The extended Bloch group and algebraic K-theory
aa r X i v : . [ m a t h . K T ] O c t THE EXTENDED BLOCH GROUP ANDALGEBRAIC K –THEORY CHRISTIAN K. ZICKERT
Abstract.
We define an extended Bloch group for an arbitrary field F , andshow that this group is naturally isomorphic to K ind3 ( F ) if F is a numberfield. This gives an explicit description of K ind3 ( F ) in terms of generators andrelations. We give a concrete formula for the regulator, and derive concretesymbol expressions generating the torsion. As an application, we show thata hyperbolic –manifold with finite volume and invariant trace field k has afundamental class in K ind3 ( k ) ⊗ Z [ ] . Introduction
The extended Bloch group b B ( C ) was introduced by Walter Neumann [12] inhis computation of the Cheeger–Chern–Simons class related to PSL(2 , C ) . It isa Q / Z –extension of the classical Bloch group B ( C ) , and was used by Neumannto give explicit simplicial formulas for the volume and Chern–Simons invariant ofhyperbolic –manifolds; see also Zickert [21]. There are two distinct versions ofthe extended Bloch group. One is isomorphic to H (PSL(2 , C ) δ ) and the other isisomorphic to H (SL(2 , C ) δ ) . The δ denotes that the groups are regarded as discretegroups, and will from now on be omitted. For a discussion of the relationshipbetween the two versions of the extended Bloch group, see Goette–Zickert [5].In Section 3, we define an extended Bloch group for an arbitrary field F . Moreprecisely, we show that there is an extended Bloch group b B E ( F ) for each extension E of F ∗ by Z , which only depends on the class of E in Ext( F ∗ , Z ) . The originalextended Bloch group is the extended Bloch group associated to the extension of C ∗ given by the exponential map. For a large class of fields, including numberfields and finite fields, the extended Bloch groups b B E ( F ) are isomorphic, and canbe glued together to form an extended Bloch group b B ( F ) which only depends on F , and admits a natural Galois action. This is studied in Section 4.By a result of Suslin [18], the classical Bloch group B ( F ) of an (infinite) field F is isomorphic to the algebraic K –group K ind ( F ) modulo torsion. More precisely,Suslin proves that there is an exact sequence(1.1) → f µ F → K ind ( F ) → B ( F ) → , where µ F denotes the roots of unity in F , and f µ F is the unique non-trivial extensionof µ F by Z / Z (in characteristic , f µ F = µ F ). Our main result is the following. Theorem 1.1.
For every number field F , there is a natural isomorphism b λ : K ind ( F ) ∼ = b B ( F ) respecting the Galois actions. (cid:3) In Section 9 we give the following geometric application generalizing a result ofGoncharov [6], who proved the existence of a fundamental class in K ind3 ( Q ) ⊗ Q . Theorem 1.2.
Let M be a complete, oriented, hyperbolic –manifold of finite vol-ume. Let K and k denote the trace field and invariant trace field of M . If M isclosed, M has a fundamental class [ M ] in K ind3 ( K ) defined up to two-torsion, andsatisfying that M ] ∈ K ind3 ( k ) . If M has cusps, there is a fundamental class [ M ] in K ind3 ( k ) ⊗ Z [ ] such that M ] is in K ind3 ( k ) . (cid:3) The result is proved using both concrete properties of the extended Bloch groupand abstract properties of K ind3 ( F ) .There is a regulator map R : K ind ( C ) → C / π Z . The regulator is equivariant with respect to complex conjugation, so if F is a numberfield, we obtain a regulator(1.2) b B : K ind ( F ) → ( R / π Z ) r ⊕ ( C / π Z ) r , where r and r are the number of real and (conjugate pairs of) complex embeddingsof F in C . This regulator fits into a diagram K ind ( F ) b B / / (cid:15) (cid:15) ( R / π Z ) r ⊕ ( C / π Z ) r (cid:15) (cid:15) B ( F ) B / / R r , where the left vertical map is the map in (1.1), and the right vertical map is pro-jection onto the imaginary part. The lower map B is known as the Borel regulatorand has been extensively studied. It is related to hyperbolic volume, and it isknown that the image in R r is a lattice whose covolume is proportional to the zetafunction of F evaluated at . We refer to Zagier [20] for a survey. The upper mapis much less understood. The real part is related to the Chern-Simons invariant,but little is known about its relations to number theory.In section 4, we give a concrete formula for b B defined on the extended Blochgroup b B ( F ) = K ind ( F ) . Elements in b B ( F ) are easy to produce, e.g. using computersoftware like PARI/GP, and our result can thus provide lots of experimental datafor studying the map b B . We give an example in Example 4.12.The torsion in K ind ( F ) is known to be cyclic of order w = 2 Q p ν p , where ν p = max { ν | ξ p ν + ξ − p ν ∈ F } . The product, which is easily seen to be finite, is over all rational primes, and ξ p ν isa primitive root of unity of order p ν . This result is due to Merkurjev–Suslin [10];see also the survey paper Weibel [19].In Section 8, we give explicit elements in b B ( F ) generating the torsion. As acorollary, this gives explicit generators of the torsion in the Bloch group. We statethis result below. Let B ( F ) p denote the elements in B ( F ) of order a power 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 } . HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 3 Theorem 1.3.
