The extended Bloch group and algebraic K-theory

THE EXTENDED BLOCH GR OUP AND ALGEBRAIC K –THEOR Y CHRISTIAN K. ZICKER T Abstract. W e deﬁne an extended Blo c h group for an arbitrar y ﬁeld F , and sho w that this g roup is natur ally isomorphic to K ind 3 ( F ) if F is a num ber ﬁeld. This giv es an explicit description of K ind 3 ( F ) in terms of generat ors and relations. W e give a concre te form ula f or the reg ulator, and de rive concrete symbol ex pressions generat ing the torsion. As an application, we sho w that a hy perb olic 3 –manifold with ﬁnite v olume and inv arian t trace ﬁeld k has a fundamen tal class in K ind 3 ( k ) ⊗ Z [ 1 2 ] . 1. Introduction The extended Blo c h gro up b B ( C ) w as introduced by W alter Neumann [1 2] in his computation of the Cheeger – Chern–Simons class related to P SL(2 , C ) . It is a Q / Z –extension of the cla ssical Blo ch group B ( C ) , a nd w as used by Neumann to g ive explicit simplicia l formulas for the volume and Chern– Simons inv ariant of h yp erbo lic 3 –manifolds; see also Zick er t [2 1]. There are tw o distinct versions o f the extended Blo ch group. O ne is is o morphic to H 3 (PSL(2 , C ) δ ) and the o ther is isomorphic to H 3 (SL(2 , C ) δ ) . The δ denotes that the g roups are regarded as discrete groups, and will from now on b e omitted. F or a discussion of the rela tionship betw een the tw o versions o f the e xtended Blo ch gr oup, se e Go ette–Zick ert [5 ]. In Section 3, we deﬁne an extended Blo ch gr o up for an arbitrar y ﬁeld F . More precisely , we show that there is an e x tended Blo c h gro up b B E ( F ) for ea c h ex tension E of F ∗ b y Z , which only depends on the class of E in Ex t ( F ∗ , Z ) . The or iginal extended Blo c h group is the extended B loch g roup asso ciated to the extension of C ∗ given by the exp onen tial map. F or a large class of ﬁelds, including num b er ﬁelds and ﬁnite ﬁelds, the extended Blo c h gr oups b B E ( F ) a re isomorphic, and can be glued tog e ther to fo r m an e x tended B lo ch gr oup b B ( F ) w hich only dep ends on F , and admits a natural Galois action. This is studied in Sectio n 4. By a result of Suslin [1 8], the classica l Blo ch gro up B ( F ) of an (inﬁnite) ﬁeld F is iso mo rphic to the algebra ic K –gr o up K ind 3 ( F ) mo dulo torsion. More precise ly , Suslin proves that there is an exact s equence (1.1) 0 → f µ F → K ind 3 ( F ) → B ( F ) → 0 , where µ F denotes the ro ots o f unity in F , and f µ F is the unique non-trivial extension of µ F b y Z / 2 Z (in characteristic 2 , f µ F = µ F ). Our ma in r esult is the fo llowing. Theorem 1.1. F or every numb er ﬁeld F , ther e is a natur al isomorphism b λ : K ind 3 ( F ) ∼ = b B ( F ) r esp e cting the Galois actions.  1 2 CHRISTIAN K. ZICKER T In Section 9 w e give the fo llo wing geometric application generalizing a result of Goncharo v [6], who prov ed the existence of a fundament al class in K ind 3 ( Q ) ⊗ Q . Theorem 1 .2. L et M b e a c omplete, oriente d, hyp erb olic 3 –manifold of ﬁnite vol- ume. L et K and k denote the t r ac e ﬁeld and invariant tr ac e ﬁeld of M . If M is close d, M has a fundamental class [ M ] in K ind 3 ( K ) deﬁne d up to two-torsion, and satisfying that 2 [ M ] ∈ K ind 3 ( k ) . If M has cusps, ther e is a fundamental class [ M ] in K ind 3 ( k ) ⊗ Z [ 1 2 ] such that 8 [ M ] is in K ind 3 ( k ) .  The res ult is proved using bo th concr ete pro perties o f the extended Blo ch group and abs tract pr o perties of K ind 3 ( F ) . There is a regulator ma p R : K ind 3 ( C ) → C / 4 π 2 Z . The regulato r is equiv a riant with res p ect to c o mplex conjugation, so if F is a n um b e r ﬁeld, we obtain a r egulator (1.2) b B : K ind 3 ( F ) → ( R / 4 π 2 Z ) r 1 ⊕ ( C / 4 π 2 Z ) r 2 , where r 1 and r 2 are the num b er of real and (conjugate pairs of ) complex embed dings of F in C . This regulator ﬁts into a diagram K ind 3 ( F ) b B / /   ( R / 4 π 2 Z ) r 1 ⊕ ( C / 4 π 2 Z ) r 2   B ( F ) B / / R r 2 , where the left vertical map is the map in (1.1), a nd the rig ht vertical map is pro- jection onto the imagina ry part. The lower map B is known a s the Borel regulator and has b een ex tensively studied. It is related to hyperb olic volume, and it is known that the image in R r 2 is a lattice whose covolume is prop ortional to the zeta function o f F ev alua ted at 2 . W e refer to Z agier [2 0] for a survey . The upp er map is muc h less understo od. The re al par t is r e la ted to the Chern- Simons inv ariant, but little is known ab out its r elations to num b er theory . In sec tion 4, we give a co ncrete for m ula for b B deﬁned on the e xtended Blo c h group b B ( F ) = K ind 3 ( F ) . Element s in b B ( F ) ar e e a sy to pro duce, e.g. using computer softw are like P ARI/GP , and our r e s ult ca n th us provide lots of exp eriment al data for s tudying the map b B . W e give a n example in Example 4.12. The torsion in K ind 3 ( F ) is known to b e c y clic of o rder w = 2 Q p ν p , where ν p = max { ν | ξ p ν + ξ − 1 p ν ∈ F } . The pro duct, which is eas ily s een to b e ﬁnite, is ov er all r ational primes, and ξ p ν is a pr imitiv e ro ot of unity of order p ν . This result is due to Mer kurjev–Suslin [10]; see a lso the survey pap er W eibel [19]. In Section 8, we giv e explicit elemen ts in b B ( F ) g enerating the torsio n. As a corollar y , this gives explicit genera tors of the torsio n in the Blo c h gr oup. W e state this result b elow. Let B ( F ) p denote the elements in B ( F ) of order a p ow er of p . By (1.1), the order of B ( F ) p is p ν ′ p , where ν ′ p = ν p − max { ν | ξ p ν ∈ F } . THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 3 Theorem 1. 3. L et F b e a numb er ﬁeld and let p b e a prime numb er with ν ′ p > 0 . L et x b e a primitive r o ot of u nity of or der p ν p . The elements β p = p ν p X k =1  ( x k +1 + x − k − 1 )( x k − 1 + x − k +1 ) ( x k + x − k ) 2  , β 2 = 2 ν 2 − 1 X k =1  ( x k +1 − x − k )( x k − 1 − x − k +2 ) ( x k − x − k +1 ) 2  gener ate B ( F ) p for o dd p and p = 2 , r esp e ctively.  Note that the torsio n in the Blo ch group comes from a to tally real ab elian subﬁeld of F . R emark 1.4 . The torsion in the Blo ch group of a num b er ﬁeld is related to conformal ﬁeld theor y , and there is a n in ter e s ting conjecture r elating torsion in the Blo c h g roup to mo dularity o f a certain q – hype r geometric function. See Nahm [11], or Zagier [20]. A ck no wl edgemen ts. I wish to tha nk Ian Agol, Johan Dupo n t, Sta vros Garo ufa- lidis, Matthias Go erner, Dylan Thurston and, in par ticular, W alter Neumann for helpful discussions. I also wish to thank W alter Neumann for his comment s on earlier drafts o f the pap er. Parts of this work was done during a visit to the Max Planck Institute of Mathematics, Bo nn. I wish to thank MPIM fo r its ho spitalit y , and for providing an exc e llent working environmen t. 2. Preliminaries F or a n ab elian g roup A , we deﬁne ∧ 2 ( A ) = A ⊗ Z A  h a ⊗ b + b ⊗ a i . Note that 2 a ∧ a = 0 , but a ∧ a is gener ally no t 0 . F or a set X , w e let Z [ X ] denote the free ab elian gro up generated by X . 2.1. The classical Bl o c h group. Let F be a ﬁeld and let F ∗ be the m ultiplicative group of units in F . Consider the set of ﬁve term r elations FT =  x, y , y x , 1 − x − 1 1 − y − 1 , 1 − x 1 − y    x 6 = y ∈ F \ { 0 , 1 }  . One ca n show that there is a chain co mplex (2.1) Z [FT] ρ / / Z [ F \ { 0 , 1 } ] ν / / ∧ 2 ( F ∗ ) , with ma ps deﬁned b y ρ ([ z 0 , . . . , z 4 ]) = [ z 0 ] − [ z 1 ] + [ z 2 ] − [ z 3 ] + [ z 4 ] , ν ([ z ]) = z ∧ (1 − z ) . R emark 2.1 . By Matsumoto’s theor em, the cokernel of ν is K 2 ( F ) . Deﬁnition 2.2. The Blo ch gr oup of F is the quo tien t B ( F ) = Ker( ν ) / Im( ρ ) . It is a subg roup o f the pr e-Blo ch gr oup P ( F ) = Z [ F \ { 0 , 1 } ] / Im( ρ ) . 4 CHRISTIAN K. ZICKER T 2.2. The extended Blo c h group of C . The orig ina l refer ence is Neumann [1 2]; see also Dupo nt–Zick ert [2] and Go ette–Zick er t [5]. W e s tr ess that our extended Blo c h gr oup is what Neumann calls the mor e extended Blo c h g roup [12, Section 8]. Consider the set b C =  ( w 0 , w 1 ) ∈ C 2   exp( w 0 ) + exp( w 1 ) = 1  . W e will refer to elements of b C as ﬂatt enings . W e can view b C as the Riemann surface for the multiv alued function (lo g( z ) , log(1 − z )) , a nd we can thu s write a ﬂattening as [ z ; 2 p , 2 q ] = (log( z ) + 2 pπ i, log(1 − z ) + 2 q π i ) . This notation dep ends on a choice of logar ithm branch which we will ﬁx once a nd for all. The map π : b C → C \ { 0 , 1 } taking a ﬂattening [ z ; 2 p, 2 q ] to z is the universal ab elian cover of C \ { 0 , 1 } . R emark 2.3 . Neumann considered the Riemann surface of (lo g ( z ) , − log(1 − z )) , and considered a ﬂattening [ z ; 2 p, 2 q ] as a triple ( w 0 , w 1 , w 2 ) , with w 0 = log( z ) + 2 pπ i , w 1 = − log(1 − z ) + 2 q π i and w 2 = − w 1 − w 0 . One tr anslates betw een the tw o deﬁnitions by changing the sign of w 1 , or equiv alently , by changing the sign of q . Let FT 0 =  ( x 0 , . . . , x 4 ) ∈ FT   0 < x 1 < x 0 < 1  , and deﬁne the set of lifte d ﬁve term r elations c FT ⊂ ( b C ) 5 to b e the co mp onent of the preimage o f FT co n taining all p oints  [ x 0 ; 0 , 0] , . . . , [ x 4 ; 0 , 0]  with ( x 0 , . . . , x 4 ) ∈ FT 0 . There is a chain complex (2.2) Z [ c FT ] b ρ / / Z [ b C ] b ν / / ∧ 2 ( C ) , with ma ps deﬁned b y b ρ ([( w 0 0 , w 0 1 ) , . . . , ( w 4 0 , w 4 1 )]) = 4 X i =0 ( − 1) i [( w i 0 , w i 1 )] , b ν ([( w 0 , w 1 )]) = w 0 ∧ w 1 . Deﬁnition 2. 4 . The extende d Blo ch gr oup is the q uo tien t b B ( C ) = Ker( b ν ) / Im( b ρ ) . It is a subgroup o f the extende d pr e-Blo ch gr oup b P ( C ) = Z [ b C ] / Im( b ρ ) . Theorem 2. 5. L et µ C denote the r o ots of unity in C ∗ . Ther e is a c ommutative diagr am as b elow with exact r ows and c olumns.  (2.3) 0   0   0   0 / / µ C / / χ   C ∗ / / χ   C ∗ /µ C / / β   0   0 / / b B ( C ) / / π   b P ( C ) b ν / / π   ∧ 2 ( C ) / / ǫ   K 2 ( C ) / / 0 0 / / B ( C ) / /   P ( C ) ν / /   ∧ 2 ( C ∗ )   / / K 2 ( C ) / /   0 0 0 0 0 THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 5 W e r efer to Go ette–Zick ert [5] o r Section 3 b elow for the deﬁnition of χ . The other ma ps are deﬁned as follows: β ([ z ]) = log( z ) ∧ 2 π i ; ǫ ( w 1 ∧ w 2 ) = exp( w 1 ) ∧ exp( w 2 ); π ([ z ; 2 p, 2 q ]) = [ z ] . R emark 2.6 . There is a similar deﬁnition of b P ( F ) and b B ( F ) for a s ubﬁeld F of C . Theorem 2.5 ho lds if one replace s ∧ 2 ( C ) by ∧ 2 ( { w ∈ C | exp( w ) ∈ F ∗ } ) and µ C with f µ F = { w ∈ C | w 2 ∈ µ F } . W e will genera lize to ar bitra ry ﬁelds b elo w. 2.2.1. The r e gulator. The function R : b C → C / 4 π 2 , given by (2.4) [ z ; 2 p, 2 q ] 7→ Li 2 ( z ) + 1 2 (log( z ) + 2 pπ i )(log (1 − z ) − 2 q π i ) − π 2 / 6 is well deﬁned and holomorphic, see e.g. Neumann [12] or Go ette–Zic kert [5]. It is well known that L ( z ) = Li 2 ( z ) + 1 2 log( z ) log(1 − z ) − π 2 / 6 satisﬁes 4 X i =1 ( − 1) i L ( z i ) = 0 for ( z 0 , . . . , z 4 ) ∈ FT 0 , and it thus follows by analy tical contin uation that R gives rise to a function (2.5) R : b P ( C ) → C / 4 π 2 Z . W e brieﬂy des cribe a mor e elegant deﬁnition o f R due to Don Zagier [2 0]: The deriv ative of Li 2 ( z ) is − log(1 − z ) /z . It follows that the function F ( x ) = Li 2 (1 − e x ) has deriv a tiv e F ′ ( x ) = xe x / (1 − e x ) . Since this function is meromorphic with simple po les at 2 π in , n ∈ Z , with corresp onding residues − 2 π in , it follows that F deﬁnes a sing le v alued function on C \ { 2 π i Z } with v a lues in C / 4 π 2 Z . W e ca n now deﬁne (2.6) R : b C → C / 4 π 2 , ( w 0 , w 1 ) 7→ F ( w 1 ) + w 0 w 1 2 − π 2 / 6 . W e leav e it to the reader to show tha t this deﬁnition of R ag rees with the one ab o ve. 2.3. Al g ebraic K –theory and homol ogy of linear groups . W e giv e a brief review of the results that we shall need. Let F b e a ﬁeld. The a lgebraic K –g roups are deﬁned by K i ( F ) = π i ( B GL( F ) + ) . The Milnor K –groups K M ∗ ( F ) a re deﬁned as the quotient of the tensor algebra of F ∗ b y the tw o-sided ideal generated b y a ⊗ (1 − a ) . There is a natural map K M i ( F ) → K i ( F ) whose cokernel, by deﬁnition, is the inde c omp osable K – group K ind i ( F ) . F or F = C , there is a r e gulator R deﬁned as the comp osition K 2 k − 1 ( C ) H / / H 2 k − 1 (GL( C )) ˆ c k / / C / (2 π i ) k Z , where H is the Hurewicz map, and ˆ c k is the universal Cheeger–Chern– Simons class. It is well k no wn that R is 0 on the image of K M 2 k − 1 ( C ) , s o R induces a re g ulator K ind 2 k − 1 ( C ) → C / (2 πi ) k Z . 6 CHRISTIAN K. ZICKER T Theorem 2.7 (Suslin [17]) . F or any ﬁeld F , t her e is an isomorphism H n (GL( n, F )) ∼ = H n (GL( F )) induc e d by inclusion.  Theorem 2.8 (Sah [16]) . K ind 3 ( C ) is a dir e ct summand of K 3 ( C ) and t he Hure wicz map H induc es an isomorphism K ind 3 ( C ) ∼ = H 3 (SL(2 , C )) .  Theorem 2.9 (Goette–Z ic kert [5]; see also Neumann [12]) . Ther e is a c anonic al isomorphi sm H 3 (SL(2 , C )) ∼ = b B ( C ) . Under this isomorphism, ˆ c 2 c orr esp onds to t he map R in (2 .4) .  Theorem 2.10 (Dupont–Sah [1]) . The diagonal map x 7→  x 0 0 x − 1  induc es an inje ct ion H 3 ( µ C ) → H 3 (SL(2 , C )) onto t he torsion su b gr oup of H 3 (SL(2 , C )) .  3. The extended Bl och group o f an extension Let F be a ﬁeld and let E : 0 / / Z ι / / E π / / F ∗ / / 0 be an extension of F ∗ b y Z . W e stre s s that the letter E is used bo th to denote the extension and the middle g roup. As w e shall see, most of the results in Neumann [12] and Go ette–Zick er t [5] can b e formulated in this purely a lgebraic setup. Deﬁnition 3.1. The set of (algebr aic) ﬂattenings is the set b F E =  ( e, f ) ∈ E × E   π ( e ) + π ( f ) = 1 ∈ F  . The map ( e , f ) 7→ π ( e ) induces a surjection π : b F E → F \ { 0 , 1 } , and we say that ( e, f ) is a ﬂatt ening of π ( e ) . Recall the set of ﬁve term relations FT =  x, y , y x , 1 − x − 1 1 − y − 1 , 1 − x 1 − y    x 6 = y ∈ F \ { 0 , 1 }  . Deﬁnition 3.2. The set of lifte d ﬁve term r elations c FT E ⊂ ( b F E ) 5 is the set o f tuples of ﬂattenings  ( e 0 , f 0 ) , . . . , ( e 4 , f 4 )  satisfying e 2 = e 1 − e 0 e 3 = e 1 − e 0 − f 1 + f 0 f 3 = f 2 − f 1 e 4 = f 0 − f 1 f 4 = f 2 − f 1 + e 0 . (3.1) If  ( e 0 , f 0 ) , . . . , ( e 4 , f 4 )  ∈ c FT E , where ( e i , f i ) is a ﬂattening o f x i ∈ F \ { 0 , 1 } , then (3 .1) implies that x 2 = x 1 x 0 , x 3 = x 1 x 0 (1 − x 0 ) (1 − x 1 ) = 1 − x − 1 0 1 − x − 1 1 , x 4 = 1 − x 0 1 − x 1 . Hence, a lifted ﬁve term relation is indeed a lift of a ﬁve term rela tion. On the other hand, if ( x 0 , . . . , x 4 ) ∈ FT it is not diﬃcult to chec k that there exist ﬂattenings ( e i , f i ) s atisfying (3.1 ). Hence, the map π : c FT E → FT is surjective. THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 7 Consider the co mplex (3.2) Z [ c FT E ] b ρ / / Z [ b F E ] b ν / / ∧ 2 ( E ) , with ma ps deﬁned b y b ρ  ( e 0 , f 0 ) , . . . , ( e 4 , f 4 )  = 4 X i =0 ( − 1) i ( e i , f i ) , b ν ( e, f ) = e ∧ f . Lemma 3.3. The c omplex (3.2) is a chain c omplex, i.e. b ν ◦ b ρ = 0 . Pr o of. Let α =  ( e 0 , f 0 ) , . . . , ( e 4 , f 4 )  ∈ c FT E . Using (3.1) w e have b ν ◦ b ρ ( α ) = 4 X i =0 ( − 1) i e i ∧ f i = e 0 ∧ f 0 − e 1 ∧ f 1 + ( e 1 − e 0 ) ∧ f 2 − ( e 1 − e 0 − f 1 + f 0 ) ∧ ( f 2 − f 1 ) + ( f 0 − f 1 ) ∧ ( f 2 − f 1 + e 0 ) = e 0 ∧ ( f 0 − f 1 ) + ( f 0 − f 1 ) ∧ e 0 = 0 ∈ ∧ 2 ( E ) .  Deﬁnition 3.4. The exten de d pr e-Blo ch gr oup b P E ( F ) is the quotien t Z [ b F E ] / Im( b ρ ) . The ext en de d Blo ch gr oup b B E ( F ) is the quo tient K er( b ν ) / Im( b ρ ) . Example 3.5 . The ex tended B lo ch group b B ( C ) is the extended Blo ch gro up as so- ciated to the extension (3.3) 0 / / Z 2 π i / / C exp / / C ∗ / / 0 . The extended g r oups ﬁt to g ether with the classical groups in a diagram b B E ( F ) / /   b P E ( F )   b ν / / ∧ 2 ( E )   B ( F ) / / P ( F ) ν / / ∧ 2 ( F ∗ ) , where the vertical maps a re surjections. 3.1. R elations in the extended Blo ch group. W e now derive some re la tions in b P E ( F ) . W e enco ur age the re a der to co mpare w ith Neumann [12] a nd Go ette– Zick ert [5] where similar r e la tions a re der iv ed in b P ( C ) using analytic co n tinuation. In the following we will r egard Z as a subgro up of E . Consider the se t (3.4) V =  ( p 0 , q 0 ) , ( p 1 , q 1 ) , ( p 1 − p 0 , q 2 ) , ( p 1 − p 0 − q 1 + q 0 , q 2 − q 1 ) , ( q 0 − q 1 , q 2 − q 1 + p 0 )    p 0 , p 1 , q 0 , q 1 , q 2 ∈ Z  ⊂ ( Z × Z ) 5 , also cons idered by Neuma nn [12, Deﬁnition 2.2]. By (3.1) it follows that comp onen twise addition gives rise to an action + : c FT E × V → c FT E , ( α, v ) 7→ α + v . Lemma 3.6. L et q , q ′ , ¯ q , ¯ q ′ b e inte gers satisfying q − q ′ = ¯ q − ¯ q ′ . F or e ach ﬂattening ( e, f ) ∈ b F E we have (3.5) ( e, f + q ) − ( e, f + q ′ ) = ( e, f + ¯ q ) − ( e, f + ¯ q ′ ) ∈ b P E ( F ) . 8 CHRISTIAN K. ZICKER T Pr o of. Let α =  ( e 0 , f 0 ) , . . . , ( e 4 , f 4 )  ∈ c FT E . F or each in teger r , consider the element v r ∈ V given by v r =  (0 , r ) , (0 , r ) , (0 , r ) , (0 , 0) , (0 , 0)  . The rela tion b ρ ( α + v q ) − b ρ ( α + v q ′ ) = 0 ∈ b P E ( F ) can be written as (3.6) ( e 0 , f 0 + q ) − ( e 0 , f 0 + q ′ ) −  ( e 1 , f 1 + q ) − ( e 1 , f 1 + q ′ )  + ( e 2 , f 2 + q ) − ( e 2 , f 2 + q ′ ) = 0 ∈ b P E ( F ) . Let β = α +  (0 , 0) , (0 , s ) , (0 , s ) , ( − s, 0) , ( − s, 0)  , where s = q − ¯ q = q ′ − ¯ q ′ . Then β ∈ c FT E , and the relation b ρ ( β + v ¯ q ) − b ρ ( β + v ¯ q ′ ) = 0 ∈ b P E ( F ) b ecomes (3.7) ( e 0 , f 0 + ¯ q ) − ( e 0 , f 0 + ¯ q ′ ) −  ( e 1 , f 1 + q ) − ( e 1 , f 1 + q ′ )  + ( e 2 , f 2 + q ) − ( e 2 , f 2 + q ′ ) = 0 ∈ b P E ( F ) . The res ult now follows by subtracting (3.7) from (3.6).  Corollary 3.7. L et e ∈ E \ Z . The element ( e, f + 1) − ( e, f ) ∈ b P E ( F ) is indep endent of f whenever ( e, f ) is in b F E .  Using Corollar y 3.7, w e can deﬁne a map (3.8) χ : E \ Z → b P E ( F ) , e 7→ ( e, f + 1) − ( e, f ) . Lemma 3.8. Supp ose e, e ′ and e + e ′ ar e elements in E \ Z . W e have (3.9) χ ( e ) + χ ( e ′ ) = χ ( e + e ′ ) . Pr o of. This follows fro m (3.6) a fter noting that e 1 = e 0 + e 2 .  The followin g is elementary . Lemma 3.9. L et G and G ′ b e gr oups and let H b e a sub gr oup of G of index gr e ater than 2 . Su pp ose φ : G \ H → G ′ is a map satisfying φ ( g 1 g 2 ) = φ ( g 1 ) φ ( g 2 ) whenever b oth sides ar e deﬁne d. Then φ extends uniquely to a homomorphi sm φ : G → G ′ .  Corollary 3.10. The map χ : E \ Z → b P E ( F ) extends to a homomorph ism deﬁne d on al l of E .  Lemma 3.11. F or any t wo ﬂattenings ( e, f ) , ( g , h ) ∈ b F E we have ( e, f ) + ( f , e ) = ( g , h ) + ( h, g ) ∈ b B E ( F ) . Pr o of. It fo llows from (3.1) that  ( e 0 , f 0 ) , ( e 1 , f 1 ) , ( e 2 , f 2 ) , ( e 3 , f 3 ) , ( e 4 , f 4 )  ∈ c FT E if a nd only if  ( f 1 , e 1 ) , ( f 0 , e 0 ) , ( e 4 , f 4 ) , ( e 3 , f 3 ) , ( e 2 , f 2 )  ∈ c FT E . Subtracting the t wo rela tio ns in b P E ( F ) y ields ( e 0 , f 0 ) − ( e 1 , f 1 ) = ( f 1 , e 1 ) − ( f 0 , e 0 ) ∈ b P E ( F ) , from whic h the cla im follows. Since e ∧ f + f ∧ e = 0 ∈ ∧ 2 ( E ) , the element lies in b B E ( F ) .  Lemma 3.12. The homomorphism χ : E → b P E ( F ) takes 2 Z ⊂ E to 0 . THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 9 Pr o of. Let e ∈ E b e any element not in Z . The result follows from the computation χ (1) = χ ( e + 1) − χ ( e ) = ( e + 1 , f + 1) − ( e + 1 , f ) − ( e, f + 1) + ( e, f ) = − ( f + 1 , e + 1) + ( f , e + 1) + ( f + 1 , e ) − ( f , e ) = − χ ( f + 1) + χ ( f ) = − χ (1) , (3.10) where the third equalit y follows fro m Lemma 3 .11.  Theorem 3.13. Ther e is an exact se quenc e (3.11) E / 2 Z χ / / b P E ( F ) π / / P ( F ) / / 0 . Pr o of. It is clear that π ◦ χ = 0 and that π is surjectiv e. Since ev er y ﬁve term relation lifts to a lifted ﬁve term relation, the kernel of π must b e gener ated b y element s o f the for m ( e + p, f + q ) − ( e, f ) . B y Lemma 3.6 w e have, (3.12) ( e, f + q ) = ( e, f + q − 1) + ( e, f + 1 ) − ( e, f ) = q ( e, f + 1 ) − q ( e, f ) + ( e, f ) = q χ ( e ) + ( e, f ) , and we see that ( e, f + q ) − ( e, f ) is in Im( χ ) . Using Lemma 3.11 we similarly obtain (3.13) ( e + p, f ) − ( e, f ) = − pχ ( f ) ∈ Im( χ ) , and the result follows.  R emark 3.14 . W e do not know if χ is injective, but w e exp ect this to be the ca se. In the next section, we show that χ is injective if F is a n um b er ﬁeld a nd E is a primitive extensio n. R emark 3.15 . If E is the ex tensio n in Example 3.5, the exact sequence (3.11) is equiv alent to the cor respo nding exac t se quence in Theo r em 2.5 us ing the identi- ﬁcation of C ∗ with E / 2 Z = C / 4 π i Z taking z ∈ C ∗ to − 2 log( z ) ∈ C / 4 π i Z . The restriction of the regulator (2.4) to C / 4 π i Z ⊂ b P ( C ) is multiplication by − π i . Lemma 3.16. The fol lowing e quality holds in b P E ( F ) . ( e + p , f + q ) − ( e, f ) = χ ( q e − pf + pq ) . Pr o of. This is a n easy consequence of (3 .12) and (3.13).  