-algebraic. Tw o structures are e quivalent if eac h of them is ﬁner than the o ther. A lar ge semi-algebr aic set is a set X together with an equiv a lence class of semi-algebraic structures on X . If X is a large semi-algebraic set, then a ny large semi-algebraic structure { X λ } in the equiv alence class deﬁning X is called a deﬁ ning structur e fo r X . If X = ( X, Λ ) and Y = ( Y , Γ) ar e large sem i-algebraic sets, then a set map f : X → Y is called a morphism if f or every λ ∈ Λ there exists a γ ∈ Γ suc h that f ( X λ ) ⊂ Y γ , and suc h that the induced 18 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM map f : X λ → Y γ is semi-algebraic and con tinu ous. W e write V ∞ for the category of large semi-algebraic sets. R emark 4.1.1 . If f : X → Y is a morphism o f large semi-algebraic sets, then w e may c ho ose structures { X n } and { Y n } indexed b y N , suc h t hat f strictly preserv es ﬁltra t io ns, i.e. f ( X n ) ⊂ Y n for all n . Ho wev er, if X = Y , then there ma y not exist a structure { X λ } suc h that f ( X λ ) ⊂ X λ . R emark 4.1 .2 . Consider the category V s,b of compact semi-algebraic sets with c ontinuous semi-algebraic mappings and its ind-category ind- V s,b . The ob jects are functors T : ( X T , ≤ ) → V s,b where ( X T , ≤ ) is a ﬁltered partially o r dered set. W e set hom( T , S ) = lim d ∈ X T co lim e ∈ X S hom V s,b ( T ( d ) , S ( e )) . The category of large semi-algebraic sets is equiv alen t to the sub categor y of those ind- ob jects whose structure maps a re all injectiv e and whose index po sets a r e archimede an. In particular, a ﬁltering colimit of an arc himedian system of injectiv e homomorphisms of large semi-algebraic sets is again a large semi-algebraic set. If X and Y are large semi-algebraic sets with structures { X λ : λ ∈ Λ } and { Y γ : γ ∈ Γ } then the cartesian pro duct X × Y is ag a in a large semi-algebraic set, with structure { X λ × Y γ : ( λ, γ ) ∈ Λ × Γ } . A lar ge semi-a l g ebr aic gr oup is a group ob ject G in V ∞ . Th us G is a group whic h is a la r ge semi-algebraic set and each of t he maps deﬁning the multiplication, unit, and in v erse, are homomorphisms in V ∞ . W e shall additionally a ssume that G admits a structure { G λ } such tha t G − 1 λ ⊂ G λ . This h yp othesis, although not strictly necessary , is v eriﬁed b y all the examples w e shall consider, and mak es pro ofs tec hnically simpler. W e shall also need the notions of large semi-algebraic v ectorspace and of large semi-algebraic ring, whic h are deﬁned similarly . Example 4.1.3 . An y semi-algebraic set S can b e considered as a large semi-algebraic set, with the structure deﬁned b y its compact semi-algebraic subsets, whic h is equiv alen t to the struc- ture deﬁned by an y exhaustiv e ﬁltr a tion of S by compact semi-algebraic subsets. In particular this applies to an y ﬁnite dimensional real v ectorspace V ; moreov er the vec torspace op era- tions are semi-algebraic and con tinuous, so that V is a (large) se mi-algebraic v ectorspace. An y linear map b et w een ﬁnite dimensional vec torspaces is semi-algebraic and contin uous, whence it is a homomorphism of semi-algebraic ve ctorspaces. Moreo ver, the same is true of an y m ultilinear map f : V 1 × · · · × V n → V n +1 b et w een ﬁnite dimensional ve ctorspaces. Deﬁnition 4.1.4. Let V b e a real v ectorspace of coun table dimension. The ﬁne larg e semi- algebraic structure F ( V ) is that give n b y all the compact semi-algebraic subsets of all the ﬁnite dimensional subspaces of V . R emark 4.1.5 . The ﬁne large semi-algebraic structure is reminiscen t of the ﬁne lo c al ly c o nvex top olo gy whic h makes ev ery complex algebra of coun table dimension into a lo cally con ve x algebra. F or details, see [ 4 ], c hap. I I, § 2, Exe rcise 5 . ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 19 Lemma 4.1.6. L et n ≥ 1 , V 1 , . . . , V n +1 b e c ountable dimens i o nal R -ve ctorsp ac es, and f : V 1 × · · · × V n → V n +1 a multiline ar map. Equip e ach V i with the ﬁne lar ge semi-a lgebr aic structur e and V 1 × · · · × V n with the pr o duct lar ge semi-algebr aic structur e. Then f is a morphism of lar ge semi-algebr ai c sets. Pr o of. In view of the deﬁnition of the ﬁne large semi-algebraic structure, the general case is immediate from the ﬁnite dimensional case.  Prop osition 4.1.7. L et A b e a c ountable dimens i o nal R -algebr a, e quipp e d with the ﬁne lar ge semi-algebr aic structur e . Then: (i) A is a lar ge s e m i-algebr aic ring. (ii) Assume A is unital. T h e n the gr oup GL n ( A ) to ge ther with the structur e F (GL n ( A )) = {{ g ∈ G L n ( A ) : g , g − 1 ∈ F } : F ∈ F ( M n ( A )) } is a lar ge sem i-algebr aic gr oup, and if A → B is an algebr a homom o rphism, then the induc e d gr oup h o momorphism GL n ( A ) → GL n ( B ) is a homom orphism of lar ge semi-algebr aic s ets. Pr o of. P art (i) is immediate from Lemma 4.1.6 . If F ∈ F ( M n ( A )), write (4.1.8) GL n ( A ) F = { g ∈ GL n ( A ) : g , g − 1 ∈ F } W e will sho w that GL n ( A ) F is a compact semi-algebraic set. W rite m, π i : M n ( A ) × M n ( A ) → M n ( A ) for the m ultiplication and pro jection maps i = 1 , 2, and τ : M n ( A ) × M n ( A ) → M n ( A ) × M n ( A ) for the p erm utation of factors. If A is unital, then GL n ( A ) F = π 1 ( m − 1 | F × F (1) ∩ τ m − 1 | F × F (1)) whic h is compact semi-alg ebraic, b y Prop osition 3.1.2 .  W e will also study the large semi-algebraic group GL S ( R ) for a subset S ⊂ Z . This is understo o d to be the group of mat rices g indexed b y Z , where g i,j = δ i,j if either i or j is not in S . Corollary 4.1.9. If A is unital and S ⊂ Z , then GL S ( A ) c arrie s a natur al lar ge semi- algebr aic gr oup structur e, nam ely that of the c olimit GL S ( A ) = S T GL T ( A ) wher e T runs among the ﬁnite subsets of S . R emark 4.1.10 . If A is an y , not necessarily unital ring, and a, b ∈ M n A , then a ⋆ b = a + b + ab is a n asso ciativ e op eration, with neutral elemen t the zero matrix; the group GL n ( A ) is deﬁned as the set of all matrices whic h are in v ertible under ⋆ . If A happens to b e unital, the resulting group is isomorphic to that of in ve rtible matrices via g 7→ g + 1. If A is an y coun table dimensional R -algebra, then part ii) of Prop osition 4.1.7 still holds if w e replace g − 1 b y the in v erse of g under the op erat ion ⋆ in the deﬁnition o f GL n ( A ) F . Corollary 4.1.9 also remains v alid in the non-unital case, and the pro of is the same. 20 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM 4.2. Construction of quotien ts of large semi- algebraic groups. Lemma 4.2.1. L et X ⊂ Y b e an inclusion of lar ge se m i-algebr aic sets. Then the fol l o wing ar e e quivalent. (i) Ther e exists a deﬁn ing structur e { Y λ } of Y such that { Y λ ∩ X } is a deﬁning structur e for X . (ii) F or every deﬁnin g structur e { Y λ } of Y , { Y λ ∩ X } is a deﬁning structu r e for X . Deﬁnition 4.2.2. Let X ⊂ Y b e an inclusion o f large semi-algebraic sets. W e sa y that X is c omp atible with Y if the equiv alen t conditions o f Lemma 4.2.1 are satisﬁed. Prop osition 4.2.3. L et H ⊂ G b e a inclusion of lar ge se mi-algebr aic gr oups, { G λ } a deﬁning structur e for G and π : G → G/H the p r oje ction. Assume that the inclusion is c omp atible in the s e n se of Deﬁnition 4.2.2 . Then ( G/H ) λ = π ( G λ ) is a l a r ge semi - a lgebr aic structur e , and the r esulting lar ge semi-algebr aic s e t G/H is the c ate goric al quotient in V ∞ . Pr o of. The map G λ → ( G/H ) λ is the set-theoretical quotient mo dulo t he relation G λ × G λ ⊃ R λ = { ( g 1 , g 2 ) : g − 1 1 g 2 ∈ H } . Let µ b e suc h that the pro duct map m sends G λ × G λ in to G µ ; write in v : G → G fo r the map inv( g ) = g − 1 . Then R λ = ( m ◦ (in v , id )) − 1 ( H ∩ G µ ) Because H ⊂ G is compatible, H ∩ G µ ⊂ G µ is closed and semi-algebraic, whence the same is true of R λ . By Theorem 3.1.4 , ( G/H ) λ is semi-algebraic and G λ → ( G/H ) λ is semi-algebraic and contin uous. It follow s that the ( G/ H ) λ deﬁne a la r g e semi-algebraic structure on G/H and that the pro jection is a morphism in V ∞ . The univ ersal prop ert y of the quotien t is straigh tforw ard.  