Let F be a number field and let p be a prime number with ν ′ p > .Let x be a primitive root of unity of order p ν p . The elements β p = p νp X k =1 (cid:20) ( x k +1 + x − k − )( x k − + x − k +1 )( x k + x − k ) (cid:21) ,β = ν − X k =1 (cid:20) ( x k +1 − x − k )( x k − − x − k +2 )( x k − x − k +1 ) (cid:21) generate B ( F ) p for odd p and p = 2 , respectively. (cid:3) Note that the torsion in the Bloch group comes from a totally real abelian subfieldof F . Remark . The torsion in the Bloch group of a number field is related to conformalfield theory, and there is an interesting conjecture relating torsion in the Bloch groupto modularity of a certain q –hypergeometric function. See Nahm [11], or Zagier [20]. Acknowledgements.
I wish to thank Ian Agol, Johan Dupont, Stavros Garoufa-lidis, Matthias Goerner, Dylan Thurston and, in particular, Walter Neumann forhelpful discussions. I also wish to thank Walter Neumann for his comments onearlier drafts of the paper. Parts of this work was done during a visit to the MaxPlanck Institute of Mathematics, Bonn. I wish to thank MPIM for its hospitality,and for providing an excellent working environment.2.
Preliminaries
For an abelian group A , we define ∧ ( A ) = A ⊗ Z A (cid:14) h a ⊗ b + b ⊗ a i . Note that a ∧ a = 0 , but a ∧ a is generally not .For a set X , we let Z [ X ] denote the free abelian group generated by X .2.1. The classical Bloch group.
Let F be a field and let F ∗ be the multiplicativegroup of units in F . Consider the set of five term relations FT = (cid:8)(cid:0) x, y, yx , − x − − y − , − x − y (cid:1) (cid:12)(cid:12) x = y ∈ F \ { , } (cid:9) . One can show that there is a chain complex(2.1) Z [FT] ρ / / Z [ F \ { , } ] ν / / ∧ ( F ∗ ) , with maps defined by ρ ([ z , . . . , z ]) = [ z ] − [ z ] + [ z ] − [ z ] + [ z ] ,ν ([ z ]) = z ∧ (1 − z ) . Remark . By Matsumoto’s theorem, the cokernel of ν is K ( F ) . Definition 2.2.
The
Bloch group of F is the quotient B ( F ) = Ker( ν ) / Im( ρ ) . It isa subgroup of the pre-Bloch group P ( F ) = Z [ F \ { , } ] / Im( ρ ) . CHRISTIAN K. ZICKERT
The extended Bloch group of C . The original reference is Neumann [12];see also Dupont–Zickert [2] and Goette–Zickert [5]. We stress that our extendedBloch group is what Neumann calls the more extended Bloch group [12, Section 8].Consider the set b C = (cid:8) ( w , w ) ∈ C (cid:12)(cid:12) exp( w ) + exp( w ) = 1 (cid:9) . We will refer to elements of b C as flattenings . We can view b C as the Riemann surfacefor the multivalued function (log( z ) , log(1 − z )) , and we can thus write a flatteningas [ z ; 2 p, q ] = (log( z ) + 2 pπi, log(1 − z ) + 2 qπi ) . This notation depends on a choiceof logarithm branch which we will fix once and for all. The map π : b C → C \ { , } taking a flattening [ z ; 2 p, q ] to z is the universal abelian cover of C \ { , } . Remark . Neumann considered the Riemann surface of (log( z ) , − log(1 − z )) , andconsidered a flattening [ z ; 2 p, q ] as a triple ( w , w , w ) , with w = log( z ) + 2 pπi , w = − log(1 − z ) + 2 qπi and w = − w − w . One translates between the twodefinitions by changing the sign of w , or equivalently, by changing the sign of q .Let FT = (cid:8) ( x , . . . , x ) ∈ FT (cid:12)(cid:12) < x < x < (cid:9) , and define the set of lifted fiveterm relations c FT ⊂ ( b C ) to be the component of the preimage of FT containingall points (cid:0) [ x ; 0 , , . . . , [ x ; 0 , (cid:1) with ( x , . . . , x ) ∈ FT .There is a chain complex(2.2) Z [ c FT] b ρ / / Z [ b C ] b ν / / ∧ ( C ) , with maps defined by b ρ ([( w , w ) , . . . , ( w , w )]) = X i =0 ( − i [( w i , w i )] , b ν ([( w , w )]) = w ∧ w . Definition 2.4.