3.2. F unctorialit y. Let F 1 and F 2 be ﬁelds a nd let E 1 and E 2 be extensions of F ∗ 1 and F ∗ 2 b y Z . Deﬁnition 3.17. A map Ψ : E 1 → E 2 of extensions is called a c overing if the ba se homomorphism Ψ : F ∗ 1 → F ∗ 2 extends to a n embedding of F 1 in F 2 . T wo cov erings are e quivalent if they cover the same embedding. A c overing Ψ : E 1 → E 2 gives rise to a chain map Z [ c FT E 1 ] b ρ 1 / / Ψ ∗   Z [ b F E 1 ] b ν 1 / / Ψ ∗   ∧ 2 ( E 1 ) Ψ ∧ Ψ   Z [ c FT E 2 ] b ρ 2 / / Z [ b F E 2 ] b ν 2 / / ∧ 2 ( E 2 ) 10 CHRISTIAN K. ZICKER T deﬁned b y taking an algebraic ﬂa ttening ( e, f ) to (Ψ ( e ) , Ψ( f )) . W e thus o btain maps (3.14) Ψ ∗ : b P E 1 ( F 1 ) → b P E 2 ( F 2 ) , Ψ ∗ : b B E 1 ( F 1 ) → b B E 2 ( F 2 ) , satisfying the usual functoriality prop erties. Lemma 3.18. L et Ψ : E 1 → E 2 b e a c overing. Ther e is a c ommutative diagr am of exact se quenc es E 1 / 2 Z χ / / Ψ ∗   b P E 1 ( F 1 ) π / / Ψ ∗   P ( F 1 ) / / Ψ ∗   0 E 2 / 2 Z χ / / b P E 2 ( F 2 ) π / / P ( F 2 ) / / 0 . The map Ψ ∗ : E 1 / 2 Z → E 2 / 2 Z take s e ∈ E 1 / 2 Z to Ψ(1 )Ψ ( e ) ∈ E 2 / 2 Z , wher e Ψ(1)Ψ( e ) is deﬁne d u sing t he natur al action of Z ⊂ E 2 on E 2 by multiplic ation. Pr o of. Exactness follows fr o m Theorem 3 .13 . Commutativit y of the right square is obvious, and commutativit y of the left square follows from the computation Ψ ∗ ( χ ( e )) =  Ψ( e ) , Ψ( f ) + Ψ(1 )  −  Ψ( e ) , Ψ( f )  = χ  Ψ(1)Ψ( e )  , where the second e qualit y follows from L emma 3.16.  W e w is h to prov e that the induced map Ψ ∗ : b B E 1 ( F 1 ) → b B E 2 ( F 2 ) of a cov er ing only dep ends on the underlying em b e dding . The res ult b elow is elementary . Lemma 3. 19. Le t F b e a ﬁ eld. The torsion sub gr oup T or( F ∗ ) is isomorphic t o a sub gr oup of Q / Z . F or any ext ension E of F ∗ by Z , the same is tru e for T o r( E ) .  Lemma 3.20. An element P i n i e i ∧ f i is zer o in ∧ 2 ( E ) if and only if we c an write (3.15) e i = k i w + X j r ij p j , f i = l i w + X j s ij p j , wher e p j ∈ E , w ∈ E is a torsion element, and the inte gers s ij , r ij , k i and l i satisfy (i) P i n i r ij s ij is even for e ach j ; (ii) P i n i ( r ij s ik − r ik s ij ) = 0 for e ach j 6 = k ; (iii) P i n i ( l i r ij − k i s ij ) is divisibl e by o rd( w ) for e ach j ; (iv) P i n i k i l i is even. Pr o of. Let α = P i n i e i ∧ f i . Since ∧ 2 commut es with direct limits, there exists a ﬁnitely gener ated subgroup H , containing the e i ’s a nd f i ’s, such that α is zero in ∧ 2 ( E ) if and only if α is zer o in ∧ 2 ( H ) . Let p j be free genera tors of H and let w be a genera tor of the tor sion (which is cyclic by Lemma 3 .19) . W rite the e i ’s and f i ’s as in (3.15). When expanding α ∈ ∧ 2 ( H ) , the co eﬃcien ts of p j ∧ p j , p j ∧ p k , p j ∧ w and w ∧ w of ∧ 2 ( H ) are given, r espectively , by (i)-(iv). Hence, α = 0 ∈ ∧ 2 ( H ) if (i)-(iv) hold. On the other hand, if α = 0 ∈ ∧ 2 ( H ) , (i )-(iii) hold, whereas (iv) may fail if w ∧ w = 0 ∈ ∧ 2 ( H ) . This ha ppens if a nd only if w is 2 –divisible, in which case, we may replace w by a half if necessar y , to make (iv) hold as well.  Theorem 3.21. If Ψ 1 , Ψ 2 : E 1 → E 2 ar e e qu ivalent c overings then (3.16) Ψ 1 ∗ = Ψ 2 ∗ : b B E 1 ( F 1 ) → b B E 2 ( F 2 ) . THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 11 Pr o of. F or notational simplicit y , we as sume that Ψ 1 = id , and drop the s ubscripts of F i and E i . This case is suﬃcient for most of the applications. W e leave the general pr oof, whic h diﬀers only in notation, to the r eader. Let φ = Ψ 2 − Ψ 1 : E 1 → Z ⊂ E 2 and let α = P n i ( e i , f i ) ∈ b B E ( F ) . W e wish to prov e that (3.17) ∆ := Ψ 2 ∗ ( α ) − Ψ 1 ∗ ( α ) = X i n i  e i + φ ( e i ) , f i + φ ( f i )  − ( e i , f i )  is zer o in b B E ( F ) . By Lemma 3.16, we have (3.18) ∆ = χ  X i n i  φ ( f i ) e i − φ ( e i ) f i + φ ( e i ) φ ( f i )   . Since α ∈ b B E ( F ) , we hav e b ν ( α ) = P n i e i ∧ f i = 0 ∈ ∧ 2 ( E ) . By Le mma 3.2 0 , we can wr ite e i = k i w + X j r ij p j , f i = l i w + X j s ij p j , where (i)-(iv) in L e mma 3 .2 0 a re s atisﬁed. L et a j = φ ( p j ) . Let T denote the s um in (3.18), i.e. χ ( T ) = ∆ . Since w is a tors ion elemen t, φ ( w ) = 0 , so b y (3.1 8 ) we hav e (3.19) T = X i n i  X j s ij a j  X j r ij p j + k i w  − X j r ij a j  X j s ij p j + l i w  + X j r ij a j X j s ij a j  . When ex pa nding the sum, the co eﬃcien t of p k is X i n i  X j s ij a j  r ik − X i n i  X j r ij a j  s ik = − X j a j X i n i ( r ij s ik − s ij r ik ) , which is zero b y Lemma 3.20, (ii) . Similar ly , the co eﬃcien t of w is X i n i  X j a j k i s ij − X j a j l i r ij  = − X j a j X i n i ( l i r ij − k i s ij ) , which , by Lemma 3.20, (iii ), is divisible by the order of w . Finally , the remaining terms sum to the integer X i n i  X j r ij a j X j s ij a j  = X i n i  X j 6 = k r ij s ik a j a k  + X j a 2 j X i n i r ij s ij , which is even by Lemma 3.2 0, (i ) and (ii ). Hence, T is z e r o in E / 2 Z , so ∆ = χ ( T ) is als o zero.  Corollary 3. 2 2. Up to c anonic al iso morphism, the extende d Blo ch gr oup b B E ( F ) dep ends only on the class of E in Ext( F ∗ , Z ) . Pr o of. If E 1 = E 2 ∈ Ext( F ∗ , Z ) , ther e must exist a cov ering Ψ : E 1 → E 2 of the iden tit y o n F . Since any tw o such cov er ings are equiv alent, the result follows.  3.3. Gene ral prop erties of extensions. Let µ F ⊂ F ∗ denote the ro ots of unity in F . F or a prime num b er p let µ p denote the p th ro ots of unity in µ F , and let µ p ∞ be the subgr oup o f ro ots o f unity of order a p ow er of p . Note that µ F = ⊕ µ p ∞ . Also note that up to is omorphism, µ p ∞ is either Z /p n Z o r Z [1 /p ] / Z . W e have the fo llo wing clas siﬁcation of Z –extensions of Z /p n Z and Z [1 /p ] / Z . The pro ofs ar e element ary a nd left to the reader . 12 CHRISTIAN K. ZICKER T Lemma 3.23. W e have Ext( Z /p n Z , Z ) = Z /p n Z . L et 0 ≤ k ≤ n − 1 and let x b e an inte ger which is not divisible by p . The non-trivial ext ensions ar e given ex plic itly by (3.20) 0 / / Z ι / / Z ⊕ Z /p k Z π / / Z /p n Z / / 0 , wher e the maps ar e deﬁne d by (3.21) ι (1) = ( p n − k , − x ) , π ( a, b ) = xa + p n − k b. The e qu ival enc e class of (3.20) only dep ends on the value of x in ( Z /p n − k Z ) ∗ , and the or der of the ex t ension in Ext( Z /p n Z , Z ) is p n − k .  Lemma 3.24. W e ha ve E xt( Z [ 1 p ] / Z , Z ) = Z p , the p –adic inte gers. L et 0 ≤ k b e an inte ger and let y b e a p –adic int e ger with discr ete valuation k . The non-trivial extensions ar e given explicitly by (3.22) 0 / / Z ι / / Z [ 1 p ] ⊕ Z /p k Z π / / Z [ 1 p ]  Z / / 0 , wher e the maps ar e deﬁne d by (3.23) ι (1) = (1 /p k , − y /p k ) , π ( a, b ) = ay + b p k .  Deﬁnition 3 . 25. An extension E is called primitive if E is torsio n free. If µ p is non-trivial, and if the restriction E µ p ∞ is torsion free, we say that E is p –primitive . W e s ta te some e le mentary c o rollaries of L emma 3.2 3 a nd Lemma 3.24. Corollary 3. 26. E is p –primitive if and only if 1 ∈ E is divisible by p . If so, 1 is divisib le by | µ p ∞ | (if | µ p ∞ | = ∞ , 1 is divisible by p inﬁnitely often).  Corollary 3.27 . Supp ose µ F is ﬁnite. Then E is primitive if and only if E µ F gener ates Ex t( µ F , Z ) . In this c ase E µ F is fr e e of r ank one. L etting ˜ x denote a gener ator, the extension is given explicitly by (3.24) E µ F : 0 / / Z ι x / / E µ F π x / / µ F / / 0 , wher e ι x takes 1 t o | µ F | ˜ x and π x takes ˜ x to a primitive r o ot of un ity x .  Lemma 3.28. Supp ose E is p –primitive for an o dd prime p . Then ∧ 2 ( E µ p ∞ ) = Z / 2 Z gener ate d by 1 ∧ 1 . If E is also 2 –primitive, the map ∧ 2 ( E µ p ∞ ) → ∧ 2 ( E ) is 0 . Otherwise it is inje ctive. Pr o of. W e assume tha t E µ p ∞ ∼ = Z [1 /p ] leaving the simpler case E µ p ∞ ∼ = Z to the reader. It is easy to see that Z [1 /p ] is 2 – torsion genera ted by elemen ts p − k ∧ p − k , and since p is o dd, p − k ∧ p − k = p 2 k ( p − k ∧ p − k ) = 1 ∧ 1 . By Corolla ry 3.2 6, 1 is 2 –divisible in E if a nd only if E is 2 –primitive. This concludes the pro of.  Note that a primitive extensio n is 2 –primitive if and only if the c haracteristic of F is not 2 . This is be c a use µ 2 is tr ivial in characteristic 2 a nd non-trivial otherwise . Prop osition 3.29. L et E b e a primitive ext ension and let E ( µ F ) = 2 E µ F if the char acteristic of F is 2 and E ( µ F ) = E µ F otherwise. W e have an exact se quenc e (3.25) 0 / / E ( µ F ) ι / / E β / / ∧ 2 ( E ) π ∧ π / / ∧ 2 ( F ∗ ) / / 0 , wher e β is the m ap taking e t o e ∧ 1 . THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 13 Pr o of. The o nly non-triv ia l par t is to sho w that the k ernel o f β is E ( µ F ) . First note that if e ∧ 1 = 0 ∈ ∧ 2 ( E ) , e and 1 m ust b e linearly dependent over Z (this follows from Lemma 3.20). Hence, e mu st be in E µ F . Let e ∈ E µ F . W e may ass ume that e ∈ E µ p ∞ for s o me prime p . Suppos e the characteristic of F is not 2 . By Lemma 3.28, the restric tio n of β to E µ p ∞ is zer o if p 6 = 2 , and by insp ection, this also holds if p = 2 . Hence, β ( e ) = 0 . Now suppo se the characteristic o f F is 2 . In pa rticular p must b e o dd. Pick in tegers k and l such that p k e = l ∈ E µ p ∞ . By Lemma 3.28, e ∧ 1 = p k e ∧ 1 = l ∧ 1 is ze r o in ∧ 2 ( E ) if and only if l is even. Hence, p k e , and therefor e also e , is 2 –divisible.  Corollary 3.30. L et E b e primitive. Ther e is an exact se quenc e f µ F χ / / b B E ( F ) π / / B ( F ) / / 0 , wher e f µ F = E ( µ F ) / 2 Z .  Note that f µ F is independent of E up to isomorphism. If the characteristic of F is not 2 , f µ F is the unique no n-trivial Z / 2 Z –extension of µ F , and in characteristic 2 , f µ F is just µ F . The notatio n f µ F th us agrees with that o f Suslin [18]. 4. The extended Bloch group o f a field W e now show that if we imp ose some conditions on F , the extended Blo c h groups b B E ( F ) a re naturally isomo rphic, and we can glue them together to form a n extended Blo ch group b B ( F ) depending only on F . This group a dmits a na tur a l action by the automorphism g roup of F . Deﬁnition 4.1. A ﬁeld F is ca lled fr e e if it satisﬁes the following tw o conditions: (i) F ∗ /µ F is a free ab elian gro up; (ii) | µ F | < ∞ . F ree ﬁelds include num b er ﬁelds and ﬁnite ﬁelds, and freenes s is pr eserved un- der ﬁnite tr anscenden tal ex tensions. F or a discussion of ﬁelds satisfying (i), see e.g. May [9]. Throug hout this section F denotes a free ﬁeld. Corollar y 3 .27 implies that primitive Z –extensions of F ∗ are in one-one c orre- sp o ndence with primitiv e ro ots o f unity . If x is a primitive ro ot of unity , we let E x denote the corres p onding primitive extension. The restr iction of E x to µ F is free of ra nk one with gener ator ˜ x ma pping to x ∈ F ∗ . Since F ∗ is free mo dulo torsion, any extension of F ∗ is uniquely determined b y its restriction to µ F . The following result is a dir ect conse q uence of this discussion and Theore m 3.21. Lemma 4.2. L et F ′ b e any ﬁeld (not ne c essarily fr e e) and let σ : F → F ′ b e an emb e dding. L et E ′ b e a Z –extension of F ′∗ and let E b e a primitive Z –exten sion of F ∗ . Ther e exists a c overing b σ : E → E ′ of σ . The induc e d map b σ ∗ : b B E ( F ) → b B E ′ ( F ′ ) dep ends only on σ and not on the choic e of c overing.  