Examples 4.2.4 . Let R b e a unital, countable dimensional R -algebra. The set P n ( R ) = { g (1 n ⊕ 0 ∞ ) g − 1 : g ∈ GL( R ) } of all ﬁnite idemp otent matrices whic h ar e conjugate to the n × n iden tit y matrix can b e written as a quotien t of a compatible inclusion of larg e semi-algebraic groups. W e ha v e P n ( R ) = GL( R ) [1 , ∞ ) GL [1 ,n ] ( R ) × GL [ n +1 , ∞ ) ( R ) . On the other hand, since P n ( R ) ⊂ M ∞ R , it also carries another large semi-algebraic struc- ture, induced b y the ﬁne structure on M ∞ R . Since g 7→ g (1 n ⊕ 0 ∞ ) g − 1 is semi-algebraic, the univ ersal prop ert y of the quotien t 4.2.3 implies that the quotient structure is ﬁner than the subspace structure. Similarly , w e may write the set of tho se idempotent matrices whic h are stably conjugate to 1 n ⊕ 0 ∞ as P ∞ n ( R ) = { g (1 ∞ ⊕ 1 n ⊕ 0 ∞ ) g − 1 : g ∈ GL Z ( R ) } = GL ( −∞ , + ∞ ) ( R ) GL ( −∞ ,n ] ( R ) × GL [ n +1 , ∞ ) ( R ) ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 21 Again this carries tw o large semi-algebraic structures; the quotien t structure, and that coming from the inclusion P ∞ n ( R ) ⊂ M 2 ( M ∞ ( R ) + ) in to the 2 × 2 matrices of t he unitalizat io n of M ∞ ( R ). 4.3. Large semi-algebraic sets as c ompactly generated spaces. A top olog ical space X is said to b e c omp actly gener ate d if it carries the inductiv e top ology with respect to its compact subsets, i.e. a map f : X → Y is con tin uous if and only if its restriction to an y compact subset of X is contin uous. In o ther w ords, a subset U ⊂ X is op en (resp. closed) if a nd only if U ∩ K is op en (r esp. closed) in K for ev ery K ⊂ X compact. Observ e tha t an y ﬁltering colimit of compact spaces is compactly generated. In particular, if X is a la rge semi-algebraic set, with deﬁning structure { X λ } , t hen X = S λ X λ equipped with the col- imit top o logy is compactly generated, and this to p ology dep ends o nly on the equiv alence class of the structure { X λ } . In what follows, whenev er we regard a large semi-algebraic set as a top ological space, we will implicitly assum e it equipp ed with the compactly generated top ology just deﬁned. Note f urt her that any morphism of large semi-algebraic sets is con- tin uous fo r the compactly generated top o logy . Lemma 4.3.1 characterizes t hose inclusions of large semi-algebraic sets whic h are closed sub spaces, and Lemma 4.3.2 concerns quotien ts of large semi-alg ebraic g roups with the compactly generated top olo gy . Both lemmas are straigh tforw ard. Lemma 4.3.1. An inclusion X ⊂ Y o f lar ge semi-algebr aic sets is c omp atible if and on l y i f X is a clo s e d subsp ac e of Y with r es p e ct to the c omp actly gener ate d top olo gies. Lemma 4.3.2. L et H ⊂ G b e a c o mp atible inclusion o f lar ge semi-algebr aic gr o ups. View G and H a s top olo gic al gr oups e quipp e d with the c omp actly ge n er ate d top olo gies. Then the c omp a c tly gener ate d top ol o gy asso c i ate d to the quotient lar ge semi-algebr aic set G /H is the quotient top olo gy. W e shall b e concerned with large semi-algebraic groups whic h a r e Hausdorﬀ for the compactly generated top ology . The main examples are coun ta ble dimensional R -v ector spaces and groups suc h as GL S ( A ), for some subset S ⊂ Z and some coun tably dimensional unital R -algebra A . Lemma 4.3.3. L et F = R or C , V a c ountable dimens i o nal F -ve ctorsp ac e, A a c ount- able dimensional F -algebr a , S ⊂ Z a subset, X a c omp act Hausdorﬀ top olo gi c al sp ac e, and C ( X ) = map( X , F ) . Equip V , A , and G L S ( A ) with the c omp actly ge ner ate d top olo gi e s. Then the natur al homomorph isms C ( X ) ⊗ F V → map( X , V ) a n d G L S (map( X , A )) → map( X, G L S ( A )) ar e bije ctive. Pr o of. It is clear that the b oth homomorphisms are injectiv e. The image of the ﬁrst one consists of those con tin uous maps whose image is contained in a ﬁnitely generated subspace of V . But since X is compact and V has the inductiv e top o logy o f a ll closed balls in a ﬁnitely generated subspace, ev ery contin uous map is of that form. Next note that GL S ( A ) = 22 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM S S ′ ,F GL S ′ ( A ) F where the union runs among the ﬁnite subsets of S and the compact semi- algebraic sets of the form F = M S ′ ( B ) where B is a compact semi-algebraic subs et of some ﬁnitely generated subspace o f A . Hence an y contin uous map f : X → G L S ( A ) sends X into some M S ′ B , and th us each of the en tries f ( x ) i,j i, j ∈ S ′ is a con tin uous function. Th us f ( x ) comes from an elemen t of GL S (map( X , A )).  4.4. The compact ﬁbration t heorem for quotien ts of large semi-algebraic groups. Recall t ha t a con tinuous map f : X → Y of top ological spaces is said to ha v e the hom o topy lifting pr op erty with resp ect to a space Z if for an y solid arro w diagram Z id × 0   / / X f   Z × [0 , 1 ] / / : : Y of con tin uous maps a con tin uous dotted arr ow exists and makes b oth tr ia ngles comm ute. The map f is a (Hurewicz) ﬁbration if it has the H LP with respect to any space Z , and is a Serre ﬁbration if it has the H LP with respect to all disk s D n n ≥ 0. Deﬁnition 4.4.1. Let X , Y b e top olog ical space and f : X → Y a con tin uous map. W e say that f is a c omp act ﬁbr ation if f or ev ery compact subspace K , the map f − 1 ( K ) → K is a ﬁbration. R emark 4.4.2 . Note that ev ery compact ﬁbratio n is a Serre ﬁbration. Also, since ev ery map p : E → B with compact B which is a lo cally trivial bundle is a ﬁbra tion, an y map f : X → Y suc h that the restriction f − 1 ( K ) → K to an y compact subs pace K ⊂ Y is a lo cally trivial bundle, is a compact ﬁbration. The not io n of compact ﬁbration comes up natura lly in the study of homogenous spaces of inﬁnite dimensional top o logical groups. Theorem 4.4.3. L et H ⊂ G b e a c omp a tible i n clusion of lar ge sem i - algebr aic gr oups. Then the quotient map π : G → G/H is a c omp act ﬁbr ation. Pr o of. Cho ose deﬁning structures { H p } and { G p } indexed o v er N and suc h that H p = G p ∩ H ; let { ( G /H ) p } b e as in Prop osition 4.2.3 . Since an y compact subspace K ⊂ G/H is contained in some ( G/H ) p , it suﬃce s to sho w that the pro jection π p = π | π − 1 (( G/H ) p ) : π − 1 (( G/H ) p ) → ( G/H ) p is a lo cally t r ivial bundle. By a w ell-known arg umen t (see e.g. [ 47 , Thm. 4.13]) if the quotient map of a group by a closed subgroup admits lo cal sections then it is a locally trivial bundle; the same argument applies in our case to sho w that if π p admits lo cal sections then it is a lo cally trivial bundle. Consider the functor F : Comp → Set + , F ( X ) = map( X , G/H ) map( X , G ) By Lemma 2.1.3 , F is split exact. By Theorem 3.2.1 applied to F and to the pro p er semi- algebraic surjection G p → ( G/H ) p , there is a triangulation of ( G/H ) p suc h that (4.4.4) k er ( F (∆ n ) → F ( π − 1 p (∆ n ))) = ∗ ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 23 for eac h simplex ∆ n in the triangulation. The diagram π − 1 (∆ n ) ∩ G p / /   G p / / π p   G π   ∆ n / / ( G/H ) p / / G/H sho ws that the class of the inclusion ∆ n ⊂ G/H is a n elemen t of that k ernel, and therefore it can b e lifted to a contin uous map ∆ n → G , b y ( 4.4.4 ). Th us π p admits a con tin uous section o v er ev ery simplex in the t riangulation. Therefore, using split-exactness of F , it admits a contin uous section o v er each of the op en stars st o ( x ) o f the vertice s of the barycen tric sub division. As the op en stars of v ertices for m an o p en co v ering of ( G /H ) p , w e conclude that π p admits lo cal sections. This ﬁnishes the pro of .  