The extended Bloch group is the quotient b B ( C ) = Ker( b ν ) / Im( b ρ ) .It is a subgroup of the extended pre-Bloch group b P ( C ) = Z [ b C ] / Im( b ρ ) . Theorem 2.5.
Let µ C denote the roots of unity in C ∗ . There is a commutativediagram as below with exact rows and columns. (cid:3) (2.3) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / µ C / / χ (cid:15) (cid:15) C ∗ / / χ (cid:15) (cid:15) C ∗ /µ C / / β (cid:15) (cid:15) (cid:15) (cid:15) / / b B ( C ) / / π (cid:15) (cid:15) b P ( C ) b ν / / π (cid:15) (cid:15) ∧ ( C ) / / ǫ (cid:15) (cid:15) K ( C ) / / / / B ( C ) / / (cid:15) (cid:15) P ( C ) ν / / (cid:15) (cid:15) ∧ ( C ∗ ) (cid:15) (cid:15) / / K ( C ) / / (cid:15) (cid:15)
00 0 0 0
HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 5 We refer to Goette–Zickert [5] or Section 3 below for the definition of χ . Theother maps are defined as follows: β ([ z ]) = log( z ) ∧ πi ; ǫ ( w ∧ w ) = exp( w ) ∧ exp( w ); π ([ z ; 2 p, q ]) = [ z ] . Remark . There is a similar definition of b P ( F ) and b B ( F ) for a subfield F of C .Theorem 2.5 holds if one replaces ∧ ( C ) by ∧ ( { w ∈ C | exp( w ) ∈ F ∗ } ) and µ C with f µ F = { w ∈ C | w ∈ µ F } . We will generalize to arbitrary fields below.2.2.1. The regulator.
The function R : b C → C / π , given by(2.4) [ z ; 2 p, q ] Li ( z ) + 12 (log( z ) + 2 pπi )(log(1 − z ) − qπi ) − π / is well defined and holomorphic, see e.g. Neumann [12] or Goette–Zickert [5].It is well known that L ( z ) = Li ( z ) + log( z ) log(1 − z ) − π / satisfies X i =1 ( − i L ( z i ) = 0 for ( z , . . . , z ) ∈ FT , and it thus follows by analytical continuation that R gives rise to a function(2.5) R : b P ( C ) → C / π Z . We briefly describe a more elegant definition of R due to Don Zagier [20]: Thederivative of Li ( z ) is − log(1 − z ) /z . It follows that the function F ( x ) = Li (1 − e x ) has derivative F ′ ( x ) = xe x / (1 − e x ) . Since this function is meromorphic with simplepoles at πin , n ∈ Z , with corresponding residues − πin , it follows that F definesa single valued function on C \ { πi Z } with values in C / π Z . We can now define(2.6) R : b C → C / π , ( w , w ) F ( w ) + w w − π / . We leave it to the reader to show that this definition of R agrees with the oneabove.2.3. Algebraic K –theory and homology of linear groups. We give a briefreview of the results that we shall need.Let F be a field. The algebraic K –groups are defined by K i ( F ) = π i ( B GL( F ) + ) .The Milnor K –groups K M ∗ ( F ) are defined as the quotient of the tensor algebra of F ∗ by the two-sided ideal generated by a ⊗ (1 − a ) . There is a natural map K Mi ( F ) → K i ( F ) whose cokernel, by definition, is the indecomposable K –group K ind i ( F ) .For F = C , there is a regulator R defined as the composition K k − ( C ) H / / H k − (GL( C )) ˆ c k / / C / (2 πi ) k Z , where H is the Hurewicz map, and ˆ c k is the universal Cheeger–Chern–Simons class.It is well known that R is on the image of K M k − ( C ) , so R induces a regulator K ind k − ( C ) → C / (2 πi ) k Z . CHRISTIAN K. ZICKERT
Theorem 2.7 (Suslin [17]) . For any field F , there is an isomorphism H n (GL( n, F )) ∼ = H n (GL( F )) induced by inclusion. (cid:3) Theorem 2.8 (Sah [16]) . K ind3 ( C ) is a direct summand of K ( C ) and the Hurewiczmap H induces an isomorphism K ind3 ( C ) ∼ = H (SL(2 , C )) . (cid:3) Theorem 2.9 (Goette–Zickert [5]; see also Neumann [12]) . There is a canonicalisomorphism H (SL(2 , C )) ∼ = b B ( C ) . Under this isomorphism, ˆ c corresponds to themap R in (2.4) . (cid:3) Theorem 2.10 (Dupont–Sah [1]) . The diagonal map x (cid:0) x x − (cid:1) induces aninjection H ( µ C ) → H (SL(2 , C )) onto the torsion subgroup of H (SL(2 , C )) . (cid:3) The extended Bloch group of an extension
Let F be a field and let E : 0 / / Z ι / / E π / / F ∗ / / be an extension of F ∗ by Z . We stress that the letter E is used both to denote theextension and the middle group. As we shall see, most of the results in Neumann [12]and Goette–Zickert [5] can be formulated in this purely algebraic setup. Definition 3.1.