Corollary 4.3. F or any p air E x , E y of primitive Z –extensions of F ∗ , ther e exists a c overing Ψ xy : E x → E y of the identity on F . The induc e d map Ψ xy ∗ : b B E x ( F ) → b B E y ( F ) is an isomorphism with inverse Ψ y x ∗ . Pr o of. Existence of Ψ xy follows from Lemma 4.3, and since the cov erings Ψ xy ◦ Ψ y x and Ψ y x ◦ Ψ xy are equiv alent to the identit ies o n E y and E x , the result fo llo ws .  14 CHRISTIAN K. ZICKER T Corollary 4.4. L et τ b e an automorphism of F . F or e ach primitive r o ot of un ity x ∈ F , ther e exists a un ique c overing b τ x : E x → E τ ( x ) of τ . The induc e d maps b τ x ∗ : b B E x ( F ) → b B E τ ( x ) ( F ) satisfy (4.1) b τ y ∗ ◦ Ψ xy ∗ = Ψ τ ( x ) τ ( y ) ∗ ◦ b τ x ∗ . Pr o of. Existence of b τ x follows from Lemma 4.3, and since b τ y ◦ Ψ xy and Ψ τ ( x ) τ ( y ) ◦ b τ x are b oth cov erings of τ , the res ult follows.  Deﬁnition 4.5. The extended Blo ch gr oup of F is deﬁned b y (4.2) b B ( F ) = lim ← − b B E x ( F ) = n ( α E x ) ∈ Y b B E x ( F )   α E y = Ψ xy ∗ ( α E x ) o , where the pro duct is ov er primitive ro ots of unit y . R emark 4.6 . If F is not free, we can still deﬁne b B ( F ) as an inv er s e limit, but in the general case, the primitive extensions do not fo r m a directed set, and we do not see how to establish the desire d connection with the classical Blo c h group. Prop osition 4.7. Ther e is a natur al action of Aut ( F ) on b B ( F ) , wher e e ach auto- morphism acts by an isomorphism. Pr o of. This follows direc tly from (4 .1).  Prop osition 4.8. Ther e is an exact se quenc e f µ F χ / / b B ( F ) π / / B ( F ) / / 0 . Pr o of. This is a n easy consequence of Coro llary 3.3 0 .  Note that the action of Aut( F ) on f µ F is through the quadratic c haracter. 4.1. Emb eddings in C and the regulator. Recall that b B ( C ) is the extended Blo c h group a ssoc ia ted to the extension o f C ∗ given by the exp onen tial map. Let σ b e an embedding of F in C . By Le mma 4 .3, ea c h primitive extension E x admits a cov er ing b σ x : E x → C of σ . Lemma 4 .9. The induc e d map b σ x ∗ : b P E x ( F ) → b P ( C ) r est ricts to an inje ction E x / 2 Z → C / 4 π i Z . Pr o of. Clearly , b σ x : E x → C is injectiv e. Since b σ x m ust take 1 to 2 π ik , where k is relatively prime to | µ F | , the res ult follows from Lemma 3.18.  Corollary 4 .10. Le t F b e a fr e e ﬁ eld admitting an emb e dding in C . The map χ : E / 2 Z → b P E ( F ) is inje ctive for al l primitive ext ensions E . Pr o of. W e may assume that E = E x . It is eno ug h to pr o ve that b σ x ∗ ◦ χ : E x / 2 Z → b P ( C ) is injective. By Lemma 3.18, b σ x ∗ ◦ χ = χ C ◦ b σ x ∗ , where χ C denotes the map χ : C / 4 π i Z → b P ( C ) . By Corollar y 4.10 (and Remark 3 .15) , this is a compo sition of injective maps, hence injectiv e.  Since b σ x and b σ y ◦ Ψ xy bo th c over σ , the induced maps s atisfy b σ x ∗ = b σ y ∗ ◦ Ψ xy ∗ , and we obtain a map (4.3) σ ∗ : b B ( F ) → b B ( C ) depending only on σ . The following is a simple consequence o f Lemma 4.9 and Corollar y 4.10. THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 15 Prop osition 4.11. L et F b e a fr e e ﬁeld admitting an emb e dding σ in C . The map σ ∗ : b B ( F ) → b B ( C ) r estricts to an inje ction f µ F → µ C . F urthermor e, the se quenc e (4.4) 0 / / f µ F χ / / b B ( F ) π / / B ( F ) / / 0 . is exact.  4.1.1. The r e gulator. A num b er ﬁeld F of type [ r 1 , r 2 ] has r 1 real em b eddings and r 2 pairs of complex em b eddings. The regulator ma p (2.4) is equiv a rian t with resp ect to the action o n b B ( C ) by co mplex c onjugation g iv e n by [ z ; p, q ] 7→ [ ¯ z ; − p , − q ] . W e therefore obtain a regulator (4.5) b B ( F ) → ( R / 4 π 2 Z ) r 1 ⊕ ( C / 4 π 2 Z ) r 2 . Example 4.12. Let x b e a r oot o f p ( x ) = x 4 − x 3 + 2 x 2 − 2 x + 1 , and consider the n umber ﬁeld F = Q ( x ) . One can chec k that µ F has order 6 and is generated b y w = x 3 + x . Let E b e the pr imitiv e extension corresp onding to w and let ˜ w b e the g enerator of E µ F . Let u = − x 3 − 2 x + 1 a nd let v = x 2 − x + 1 . The re la tions 1 − u = u 2 w 4 , v = w 3 u − 2 , 1 − v = u − 3 w, are ea s ily veriﬁed, and it fo llows that α = [ u ] + 2[ v ] ∈ B ( F ) . Let ˜ u b e a lift o f u , and co ns ider the lift (4.6) ˜ α = ( ˜ u, 2 ˜ u + 4 ˜ w ) + 2( − 2 ˜ u + 3 ˜ w , − 3 ˜ u + ˜ w ) − 3 χ ( ˜ u ) ∈ b P E ( F ) of α . One easily chec ks that b ν ( ˜ α ) = 0 , so ˜ α is in b B E ( F ) . O ne can c heck, e.g. using Lemma 3 .1 6 , that ˜ α is independent o f the particular choice o f ˜ u . Let z b e the ro ot of p given b y z = − 0 . 121 7 . . . + i 1 . 30 6 6 . . . , and let σ denote the corresp onding embedding. Then σ ( w ) = exp( − π i/ 3 ) . Let b σ : E → C b e a cov e r ing of σ . Letting log denote the principal branch of logarithm, we may assume that b σ ( ˜ w ) = − π i/ 3 , b σ ( ˜ u ) = log ( σ ( u )) = − 0 . 27 17 . . . − i 0 . 61 65 . . . . Using Lemma 3.18, we see that b σ ∗ takes χ ( ˜ u ) to − χ (log( σ ( u ))) . W e now have σ ∗ ( α ) = ( − 0 . 2717 . . . − i 0 . 6165 . . . , − 0 . 543 5 . . . − i 5 . 4218 . . . ) + 2(0 . 5 435 − i 1 . 9085 . . . , 0 . 81 53 . . . + i 0 . 8023 . . . ) + 3 χ ( − 0 . 2717 . . . − i 0 . 6 1 65 . . . ) . Using (2.4) or (2.6), (and Rema r k 3.15), we obtain R ( σ ∗ ( α )) = − 7 . 4532 . . . − i 2 . 3126 . . . ∈ C / 4 π 2 Z . R emark 4.1 3 . Examples like the ab o ve can b e pro duced in a bundance using com- puter so ftw a re like P ARI/GP . 5. The other version of the extended Bl och group As men tioned in the introduction there are tw o versions, b B ( C ) SL and b B ( C ) PSL , o f the extended Blo ch group. They are isomorphic to H 3 (SL(2 , C )) and H 3 (PSL(2 , C )) , resp ectiv ely . In this section we deﬁne the alg ebraic version of b B ( C ) PSL , a nd dis- cuss its relationship with hyperb olic geometry . W e stress that this version is o nly deﬁned when the extension E of F is 2 –primitive. 16 CHRISTIAN K. ZICKER T Let F b e a ﬁeld and let E b e a 2 –primitive extension of F ∗ b y Z . By Coro l- lary 3.26, 1 ∈ Z ⊂ E is uniquely t wo-divisible, so 1 2 ∈ E is well deﬁned. C o nsider the s et of o dd ﬂattenings (5.1) F E =  ( e, f ) ∈ E × E   ± π ( e ) ± π ( f ) = 1 ∈ F  , Since knowledge of z and 1 − z up to a sig n deter mines z , we hav e a map π : F E → F \ { 0 , 1 } . Deﬁne FT E as in Deﬁnition 3 .2, and deﬁne b P E ( F ) PSL to b e the ab elian group ge ner ated by F E sub ject to the rela tions 4 X i =0 ( − 1) i ( e i , f i ) = 0 for  ( e 0 , f 0 ) , . . . ( e 4 , f 4 )  ∈ FT E ( e + 1 2 , f + 1 2 ) + ( e, f ) = ( e + 1 2 , f ) + ( e, f + 1 2 ) . The second rela tion is the analog of the tr ansfer r elation ; see Go ette–Zick ert [5 ] or Neumann [12, Prop osition 7.2 ]. The ex tended Blo ch gr oup b B E ( F ) PSL is deﬁned as in Deﬁnition 4.5. Note that π : F E → F \ { 0 , 1 } induces ma ps from the extended groups b P E ( F ) PSL and b B E ( F ) PSL to the classica l ones. F or a ∈ E \ Z let χ ( a ) = ( e, f + 1 / 2) − ( e, f ) , w her e ( e, f ) is any ﬂattening of π ( a ) . The ana log of L e mma 3 .6 ho lds, proving indep endence o f f , and indep en- dence of e follo ws from the transfer relation. As in Co rollary 3.1 0, χ extends to a homomor phism χ : E → b P E ( F ) PSL , and a computation as in (3 .1 0) shows that χ (1) = 2 χ ( 1 2 ) = 0 . Lemma 5.1. Ther e is a c ommutative diagr am of ex act se quenc es. (5.2) E / 2 Z χ / / 2   b P E ( F ) p   / / P ( F ) / / 0 E / Z χ / / b P E ( F ) PSL / / P ( F ) / / 0 , Pr o of. Exactness o f the b ottom sequence is prov ed as in Theorem 3.1 3. F or com- m utativity of the left dia g ram, note that in b P E ( F ) PSL , we have (5.3) ( e, f + 1) − ( e, f ) = ( e, f + 1) − ( e, f + 1 2 ) + ( e, f + 1 2 ) − ( e, f ) = 2 χ ( e ) = χ (2 e ) . This proves the result.  It fo llows fr o m Prop osition 3.2 9 that we hav e a n exa ct s equence (5.4) µ F χ / / b B E ( F ) PSL / / B ( F ) / / 0 If F is a free ﬁeld, w e can form b B ( F ) PSL as in Section 4 . An embedding F → C induces a map b B ( F ) PSL → b B ( C ) PSL restricting to an injection µ F → µ C . In particular, χ is injective, and it follows from Lemma 5.1 that there is an exact sequence (5.5) 0 / / Z / 4 Z / / b B ( F ) p / / b B ( F ) PSL / / Z / 2 Z / / 0 . THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 17 R emark 5.2 . O ne can c o ncretely determine if an ele ment α ∈ b B E ( F ) PSL lifts: Pic k any element τ ∈ b P E ( F ) lifting π ( α ) ∈ B ( F ) . Then x = α − pτ ∈ b P E ( F ) PSL is in E / Z = F ∗ , and it follows fr om Lemma 5.1 that α lifts if and only if x is a sq ua re in F ∗ . 5.1. Par ab olic represe n tations. The res ult b elow g eneralizes the main res ult in Zick ert [2 1]. Theorem 5.3. L et M b e a tame manifold , and let F b e a fr e e ﬁeld. A p ar ab olic r epr esentation ρ : π 1 ( M ) → PSL(2 , F ) deﬁnes a fun damental class [ ρ ] ∈ b B ( F ) PSL . If σ : F → C is an emb e dding, σ ∗ ([ ρ ]) = [ σ ◦ ρ ] ∈ b B ( C ) PSL . Pr o of. In Zick ert [21] we deﬁned a homomorphism H 3 (PSL(2 , C ) , P ) → b B ( C ) PSL , where P = { ( 1 ∗ 0 1 ) } . This ma p can b e deﬁned ov er any free ﬁeld by replacing the logarithms in Z ickert [21, (3.6)] by a lift of E → F ∗ . By Zick ert [2 1, Theorem 5 .13] a decorated para bolic representation ρ : π 1 ( M ) → PSL(2 , F ) of M has a fundamental class H 3 (PSL(2 , F ) , P ) , a nd a s in Zick ert [21, Theorem 6.1 0 ], the imag e of the fundamen tal class in b B ( F ) PSL is independent o f the decoration. This prov e s the ﬁrst statement . The second statement is an immediate consequenc e of the deﬁnition.  R emark 5.4 . W e stres s that b B ( F ) PSL is not isomor phic to H 3 (PSL(2 , F )) in general. 