5. Algebraic comp actness, bounded s e quences and algebraic appro xima tion. 5.1. Algebraic compactness. Let R b e a countable dimensional unital R -alg ebra, equipp ed with the ﬁne large semi-algebraic structure. Consider the large semi-algebraic sets M ∞ R ⊃ P n ( R ) = GL( R ) [1 , ∞ ) GL [1 ,n ] ( R ) × GL [ n +1 , ∞ ) ( R ) (5.1.1) M 2 (( M ∞ R ) + ) ⊃ P ∞ n ( R ) = GL ( −∞ , + ∞ ) ( R ) GL ( −∞ ,n ] ( R ) × GL [ n +1 , ∞ ) ( R ) (5.1.2) in tro duced in 4.2.4 . Recall tha t eac h of P n ( R ), P n ( R ) ∞ carries t w o larg e semi-algebra struc- tures: the homogenous ones, coming from the quotien ts, and t ho se induced b y the inclusions ab ov e. As the homogeneous structures are ﬁner than t he induced ones, the same is true of the corresp onding compactly generated to p ologies; they agree if and only if the corresp onding large semi-algebraic structures are equiv alent, or, what is the same, if ev ery subset whic h is compact in the homogeneous top ology is also compact in t he induced one. This motiv a tes the follo wing deﬁnition. Deﬁnition 5.1.3. Let R b e a coun table dimensional unital R - algebra, equipp ed with the ﬁne large semi-alg ebraic structure. W e sa y that R has the algebr aic c omp actness prop ert y if for eve ry n ≥ 1 the homogeneous a nd the induced large semi-algebraic structure of P ∞ n ( R ) agree. W e show in Propo sition 5.1.5 below that if R satisﬁe s algebraic compactness , then the t w o top ologies in ( 5.1.1 ) also agree. F o r this w e need some prop erties of compactly generated top ological groups. All to p ological groups under consideration ar e assumed to b e Hausdorﬀ. Lemma 5.1.4. L et H and H ′ b e close d sub gr o ups o f a Hausdorﬀ c omp actly gene r ate d gr oup G . Then the quotient top olo gy on H /H ∩ H ′ is the subsp ac e top olo gy i n herite d fr om the quotient top olo gy on G/H ′ . 24 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM Pr o of. First of all, it is clear that the canonical inclusion map ι : H/H ∩ H ′ → G/H ′ is con tin uous. Indeed, let π : G → G/H ′ b e the pro jection; iden tify H → H/H ∩ H ′ with the restriction of π . A subset A ⊂ G/H ′ is closed if and only if π − 1 ( A ) ∩ K is closed fo r K ⊂ G compact. Hence, π − 1 ( A ) ∩ H ∩ K = π − 1 ( A ∩ π ( H )) ∩ K is closed and the claim follows since compact subsets of H a r e also compact in G . Let now A ⊂ H /H ∩ H ′ b e closed, i.e. π − 1 ( A ) ∩ K ′ closed for ev ery compact K ′ ⊂ H . F or compact K ⊂ G , the set K ′ = K ∩ H is compact in H and w e get that π − 1 ( A ) ∩ K closed in H and hence in G . This ﬁnishes the pro of.  Prop osition 5.1.5. L et R b e a unital, c ountable dimen sional R -algebr a , e quipp e d with the ﬁne lar ge semi-algebr aic structur e. Assume that R has the algebr aic c omp a ctness pr op erty. Then the homo gene ous and induc e d lar ge semi-algebr a ic structur es of P n ( R ) agr e e. Pr o of. Consider the diagram GL ( −∞ ,n ] ( R ) × GL [ n +1 , ∞ ) ( R ) / / GL ( −∞ , ∞ ) ( R ) / / P ∞ n ( R ) GL [1 ,n ] ( R ) × GL [ n +1 , ∞ ) ( R ) O O / / GL [1 , ∞ ) ( R ) / / O O P n ( R ) . O O No w apply Lemma 5.1.4 .  5.2. Bounded sequences, algebraic compactness and K 0 -trivialit y . Let X b e a large semi-algebraic set and let { X λ } b e a deﬁning structure. The space of b ounde d se quenc es in X is ℓ ∞ ( X ) = ℓ ∞ ( N , X ) = { z : N → X | ∃ λ : z ( N ) ⊂ X λ } Note t ha t with our deﬁnition, the ob jects ℓ ∞ ( R ) and ℓ ∞ C coincide with the w ell-kno wn spaces of b ounded sequences. Lemma 5.2.1. L et F = R or C and V a c ountable d i mensional F -ve ctorsp ac e e quipp e d with the ﬁne lar ge semi-algeb r aic structur e. T h en the natur al map ℓ ∞ ( F ) ⊗ F V → ℓ ∞ ( V ) is an isomorphi s m . Pr o of. Cho ose a basis { v q } of V . An y elemen t of ℓ ∞ ( F ) ⊗ F V can b e written uniquely as a ﬁnite sum P q λ q ⊗ v q ; this gets mapp ed to the sequence { n 7→ P q λ q ( n ) · v q } , whic h v anishes if and only if all the λ q are zero. This prov es the injectivit y statemen t. Let z ∈ ℓ ∞ ( V ); b y deﬁnition, there is a ﬁnite dimensional subspace W ⊂ V and a b ounded closed semi-algebraic subset S ⊂ W suc h that z ( N ) ⊂ S . W e may assume that S is a closed ball cen tered at zero, and that W is the smallest subspace containing z ( N ) . Hence there exist i 1 < · · · < i p ∈ N suc h that B = { z i 1 , . . . , z i p } is a basis of W . The map W → R p , w 7→ [ w ] B that sends a v ector w to the p -tuple of its co ordinates with respect to B is linear and therefore b o unded. In particular there exists a C > 0 suc h t hat || [ w ] B || ∞ < C for all w ∈ S . Thu s w e ma y write z = P p j = 1 λ i z i j with λ i ∈ ℓ ∞ ( F ). This pro v es the surjectivit y assertion of the lemma.  ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 25 Prop osition 5.2.2. L et R b e a unital, c ountable dimen sional R -algebr a , e quipp e d with the ﬁne la r ge semi-algebr aic structur e. Then the fol low ing ar e e quivale nt (i) R has the algebr ai c c omp actness pr op erty. (ii) F or every n the m ap (5.2.3) P ∞ n ( ℓ ∞ ( R )) → ℓ ∞ ( P ∞ n ( R )) is surje ctive. (iii) The map K 0 ( ℓ ∞ ( R )) → Q r ≥ 1 K 0 ( R ) is inje ctive. Pr o of. Cho ose countable indexed structures { G n } on G = GL Z ( R ), and { X n } on X = M 2 (( M ∞ R ) + ). Let π : G → G/H = P ∞ n ( R ) b e the pro jection. W e kno w already (see 4.2.4 ) that the induced structure on P ∞ n ( R ) is coarser than the homogeneous o ne, i.e. that eac h ( G/H ) r = π ( G r ) is con tained in some X m . Assertion (i) is therefore equiv alen t to sa ying that eac h P ∞ n ( R ) ∩ X m is contained in some π ( G r ). Negating this we o btain a b ounded sequence e = ( e r ) of idemp ot ent mat rices, i.e. e ∈ ℓ ∞ ( P ∞ n ( R )) with r esp ect to the induced la r ge semi- algebraic structure on P ∞ n ( R ), eac h e r is equiv alen t to 1 ∞ ⊕ (1 n ⊕ 0 ∞ ) in M 2 ( M ∞ ( R ) + ), but there is no sequence ( g r ) of in v ertible matrices in GL 2 ( M ∞ ( R ) + ) suc h that g r e r g − 1 r = 1 ∞ ⊕ (1 n ⊕ 0 ∞ ) and b oth ( g r ) and ( g − 1 r ) are b ounded. In other w ords, e is not in P ∞ n ( ℓ ∞ ( R )). W e ha v e sho wn that the negation o f (i) is equiv alen t to that o f (ii). Next note that ev ery elemen t x ∈ K 0 ( ℓ ∞ ( R )) can be written as a diﬀerence x = [ e ] − [1 ∞ ⊕ 0 ∞ ] with e ∈ M 2 ( M ∞ ( R ) + ) idemp oten t and e ≡ 1 ∞ ⊕ 0 ∞ mo dulo the ideal M 2 M ∞ ( R ). The idemp otent e is determined up to conjugat io n by GL 2 ( M ∞ ( R ) + ). The elemen t x go es to zero in Q r ≥ 1 K 0 ( R ) if and o nly if eac h e r is conjugate to 1 ∞ ⊕ 0 ∞ . Hence condition (iii) is satisﬁed if (ii) is. The con v erse follo ws easily . Indee d, for any seq uence ( e r ) as ab ov e, w e see that the image of the classe s [ e ] − [1 ∞ ⊕ 0 ∞ ] and [1 ∞ ⊕ (1 n ⊕ 0 ∞ )] − [1 ∞ ⊕ 0 ∞ ] in Q p ≥ 1 K 0 ( R ) coincide. Hence, b y injectivity of the comparison map, e is conjugate to 1 ∞ ⊕ (1 n ⊕ 0 ∞ ) and w e get a sequence of in v ertible elemen ts ( g r ) in GL 2 ( M ∞ ( R ) + ) suc h that g r conjugates e r to 1 ∞ ⊕ (1 n ⊕ 0 ∞ ) and the sequences ( g r ) and ( g − 1 r ) are b ounded. This completes the pro o f.  Example 5.2.4 . Bo th R and C ha v e the algebraic compactness prop erty since the third condition is well-kno wn to b e satisﬁed. Indeed, ℓ ∞ ( R ) and ℓ ∞ ( C ) are (r eal) C ∗ -algebras, and one can easily compute that K 0 ( ℓ ∞ ( C )) = ℓ ∞ ( Z ) ⊂ Y n ≥ 1 Z = Y n ≥ 1 K 0 ( C ) . The same computation applies to R in place of C . 