The set of (algebraic) flattenings is the set b F E = (cid:8) ( e, f ) ∈ E × E (cid:12)(cid:12) π ( e ) + π ( f ) = 1 ∈ F (cid:9) . The map ( e, f ) π ( e ) induces a surjection π : b F E → F \ { , } , and we say that ( e, f ) is a flattening of π ( e ) .Recall the set of five term relations FT = (cid:8)(cid:0) x, y, yx , − x − − y − , − x − y (cid:1) (cid:12)(cid:12) x = y ∈ F \ { , } (cid:9) . Definition 3.2.
The set of lifted five term relations c FT E ⊂ ( b F E ) is the set oftuples of flattenings (cid:0) ( e , f ) , . . . , ( e , f ) (cid:1) satisfying e = e − e e = e − e − f + f f = f − f e = f − f f = f − f + e . (3.1)If (cid:0) ( e , f ) , . . . , ( e , f ) (cid:1) ∈ c FT E , where ( e i , f i ) is a flattening of x i ∈ F \ { , } ,then (3.1) implies that x = x x , x = x x (1 − x )(1 − x ) = 1 − x − − x − , x = 1 − x − x . Hence, a lifted five term relation is indeed a lift of a five term relation. On the otherhand, if ( x , . . . , x ) ∈ FT it is not difficult to check that there exist flattenings ( e i , f i ) satisfying (3.1). Hence, the map π : c FT E → FT is surjective. HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 7 Consider the complex(3.2) Z [ c FT E ] b ρ / / Z [ b F E ] b ν / / ∧ ( E ) , with maps defined by b ρ (cid:0) ( e , f ) , . . . , ( e , f ) (cid:1) = X i =0 ( − i ( e i , f i ) , b ν ( e, f ) = e ∧ f. Lemma 3.3.
The complex (3.2) is a chain complex, i.e. b ν ◦ b ρ = 0 .Proof. Let α = (cid:0) ( e , f ) , . . . , ( e , f ) (cid:1) ∈ c FT E . Using (3.1) we have b ν ◦ b ρ ( α ) = X i =0 ( − i e i ∧ f i = e ∧ f − e ∧ f + ( e − e ) ∧ f − ( e − e − f + f ) ∧ ( f − f ) + ( f − f ) ∧ ( f − f + e )= e ∧ ( f − f ) + ( f − f ) ∧ e = 0 ∈ ∧ ( E ) . (cid:3) Definition 3.4.
The extended pre-Bloch group b P E ( F ) is the quotient Z [ b F E ] / Im( b ρ ) .The extended Bloch group b B E ( F ) is the quotient Ker( b ν ) / Im( b ρ ) . Example 3.5.
The extended Bloch group b B ( C ) is the extended Bloch group asso-ciated to the extension(3.3) / / Z πi / / C exp / / C ∗ / / . The extended groups fit together with the classical groups in a diagram b B E ( F ) / / (cid:15) (cid:15) b P E ( F ) (cid:15) (cid:15) b ν / / ∧ ( E ) (cid:15) (cid:15) B ( F ) / / P ( F ) ν / / ∧ ( F ∗ ) , where the vertical maps are surjections.3.1. Relations in the extended Bloch group.
We now derive some relationsin b P E ( F ) . We encourage the reader to compare with Neumann [12] and Goette–Zickert [5] where similar relations are derived in b P ( C ) using analytic continuation.In the following we will regard Z as a subgroup of E . Consider the set(3.4) V = (cid:8)(cid:0) ( p , q ) , ( p , q ) , ( p − p , q ) , ( p − p − q + q , q − q ) , ( q − q , q − q + p ) (cid:1) (cid:12)(cid:12) p , p , q , q , q ∈ Z (cid:9) ⊂ ( Z × Z ) , also considered by Neumann [12, Definition 2.2].By (3.1) it follows that componentwise addition gives rise to an action + : c FT E × V → c FT E , ( α, v ) α + v. Lemma 3.6.
Let q, q ′ , ¯ q, ¯ q ′ be integers satisfying q − q ′ = ¯ q − ¯ q ′ . For each 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 ) . CHRISTIAN K. ZICKERT
Proof.