6. Ideal cochains and fla ttenings of 3 –cycles Fix a ﬁeld F and a primitiv e extension E o f F ∗ b y Z . By a simplex we will alwa ys mean a standard simplex together with a ﬁxed vertex ordering. Unless otherwise sp e ciﬁed, a s implex means a 3 –s implex. Deﬁnition 6.1. An (algebr aic) ﬂattening of a simplex ∆ is a n asso ciation of an algebraic ﬂattening ( e , f ) ∈ b F E to ∆ . If ( e , f ) is a ﬂattening of z ∈ F \ { 0 , 1 } , we refer to z as the cr oss-r atio o f the ﬂattened simplex. R emark 6.2 . Deﬁnition 6.1 is a generalization of even ﬂattenings , i.e. ﬂattenings [ z ; p, q ] , with p and q even. Neumann [12] also a llo ws o dd v alues of p a nd q . W e w ill asso ciate elements in E to edges of a ﬂattened simplex as indicated in Figure 1. W e will refer to these element s a s lo g-p ar ameters . Deﬁnition 6.3. An (or der e d, oriente d) 3 –cycl e is a space K obtained from a collection o f simplices by gluing together pairs of faces using simplicial attaching maps preser ving the vertex order ings. If a ll faces hav e been glued, we say that K is close d . W e assume that the manifold K 0 with bo undary (and co rners) obtained b y removing disjoint reg ular neig h b orho o ds of the 0 –cells is oriented. If ∆ i is a simplex in K , w e let ε i be a s ig n enco ding whether or not the orientation o f ∆ i coming from the vertex or dering a grees with the o rien tation of K 0 . R emark 6.4 . Neumann [12] only considers closed 3 –cycles . With our deﬁnition, a single s implex is a 3 – cycle. The deﬁnition b elow is the analog of Neumann [1 2, Deﬁnition 4.4], which the reader may co nsult fo r further details. Deﬁnition 6.5. Let K b e a closed 3 –c y cle. A ﬂattening of K is a choice of ﬂat- tening of ea c h simplex of K such that the total log-par ameter (summed a ccording 18 CHRISTIAN K. ZICKER T to the sign conv entions o f Neumann [12, Deﬁnition 4.3 ]) ar o und each edge is zer o. If the total log- parameter along a n y no r mal curve in the sta r of e ac h zero- cell is zero, it is called a str ong ﬂ att ening . R emark 6.6 . W e do not need any co nditions on the pa rit y . This is b ecause th e parity condition is automatically s atisﬁed for ev en ﬂattenings. The proo f of this fact is identical to the pro of o f Neumann [12, Propo sition 5.3 ]. If K is a 3 –cycle, a nd G is an abelian group, we let C 1 ( K ; G ) denote the set of cellular 1 –co ch ains in K with v a lues in G . A co chain c ∈ C 1 ( K ; G ) natura lly asso ciates to each edge of each simplex of K an element in G . Edges that ar e iden tiﬁed in K acquire the same lab eling. If ∆ is a simplex of K , w e let c ij (∆) denote the lab eling of the edge joining the v ertices i and j in ∆ , see Figure 2. 0 1 2 3 e e − f − f − e + f − e + f Figure 1. Associat- ing log-parameters to edges of a ﬂattened simplex. 0 1 2 3 c 01 c 23 c 03 c 12 c 02 c 13 Figure 2. Edge labelings arising from a co c h ain. Deﬁnition 6 . 7. Let K b e a 3 –cycle. A co c ha in c ∈ C 1 ( K ; F ∗ ) is called an ide al c o chain if for each simplex ∆ i in K , there is a n element z i ∈ F ∗ \ { 0 , 1 } , such that the a sso c iated lab eling o f edges satisﬁes (6.1) c i 03 c i 12 c i 02 c i 13 = z i , c i 01 c i 23 c i 02 c i 13 = 1 − z i . An idea l co c hain thus asso ciates cr oss-ratios to ea c h simplex. R emark 6 .8 . No t all 3 –cycles admit ideal co c hains. The fact that 3 –cycles admitting ideal co chains ex is t follows from Remark 6.1 6 below. W e wish to pr ove that each lift ˜ c ∈ C 1 ( K ; E ) o f an idea l co chain c determines an element in b σ ( ˜ c ) ∈ b P E ( F ) such that if K is closed, b σ (˜ c ) is in b B E ( F ) and is independent of the choice of lift. In other words, an ideal co ch ain o n a clo sed 3 –cycle determines an elemen t in b B E ( F ) . Let I n be the free a belian gro up on cochains ˜ c ∈ C 1 (∆ n ; E ) on an n – simplex ∆ n , whose restriction to e ac h 3 – dimensional face is the lift o f an ideal co c hain. The usual b oundary map induces bo undary maps ∂ : I n → I n − 1 , making I ∗ in to a chain complex. A lift ˜ c of an ideal co c ha in c on K determines an element in I 3 given by P ε i ˜ c i . W e may th us reg ard lifts of ideal co chains as elements in I 3 . Note that if K is clo sed, ˜ c is a cycle, i.e. ∂ ˜ c = 0 ∈ I 2 . THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 19 Consider the maps b σ : I 3 → Z [ b F E ] and µ : I 2 → ∧ 2 ( E ) deﬁned on gener ators by b σ (˜ c ) = ( ˜ c 03 + ˜ c 12 − ˜ c 02 − ˜ c 13 , ˜ c 01 + ˜ c 23 − ˜ c 02 − ˜ c 13 ) , (6.2) µ (˜ c ) = − ˜ c 01 ∧ ˜ c 02 + ˜ c 01 ∧ ˜ c 12 − ˜ c 02 ∧ ˜ c 12 + ˜ c 02 ∧ ˜ c 02 . (6.3) Lemma 6.9. Ther e is a c ommutative diagr am (6.4) I 4 ∂ / / b σ   I 3 ∂ / / b σ   I 2 µ   Z [ c FT E ] b ρ / / Z [ b F E ] b ν / / ∧ 2 ( E ) . Pr o of. Let ˜ c ∈ I 4 be a generato r , and supp o se b σ ◦ ∂ (˜ c ) = P ( − 1) i ( e i , f i ) . W e must prov e tha t the ﬂa ttenings ( e i , f i ) satisfy the ﬁve equations of Deﬁnition 3.2. W e chec k the ﬁrst e q uation, a nd leave the veriﬁcation of the four others to the reader. By (6.2) we have e 0 = ˜ c 14 + ˜ c 23 − ˜ c 13 − ˜ c 24 e 1 = ˜ c 04 + ˜ c 23 − ˜ c 03 − ˜ c 24 e 2 = ˜ c 04 + ˜ c 13 − ˜ c 14 − ˜ c 03 , and it follows that e 2 = e 1 − e 0 . Letting ˜ c ∈ I 3 be a generator , we s e e that b ν ◦ b σ (˜ c ) and µ ◦ ∂ ( ˜ c ) ar e b oth a sum of 12 terms o f the form ˜ c ij ∧ ˜ c kl with { i, j } 6 = { k , l } and 4 t wo-torsion terms summing to ˜ c 02 ∧ ˜ c 02 + ˜ c 13 ∧ ˜ c 13 (the tw o other terms ca ncel out). The tw o-tor sion terms thus match up, and o ne easily checks that the other terms matc h up a s well, pr o v ing commut ativity o f the rig ht squar e.  Corollary 6.10. If K is close d and ˜ c is a lift of an ide al c o chain on K , b σ ( ˜ c ) is in b B E ( F ) .  Prop osition 6. 11. Le t K b e a close d 3 –cycl e. The set of ﬂattenings c oming fr om a lift of an ide al c o chain is a str ong ﬂattening of K . Pr o of. The pro of is identical to the pr oof of Zick er t [2 1, Theo rem 6.5], so we omit some details. Consider a curve α in the sta r of a 0 – cell as shown in Figure 3. When α passes through a simplex, it pic ks up a log- parameter, whic h is a signed s um of four terms. The signs ar e shown in the ﬁgure. If α is a closed cur v e, it is not diﬃcult to see that all terms m ust cancel out.  α + − + − + − + − + − − + − + − + − + − + Figure 3. A normal curve in the star of a 0 –cell. Eac h edge and each vertex corresponds to a 1 –cell in K . 20 CHRISTIAN K. ZICKER T Lemma 6.12. L et K b e a 3 –cycle and let c b e an ide al c o chain on K . L et e b e an interior 1 –c el l of K , and let α e ∈ C 1 ( K ; Z ) b e the c o chain taking e to 1 and al l other 1 –c el ls to 0 . F or every lift ˜ c of c , we have (6.5) b σ (˜ c + α e ) = b σ ( ˜ c ) ∈ b P E ( F ) . Pr o of. The map b σ asso ciates ﬂattenings to the simplices o f K , and using (6.2) one chec ks that the ﬂattenings coming from ˜ c and ˜ c + α diﬀer by Neumann’s cycle relation [1 2, Section 6] ab out e (o r r ather the o bvious g e ner alization of this relation to algebra ic ﬂattenings). Neumann’s pro of that the cycle r elation is a consequence of the lifted ﬁve term r e la tion car ries over to the alg ebraic setup word by word.  Corollary 6.13. If K is a close d 3 –cycle, b σ (˜ c ) = b σ ( ˜ c + α ) ∈ b B E ( F ) for any α ∈ C 1 ( K ; Z ) . Hen c e, an id e al c o chain c on K determines an element in b σ ( c ) ∈ b B E ( F ) .  6.1. The action of Z 1 ( K ; Z / 2 Z ) on ide al co c hains. Let K b e a closed 3 –cycle, and s upp ose that the c hara cteristic of F is not 2 . T he gro up Z 1 ( K ; Z / 2 Z ) of cellular 1 –co cycles on K acts on the set of ideal c o chains by mult iplication. Note that the action do es no t change the cr o ss-ratios. A co chain α ∈ Z 1 ( K ; Z / 2 Z ) determines a map B α : K → B ( Z / 2 Z ) = R P ∞ , and w e wish to prov e that the e le ments in b B E ( F ) a sso c ia ted to ideal co c ha ins c and αc diﬀer by a tw o- torsion elemen t which is zero if and only if B α ∗ ([ K ]) is zero in H 3 ( R P ∞ ) = Z / 2 Z . The homology of a g roup G is the homology of the complex B ∗ ( G ) where B n ( G ) is gener ated by symbols h g 1 | . . . | g n i with g i ∈ G . Suc h tuples are in one-one corr e- sp o ndence with G –co cycles on ∆ n ; a co cycle is uniquely given by its v alues on the edges be tw een vertices i a nd i + 1 . Under this corresp ondence, the b oundary ma ps are induced b y the standard o nes. Giv en a co cycle α ∈ Z 1 ( K ; G ) the restr iction of α to ∆ i determines a tuple h g i 1 | g i 2 | g i 3 i a nd by Zick er t [21, Prop osition 5.7 ] we have B α ∗ ([ K ]) = X ε i h g i 1 | g i 2 | g i 3 i . Let α ∈ Z 1 (∆ 3 ; Z / 2 Z ) a nd let c ∈ C 1 (∆ 3 ; F ∗ ) be a n ideal co c ha in. Let c ′ = αc , and pick a lift ˜ c of c . Then ˜ c endows ∆ 3 with a ﬂattening b σ (˜ c ) g iv en by ( e, f ) = (˜ c 03 + ˜ c 12 − ˜ c 02 − ˜ c 13 , ˜ c 01 + ˜ c 23 − ˜ c 02 − ˜ c 13 ) . Let w ∈ E b e the sum of the lo g-parameters at the edges wher e α ij = − 1 . O ne easily chec k s that w is a lw ays (uniquely) tw o -divisible, e.g. if α = h− 1 | 1 | − 1 i , w = 2 e + 2( − e + f ) = 2 f . Let ˜ c ′ be the lift of c ′ deﬁned by (6.6) ˜ c ′ ij = ˜ c ij + ( 1 2 if α ij = − 1 0 otherwise. Lemma 6. 14. L et δ ∈ Z ⊂ E b e 1 if α = h− 1 | − 1 | − 1 i and 0 otherwise. W e have (6.7) b σ (˜ c ′ ) − b σ ( ˜ c ) = χ ( 1 2 w ) + χ ( δ ) ∈ b P E ( F ) . Pr o of. This is do ne case by case using Lemma 3.16. If e.g. α = h 1 | − 1 | 1 i , w = − 2 e and we hav e b σ (˜ c ′ ) − b σ ( ˜ c ) = ( e , f − 1) − ( e, f ) = χ ( − e ) = χ ( 1 2 w ) + χ ( δ ) . THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 21 The other seven cases a re similar and left to the r eader.  Theorem 6.15. L et α ∈ Z 1 ( K ; Z / 2 Z ) . F or any ide al c o chain c , b σ ( αc ) − b σ ( c ) is two-torsion in b B E ( F ) , which is t rivia l if and only if B α ∗ ([ K ]) ∈ H 3 ( R P ∞ ) = Z / 2 Z is trivial. Pr o of. Let ˜ c b e a lift of c and let ˜ c ′ be the lift o f αc deﬁned by (6.6). F or each simplex ∆ i of K , we hav e elements w i and δ i as ab ov e. Since the ﬂattening of K deﬁned b y ˜ c is strong, P ε i w i = 0 ∈ E , and since E ha s no t wo-torsio n, Lemma 6.14 implies that b σ ( ˜ c ′ ) − b σ ( ˜ c ) = X ε i χ ( δ i ) . One eas ily c hecks that the iso morphism H 3 ( B Z / 2 Z ) ∼ = Z / 2 Z is induced by the map B 3 ( Z / 2 Z ) → Z / 2 Z taking h− 1 | − 1 | − 1 i to 1 a nd all other genera tors to 0 . This prov es the result.  6.2. Ideal co c hains and homo logy of li ne ar groups. In this se c tio n we deﬁne a ca nonical map b λ : H 3 (SL(2 , F )) → b B E ( F ) . This is purely algebraic and follows Dupont–Zick ert [2]. W e assume that F is inﬁnite. Let C ∗ ( F 2 ) b e the chain complex generated in dimension n by ( n + 1) –tuples of vectors in F 2 \ { 0 } in g eneral pos itio n, together with the usual bo undary map. Letting p : F 2 \ { 0 } → P 1 ( F ) be the ca nonical pro jection, a simple computation (as in Dup on t– Zic kert [2, Section 3 .1]) shows that (6.8) z = det( v 0 , v 3 ) de t ( v 1 , v 2 ) det( v 0 , v 2 ) de t ( v 1 , v 3 ) , 1 − z = det( v 0 , v 1 ) de t ( v 2 , v 3 ) det( v 0 , v 2 ) de t ( v 1 , v 3 ) , where z is the cross-ra tio of the tuple ( p ( v 0 ) , p ( v 1 ) , p ( v 2 ) , p ( v 3 )) . It follows that there is a chain ma p (6.9) Γ : C ∗ ( F 2 ) → I ∗ , Γ( v 0 , . . . , v n ) ij = log det( v i , v j ) . Here log denotes a ﬁxed section of π : E → F ∗ . W e refer to it as a lo garithm . Let b λ = b σ ◦ Γ . Then b λ is SL(2 , F ) –inv aria n t, and by Lemma 6.9 it induces a map H 3 ( C ∗ ( F 2 ) SL(2 ,F ) ) → b B E ( F ) . Recall that the homology o f a group G is the homology of ( F ∗ ) G = F ∗ ⊗ Z [ G ] Z , where F ∗ is any free resolution of Z b y G -mo dules. One such r esolution is the complex C ∗ ( G ) of tuples in G . Note that C ∗ ( G ) G equals the complex B ∗ ( G ) considered in Section 6 .1. If G = SL (2 , F ) we may as sume (see e.g. Dupo n t– Zick ert [2, Section 3.2]) that all tuples ( g 0 , . . . g n ) are in general po sition in the sense that ( g 0 v , . . . , g n v ) ∈ C n ( F 2 ) for so me ﬁxed v 6 = 0 ∈ F 2 (the particular choice is inessential). It follows that λ c anonically extends to a map (6.10) b λ : H 3 (SL(2 , F )) → b B E ( F ) , ( g 0 , g 1 , g 2 , g 3 ) 7→ b σ ◦ Γ( g 0 v , g 1 v , g 2 v , g 3 v ) . R emark 6.16 . Note that for eac h α ∈ H 3 (SL(2 , F )) , b λ ( α ) is induced b y an ideal co c ha in on a 3 –c y cle. The fact that b λ is independent of the c hoice of logarithm follows from Corollar y 6.13. R emark 6.17 . The map π ◦ b λ : C 3 ( F 2 ) → P ( F ) is GL(2 , F ) –inv ar ian t, and it follows that there is an induced map H 3 (GL(2 , F )) → B ( F ) . This map factors through H 3 ( C ∗ ( P 1 ( F )) GL(2 ,F ) ) , a nd thus agrees with that of Suslin [18]. 