5.3. Algebraic appro ximation and b ounded sequences . Theorem 5.3.1. L e t F and G b e f unctors f r om c ommutative C -algebr as to sets. Assume that b oth F a nd G pr eserve ﬁltering c olimits. L et τ : F → G b e a natur al tr ansfo rm ation. Assume that τ ( O ( V )) is inje ctive (r esp. surje ctive) fo r e ach smo oth aﬃne algebr aic variety V over C . Then τ ( ℓ ∞ ( C )) is inje c tive (r esp. surje ctive). 26 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM Pr o of. Let F ⊂ ℓ ∞ C b e a ﬁnite subse t. Put A F = C hF i ⊂ ℓ ∞ ( C ) fo r the unital subalgebra generated b y F . Because A F is reduced, it corresp onds to a n aﬃne alg ebraic v a riet y V F and the inclusion A F ⊂ ℓ ∞ C is dual to a map with pre-compact image ι F : N → ( V F ) an to the analytic v ariet y asso ciated to V F . Th us w e ma y write ℓ ∞ ( C ) as the ﬁltering colimit (5.3.2) ℓ ∞ ( C ) = colim N → V an O ( V ) Here the colimit is tak en o v er all maps ι : N → V an whose co domain is the asso ciated analytic v ariet y of the closed p oin ts of some aﬃne algebraic v ariet y V , and whic h ha v e precompact image in the euclidean to p ology . W e claim tha t ev ery suc h map factors through a map V ′ an → V an with V ′ smo oth and aﬃne. Note that the claim implies t ha t we may write ( 5.3.2 ) as a colimit of smo oth algebras; the theorem is immediate from this. R ecall Hironak a’s desingularization (see [ 23 ]) pro vides a prop er surjectiv e homomorphism of algebraic v arieties π : ˜ V → V fr o m a smo oth quasi-pro j ective v ariet y . Th us the induced map π an : ˜ V an → V an b et w een the asso ciated analytic v arieties is prop er and surjectiv e for t he usual euclidean top ologies. It follo ws fro m this that w e can lift ι along π an . Next, Jouanoulou’s devic e (see [ 25 ]) pro vides a smooth aﬃne v ector bundle torsor σ : V ′ → ˜ V ; the asso ciated map σ an is also a bundle torsor, and in particular a ﬁbration and w eak equiv a lence. Because ˜ V an is a C W -complex, σ an admits a con tin uous section. Thus ι F ﬁnally f a ctors thro ugh the smo oth aﬃne v ariet y V ′ F .  R emark 5.3.3 . The pro of a b o v e do es not w ork in the real case, since a desingularization ˜ V → V of real algebraic v arieties need not induce a surjectiv e map b et w een the corresp onding real analytic (or semi-algebraic) v arieties. F or example, consider R = R [ x, y ] / h x 2 + y 2 − x 3 i . The homomorphism f : R → R [ t ], f ( x, y ) 7→ f ( t 2 + 1 , t ( t 2 + 1)), is injectiv e and R [ t ] is in tegral ov er R . Th us the induced sc heme homomorphism f # : A 1 R = Sp ec R [ t ] → V = Sp ec R [ x, y ] / h x 2 + y 2 − x 3 i is a desingularization; it is ﬁnite (whence prop er) and surjectiv e, and an isomorphism outside of the p oint zero, represen ted b y the maximal ideal M = h x, y i ∈ V . But note that the pre-image of M consists just of the maximal ideal h t 2 + 1 i , whic h has residue ﬁeld C ; this means that the pre-image of zero has no r eal p oints. Therefore the restriction of f # to real p oin ts is not surjectiv e. 5.4. The algebraic compactness theorem. Theorem 5.4.1. L e t R b e a c ountable dimensional unital C -algebr a. Assume that the map K 0 ( O ( V )) → K 0 ( O ( V ) ⊗ R ) is an isomorphism fo r ev ery aﬃ n e smo oth algebr aic variety V over C . Then R has the algebr aic c omp actness pr op erty. Pr o of. W e ha v e a comm utativ e diag r am K 0 ( ℓ ∞ ( R )) / / Q p ≥ 1 K 0 ( R ) K 0 ( ℓ ∞ ( C )) O O / / Q p ≥ 1 K 0 ( C ) O O ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 27 The b ottom ro w is a mo no morphism b y Example 5.2.4 . Our h yp othesis on R together with Theorem 5.3.1 applied to the natural transformatio n K 0 ( − ) → K 0 ( − ⊗ C R ) imply that b oth columns are isomorphisms. It follows t ha t the to p row is injectiv e, whic h b y Prop o sition 5.2.2 sa ys that R satisﬁes algebraic compactness.  6. App lica tions: pro jective modules, lo wer K -theor y, and bundle theor y. 6.1. P arametrized Gub eladze’s t heorem and Rosen b erg’s conjecture. All monoids considered are comm utativ e, cancellativ e and without nonzero torsion elemen ts. If M is cacellativ e then it em b eds into its total quotien t gro up G ( M ). A cancellativ e monoid M is said to b e seminormal if fo r ev ery elemen t x of the tota l quotien t group G ( M ) for whic h 2 x and 3 x are con tained in the monoid M , it follows that x is containe d in the monoid M . The following is a particular case of a theorem of Joseph G ub eladze, whic h in turn generalized the celebrated theorem of Daniel Q uillen [ 35 ] a nd Andrei Suslin [ 42 ] whic h settled Serre’s conjecture: eve ry ﬁnitely generated pro j ective mo dule ov er a p olynomial ring ov er a ﬁeld is free. Theorem 6.1.1 (see [ 20 ],[ 19 ]) . L et D b e a princip al ide al doma i n , and M a c ommutative, c anc el l a tive, torsion fr e e , seminormal monoid. Then every ﬁn itely gener ate d pr oje ctive mo d- ule over the monoid alg ebr a D [ M ] is fr e e. W e shall also need the f o llo wing g eneralization of G ub eladze’s theorem, due to Rich ard G. Swan. Recall tha t if R → S is a homomorphism o f unital rings, and M is an S -mo dule, then w e say that M is extende d from R if there exists an R -mo dule N suc h that M ∼ = S ⊗ R N as S -mo dules. Theorem 6.1.2 (see [ 46 ]) . L e t R = O ( V ) b e the c o or dinate ring of a smo o th a ﬃ n e algebr aic variety over a ﬁeld, and let d = dim V . Also let M b e a torsion-f r e e, seminormal, c an c el lative monoid. Then al l ﬁnitely gener ate d pr oje ctive R [ M ] -mo d ules of r ank n > d ar e extende d fr om R . In t he next t heorem and elsewhere b elo w, w e shall consider only the complex case; thus in what follow s, C ( X ) shall alw a ys means map( X , C ). Theorem 6.1.3. L et X b e a c ontr ac tible c omp act sp ac e, and M an a b elian, c ountable, torsion-fr e e, seminormal, c anc el lative mono id. Then every ﬁnitely gener ate d pr oj e ctive mo d- ule over C ( X )[ M ] is fr e e. Pr o of. The a ssertion of the theorem is equiv alen t to the assertion that ev ery idemp oten t matrix with co eﬃcien ts in C ( X )[ M ] is conjugate to a diagonal matrix with o nly zero es and ones in the diago na l. By Lemma 4.3.3 , an idemp oten t matrix with co eﬃcien ts in C ( X )[ M ] is the same as a contin uous map fro m X to the space Idem ∞ ( C [ M ]) of all idemp oten t matrices in M ∞ ( C [ M ]), equipp ed with the induced top ology . No w observ e that, since the trace map M ∞ C [ M ] → C [ M ] is contin uous, so is the rank map Idem ∞ ( C [ M ]) → N 0 . Hence b y Theorem 6.1.1 , the space Idem ∞ ( C [ M ]) is the top ological copro duct Idem ∞ ( C [ M ]) = ` n P n ( C [ M ]), and thus an y contin uous map e : X → Idem ∞ ( C [ M ]) factors thro ugh a 28 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM map e : X → P n ( C [ M ]). By Theorem 6.1.2 and Theorem 5 .4 .1 , the induced to p ology of P n ( C [ M ]) = GL( C [ M ]) / GL [1 ,n ] ( C [ M ]) × GL [ n +1 , ∞ ) ( C [ M ]) coincides with the quotient top ology . By Theorem 4.4.3 , e lifts to a contin uous map g : X → GL( C [ M ]). By Lemma 4.3.3 , g ∈ GL( C ( X )[ M ]) and conjugates e to 1 n ⊕ 0 ∞ . This concludes the pro of.  Theorem 6.1.4. The functor Co mp → Ab , X 7→ K 0 ( C ( X )[ M ]) is h o m otopy invarian t. Pr o of. Immediate from Theorem 6.1.3 and Prop osition 2.1.5 .  Theorem 6.1.5 (Ro sen b erg’s conjecture) . T h e functor Co mp → Ab , X 7→ K − n ( C ( X )) is homotopy in v a riant for n > 0 . Pr o of. By ( 2.2.2 ), K − n ( C ( X )) is naturally a direct summand of K 0 ( C ( X )[ Z n ]), whence it is homotop y in v arian t by Theorem 6.1.4 .  