Let α = (cid:0) ( e , f ) , . . . , ( e , f ) (cid:1) ∈ c FT E . For each integer r , consider theelement v r ∈ V given by v r = (cid:0) (0 , r ) , (0 , r ) , (0 , r ) , (0 , , (0 , (cid:1) . The relation b ρ ( α + v q ) − b ρ ( α + v q ′ ) = 0 ∈ b P E ( F ) can be written as(3.6) ( e , f + q ) − ( e , f + q ′ ) − (cid:0) ( e , f + q ) − ( e , f + q ′ ) (cid:1) + ( e , f + q ) − ( e , f + q ′ ) = 0 ∈ b P E ( F ) . Let β = α + (cid:0) (0 , , (0 , s ) , (0 , s ) , ( − s, , ( − s, (cid:1) , 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 ) becomes(3.7) ( e , f + ¯ q ) − ( e , f + ¯ q ′ ) − (cid:0) ( e , f + q ) − ( e , f + q ′ ) (cid:1) + ( e , f + q ) − ( e , f + q ′ ) = 0 ∈ b P E ( F ) . The result now follows by subtracting (3.7) from (3.6). (cid:3)
Corollary 3.7.
Let e ∈ E \ Z . The element ( e, f + 1) − ( e, f ) ∈ b P E ( F ) is independent of f whenever ( e, f ) is in b F E . (cid:3) Using Corollary 3.7, we can define a map(3.8) χ : E \ Z → b P E ( F ) , e ( e, f + 1) − ( e, f ) . Lemma 3.8.
Suppose e, e ′ and e + e ′ are elements in E \ Z . We have (3.9) χ ( e ) + χ ( e ′ ) = χ ( e + e ′ ) . Proof.
This follows from (3.6) after noting that e = e + e . (cid:3) The following is elementary.
Lemma 3.9.
Let G and G ′ be groups and let H be a subgroup of G of index greaterthan . Suppose φ : G \ H → G ′ is a map satisfying φ ( g g ) = φ ( g ) φ ( g ) wheneverboth sides are defined. Then φ extends uniquely to a homomorphism φ : G → G ′ . (cid:3) Corollary 3.10.
The map χ : E \ Z → b P E ( F ) extends to a homomorphism definedon all of E . (cid:3) Lemma 3.11.
For any two flattenings ( e, f ) , ( g, h ) ∈ b F E we have ( e, f ) + ( f, e ) = ( g, h ) + ( h, g ) ∈ b B E ( F ) . Proof.
It follows from (3.1) that (cid:0) ( e , f ) , ( e , f ) , ( e , f ) , ( e , f ) , ( e , f ) (cid:1) ∈ c FT E if and only if (cid:0) ( f , e ) , ( f , e ) , ( e , f ) , ( e , f ) , ( e , f ) (cid:1) ∈ c FT E . Subtracting thetwo relations in b P E ( F ) yields ( e , f ) − ( e , f ) = ( f , e ) − ( f , e ) ∈ b P E ( F ) , from which the claim follows. Since e ∧ f + f ∧ e = 0 ∈ ∧ ( E ) , the element lies in b B E ( F ) . (cid:3) Lemma 3.12.
The homomorphism χ : E → b P E ( F ) takes Z ⊂ E to . HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 9 Proof.
Let e ∈ E be 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 equality follows from Lemma 3.11. (cid:3) Theorem 3.13.
There is an exact sequence (3.11) E/ Z χ / / b P E ( F ) π / / P ( F ) / / . Proof.
It is clear that π ◦ χ = 0 and that π is surjective. Since every five termrelation lifts to a lifted five term relation, the kernel of π must be generated byelements of the form ( e + p, f + q ) − ( e, f ) . By Lemma 3.6 we have,(3.12) ( e, f + q ) = ( e, f + q −
1) + ( e, f + 1) − ( e, f )= q ( e, f + 1) − q ( e, f ) + ( e, f ) = qχ ( 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. (cid:3) Remark . We do not know if χ is injective, but we expect this to be the case.In the next section, we show that χ is injective if F is a number field and E is aprimitive extension. Remark . If E is the extension in Example 3.5, the exact sequence (3.11) isequivalent to the corresponding exact sequence in Theorem 2.5 using the identi-fication of C ∗ with E/ Z = C / πi Z taking z ∈ C ∗ to − z ) ∈ C / πi Z . Therestriction of the regulator (2.4) to C / πi Z ⊂ b P ( C ) is multiplication by − πi . Lemma 3.16.
The following equality holds in b P E ( F ) . ( e + p, f + q ) − ( e, f ) = χ ( qe − pf + pq ) . Proof.
This is an easy consequence of (3.12) and (3.13). (cid:3)
Functoriality.
Let F and F be fields and let E and E be extensions of F ∗ and F ∗ by Z . Definition 3.17.