22 CHRISTIAN K. ZICKER T 7. The extended Bloch group a n d algebraic K –theor y In this s ection, we pr o ve our ma in result Theorem 1.1. W e do this in three steps. The ﬁrst and mos t diﬃcult step is to extend the map b λ fro m Section 6 .2 to a map b λ : H 3 (GL(3 , F )) → b B E ( F ) . Once this ha s b een done, we o btain a map (7.1) K 3 ( F ) H / / H 3 (GL( F )) ∼ = / / H 3 (GL(3 , F )) b λ / / b B E ( F ) , where H is the Hurewicz map a nd the middle iso mo rphism is Suslin’s stability result Theo rem 2.7. In the ﬁrst step, we only require that F b e inﬁnite and that E be primitive. T he second step is to prov e that this ma p takes the image o f K M 3 ( F ) to 0 . T o do this, we need to assume that F is a num b er ﬁeld, o r mo re g enerally , a free ﬁeld. The third and ﬁnal step is to show that b λ induces a map betw een the diagrams (1.1) and (4.4). The res ult then follows from the ﬁve-lemma. 7.1. Step one : Extension of b λ to H 3 (GL(3 , F )) . W e sta r t by constructing b λ on H 3 (SL(2 , F )) . W e ass ume that F is inﬁnite and that E is primitiv e. Much of the construction dr a ws inspira tion from Igusa [7], F o ck–Gonc harov [3], and pri- v a te discussio ns with Dylan Thurston and Stavros Garoufa lidis . In Garoufalidis– Thu rston–Zickert [4] we genera lize to SL( n, F ) and dis c us s some of the underlying geometric ideas motiv ating the cons truction. In Section 6.2 we ass ociated a n ideal co c hain to a quadruple of vectors in F 2 . W e now g eneralize this to tuples of vectors in F 3 . Let v = ( v 0 , . . . , v n ) b e a tuple of vectors in F 3 in genera l p osition, and let w ∈ F 3 be in genera l p osition with resp ect to the v i ’s. F or each i ∈ { 0 , . . . , n + 1 } w e hav e a co c hain ˜ c i w ∈ C 1 (∆ n ; E ) given by (7.2) ˜ c i w ( v ) j k =      log det( w, v j , v k ) if i ≤ j < k log det( v j , w, v k ) if j < i ≤ k log det( v j , v k , w ) if j < k < i, where, a s in Section 6.2, log is a ﬁxed section of π : E → F ∗ . Lemma 7. 1. Each ˜ c i w ( v ) is in I n , and for e ach r estriction to a 3 –dimensional fac e of ∆ n , the cr oss-r atio is indep endent of i . Pr o of. W e may assume that v = ( v 0 , v 1 , v 2 , v 3 ) . Let c i w ( v ) b e the pro jection of ˜ c i w ( v ) to C 1 (∆ 3 ; F ∗ ) , and let p : F 3 \ { w } → P ( F 3 / h w i ) denote the map induced b y pro jection. By applying a linear trans fo r mation if necessa r y , we may a ssume that w = (1 , 0 , 0) , and iden tify F 3 / h w i with F 2 . It now follo ws from (6.8) that the cr oss- ratio z of the tuple ( p ( v 0 ) , p ( v 1 ) , p ( v 2 ) , p ( v 3 )) o f elements in P ( F 3 / h w i ) ≈ P 1 ( F ) satisﬁes z = det( w, v 0 , v 3 ) de t( w , v 1 , v 2 ) det( w, v 0 , v 2 ) de t( w , v 1 , v 3 ) , 1 − z = det( w, v 0 , v 1 ) de t( w , v 2 , v 3 ) det( w, v 0 , v 2 ) de t( w , v 1 , v 3 ) . It fo llows that c 0 w ( v ) is an ideal co chain with cross-ra tio z . Since the e x pressions c i w ( v ) 03 c i w ( v ) 12 c i w ( v ) 02 c i w ( v ) 13 , c i w ( v ) 01 c i w ( v ) 23 c i w ( v ) 02 c i w ( v ) 13 are indep enden t of i , it follo ws that the same is true for all the c i w ( v ) ’s.  THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 23 If v = ( v 0 , v 1 , v 2 , v 3 ) , we let ( v 0 , v 1 , v 2 , v 3 ) i w denote the ﬂattening b σ ( ˜ c i w ( v )) . T he log-par a meters are given by (7.3) e = ˜ c i w ( v ) 03 + ˜ c i w ( v ) 12 − ˜ c i w ( v ) 02 − ˜ c i w ( v ) 13 f = ˜ c i w ( v ) 01 + ˜ c i w ( v ) 23 − ˜ c i w ( v ) 02 − ˜ c i w ( v ) 13 . Lemma 7.2. The fol lowing formulas hold in b P E ( F ) (note the sup erscripts). ( v 1 , v 2 , v 3 , v 4 ) 0 w − ( v 0 , v 2 , v 3 , v 4 ) 0 w + ( v 0 , v 1 , v 3 , v 4 ) 0 w − ( v 0 , v 1 , v 2 , v 4 ) 0 w + ( v 0 , v 1 , v 2 , v 3 ) 0 w = 0 ( v 1 , v 2 , v 3 , v 4 ) 0 w − ( v 0 , v 2 , v 3 , v 4 ) 1 w + ( v 0 , v 1 , v 3 , v 4 ) 1 w − ( v 0 , v 1 , v 2 , v 4 ) 1 w + ( v 0 , v 1 , v 2 , v 3 ) 1 w = 0 ( v 1 , v 2 , v 3 , v 4 ) 1 w − ( v 0 , v 2 , v 3 , v 4 ) 1 w + ( v 0 , v 1 , v 3 , v 4 ) 2 w − ( v 0 , v 1 , v 2 , v 4 ) 2 w + ( v 0 , v 1 , v 2 , v 3 ) 2 w = 0 ( v 1 , v 2 , v 3 , v 4 ) 2 w − ( v 0 , v 2 , v 3 , v 4 ) 2 w + ( v 0 , v 1 , v 3 , v 4 ) 2 w − ( v 0 , v 1 , v 2 , v 4 ) 3 w + ( v 0 , v 1 , v 2 , v 3 ) 3 w = 0 ( v 1 , v 2 , v 3 , v 4 ) 3 w − ( v 0 , v 2 , v 3 , v 4 ) 3 w + ( v 0 , v 1 , v 3 , v 4 ) 3 w − ( v 0 , v 1 , v 2 , v 4 ) 3 w + ( v 0 , v 1 , v 2 , v 3 ) 4 w = 0 . W e wil l r efer to the left hand sides of the ﬁve e quations ab ove as b oundaries , and denote them by ∂ i ( v 0 , v 1 , v 2 , v 3 , v 4 ) w , i ∈ { 0 , . . . , 4 } . Pr o of. W e will show that each of the b oundaries cor respo nds to a lifted ﬁve term relation. T o do this we m ust prov e that the ﬂattenings sa tisfy the ﬁve equa tio ns of Deﬁnition 3 .2. Supp ose w e wish to verify that f 3 = f 2 − f 1 for the bo undary ∂ 2 . The r e le v a n t terms in volv ed in this are ( v 0 , v 1 , v 2 , v 4 ) 2 w , ( v 0 , v 1 , v 3 , v 4 ) 2 w and ( v 0 , v 2 , v 3 , v 4 ) 1 w . If we denote their ﬂa ttenings b y ( e 3 , f 3 ) , ( e 2 , f 2 ) and ( e 1 , f 1 ) , it follows from (7.3) and (7.2) that f 3 = log( v 0 , v 1 , w ) + log( w , v 2 , v 4 ) − log ( v 0 , w, v 2 ) − log( v 1 , w, v 4 ) f 2 = log( v 0 , v 1 , w ) + log( w , v 3 , v 4 ) − log ( v 0 , w, v 3 ) − log( v 1 , w, v 4 ) f 1 = log( v 0 , w, v 2 ) + log( w, v 3 , v 4 ) − log ( v 0 , w, v 3 ) − log( w, v 2 , v 4 ) , where lo g( u, v, w ) deno tes log(det ( u, v , w )) . Hence, f 3 = f 2 − f 1 as des ired. The veriﬁcation o f the o ther formulas are similar a nd a re thus left to the reader .  Lemma 7.3. W e have (7.4) ( v 1 , v 2 , v 3 , v 4 ) 0 v 0 − ( v 0 , v 2 , v 3 , v 4 ) 1 v 1 + ( v 0 , v 1 , v 3 , v 4 ) 2 v 2 − ( v 0 , v 1 , v 2 , v 4 ) 3 v 3 + ( v 0 , v 1 , v 2 , v 3 ) 4 v 4 = 0 ∈ b P E ( F ) . W e wil l denote t he left hand side by ∂ ( v 0 , v 1 , v 2 , v 3 , v 4 ) . Pr o of. As in the pro of of Lemma 7 .2, we can verify that the ﬂattenings s atisfy the ﬁve equa tions in Deﬁnition 3.2. W e leave this to the r e a der.  If F is an ordered basis o f F k , we let F i denote the i th basis vector. A set S o f ordered bases is in gener al p osition if an y set of k basis vectors from S is linearly indep enden t. Let C F ∗ be the chain complex genera ted in dimension n b y tuples ( F 0 , . . . , F n ) of ordered ba ses of F 3 in general position, together with the usual b oundary ma p. Left m ultiplication makes C F ∗ in to a chain complex o f free GL(3 , F ) –mo dules. Since F is assumed to be inﬁnite, it is ea s y to s ee that C F ∗ is acyclic. Hence, the complexes ( C F ∗ ) SL(3 ,F ) and ( C F ∗ ) GL(3 ,F ) compute the homology gr o ups H ∗ (SL(3 , F )) and H ∗ (GL(3 , F )) , respe c tively . 24 CHRISTIAN K. ZICKER T Consider the SL(3 , F ) –inv ar ian t ma p b λ : C F 3 → b P E ( F ) given by sending a g en- erator ( F 0 , F 1 , F 2 , F 3) to (7.5) ( F 0 2 , F 1 1 , F 2 1 , F 3 1 ) 0 F 0 1 + ( F 0 1 , F 1 2 , F 2 1 , F 3 1 ) 1 F 1 1 + ( F 0 1 , F 1 1 , F 2 2 , F 3 1 ) 2 F 2 1 + ( F 0 1 , F 1 1 , F 2 1 , F 3 2 ) 3 F 3 1 . W e will often abbreviate the notation by omitting the F, and writing a subscr ipt F i 1 as i , e.g. we a bbreviate ( F 0 2 , F 1 1 , F 2 1 , F 3 1 ) 0 F 0 1 to (0 2 , 1 1 , 2 1 , 3 1 ) 0 0 . T o help the reader vis ualize the arguments that follow, we g iv e a geometric wa y of viewing the map b λ : A generator o f C F 3 can be thought o f as a simplex ∆ together with a n as s ociatio n o f a n or dered basis F i to ea c h vertex. W e may think o f each o f the four terms in (7.5) as a standard simplex endow ed with an ideal co chain. W e mark ∆ with tw o p oin ts on each edge and a p oint on each face as shown in Figure 4. W e will r efer to these p oint s as e dge p oints and fac e p oints resp ectively . Each p oin t is given uniquely by a tuple β = ( x 0 , x 1 , x 2 , x 3 ) w ith x 0 + x 1 + x 2 + x 3 = 3 , where the co ordinate x i measures the “distance” to the face opp osite vertex i . F or ea c h such β , let β i be the or der ed set { F i 1 , . . . , F i x i } a nd let S β = β 0 ∪ β 1 ∪ β 2 ∪ β 3 . Note that S β alwa ys has exactly 3 elemen ts. Hence, det( S β ) is well deﬁned a nd gives a labeling of ea c h marked p oin t of ∆ . As an example, the edge p oint, clos est to vertex 1 , b et ween vertices 1 and 2 is lab eled by det( F 1 1 , F 1 2 , F 2 1 ) . W e can think of ∆ as a union of four simplices ∆ i , where ∆ i is the simplex spanned b y the i th vertex of ∆ and the marked p oin ts with x i 6 = 0 . W e think of the ∆ i ’s as b eing disjoint. The la belings of the marked p oints in ∆ g iv es rise to co ch ains on ∆ i , and using (7.2), one can chec k that these a re exactly the ideal co c ha ins of the terms in (7.5). F0 F1 F2 F3 0 1 2 3 c 1 01 c 1 23 c 1 03 c 1 12 c 1 02 c 1 13 0 1 2 3 c 0 01 c 0 23 c 0 03 c 0 12 c 0 02 c 0 13 0 1 2 3 c 3 01 c 3 23 c 3 03 c 3 12 c 3 02 c 3 13 0 1 2 3 c 2 01 c 2 23 c 2 03 c 2 12 c 2 02 c 2 13 Figure 4. The ideal co c h ains on the simplices ∆ i arising from a lab eling of marked p oin ts in ∆ . The dashed lines mark t he b ottom of ∆ 1 . R emark 7.4 . An o rdered basis determines an aﬃne ﬂag , and o ne easily checks that b λ o nly dep ends on the under lying aﬃne ﬂags. If τ ∈ ( C F 3 ) SL(3 ,F ) is a cycle, we ca n represent τ by a 3 –cycle K tog ether with a labeling of the mark ed p oin ts in e a c h of the simplices of K . Iden tiﬁed po in ts acquire the same lab eling. F rom the g eometric description of the map b λ , it follows that we ca n repr esen t b λ ( τ ) b y a 3 –cycle C with b oundary tog e ther with an ideal co c ha in on C . Note that C is homeo morphic to the disjoint union o f the cones on the links of the 0 – cells of K . THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 25 Lemma 7.5 . The r estriction of b λ to cycles in ( C F 3 ) SL(3 ,F ) is indep endent of t he choic e of lo garithm. In fact, we may cho ose diﬀer ent lo garithms for e ach of the marke d p oints as long as we use t he same lo garithm for identiﬁe d p oints. Pr o of. F or edge p oin ts this follows from Lemma 6 .12 . A face point o ccurs in exactly 3 of the simplices ∆ i . Consider the face p oint opp osite vertex 0 , and supp ose the ﬂa ttenings of ∆ 1 , ∆ 2 and ∆ 3 are ( e, f ) , ( e ′ , f ′ ) a nd ( e ′′ , f ′′ ) . If we add 1 to the lo garithm of the face p oin t, it follows from (7.3) that the ﬂattenings b ecome ( e, f + 1) , ( e ′ − 1 , f ′ − 1) and ( e ′′ + 1 , f ′′ ) . By Lemma 3.16, this changes the e lement in b P E ( F ) by χ ( e − e ′ + f ′ − f ′′ + 1 ) . Keep in mind that χ (1) = − χ (1) . Using (7.3), we have (7.6) e − e ′ + f ′ − f ′′ = (0 1 , 1 1 , 3 1 ) + (1 1 , 1 2 , 2 1 ) − (0 1 , 1 1 , 2 1 ) − (1 1 , 1 2 , 3 1 ) −  (0 1 , 2 1 , 3 1 ) + (1 1 , 2 1 , 2 2 ) − (0 1 , 2 1 , 2 2 ) − (1 1 , 2 1 , 3 1 )  + (0 1 , 1 1 , 2 1 ) + (2 1 , 3 1 , 3 2 ) − (0 1 , 2 1 , 2 2 ) − (1 1 , 2 1 , 3 1 ) −  (0 1 , 1 1 , 3 1 ) + (2 1 , 3 1 , 3 2 ) − (0 1 , 2 1 , 3 1 ) − (1 1 . 