R emark 6.1.6 . Let X b e a compact to p ological space, S 1 the circle and j ≥ 0. By ( 2.2.2 ), Theorem 6.1.5 and excision, w e hav e K − j ( C ( X × S 1 )) = K − j ( C ( X )) ⊕ K − j − 1 ( C ( X )) = K − j ( C ( X )[ t, t − 1 ]) Th us the eﬀect o n negative K -theory of the cartesian pro duct of the maximal ideal sp ectrum X = Max( C ( X )) with S 1 is the same as that o f taking the pro duct of the prime ideal sp ec- trum Sp ec C ( X ) with the algebraic circle Sp ec ( C [ t, t − 1 ]). More generally , for the C ∗ -algebra tensor pro duct ⊗ min and an y commutativ e C ∗ -algebra A , we ha v e K − j ( A ⊗ min C ( S 1 )) = K − j ( A ⊗ C C [ t, t − 1 ]) ( j > 0). R emark 6.1.7 . Theorem 6.1.5 was stated by Jonathan Rosen b erg in [ 36 , Thm. 2.4] and aga in in [ 37 , Thm. 2.3] for the real case. Later, in [ 38 ], R osen b erg ack no wledges that t he pro of w as fault y , but conjectures the statemen t to b e true. Indeed, a mistak e was p oin ted out b y Mark E. W alk er (see line 8 on page 799 in [ 15 ] or line 12 on page 26 in [ 38 ]). In their w ork on semi-top olo gical K - theory , Eric F riedlander and Mark E. W alk er pro v e [ 15 , Thm. 5.1] that the negative algebraic K -t heory of the ring C (∆ n ) of complex-v alued con tin uous functions on t he simplex v anishes for all n . W e sho w in Subsection 7.2 ho w another pro of of Rosen b erg’s conjecture can b e obtained using t he F riedlander-W alk er result. R emark 6.1.8 . The pro o f of Theorem 6.1.5 do es not need the detour of the pro of of our ma in results in the case n = 1. Indeed, the ring of germs of con t inuous functions at a p oint in X is a Hensel lo cal ring with residue ﬁeld C and Vla dimir Drinfel’d pro ves that K − 1 v anishes for Hensel lo cal rings with residue ﬁeld C , see [ 12 , Thm. 3.7 ]. This solv es the problem lo cally and reduces the remaining complications to bundle theory . (This w as observ ed by the second author in discuss ions with Charles W eib el at Institut Henri P oincar ´ e, P aris in 2004.) No direct approac h lik e this is kno wn for K − 2 or in lo w er dimensions. Already in [ 36 ], Rosen b erg computed the v alues of negativ e algebraic K -theory on com- m utativ e unital C ∗ -algebras, assuming the homotopy inv ariance result. ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 29 Corollary 6.1.9 (Rosen b erg, [ 3 8 ]) . L et X b e a c o mp act top olo gic al sp ac e. L et bu denote the c onne c tive K - the ory s p e ctrum. Then, K − i ( C ( X )) = bu i ( X ) = [Σ i X , bu ] , i ≥ 0 . The fact that connectiv e K -theory sho ws up in this con text w as further explored and clariﬁed in the thesis of the second author [ 48 ], whic h was also par t ia lly built on the v alidity of Theorem 6.1.5 . 6.2. Application to bun dle t heory: lo cal tr iviality . Let R b e a coun ta ble dimensional R -algebra. An y ﬁnitely generated R - mo dule M is a coun table dimensional v ectorspace, and th us it can b e regarded as a compactly generated top ological space. W e consider not neces- sarily lo cally trivial bundles of ﬁnitely generated free R -mo dules o ve r compact spaces, suc h that each ﬁb er is equipp ed with the compactly generated top olog y just recalled. W e call such suc h a ga dget a quasi-bund le of ﬁnitely gener ate d f r e e R -mo dules Theorem 6.2.1. L et X b e a c omp a c t sp ac e, M a c ountable, torsion-fr e e, s e minormal, c an- c el lative monoid. L et E → X b e a quasi-bund le of ﬁnitely gener a te d fr e e C [ M ] -mo dules. Assume that ther e exist an n ≥ 1 , another quasi-bund le E ′ , and a quasi-bund l e isomorphism E ⊕ E ′ ∼ = X × C [ M ] n . Then E i s lo c al ly trivial. Pr o of. Put R = C [ M ]. The isomorphism E ⊕ E ′ ∼ = X × R n giv es a con tin uous function e : X → P n ( R ); E is lo cally trivial if e is lo cally conjugate to an idempo ten t of the form 1 r ⊕ 0 ∞ , i.e. if it can b e lifted lo cally along the pro jection GL( R ) → P n ( R ) to a contin uous map X → GL( R ). O ur h yp othesis o n M together with Theorems 6.1.3 , 5 .4.1 , 6.1.2 , and 4.4.3 imply that suc h lo cal liftings exist.  7. Homotopy inv ariance . 7.1. F r om compact polyhedra to compact spaces: a result of Calder-Siegel. Con- sider the category Co mp of compact Hausdorﬀ t o p ological spaces with contin uous maps and its full sub category Pol ⊂ Comp formed by those spaces whic h ar e compact p olyhe- dra. In this subse ction we show tha t for a f unctor whic h comm utes with ﬁltering colimits and is split exact on C ∗ -algebras, homot o p y inv ariance on Pol implies homotopy in v ariance on Comp . F or this w e shall need a particular case of a result of Allan Calder and Jerrold Siegel [ 6 , 7 ] that w e shall presen tly recall. W e p oin t o ut that the Calder-Siegel results ha v e b een further generalized b y Armin F r ei in [ 14 ]. F o r eac h ob j ect X ∈ Comp w e consider the comma categor y ( X ↓ Pol ) whose ob jects ar e morphisms f : X → co d( f ) where the co domain co d( f ) is a compact p olyhedron. Morphisms are comm utativ e diagr a ms as usual. Let G : Pol → Ab be a (con trav aria n t) functor to the category of ab elian groups. Its righ t Kan extension G Pol : Comp → Ab is deﬁned b y G Pol ( X ) = colim f ∈ ( X ↓ Pol ) G (co d( f )) , ∀ X ∈ Comp . The result of Calder-Siegel (see Corolla ry 2.7 and Theorem 2.8 in [ 7 ]) sa ys that homotopy in v ariance prop erties of G giv e rise to ho mo t o p y in v ariance prop erties of G Pol . More precisely: 30 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM Theorem 7.1.1 (Calder-Siegel) . If G : Pol → Ab is a (c ontr avariant) homotopy invaria n t functor, then the functor G Pol : Comp → Ab i s homotopy invariant. W e w ant to apply the theorem when G is of the form D 7→ E ( C ( D )), the functor E comm utes with (algebraic) ﬁltering colimits, and is split exact o n C ∗ -algebras. F or this w e ha v e to compare E with the right Ka n extension of G ; we need some preliminaries. Let X ∈ Comp , D ⊂ C the unit disk , and F the se t of all ﬁnite subsets of C ( X, D ). Since X is compact, for eac h f ∈ C ( X ) there is an n ∈ N suc h that (1 /n ) f ∈ C ( X , D ). Th us an y ﬁnitely generated subalgebra A ⊂ C ( X ) is generated b y a ﬁnite subset F ⊂ C ( X , D ). Let F b e the set of all ﬁnite subsets of C ( X , D ); since C ( X ) is the colimit o f its ﬁnitely generated subalgebras C h F i , w e ha v e colim F ∈F C h F i = C ( X ) . F or F ∈ F , write Y F ⊂ C F for the Zariski closure of the image of the map α F : X → C F , x 7→ ( f ( x )) f ∈ F . The image of α F is con tained in the compact semi-algebraic set P F = D ∩ Y F In particular, P F is a compact po lyhedron. Note that α F induces an isomorphism b et w een the ring O ( Y F ) of regular p o lynomial functions and the subalgebra C h F i ⊂ C ( X ) generated b y F . Hence the inclusion P F ⊂ Y F induces a ho momorphism β F whic h mak es the follo wing diagram comm ut e C h F i β F $ $ I I I I I I I I I / / C ( X ) C ( P F ) : : u u u u u u u u u T aking colimits w e obtain (7.1.2) C ( X ) β ' ' O O O O O O O O O O O C ( X ) colim F ∈F C ( P F ) π 7 7 o o o o o o o o o o o Th us the map π is a split surjection. W e can no w dra w the desired conclusion. Theorem 7.1.3. L e t E : Co mm → Ab b e a functor. Assume that F satisﬁes e ach of the fol lowing c onditions. (1) E c omm utes with ﬁ l ter e d c olimits. (2) Pol → Ab , D 7→ E ( C ( D )) , is homo topy invariant. Then the functor Comp → Ab , X 7→ E ( C ( X )) , is homotopy invariant on the c ate gory of c omp act top olo gic al sp ac es. ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 31 Pr o of. By ( 7.1.2 ) and the ﬁrst hypothesis, the map E ( β ) is a righ t in v erse of the map E ( π ) : E (colim F ∈F C ( P F )) = colim F ∈F E ( C ( P F )) → E ( C ( X )) . On the other hand, w e ha v e a commutativ e diagram (7.1.4) colim F ∈F E ( C ( P F )) θ / / E ( π ) * * V V V V V V V V V V V V V V V V V V V colim f ∈ ( X ↓ Pol ) E ( C (co d( f ))) π ′   E ( C ( X )) Hence the map π ′ in the diag ram a b o v e is split by the comp osite θ E ( β ), and therefore E ( C ( − )) is naturally a direct summand of the functor (7.1.5) G Pol : Comp → Ab , X 7→ colim f ∈ ( X ↓ Pol ) E ( C (co d( f ))) But ( 7.1.5 ) is the right Kan extension of the f unctor G : Pol → Ab , G ( K ) = E ( C ( K )), and th us it is homotop y inv ariant by t he second hypothesis and Calder-Siegel’s theorem. It follo ws that E ( C ( − )) is homotopy inv aria nt, as w e had to prov e.  