A map
Ψ : E → E of extensions is called a covering if the basehomomorphism Ψ : F ∗ → F ∗ extends to an embedding of F in F . Two coveringsare equivalent if they cover the same embedding.A covering Ψ : E → E gives rise to a chain map Z [ c FT E ] b ρ / / Ψ ∗ (cid:15) (cid:15) Z [ b F E ] b ν / / Ψ ∗ (cid:15) (cid:15) ∧ ( E ) Ψ ∧ Ψ (cid:15) (cid:15) Z [ c FT E ] b ρ / / Z [ b F E ] b ν / / ∧ ( E ) defined by taking an algebraic flattening ( e, f ) to (Ψ( e ) , Ψ( f )) . We thus obtainmaps(3.14) Ψ ∗ : b P E ( F ) → b P E ( F ) , Ψ ∗ : b B E ( F ) → b B E ( F ) , satisfying the usual functoriality properties. Lemma 3.18.
Let
Ψ : E → E be a covering. There is a commutative diagram ofexact sequences E / Z χ / / Ψ ∗ (cid:15) (cid:15) b P E ( F ) π / / Ψ ∗ (cid:15) (cid:15) P ( F ) / / Ψ ∗ (cid:15) (cid:15) E / Z χ / / b P E ( F ) π / / P ( F ) / / . The map Ψ ∗ : E / Z → E / Z takes e ∈ E / Z to Ψ(1)Ψ( e ) ∈ E / Z , where Ψ(1)Ψ( e ) is defined using the natural action of Z ⊂ E on E by multiplication.Proof. Exactness follows from Theorem 3.13. Commutativity of the right square isobvious, and commutativity of the left square follows from the computation Ψ ∗ ( χ ( e )) = (cid:0) Ψ( e ) , Ψ( f ) + Ψ(1) (cid:1) − (cid:0) Ψ( e ) , Ψ( f ) (cid:1) = χ (cid:0) Ψ(1)Ψ( e ) (cid:1) , where the second equality follows from Lemma 3.16. (cid:3) We wish to prove that the induced map Ψ ∗ : b B E ( F ) → b B E ( F ) of a coveringonly depends on the underlying embedding. The result below is elementary. Lemma 3.19.
Let F be a field. The torsion subgroup Tor( F ∗ ) is isomorphic to asubgroup of Q / Z . For any extension E of F ∗ by Z , the same is true for Tor( E ) . (cid:3) Lemma 3.20.
An element P i n i e i ∧ f i is zero in ∧ ( E ) if and only if we can 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 , where p j ∈ E , w ∈ E is a torsion element, and the integers s ij , r ij , k i and l i satisfy(i) P i n i r ij s ij is even for each j ;(ii) P i n i ( r ij s ik − r ik s ij ) = 0 for each j = k ;(iii) P i n i ( l i r ij − k i s ij ) is divisible by ord( w ) for each j ;(iv) P i n i k i l i is even.Proof. Let α = P i n i e i ∧ f i . Since ∧ commutes with direct limits, there exists afinitely generated subgroup H , containing the e i ’s and f i ’s, such that α is zero in ∧ ( E ) if and only if α is zero in ∧ ( H ) . Let p j be free generators of H and let w bea generator of the torsion (which is cyclic by Lemma 3.19). Write the e i ’s and f i ’sas in (3.15). When expanding α ∈ ∧ ( H ) , the coefficients of p j ∧ p j , p j ∧ p k , p j ∧ w and w ∧ w of ∧ ( H ) are given, respectively, by (i)-(iv). Hence, α = 0 ∈ ∧ ( H ) if(i)-(iv) hold. On the other hand, if α = 0 ∈ ∧ ( H ) , (i)-(iii) hold, whereas (iv) mayfail if w ∧ w = 0 ∈ ∧ ( H ) . This happens if and only if w is –divisible, in whichcase, we may replace w by a half if necessary, to make (iv) hold as well. (cid:3) Theorem 3.21. If Ψ , Ψ : E → E are equivalent coverings then (3.16) Ψ ∗ = Ψ ∗ : b B E ( F ) → b B E ( F ) . HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 11 Proof.