3 1 , 3 2 )  =  (1 1 , 1 2 , 2 1 ) − (1 1 , 2 1 , 2 2 )  +  (1 1 , 3 1 , 3 2 ) − (1 1 , 1 2 , 3 1 )  +  (2 1 , 2 2 , 3 1 ) − (2 1 , 3 1 , 3 2 )  , where ( i j , k l , m n ) denotes log de t( F i j , F k l , F m n ) . Note that each ter m is a log a rithm of one of the six edge p oint s on the face opp osite vertex 0 . By a similar ca lculation one can c heck that this holds in gener al, i.e. the change in the b P E ( F ) ele ment when adding 1 to the log arithm of a face p oint, is a signed sum of logarithms of the edge po in ts on that face (plus χ (1) ). The sig ns are shown in Figure 5. 1 3 2 − − − + + + 3 0 2 − − − + + + 1 0 3 − − − + + + 1 2 0 − − − + + + Figure 5 . Change in th e b P E ( F ) element when adding 1 to th e loga- rithm of a face p oin t. There is a con tribution for each edge p oint on the giv en face. In a cycle, each face po in t lies in exactly t wo s implices, and since the face pairings preserve orderings, it follows from Figure 5 that the changes in the element in b P E ( F ) , r esulting from adding 1 to the loga rithm, app ear with oppo s ite signs.  Lemma 7.6. b λ takes b oundaries in C F 3 to 0 ∈ b P E ( F ) . Pr o of. Using (7.5), we s e e that b λ ( ∂ ( F 0 , . . . , F 4)) ∈ b P E ( F ) equals + (1 2 , 2 1 , 3 1 , 4 1 ) 0 1 − (0 2 , 2 1 , 3 1 , 4 1 ) 0 0 + (0 2 , 1 1 , 3 1 , 4 1 ) 0 0 − (0 2 , 1 1 , 2 1 , 4 1 ) 0 0 + (0 2 , 1 1 , 2 1 , 3 1 ) 0 0 + (1 1 , 2 2 , 3 1 , 4 1 ) 1 2 − (0 1 , 2 2 , 3 1 , 4 1 ) 1 2 + (0 1 , 1 2 , 3 1 , 4 1 ) 1 1 − (0 1 , 1 2 , 2 1 , 4 1 ) 1 1 + (0 1 , 1 2 , 2 1 , 3 1 ) 1 1 + (1 1 , 2 1 , 3 2 , 4 1 ) 2 3 − (0 1 , 2 1 , 3 2 , 4 1 ) 2 3 + (0 1 , 1 1 , 3 2 , 4 1 ) 2 3 − (0 1 , 1 1 , 2 2 , 4 1 ) 2 2 + (0 1 , 1 1 , 2 2 , 3 1 ) 2 2 + (1 1 , 2 1 , 3 1 , 4 2 ) 3 4 − (0 1 , 2 1 , 3 1 , 4 2 ) 3 4 + (0 1 , 1 1 , 3 1 , 4 2 ) 3 4 − (0 1 , 1 1 , 2 1 , 4 2 ) 3 4 + (0 1 , 1 1 , 2 1 , 3 2 ) 3 3 . 26 CHRISTIAN K. ZICKER T Using Lemma 7.2, this simpliﬁes to − (1 1 , 2 1 , 3 1 , 4 1 ) 0 0 + (0 1 , 2 1 , 3 1 , 4 1 ) 1 1 − (0 1 , 1 1 , 3 1 , 4 1 ) 2 2 + (0 1 , 1 1 , 2 1 , 4 1 ) 3 3 − (0 1 , 1 1 , 2 1 , 3 1 ) 4 4 , which by Lemma 7.3 is 0 ∈ b P E ( F ) .  W e thus o btain an induced map b λ : H 3 (SL(3 , F )) → b P E ( F ) . Lemma 7.7. The image of b λ is in b B E ( F ) . Pr o of. Consider the sequence of maps J n : C F n → I n given by (7.7) ( F 0 , . . . , F n ) 7→ n X i =0 ( F 0 1 , . . . , F i 2 , . . . , F n 1 ) i F i 1 . Note that J is not a chain ma p. By deﬁnition, b λ : C F 3 → b P E ( F ) is equal to b σ ◦ J 3 , where b σ : I 3 → b P E ( F ) is the map given by (6.2). Consider the diagram (7.8) C F 3 J 3 / / ∂   I 3 b σ / / ∂   b P E ( F ) b ν   C F 2 J 2 / / I 2 µ / / ∧ 2 ( E ) . By Lemma 6.9 the r ig h t sq ua re is comm utative. Using the usual no ta tional a bbre- viations, i.e. o mitting the F, and shor tening subscripts to i , a direct computation shows that (7.9) δ := ( ∂ J 3 − J 2 ∂ )( F 0 , F 1 , F 2 , F 3) = (1 1 , 2 1 , 3 1 ) 0 0 − (0 1 , 2 1 , 3 1 ) 1 1 + (0 1 , 1 1 , 3 1 ) 2 2 − (0 1 , 1 1 , 2 1 ) 3 3 . One ea sily chec ks that µ takes δ to 0 ∈ ∧ 2 ( E ) , and the result follows.  Lemma 7.8. The r estriction of b λ to H 3 (SL(2 , F )) agr e es with the map fr om the pr evious se ction. Pr o of. W e co nsider F 2 as a subspac e of F 3 using the inclusion ( x, y ) 7→ (0 , x, y ) . Let p : F 3 → F 2 be the natural pro jection, and let D ∗ be the sub complex of C F ∗ consisting of tuples ( F 0 , . . . , F n ) such that ( p F 0 1 , . . . , p F n 1 ) ∈ C n ( F 2 ) . Note that D ∗ is a n acyclic SL(2 , F ) –co mplex, wher e SL(2 , F ) is r egarded as a subg r oup of SL(3 , F ) in the natural wa y . Consider the GL(2 , F ) –equiv aria n t map (7.10) Ψ : D ∗ → C ∗ ( F 2 ) , ( F 0 , . . . , F n ) 7→ ( p F 0 1 , . . . , p F n 1 ) . Let b τ denote the map C 3 ( F 2 ) → b P E ( F ) from Section 6.2. W e wish to prov e that b τ ◦ Ψ a nd b λ diﬀ er by a coboundar y . Note that (0 1 , 1 1 , 2 1 , 3 1 ) 0 w = b τ ◦ Ψ( F 0 , F 1 , F 2 , F 3) . By deﬁnition, b λ takes ( F 0 , F 1 , F 2 , F 3) ∈ D 3 to (0 2 , 1 1 , 2 1 , 3 1 ) 0 0 + (0 1 , 1 2 , 2 1 , 3 1 ) 1 1 + (0 1 , 1 1 , 2 2 , 3 1 ) 2 2 + (0 1 , 1 1 , 2 1 , 3 2 ) 3 3 . W e may subtra ct b oundaries without eﬀecting the image in b P E ( F ) , and after sub- tracting ∂ 1 ( w, 0 2 , 1 1 , 2 1 , 3 1 ) 0 + ∂ 2 ( w, 0 1 , 1 2 , 2 1 , 3 1 ) 1 + ∂ 3 ( w, 0 1 , 1 1 , 2 2 , 3 1 ) 2 + ∂ 4 ( w, 0 1 , 1 1 , 2 1 , 3 2 ) 3 , THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 27 the r emaining terms b ecome ( w, 1 1 , 2 1 , 3 1 ) 1 0 − ( w , 0 2 , 2 1 , 3 1 ) 1 0 + ( w , 0 2 , 1 1 , 3 1 ) 1 0 − ( w , 0 2 , 1 1 , 2 1 ) 1 1 ( w, 1 2 , 2 1 , 3 1 ) 1 1 − ( w , 0 1 , 2 1 , 3 1 ) 2 1 + ( w , 0 1 , 1 2 , 3 1 ) 2 1 − ( w , 0 1 , 1 2 , 2 1 ) 2 1 ( w, 1 1 , 2 2 , 3 1 ) 2 2 − ( w , 0 1 , 2 2 , 3 1 ) 2 2 + ( w , 0 1 , 1 1 , 3 1 ) 3 2 − ( w , 0 1 , 1 1 , 2 2 ) 3 2 ( w, 1 1 , 2 1 , 3 2 ) 3 3 − ( w , 0 1 , 1 1 , 3 2 ) 3 3 + ( w , 0 1 , 1 1 , 3 2 ) 3 3 − ( w , 0 1 , 1 1 , 2 1 ) 4 3 . By Lemma 7.3 the diagonal terms sum to (0 1 , 1 1 , 2 1 , 3 1 ) 0 w = b τ ◦ Ψ( F 0 , F 1 , F 2 , F 3) . If we deﬁne φ : D 2 → b P E ( F ) by φ ( F 0 , F 1 , F 2) = ( w , 0 2 , 1 1 , 2 1 ) 1 0 + ( w , 0 1 , 1 2 , 2 1 ) 2 1 + ( w , 0 1 , 1 1 , 2 2 ) 3 2 , the r emaing terms a re easily seen to equal φ ◦ ∂ ( F 0 , F 1 , F 2 , F 3) . Hence, b τ ◦ Ψ and b λ diﬀer by a cobo undary as desired.  R emark 7.9 . As in Remark 6.17, the map π ◦ b λ : C F 3 → P ( F ) is GL(3 , F ) inv ariant, and induces a ma p H 3 (GL(3 , F )) → B ( F ) , whic h factors thro ugh the homolog y of the c o mplex P C F ∗ of pro jective bases o f F 3 . The pro of of Lemma 7 .8 s hows tha t the map H 3 (GL(3 , F )) → B ( F ) agr e e s with the map in Remar k 6.17, and that the map H 3 (GL(2 , F )) → B ( F ) lifts to b B E ( F ) via H 3 (SL(3 , F )) and the stabilization map GL(2 , F ) → SL(3 , F ) . 7.1.1. Extension to H 3 (GL( n, F )) . In Gar oufalidis–Thurston–Zic kert [4 ] we con- struct maps H 3 (SL( n, F )) → b B E ( F ) commut ing with the stabilization maps. These maps a re induced by an SL( n, F ) –inv a r ian t ma p b λ : C F n 3 → b P E ( F ) , where C F n ∗ is the complex of ordered bases of F n (or aﬃne ﬂags, c.f. Remark 7.4). Remark 7 .9 generalizes, i.e. the maps (7.11) H 3 (GL( n, F )) → H 3 (SL( n + 1 , F )) → b B E ( F ) commut e with stabilization. Hence, by (7.1), we obtain a map K 3 ( F ) → b B E ( F ) . If F is free (a nd inﬁnite), b λ co mm utes with the maps Ψ xy from Prop osition 4 .3, so b λ induces a map K 3 ( F ) → b B ( F ) . The map b λ c o mm utes with Galo is actions, and resp ects the maps induced by embeddings in C . This implies tha t the regulator s (1.2) a nd (4.5) agree. 7.2. Step t wo: K M 3 ( F ) m aps to zero. F rom now o n, we assume that F is a free ﬁeld a dmitting an em b edding in C . This is used in Prop osition 7.11 but not in Lemma 7 .1 0 . Lemma 7.10. The c omp osition H 3 (GL(3 , F )) b λ / / b B E ( F ) π / / B ( F ) agr e es with the map c onstructe d by Suslin [18, Section 3] . Pr o of. By a result of Suslin [17], H 3 (GL(3 , F )) is genera ted by H 3 (GL(2 , F )) and H 3 ( T ) . By Remark 6.17 the tw o maps on H 3 (GL(2 , F )) . B y Remark 7.9 the map π ◦ b λ factors through the complex P C F ∗ of pro jective bases . Since T a cts trivia lly on pro jectiv e bases, P C F 0 → Z has a T –equiv ariant section, and it follows that π ◦ b λ is 0 on H 3 ( T ) . By Suslin [18, Proposition 3 .1], this also holds for Suslin’s map. Hence, the tw o maps a g ree.  28 CHRISTIAN K. ZICKER T Prop osition 7.11. The c omp osition K M 3 ( F ) → K 3 ( F ) → b B ( F ) is 0 . Pr o of. Let σ : F → C b e a n embedding and let σ ∗ : b B ( F ) → b B ( C ) b e the induced map. B y Lemma 7.10, the ima ge of K M 3 ( F ) in b B ( F ) is in f µ F . By Prop osition 4.11, σ ∗ maps f µ F injectiv ely to µ C , and since the re gulator R is injectiv e on µ C , it is enough to prov e that the co mposition K M 3 ( F ) / / f µ F / / µ C R / / C / 4 π 2 Z is z ero. Since this factors through K M 3 ( C ) , the result follows from Theorem 2.8 and Theorem 2.9.  7.3. Step three: A ﬁv e lemm a argument. Lemma 7.12. The exact se quenc es (1.1) and (4.4) ﬁt to gether in a diagr am 0 / / f µ F / / K ind 3 ( F ) / / b λ   B ( F ) / / 0 0 / / f µ F / / b B ( F ) / / B ( F ) / / 0 . Pr o of. Commutativit y of the right square follows fro m Lemma 7.10. T o prove commut ativity of the left squar e, we pr oceed a s in the pro of of Lemma 7.11. Since σ ∗ is injective on f µ F , it is enough to prov e the co rresp onding r esult with F r eplaced b y C . The result now fo llo ws from Theor em 2.8 and Theor e m 2.9.  The theorem b elow s umma r izes o ur results. Theorem 7.13. L et F b e a fr e e ﬁeld admitting an emb e dding in C . Ther e is a natur al isomorphism b λ : K ind 3 ( F ) ∼ = b B ( F ) c ommuting with Galois actions.  If F ⊂ E is a ﬁeld extension, the natura l map K ind 3 ( F ) → K ind 3 ( E ) is a n inclu- sion. F urthermor e, if F ⊂ E is Galo is, w e hav e K ind 3 ( E ) Gal( E ,F ) = K ind 3 ( F ) . This prop ert y is called Ga lois des cen t. W e refer to Merkur jev–Suslin [1 0] for pro ofs. Corollary 7 .14. F or any fr e e su bﬁeld F of C , the map b B ( F ) → b B ( C ) induc e d by inclusion is inje ctive.  Corollary 7.15. The ext ende d Blo ch gr oup of a numb er ﬁeld satisﬁes Galois de- sc ent.  8. Torsion in the extended Bloch group In this section we give a concr ete descr iption of the tor sion in b B ( F ) . W e s ta rt b y r eviewing s ome elementary prop erties of homolog y o f cyclic g roups. Prop osition 8.1. L et G b e a cyclic gr oup of or der n gener ate d by an element g ∈ G . The homolo gy gr oup H 3 ( G ) is cyclic of or der n and is gener ate d by the cycle n X k =1 h g | g k | g i . W e may t hus identify G with H 3 ( G ) .  THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 29 W e r efer to P arry– Sah [15, Pro position 3.25 ] fo r a n alg e braic pro of, and to Neu- mann [12] for a geo metric pro of using the lens space L ( n, 1) . Let p b e a prime num b er. As expla ined in the in tr o duction, K ind 3 ( F ) p has order p ν p for p o dd and 2 p ν p for p = 2 , where ν p = max { ν | ξ p ν + ξ − 1 p ν ∈ F } . T o co nstruct the to rsion in b B ( F ) , it is thus enough to exhibit elements in b B ( F ) p of order p ν p for p o dd and 2 p ν p for p = 2 . Let n = p ν p , and let x b e a pr imitive n th ro ot of unit y . Co nsider the matrices (8.1) g =  x + x − 1 − 1 1 0  , µ =  x 0 0 x − 1  , X =  x 1 1 x  . Note that g ∈ SL(2 , F ) and that g = X µX − 1 . Hence, g generates a cyclic subgroup of SL(2 , F ) of order n . Let [ g ] ∈ H 3 (SL(2 , F )) deno te the homology cla ss o f the cycle P n k =1 h g | g k | g i . Lemma 8.2. The element b λ ([ g ]) ∈ b B ( F ) has or der n = p ν p . Pr o of. It follows from Pro position 8 .1 that b λ ([ g ]) has order at mo st n . If we ﬁx an embedding of F ( x ) in C , we can v ie w g and µ as elements in SL(2 , C ) . Since g and µ are conjugate in SL(2 , C ) , it follows from Theorem 2.10 that b λ ([ g ]) has o rder at least n . Hence, b λ ([ g ]) has o r der n .  Corollary 8.3. F or p o dd, b B ( F ) p is gener ate d by b λ ([ g ]) .  8.1. Expl i cit computations. W e now give an explicit e x pression for b λ ([ g ]) . As- sume for now that p is o dd. F or any h 1 and h 2 in SL(2 , F ) , there is a homogeneo us representativ e of [ g ] of the fo r m (8.2) n X i =1 ( h 1 , g h 1 , g k h 2 , g k +1 h 2 ) , see e.g. the exa mple in Neuma nn [12, Section 12]. Using (6.1 0), we see that b λ takes a ter m ( h 1 , g h 1 , g k h 2 , g k +1 h 2 ) to a ﬂattening ( e k , f k ) , with (8.3) e k = log (det( v 1 , g k +1 v 2 )) + log(det( v 1 , g k − 1 v 2 )) − 2 log (det( v 1 , g k v 2 )) f k = log(det( v 1 , g v 1 )) + log(det( v 2 , g v 2 )) − 2 log (det( v 1 , g k v 2 )) , where v 1 = h 1  1 0  and v 2 = h 2  1 0  . It follo ws that b λ ([ g ]) = P n i =1 ( e k , f k ) ∈ b B ( F ) . Since the cycles (8.2) all represent [ g ] , w e may cho ose v 1 and v 2 as we plea se (as long as the vectors v 1 , g v 1 , g k v 2 , g k +1 v 2 in F 2 are in genera l p osition) without eﬀecting the element in b B ( F ) . If we let v 1 =  1 − 1  and v 2 =  1 1  , w e hav e (8.4) det( v 1 , g k v 2 ) = det( v 1 , X µ k X − 1 v 2 ) = ( x 2 − 1 ) det( X − 1 v 1 , µ k X − 1 v 2 ) = 1 x 2 − 1 det  x +1 − x − 1  , µ k  x − 1 x − 1  = det  1 − 1  ,  x k 0 0 x − k  1 1  = x k + x − k . 30 CHRISTIAN K. ZICKER T Letting z k denote the co rresp onding c r oss-ratio of ( e k , f k ) , it follows from (8.3) that z k = ( x k +1 + x − k − 1 )( x k − 1 + x − k +1 ) ( x k + x − k ) 2 . Since p is assumed to b e odd, z k ∈ F \ { 0 , 1 } . This prov es Theorem 1.3 for p o dd. Suppos e p = 2 . By computations similar to (8 .4) using v 1 =  1 0  and v 2 =  1 − 1  we obta in (8.5) det( v 1 , g k v 2 ) = x k − x − k +1 x − 1 , det( v 1 , g v 1 ) = 1 , det( v 2 , g v 2 ) = 2 + x + x − 1 . W e wis h to prov e that b λ ([ g ]) ∈ b B ( F ) is 2 –divisible. Let c k = det( v 1 , g k v 2 ) and let ˜ c k = lo g( c k ) . Also , let a = 2 + x + x − 1 and let ˜ a = log( a ) . By (8.5), we see that c k = c n − k +1 and c k = − c k + n/ 2 . By (8.3), ( e k , f k ) = ( ˜ c k +1 + ˜ c k − 1 − 2 ˜ c k , ˜ a − 2 ˜ c k ) . W e may choo se diﬀerent lo garithms for eac h k without eﬀecting the element b λ ([ g ]) = P n i =1 ( e k , f k ) . W e will choose them such that ˜ c k − ˜ c n − k +1 and ˜ c k + n/ 2 − ˜ c k are independent o f k and such that 2 ˜ c k = 2 ˜ c k + n/ 2 . With these particular choices, it is easy to see that b λ ([ g ]) is 2 – divisible. Indeed, b λ ([ g ]) = 2 Q , wher e (8.6) Q = n/ 2 X i =1 ( e k , f k ) ∈ b P ( F ) . W e now only need to prove that Q is in b B ( F ) . This follows from the computation b ν ( Q ) = X n/ 2 k =1 (˜ c k +1 + ˜ c k − 1 − 2 ˜ c k ) ∧ (˜ a − 2˜ c k ) = X n/ 2 k =1 2 ˜ c k ∧ ( ˜ c k +1 + ˜ c k − 1 ) − X n/ 2 k =1 2 ˜ c k ∧ ˜ a + X n/ 2 k =1 (˜ c k +1 + ˜ c k − 1 ) ∧ ˜ a = X n/ 2 k =1 (2 ˜ c k ∧ ˜ c k +1 − 2 ˜ c k − 1 ∧ ˜ c k ) + X n/ 2 k =1  (˜ c k +1 − ˜ c k ) − (˜ c k − ˜ c k − 1 )  ∧ ˜ a = 2 ˜ c n/ 2 ∧ ˜ c n/ 2+1 − 2 ˜ c 0 ∧ ˜ c 1 + ( ˜ c n/ 2+1 − ˜ c 1 ) ∧ ˜ a − (˜ c n/ 2 − ˜ c 0 ) ∧ ˜ a = 0 ∈ ∧ 2 ( E ) . Since z k = c k +1 c k − 1 c 2 k = ( x k +1 − x − k )( x k − 1 − x − k +2 ) ( x k − x − k +1 ) 2 ∈ F , this proves Theorem 1.3 for p = 2 . W e give some examples b elow. The computationa l details are left to the r eader. Example 8.4. F or any num b er ﬁeld F , which do es not contain a 3 rd ro ot of unity , the e lement 2[ − 2 ] + [ 1 4 ] ∈ B ( F ) ha s order 3 . Example 8.5. Let F = Q ( √ 2) . Doing the ab ov e co mputations, we o btain that Q = [ √ 2 − 1; 0 , 0] + [ √ 2 − 1; 0 , − 2] + [ − √ 2 − 1; 0 , 0] + [ − √ 2 − 1; − 2 , − 2 ] ∈ b B ( F ) . It follows that the element β 2 = 2[ √ 2 − 1] + 2[ − √ 2 − 1] ∈ B ( F ) has order 4 a nd generates B ( F ) 2 . Note that β 2 is not 2 –divisible. Applying the r e g ulator (2.4), w e get R ( Q ) = π 2 / 4 ∈ C / 4 π 2 , whic h has order 16 = 2 ν 2 +1 as e x pected. THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 31 R emark 8.6 . The order of the torsion in K ind 3 ( F ) is alwa y s divible by 24 . A partic- ular gener ator o f the 24 –tor sion in b B ( F ) is given by the element ( e, f ) + ( f , e ) from Lemma 3 .1 1 . W e omit the pro of o f this. 9. Hyperbolic 3 –manifolds Let M b e a complete, or iented, hyperb olic 3 –ma nifold with ﬁnite volume, a nd let K and k denote the tr ace ﬁeld and inv a rian t trace ﬁeld o f M . By a res ult of Goncharo v [6 , Theorem 1.1 ], M deﬁnes an element [ M ] in K ind 3 ( Q ) ⊗ Q , which equals the B loch inv ariant o f M (see e.g . Neumann–Y ang [14]) under the isomorphism K ind 3 ( Q ) ⊗ Q ∼ = B ( Q ) . Recall that a spin structure on M is equiv a len t to a lift of the g eometric rep- resentation to SL(2 , C ) , a nd that the set of spin structures is an aﬃne space ov er H 1 ( M ; Z / 2 Z ) . W e thank W a lter Neumann for assista nce with the pro of of the result b elow. Theorem 9.1. Supp ose M is close d. A spin structu r e ρ on M determines a fundamental class [ M ρ ] in K ind 3 ( K ) lifting the Blo ch invariant. F or e ach α ∈ H 1 ( M ; Z / 2 Z ) , the element [ M ρ ] − [ M αρ ] is two-torsion, which is trivial if and only if the induc e d map B α ∗ : H 3 ( M ) → H 3 ( B ( Z / 2 Z )) = Z / 2 Z is trivial. In p articular, 2[ M ρ ] is indep en dent of ρ . Mor e over, 2[ M ρ ] is in K ind 3 ( k ) . Pr o of. By Reid–Ma clac hlan [8, Co rollary 3.2.4], we may assume that ρ has image in SL(2 , K ( λ )) , where λ is a n a lgebraic element o f degr ee at mo st 2 over K . Endo w M with the structure o f a closed 3 –cycle, and ﬁx an SL(2 , K ( λ )) –co cycle α on M representing the fundamental class [ ρ ] o f ρ in H 3 (SL(2 , K ( λ ))) ; see e.g. Zick ert [2 1, Section 5]. Let [ M ρ ] = b λ ([ ρ ]) . Then [ M ρ ] , is giv en b y the ideal co chain c o n M deﬁned by α using (6.10) and (6.9). If λ has degr ee 1 , [ M ρ ] is o b vio usly in b B ( K ) . If λ has degree 2 , the non-tr ivial element in Ga l( K ( λ ) , K ) preser v es traces of ρ ( K is the trace ﬁeld), and therefor e ta kes ρ to a r epresen tation which is conjuga te over C . It follows that the image of [ M ρ ] in b B ( C ) is inv ariant under Gal( K ( λ ) , K ) , so b y the Coro llaries 7 .14 and 7 .1 5, [ M ρ ] is in b B ( K ) . The second statement follows from Theorem 6.15, so we now only need to prove that 2[ M ρ ] is in b B ( k ) . By Neumann–Reid [13, Theo rem 2.1], K is Galo is ov er k . Let σ ∈ Gal( K, k ) . Since k is the ﬁeld of squar es of traces of ρ , it follows that σ ρ a s a r epresen tation in PSL(2 , C ) is c o njugate to the geometric repres en tation. After a conjugation (which do es not change the fundamental cla ss), we may thu s assume that ρ and σ ρ are equal as representations in P SL(2 , C ) . Hence, c and σ ( c ) diﬀer b y a Z / 2 Z –co cycle, so b y Theor em 6.1 5 , 2 σ ([ M ρ ]) = 2[ M ρ ] ∈ b B ( C ) . As ab o ve, this implies that 2[ M ρ ] is in b B ( k ) .  9.1. Cus p ed manifolds. If M has cusps, Reid–Maclachlan [8] shows that the geometric repr e sen tatio n ha s imag e in P SL(2 , K ) . It thus follows from Theor e m 5.3 that M has a fundamental class [ M ] ∈ b B ( K ) PSL . Neumann–Y ang [14] show that the Blo c h in v a riant o f M is a lw ays in B ( k ) , but they deﬁne B ( k ) as the kernel of z 7→ 2 z ∧ (1 − z ) . With our deﬁnition, only 2[ M ] is in B ( k ) . An explicit exa mple whith [ M ] / ∈ B ( k ) is given by the manifold m 0 09 in the SnapPea census. Similarly , only 2[ M ] is in b B ( k ) PSL . Using remark 5.2 one chec ks that 2[ M ] alw ays lifts to 32 CHRISTIAN K. ZICKER T b B ( k ) , and by Lemma 5.1, 8[ M ] lifts canonically . W e do not believe that a canonica l lift of 2[ M ] is p ossible, s o this result is likely to be optimal. 9.1.1. Knot c omplements. If M is a knot complement, Reid–Machlachlan [8, Coro l- lary 4.2.2] implies that K = k . The obstruction to a lift of [ M ] ∈ b B ( k ) PSL to b B ( k ) is a Z / 2 Z –v alued knot inv a r ian t, whic h b y Remark 5.2 is explicitly computable. F or e x ample, [ M ] lifts for the ﬁgur e 8 knot complement and the 5 2 knot comple- men t, but not for the 6 1 knot complement . Since the sig niﬁcance of this inv ariant is unclear at this moment, we spar e the re ader for the computations . References [1] Joha n L. Dupont and Chih Han Sah. Scissors congruences. I I. J. Pur e Appl. Algebr a , 25(2):159 –195, 1982. [2] Joha n L. D upont and Christian K. Zick ert. A dil ogarithmic formula f or the Cheeger-Cher n- Simons class . Geo m. T op ol. , 10:1347–1372 (electronic), 2006. [3] Vladimir F o c k and Alexander Gonc harov. Mo duli spaces of lo cal systems and higher T eic h- müller theory . Publ. Math. Inst. Hautes Études Sci. , (103):1–211, 2006. [4] Sta vros Garoufalidis, Dylan Thur ston, and Christian K. Zick ert. In pr ep ar ation . 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Springer, Berlin, 1984. [18] A . A. Suslin. K 3 of a ﬁeld, and the Blo c h group. T rudy M at. Inst. Steklov. , 183:180–199, 229, 1990. T ranslated in Proc. Steklov Inst. Math. 199 1 , no. 4, 217–239, Galois theory , rings, algebraic groups and their applications (Russian). [19] C harles W eibel. Algebraic K -theory of rings of in tegers in lo cal and global ﬁelds. In Handb o ok of K -the ory. V ol. 1, 2 , pages 139–190. Springer, Berlin, 2005. [20] D on Zagier. The dilogarithm function. In F r ontiers in numb er the ory, physics, and ge ometry. II , pages 3–65. Springer, Berlin, 2007. [21] C hristian K. Zick ert. The volume and Chern-Simons inv ariant of a represen tation. Duke Math. J. , 150(3):489–532, 2009. THE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEOR Y 33 Dep ar tment of Ma thema tics, Unive rsity of California, Berkele y, CA 94720 -3840, USA E-mail addr ess : zickert@ma th.berkel ey.edu

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