R emark 7.1.6 . If in Theorem 7.1.3 t he functor E is split exact, then A 7→ E ( A ) is homotopy in v arian t o n the categor y of comm uta tiv e C ∗ -algebras. Indeed, since ev ery comm utativ e unital C ∗ -algebra is of the form C ( X ) for some compact space X , it follow s that F is homotop y in v arian t on unital commu tative C ∗ -algebras. Using this and split exactness , w e get that it is also homotop y in v arian t o n all commutativ e C ∗ -algebras. R emark 7.1.7 . In general, one cannot exp ect that the homomorphism π ′ of ( 7.1.4 ) b e an isomorphism. F or the injectivit y one w ould need the followin g implication: If D is a compact p olyhedron, and f : X → D a nd s : D → C are contin uous maps suc h that 0 = s ◦ f : X → C , then there exis t a compact p olyhedron D ′ and contin uous maps g : X → D ′ , h : D ′ → D suc h that h ◦ g = f : X → D a nd 0 = h ◦ s : D ′ → C . That is to o strong if X is a pathological space. T o giv e a concrete example : let X b e a Can tor set inside [0 , 1], f the natural inclusion, and s the distance function to the Can tor set, and supp ose g and h as ab ov e exist. If 0 = h ◦ s : D ′ → C , then the image of D ′ in [0 , 1] has to b e con tained in X . But the image has only ﬁnitely many connected comp onen ts, since D ′ has this prop ert y . Hence, since X is totally disconnected, the image of D ′ in [0 , 1] cannot b e a ll of X . This is a con tr adiction. 7.2. Second pro of of Rosenberg’s conjecture. A second pro of of Rosen b erg’s conjecture can b e obtained b y com bining Theorem 7.1.3 with the follo wing theorem, whic h is due to Eric F riedlander and Mark E. W alk er. Theorem 7.2.1 (Theorem 5 . 1 in [ 15 ]) . I f n > 0 a n d q ≥ 0 , then K − n ( C (∆ q )) = 0 Se c ond pr o of of R osenb er g’s c onje c tur e . By Prop osition 2.1.5 and Theorem 7.1.3 it suﬃces to sho w tha t K n ( C ( D )) = 0 for contractible D ∈ Pol . If D is con t r a ctible, then t he iden tity 1 D : D → D factors ov er the cone cD . Hence, it is suﬃcien t to show that K n ( C ( cD )) = 0. 32 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM The cone cD is a star-like simplicial complex a nd for an y sub complexes A, B ⊂ D with A ∪ B = D , we get a Milnor squ are C ( cD ) / /   C ( cA )   C ( cB ) / / C ( c ( A ∩ B ) ) . Since cA is contractible , it retracts on to c ( A ∩ B ) and therefore, the square ab ov e is split. Using excision, w e obtain split-exact sequence of ab elian g roups as fo llo ws: 0 → K n ( C ( cD )) → K n ( C ( cA )) ⊕ K n ( C ( cB )) → K n ( C ( c ( A ∩ B ))) → 0 . Decomp osing cD lik e this, w e see that the result of Theorem 7.2.1 is suﬃcien t for the v an- ishing of K n ( C ( cD )).  7.3. The homotopy inv ariance theorem. The a im of this subsection is to prov e the follo wing result. Theorem 7.3.1. L et F b e a functor on the c ate gory of c ommutative C -algebr as with values in ab elian gr oups. Assume that the fol lowing thr e e c onditions ar e satisﬁe d. (i) F is split-exact on C ∗ -algebr as. (ii) F van i s h es on c o or din a te rings of sm o oth aﬃn e varieties. (iii) F c ommutes with ﬁltering c olimits. Then the functor Comp → Ab , X 7→ F ( C ( X )) is homotopy invariant on the c ate go ry of c o m p act Hausdorﬀ top olo gic a l sp ac es and F ( C ( X ) ) = 0 for X c ontr actible. Pr o of. Note that, since a p o in t is a smo oth algebraic v ariety , o ur h yp othesis imply that F ( C ) = 0. Th us if F is homotop y in v a r ia n t a nd X is contractible, then F ( X ) = F ( C ) = 0. Let us pro ve then that X 7→ F ( C ( X )) is homotop y inv ariant on the category of compact Hausdorﬀ top ological spaces. Pro ceeding as in the pro o f of Theorem 7.2.1 , w e see t ha t it is suﬃcien t to sho w that F ( C (∆ n )) = 0, for all n ≥ 0. Any ﬁnitely generated subalgebra of C (∆ n ) is reduced and hence corresponds to an algebraic v ariety o v er C . Since F comm utes with ﬁltered colimits, w e obtain: F ( C (∆ n )) = colim ∆ n → Y an F ( O ( Y )) , where the colimit runs ov er a ll con tinuous maps from ∆ n to the analytic v ariety Y an equipped with the usual euclidean top ology . F or ease of notation, w e will from now on just write Y for b oth the algebraic v ariet y and the ana lytic v a riet y asso ciated to it. Let ι : ∆ n → Y b e a con tin uous map. As in the pro of of the a lgebraic appro ximation theorem ( 5.3.1 ), w e consider Hironak a’s desingularization π : ˜ Y → Y and Jouanoulou’s aﬃne bundle torsor σ : Y ′ → ˜ Y . Let T ⊂ Y b e a compact semi-algebraic subset such that ι (∆ n ) ⊂ T . Because π is a prop er morphism, ˜ T = π − 1 ( T ) is compact and semi-algebraic. By deﬁnition of vec tor bundle to rsor ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 33 ([ 51 ]), there is a Zariski co v er of ˜ Y suc h that the pull-bac k of σ o v er eac h op en subs c heme U ⊂ ˜ Y of the co v ering is isomorphic, as a sc heme ov er U , t o an algebraic trivial vec tor bundle. Th us σ is lo cally trivial ﬁbration for the euclidean top olog ies, and the trivialization maps are (semi)-algebraic. Hence b ecause ˜ T , b eing compact, is lo cally compact, w e ma y ﬁnd a ﬁnite cov ering { ˜ T i } of ˜ T b y closed semi-algebraic subsets suc h that σ is a trivial ﬁbration ov er eac h ˜ T i , and compact semi-algebraic subsets S i ⊂ Y ′ suc h that σ ( S i ) = ˜ T i . Put S = S i S i ; then S is compact semi-algebraic, and f = ( π ◦ σ ) | S : S → T is a con tin uous semi-algebraic surjection. By Theorem 3.2.1 , t here exists a semi-algebraic triangulation of T suc h that ke r( F (∆ m ) → F ( f − 1 (∆ m )) = 0 for eac h simplex ∆ m in the triangulatio n. Consider the diagram F ( C ( f − 1 (∆ m )) F ( C ( S )) o o F ( O ( Y ′ )) o o F ( C (∆ m )) O O F ( C ( T )) O O o o F ( O ( Y )) o o O O If α ∈ F ( O ( Y )), then its image in F ( C ( f − 1 (∆ m ))) v anishes since f − 1 (∆ m ) → ∆ m factors through the smo oth aﬃne v ariet y Y ′ , and F ( O ( Y ′ )) = 0. Hence, by Theorem 3.2.1 , α | ∆ m = 0, for eac h simplex in the triangulatio n. Coming back to the map ι : ∆ n → Y , w e hav e a diagram ∆ n ι / /  ! ! C C C C C C C C Y ∆ m θ / / T O O Here θ : ∆ m → T is the inclusion of a simplex in the tria ng ulation, and  is the corr estriction of ι . W e need to conclude that ι ∗ ( α ) = 0, know ing only tha t θ ∗ ( α ) = 0 for eac h simplex in a triangulation of T . This is done using split-exactness and barycen tric sub divisions. Indeed, we p erform the barycen tric sub division of ∆ n suﬃcien tly man y times so tha t eac h n - dimensional simplex is mapp ed to the closed star st( x ) of some some v ertex x in the tria ng ulation of T . Since ∆ n is star-lik e, the reduction argumen t of the pro of of Theorem 5.3.1 sho ws that it is enough to show the v anishing of ι ∗ ( α ) for the (top- dimensional) simplices in this sub division of ∆ n . If ∆ ′ n is one of these top dimens ional simplices, and ∆ m ⊂ st( x ), w e can complete the diagram ab o v e to a diagram ∆ ′ n / / " " F F F F F F F F ∆ n ι / /  ! ! C C C C C C C C C Y ∆ m / / st( x ) / / T O O Hence it suﬃces to show that the pullback of α to st( x ) v anishes. But since st( x ) is star- lik e, then b y the same reduction argumen t as b efore, t he v anishing of the pullback of α to each of the top simplices ∆ m ⊂ st( x ) is suﬃ cien t to conclude that α | st( x ) = 0. This ﬁnishes the pro of.  As an application, w e obtain the following. 