For notational simplicity, we assume that Ψ = id , and drop the subscriptsof F i and E i . This case is sufficient for most of the applications. We leave thegeneral proof, which differs only in notation, to the reader.Let φ = Ψ − Ψ : E → Z ⊂ E and let α = P n i ( e i , f i ) ∈ b B E ( F ) . We wish toprove that(3.17) ∆ := Ψ ∗ ( α ) − Ψ ∗ ( α ) = X i n i (cid:0)(cid:0) e i + φ ( e i ) , f i + φ ( f i ) (cid:1) − ( e i , f i ) (cid:1) is zero in b B E ( F ) . By Lemma 3.16, we have(3.18) ∆ = χ (cid:16) X i n i (cid:0) φ ( f i ) e i − φ ( e i ) f i + φ ( e i ) φ ( f i ) (cid:1)(cid:17) . Since α ∈ b B E ( F ) , we have b ν ( α ) = P n i e i ∧ f i = 0 ∈ ∧ ( E ) . By Lemma 3.20, wecan write 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 Lemma 3.20 are satisfied. Let a j = φ ( p j ) . Let T denote the sumin (3.18), i.e. χ ( T ) = ∆ . Since w is a torsion element, φ ( w ) = 0 , so by (3.18) wehave(3.19) T = X i n i (cid:16) X j s ij a j (cid:0) X j r ij p j + k i w (cid:1) − X j r ij a j (cid:0) X j s ij p j + l i w (cid:1) + X j r ij a j X j s ij a j (cid:17) . When expanding the sum, the coefficient of p k is X i n i (cid:0) X j s ij a j (cid:1) r ik − X i n i (cid:0) X j r ij a j (cid:1) s ik = − X j a j X i n i ( r ij s ik − s ij r ik ) , which is zero by Lemma 3.20, (ii). Similarly, the coefficient of w is X i n i (cid:0) X j a j k i s ij − X j a j l i r ij (cid:1) = − 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 remainingterms sum to the integer X i n i (cid:16) X j r ij a j X j s ij a j (cid:17) = X i n i (cid:16) X j = k r ij s ik a j a k (cid:17) + X j a j X i n i r ij s ij , which is even by Lemma 3.20, (i) and (ii). Hence, T is zero in E/ Z , so ∆ = χ ( T ) is also zero. (cid:3) Corollary 3.22.
Up to canonical isomorphism, the extended Bloch group b B E ( F ) depends only on the class of E in Ext( F ∗ , Z ) .Proof. If E = E ∈ Ext( F ∗ , Z ) , there must exist a covering Ψ : E → E of theidentity on F . Since any two such coverings are equivalent, the result follows. (cid:3) General properties of extensions.
Let µ F ⊂ F ∗ denote the roots of unityin F . For a prime number p let µ p denote the p th roots of unity in µ F , and let µ p ∞ be the subgroup of roots of unity of order a power of p . Note that µ F = ⊕ µ p ∞ .Also note that up to isomorphism, µ p ∞ is either Z /p n Z or Z [1 /p ] / Z . We havethe following classification of Z –extensions of Z /p n Z and Z [1 /p ] / Z . The proofs areelementary and left to the reader. Lemma 3.23.
We have
Ext( Z /p n Z , Z ) = Z /p n Z . Let ≤ k ≤ n − and let x bean integer which is not divisible by p . The non-trivial extensions are given explicitlyby (3.20) / / Z ι / / Z ⊕ Z /p k Z π / / Z /p n Z / / , where the maps are defined by (3.21) ι (1) = ( p n − k , − x ) , π ( a, b ) = xa + p n − k b. The equivalence class of (3.20) only depends on the value of x in ( Z /p n − k Z ) ∗ , andthe order of the extension in Ext( Z /p n Z , Z ) is p n − k . (cid:3) Lemma 3.24.
We have
Ext( Z [ p ] / Z , Z ) = Z p , the p –adic integers. Let ≤ k bean integer and let y be a p –adic integer with discrete valuation k . The non-trivialextensions are given explicitly by (3.22) / / Z ι / / Z [ p ] ⊕ Z /p k Z π / / Z [ p ] (cid:14) Z / / , where the maps are defined by (3.23) ι (1) = (1 /p k , − y/p k ) , π ( a, b ) = ay + bp k . (cid:3) Definition 3.25.
An extension E is called primitive if E is torsion free. If µ p isnon-trivial, and if the restriction E µ p ∞ is torsion free, we say that E is p –primitive .We state some elementary corollaries of Lemma 3.23 and Lemma 3.24. Corollary 3.26. E is p –primitive if and only if ∈ E is divisible by p . If so, isdivisible by | µ p ∞ | (if | µ p ∞ | = ∞ , is divisible by p infinitely often). (cid:3) Corollary 3.27.
Suppose µ F is finite. Then E is primitive if and only if E µ F generates Ext( µ F , Z ) . In this case E µ F is free of rank one. Letting ˜ x denote agenerator, the extension is given explicitly by (3.24) E µ F : 0 / / Z ι x / / E µ F π x / / µ F / / , where ι x takes to | µ F | ˜ x and π x takes ˜ x to a primitive root of unity x . (cid:3) Lemma 3.28.