34 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM Third pro of of Rosenberg’s conjecture. If n < 0 then K n is split-exact and v anishes on co or dina t e rings of smo oth aﬃne algebraic v arieties. By Theorem 7.3.1 , this implies tha t X 7→ K n ( C ( X )) is homotop y in v arian t.  7.4. A v anishing theorem for homology theories. Theorem 7.4.1. L et E : Comm / C → S pt b e a homolo g y the ory of c o mmutative C -algebr as and n 0 ∈ Z . Assume (i) E is excisive o n c ommutative C ∗ -algebr as. (ii) E n c ommutes with algeb r aic ﬁltering c olimits for n ≥ n 0 . (iii) E n ( O ( V ) ) = 0 f o r e ach sm o oth aﬃn e algebr aic varie ty V for n ≥ n 0 . Then E n ( A ) = 0 for every c ommutative C ∗ -algebr a A for n ≥ n 0 . Pr o of. Let n ≥ n 0 . W e hav e to sho w tha t E n ( A ) = 0 for ev ery comm utativ e C ∗ -algebra A . Because by i), eac h E n is split exact on comm utat ive C ∗ -algebras, it suﬃces to sho w that E n ( A ) = 0 for unital A , i.e. for A = C ( X ), X ∈ Comp . Since by ii) E n preserv es ﬁltering colimits, the pro of of Theorem 7.1.3 sho ws that E n ( C ( X )) is a direct summand of colim f ∈ ( X ↓ Pol ) E ( C (co d( f ))) Hence it suﬃces t o sho w that E n ( C ( D )) = 0 for ev ery compact p olyhedron D . By iii) and excision, this is tr ue if dim D = 0. Let m ≥ 1 a nd assume the assertion of the theorem holds for compact p olyhedra of dimension < m . By Theorem 7.3.1 , D 7→ E n ( C ( D )) is homotop y inv aria nt; in part icular E n ( C (∆ m )) = 0. If dim D = m and D is not a simple x, write D = ∆ m ∪ D ′ , as the union of an m -simplex and a sub complex D ′ whic h has few er m -dimensional simplices. Put L = ∆ m ∩ D ′ ; then dim L < m , a nd w e hav e an exact sequence E n +1 ( C ( L )) → E n ( C ( D )) → E n ( C (∆ m )) ⊕ E ( C ( D ′ )) → E n ( C ( L )) W e ha ve seen abov e tha t E n ( C (∆ m )) = 0; moreo v er E n ( C ( L )) = E n +1 ( C ( L )) = 0 b ecause dim L < m , and E n ( C ( D ′ )) = 0 b ecause D ′ has f ewer m -dimensional simplices than D . This concludes the pro of.  8. Applica t ions of the homotopy inv ariance and v anishing homology theorems. 8.1. K -regularit y for comm utativ e C ∗ -algebras. Theorem 8.1.1. L et V b e a smo o th aﬃne al g ebr aic variety over C , R = O ( V ) , and A a c ommutative C ∗ -algebr a. Then A ⊗ C R is K -r e g ular. Pr o of. F or eac h ﬁxed p ≥ 1 and i ∈ Z , write F p ( A ) = ho coﬁb er( K ( A ⊗ R ) → K ( A [ t 1 , . . . , t p ] ⊗ R )) for the homotop y coﬁb er. It suﬃces to prov e tha t the homology theory F p : Comm / C → Spt satisﬁes the hypothesis of Theorem 7.4.1 . By [ 52 , Corollary 9.7 ], A [ t 1 , . . . , t p ] ⊗ C R is K - excisiv e for eve ry C ∗ -algebra A and ev ery p ≥ 1. It fo llows that the homology theory F p : Ass / C → Sp t is excisiv e on C ∗ -algebras. In particular its restriction to Comm / C is excisiv e on commutativ e C ∗ -algebras. Moreov er, if W is any smo oth aﬃne algebraic v ariety , ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 35 then R ⊗ C O ( W ) = O ( V × W ), is regular no etherian, and therefore K -r egula r . Finally , F p ∗ preserv es ﬁltering colimits b ecause b oth K ∗ and ( − ) ⊗ Z [ t 1 , . . . , t p ] ⊗ R do.  R emark 8.1.2 . The case R = C of the previous theorem w as disco ve red by Jonathan R osen- b erg. Unfortunately , the t w o pro ofs he ha s giv en, in [ 37 , Thm. 3 .1] and [ 38 , p. 866 ] turned out to be problematic. A vers ion of Theorem 8.1.1 fo r A = C ( D ), D ∈ Pol , w as given b y F riedlander and W alk er in [ 15 , Thm. 5.3]. F urthermore, R o sen b erg ackno wledges in [ 38 , p. 24] tha t W alk er also fo und a pro o f of this in the general case, but that he did not publish it. An yho w, as Rosen b erg observ ed in [ 37 , p. 91], in t his situation, the p olyhedral case implies the g eneral case by a short reduction argumen t (this a lso fo llo ws from Theorem 7.1.3 ab ov e). Hence, an essen tially complete argumen t for the proo f of Theorem 8.1.1 existed already in the literature, although it w as scattered in v arious sources. The follow ing corollary compares Quillen’s algebraic K -theory with Charles A. W eib el’s homotop y algebraic K -theory , K H , in tro duced in [ 51 ]. Corollary 8.1.3. If A is a c ommutative C ∗ -algebr a, then the map K ∗ ( A ) → K H ∗ ( A ) is an isomorphism. Pr o of. W eib el pro v ed in [ 51 , Pro p osition 1.5] that if A is a unital K -regular ring, then K ∗ ( A ) ∼ → K H ∗ ( A ). Using excision, it follow s that this is true for all comm uta t iv e C ∗ - algebras. No w apply Theorem 8.1.1 .  8.2. Ho chsc hild and cyclic homology of comm utativ e C ∗ -algebras. In the following paragraph we recall some basic facts ab out Ho c hsc hild and cyclic homology whic h w e shall need; the standard reference for these topics is Jean-Louis Lo day’s b o ok [ 29 ]. Let k b e a ﬁeld of characteristic zero. Recall a mixe d c omp lex of k -vec torspaces is a gra ded v ectorspace { M n } n ≥ 0 together with maps b : M ∗ → M ∗− 1 and B : M ∗ → M ∗ +1 satisfying b 2 = B 2 = bB + B b = 0. One can associate v a r ious c hain complexes to a mixed complex M , giving rise to the Ho chs c hild, cyclic, negativ e cyclic a nd p erio dic cyclic homologies of M , denoted resp ectiv ely H H ∗ , H C ∗ , H N ∗ and H P ∗ . F or example H H ∗ ( M ) = H ∗ ( M , b ). A map o f mixed complexes is a homogeneous map whic h commutes with b oth b and B . It is called a quasi-isom orphism if it induces an isomorphism at the leve l o f Ho c hsc hild homology; this automat ically implies that it also induces an isomorphism for H C and all the other homologies mentioned a b o v e. F or a k -algebra A there is deﬁned a mixed complex ( C ( A/k ) , b, B ) with C n ( A/k ) =  ˜ A k ⊗ k A ⊗ k n n > 0 A n = 0 W e write H H ∗ ( A/k ), H C ∗ ( A/k ), etc. for H H ∗ ( C ( A/k )), H C ∗ ( C ( A/k )), etc. If furthermore A is unital and ¯ A = A/k , then t here is also a mixed complex ¯ C ( A/k ) with ¯ C n ( A/k ) = A ⊗ k ¯ A ⊗ k n and the natural surjection C ( A/k ) → ¯ C ( A/k ) is a quasi-isomorphism. Note also that (8.2.1) k er( ¯ C ( ˜ A k /k ) → ¯ C ( k /k )) = C ( A/ k ) 36 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM If A comm utative and unital, w e hav e a third mixed complex (Ω A/k , 0 , d ) giv en in degree n b y Ω n A/k , the mo dule of n -K¨ ahler diﬀeren tial f o rms, and where d is the exterior deriv a tion of forms. A natural map of mixed complexes µ : ¯ C ( A/k ) → Ω A/k is deﬁned b y (8.2.2) µ ( a 0 ⊗ k ¯ a 1 ⊗ k · · · ⊗ k ¯ a n ) = 1 n ! a 0 da 1 ∧ · · · ∧ da n It w a s sho wn b y Lo da y and Quillen in [ 30 ] (using a classical result of Ho c hsc hild-Kostan t- Rosen b erg [ 24 ]) that µ is a quasi-isomorphism if A is a smo oth k algebra, i.e. A = O ( V ) for some smo oth aﬃne algebraic v ariet y o v er k . It follows fro m this (see [ 29 ]) tha t for Z Ω n A/k = k er( d : Ω n A/k → Ω n +1 A/k ) and H ∗ dR ( A/k ) = H ∗ (Ω A/k , d ), w e ha v e ( n ∈ Z ) H H n ( A/k ) = Ω n A/k H C n ( A/k ) = Ω n A/k /d Ω n − 1 A/k ⊕ M 0 ≤ 2 i 0 H n +2 p dR ( A/k ) H P n ( A/k ) = Y p ∈ Z H 2 p − n dR ( A/k ) F or arbitrary comm uta tiv e unital A , there is a decomp osition C n ( A/k ) = ⊕ n p =0 C ( p ) ( A/k ) suc h that b maps C ( p ) to itself, while B ( C ( p ) ) ⊂ C ( p +1) (see [ 29 ]). One deﬁnes H H ( p ) n ( A/k ) = H n C ( p ) ( A/k ) W e hav e H H ( p ) q ( A/k ) = 0 for q < p , H H ( p ) p ( A/k ) = Ω p A/k , but in general for q > 0, H H ( p ) p + q ( A/k ) 6 = 0. The map ( 8.2.2 ) is still a quas i-isomorphism if A is smo oth ov er a ﬁeld F ⊃ k ; this follows fr o m the Lo day-Quillen result using the base c hange sp ectral sequence of Kassel-Sletsjøe, whic h w e recall b elo w. Lemma 8.2.4. (Kass e l-Sletsjøe, [ 27 , 4.3a] ) L et k ⊆ F b e ﬁelds of char acteristic zer o. F or e ach p ≥ 1 ther e is a b ounde d se c ond quadr ant homo l o gic al sp e ctr al se q uen c e ( 0 ≤ i < p , j ≥ 0 ): p E 1 − i,i + j = Ω i F /k ⊗ F H H ( p − i ) p − i + j ( R/F ) ⇒ H H ( p ) p + j ( R/k ) Corollary 8.2.5. If A is a smo oth F -algebr a, then ( 8.2.2 ) is a quasi-isomorphism. Theorem 8.2.6. L et X b e a c omp a ct top o l o gic al sp ac e, A = C ( X ) , and k ⊂ C a subﬁeld. Then the map ( 8.2.2 ) is a quasi-isomorphi s m, and we have the identities ( 8.2.3 ) . Pr o of. Extend C ( p ) ( /k ) (and H H ( p ) n ( /k )) to non-unita l a lg ebras b y C ( p ) ( A/k ) = k er( C ( p ) ( ˜ A k /k ) → C ( p ) ( k /k ) ) . Let E ( p ) ( A/k ) b e the sp ectrum the Dold-Ka n cor r esp ondence asso ciates to C ( p ) ( A/k ). Regard E ( p ) as a homology theory of C -algebras. Then E ( p ) is excis iv e on C ∗ -algebras, b y Remark 2.3.6 and naturality . F urther E ( p ) n ( A/k ) = H H ( p ) n ( A/k ) = 0 whenev er n > p and A is smo oth o v er C , b y Corollary 8.2.5 . It is also clear that H H ( p ) ∗ ( /k ) preserv es ﬁltering colimits, since H H ∗ ( /k ) do es. Th us w e may apply Theorem 7.4.1 to conclude the pro o f .  ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 37 8.3. The F arrell-Jones Isomorphism Conjecture. Let A b e a ring, Γ a group, and q ≤ 0. Put W h A q (Γ) = cok er ( K q ( A ) → K q ( A [Γ])) for the cok ernel of the map induced by the natural inclusion A ⊂ A [Γ]. Recall [ 31 , Conjec- ture 1, pp. 708] that the F arrell-Jones conjecture with co eﬃcien ts f or a torsion free gro up implies that if Γ torsion free and A a no etherian regular unital ring, then (8.3.1) W h A q (Γ) = 0 ( q ≤ 0 ) Note that the conjecture in particular implies that K q ( A [Γ]) = 0 for q < 0 if A is no etherian regular, for in this case w e ha v e K q ( A ) = 0 for q < 0. Theorem 8.3.2. L et Γ b e torsion fr e e gr oup which satisﬁes ( 8.3.1 ) fo r every c omm utative smo oth C -algebr a A . Also l e t q ≤ 0 . Th e n the functor W h ? q (Γ) is homotopy invaria nt on c ommutative C ∗ -algebr as. Pr o of. It follows from Theorem 7.3.1 applied to W h ? q (Γ).  Corollary 8.3.3. L et Γ b e as ab ove and X a c ontr actible c omp act sp ac e. T h en K 0 ( C ( X )[Γ]) = Z a n d K q ( C ( X )[Γ]) = 0 for q < 0 . Corollary 8.3.4. L et Γ b e as ab ove. Then, the functor X 7→ K q ( C ( X )[Γ]) is homotopy invariant on the c ate gory of c omp act top olo gic al sp ac es for q ≤ 0 . Pr o of. This follows directly from Prop osition 2.1.5 and the preceding corollary .  The general case of the F arr ell-Jones conjecture predicts that for an y group Γ and an y unital ring R , the assem bly map [ 31 ] (8.3.5) A Γ ( R ) : H Γ ( E V C (Γ) , K ( R )) → K ( R [Γ]) is an equ iv alence. Here H Γ ( − , K ( R )) is the equiv arian t homology theory asso ciated to the sp ectrum K ( R ) and E V C ((Γ)) is the classifying space with resp ect to the class of virtually cyclic subgroups (see [ 31 ] for deﬁnitions of these ob jects). W e also get: Theorem 8.3.6. L et Γ b e a gr oup such that the map ( 8.3.5 ) i s an e quivalenc e for every smo oth c ommutative C -algebr a A . Then ( 8.3.5 ) is an e quivalenc e for every C ∗ -algebr a A . Pr o of. It follows from Theorem 7.4.1 applied to E ( R ) = ho coﬁb er( A Γ ( R )).  R emark 8.3.7 . W e hav e K q ( C ( X )[Γ]) = π S q  map + ( X + , K D ( C [Γ]))  , ( q ≤ 0 ) where K D ( C [Γ]) denotes the diﬀeotop y K -theory sp ectrum, see Deﬁnition 4 . 1 . 3 in [ 10 ]. This follows f rom the study of a suitable co-assem bly map and is not carried out in detail here. The homotop y groups of K D ( C [Γ ]) can b e computed from the equiv ariant connectiv e K - homology of EΓ using the F arrell-Jones assem bly map. 38 GUILLERMO COR TI ˜ NAS AND A NDREAS THOM 8.4. Adams op er ations and the decomp osition of rational K -theory . The ratio nal K - theory of a unital comm uta tiv e ring A carries a natural decomp osition K n ( A ) ⊗ Q = ⊕ i ≥ 0 K n ( A ) ( i ) Here K n ( A ) ( i ) = T k 6 =0 { x ∈ K n ( A ) | ψ k ( x ) = k i x } , where ψ k is the Adams op eration. F or example, K (0) 0 ( A ) = H 0 (Sp ec A, Q ) is the rank comp onen t, and K (0) n ( A ) = 0 for n > 0 ([ 28 , 6.8]). A conjecture of Alexander Be ˘ ılinson and Christophe Soul ´ e (see [ 3 ], [ 40 ]) asserts that (8.4.1) K ( i ) n ( A ) = 0 for n ≥ max { 1 , 2 i } . The conjecture as stated was pro ved wrong for non-regular A (see [ 16 ] and [ 13 , 7.5.6]) but no regular coun terexamples hav e b een fo und. Moreov er, the original statemen t has b een form ulated in terms of motivic cohomology (with rational, tor sion and in tegral co eﬃcien ts) and generalized to regular no etherian sc hemes [ 26 , 4.3.4]. F or example if X = Sp ec R is smo oth then K ( i ) n ( R ) = H 2 i − n ( X , Q ( i )) is the motivic cohomology o f X with co eﬃcien ts in the t wisted sheaf Q ( i ). W e shall need the w ell-known fact tha t the v alidit y of ( 8.4.1 ) for C implies its v alidit y for all smo oth C -alg ebras; this is Prop osition 8.4.3 b elo w. In turn this uses the also w ell-known fact that ra tional K -theory sends ﬁeld inclusions to monomorphisms. W e include pro ofs of b oth facts for completeness sak e. Lemma 8.4.2. L et F ⊂ E b e ﬁelds. T hen K ∗ ( E ) ⊗ Q → K ∗ ( F ) ⊗ Q is inje ctive. Pr o of. Since K -theory comm utes with ﬁltering colimits, w e ma y assume that E / F is a ﬁnitely generated ﬁeld ex tension, whic h w e ma y write as a ﬁnite extension of a ﬁnitely generated purely transcenden ta l extension. If E /F is purely transcenden tal, then b y induction we are reduced t o the case E = F ( t ), whic h follo ws fro m [ 18 , Thm. 1.3]. If d = dim F E is ﬁnite, then the transfer map K ∗ ( E ) → K ∗ ( F ) [ 34 , pp. 111] splits K ∗ ( F ) → K ∗ ( E ) up to d - t o rsion.  Prop osition 8.4.3. If ( 8.4.1 ) holds for C , then it holds for al l smo oth C -a l g ebr as. Pr o of. The Gysin sequence argumen t a t the b eginning of [ 26 , 4 .3 .4] show s that if ( 8.4.1 ) is an isomorphism for all ﬁnitely generated ﬁeld extensions of C then it is an isomorphism for all smo oth R . If E ⊃ C is a ﬁnitely generated ﬁeld extension, then w e may write E = F [ α ] for some purely transcenden ta l ﬁeld extension F ∼ = C ( t 1 , . . . , t n ) ⊃ C and some a lg ebraic elemen t α . F rom this and the fact that C is alg ebraically closed and of inﬁnite transcendence degree o v er Q , we see that E is isomorphic to a subﬁeld of C . No w apply Lemma 8.4.2 .  Theorem 8.4.4. Assume that ( 8.4.1 ) holds for the ﬁeld C . Then i t also holds for al l c om- mutative C ∗ -algebr as. Pr o of. By Prop osition 8.4.3 , our curren t hypothesis imply that ( 8 .4 .1 ) holds for smo oth A . In particular the homolog y theory K ( i ) v anishes on smo oth A for n ≥ n 0 = max { 2 i, 1 } . Because K - theory satisﬁes exc ision for C ∗ -algebras and commutes with algebraic ﬁltering colimits, the same is true of K ( i ) . Hence w e can a pply Theorem 7.4.1 , concluding the pro of .  ALGEBRAIC GEOMETR Y OF TOPOLOGICAL S P ACES I 39 A cknow le d g ements. P a rt o f the researc h for this article was carried out during a visit o f the ﬁrst named author to Univ ersit¨ at G¨ otting en. He is indebted to this institution f o r their hospitalit y . He also wishes to thank Ch uck W eibel for a useful e-mail discussion of Be ˘ ılinson- Soul ´ e’s conjecture. A previous v ersion of this article con tained a tec hnical mistak e in the pro of of Theorem 7.1.3 ; w e are thankful to Emanue l Ro dr ´ ıguez Cirone for bring ing t his to our atten tion. Reference s [1] Ba ss, H., Some problems in “classical” algebraic K -theo ry . In Alg ebr aic K -the ory, II: “Classic al” alge- br aic K -the ory and c onne ctions with arithmetic (Pr o c. Conf., Battel le Memorial Inst., Se attle, Wash., 1972) , pages 3–73 . Lecture Notes in Math., V ol. 342 . Springer , B erlin, 1973. [2] Ba su, S., Pollack, R., & Roy , M-F., A lgorithms in r e al algebr aic ge ometry , volume 10 o f Alg orithms and Computation in Mathematics . Spring er-V er lag, B erlin, 2003. 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