Suppose E is p –primitive for an odd prime p . Then ∧ ( E µ p ∞ ) = Z / Z generated by ∧ . If E is also –primitive, the map ∧ ( E µ p ∞ ) → ∧ ( E ) is . Otherwise it is injective.Proof. We assume that E µ p ∞ ∼ = Z [1 /p ] leaving the simpler case E µ p ∞ ∼ = Z to thereader. It is easy to see that Z [1 /p ] is –torsion generated by elements p − k ∧ p − k ,and since p is odd, p − k ∧ p − k = p k ( p − k ∧ p − k ) = 1 ∧ . By Corollary 3.26, is –divisible in E if and only if E is –primitive. This concludes the proof. (cid:3) Note that a primitive extension is –primitive if and only if the characteristic of F is not . This is because µ is trivial in characteristic and non-trivial otherwise. Proposition 3.29.
Let E be a primitive extension and let E ( µ F ) = 2 E µ F if thecharacteristic of F is and E ( µ F ) = E µ F otherwise. We have an exact sequence (3.25) / / E ( µ F ) ι / / E β / / ∧ ( E ) π ∧ π / / ∧ ( F ∗ ) / / , where β is the map taking e to e ∧ . HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 13 Proof.
The only non-trivial part is to show that the kernel of β is E ( µ F ) . Firstnote that if e ∧ ∈ ∧ ( E ) , e and must be linearly dependent over Z (thisfollows from Lemma 3.20). Hence, e must be in E µ F .Let e ∈ E µ F . We may assume that e ∈ E µ p ∞ for some prime p . Suppose thecharacteristic of F is not . By Lemma 3.28, the restriction of β to E µ p ∞ is zero if p = 2 , and by inspection, this also holds if p = 2 . Hence, β ( e ) = 0 . Now supposethe characteristic of F is . In particular p must be odd. Pick integers k and l suchthat p k e = l ∈ E µ p ∞ . By Lemma 3.28, e ∧ p k e ∧ l ∧ is zero in ∧ ( E ) ifand only if l is even. Hence, p k e , and therefore also e , is –divisible. (cid:3) Corollary 3.30.
Let E be primitive. There is an exact sequence f µ F χ / / b B E ( F ) π / / B ( F ) / / , where f µ F = E ( µ F ) / Z . (cid:3) Note that f µ F is independent of E up to isomorphism. If the characteristic of F is not , f µ F is the unique non-trivial Z / Z –extension of µ F , and in characteristic , f µ F is just µ F . The notation f µ F thus agrees with that of Suslin [18].4. The extended Bloch group of a field
We now show that if we impose some conditions on F , the extended Blochgroups b B E ( F ) are naturally isomorphic, and we can glue them together to form anextended Bloch group b B ( F ) depending only on F . This group admits a naturalaction by the automorphism group of F . Definition 4.1.
A field F is called free if it satisfies the following two conditions:(i) F ∗ /µ F is a free abelian group;(ii) | µ F | < ∞ .Free fields include number fields and finite fields, and freeness is preserved un-der finite transcendental extensions. For a discussion of fields satisfying (i), seee.g. May [9]. Throughout this section F denotes a free field.Corollary 3.27 implies that primitive Z –extensions of F ∗ are in one-one corre-spondence with primitive roots of unity. If x is a primitive root of unity, we let E x denote the corresponding primitive extension. The restriction of E x to µ F is freeof rank one with generator ˜ x mapping to x ∈ F ∗ . Since F ∗ is free modulo torsion,any extension of F ∗ is uniquely determined by its restriction to µ F . The followingresult is a direct consequence of this discussion and Theorem 3.21. Lemma 4.2.
Let F ′ be any field (not necessarily free) and let σ : F → F ′ be anembedding. Let E ′ be a Z –extension of F ′∗ and let E be a primitive Z –extensionof F ∗ . There exists a covering b σ : E → E ′ of σ . The induced map b σ ∗ : b B E ( F ) → b B E ′ ( F ′ ) depends only on σ and not on the choice of covering. (cid:3) Corollary 4.3.
For any pair E x , E y of primitive Z –extensions of F ∗ , there existsa covering Ψ xy : E x → E y of the identity on F . The induced map Ψ xy ∗ : b B E x ( F ) → b B E y ( F ) is an isomorphism with inverse Ψ yx ∗ .Proof. Existence of Ψ xy follows from Lemma 4.3, and since the coverings Ψ xy ◦ Ψ yx and Ψ yx ◦ Ψ xy are equivalent to the identities on E y and E x , the result follows. (cid:3) Corollary 4.4.
Let τ be an automorphism of F . For each primitive root of unity x ∈ F , there exists a unique covering b τ x : E x → E τ ( x ) of τ . The induced 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 ∗ . Proof.
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 both coverings of τ , the result follows. (cid:3) Definition 4.5.
The extended Bloch group of F is defined by(4.2) b B ( F ) = lim ←− b B E x ( F ) = n ( α E x ) ∈ Y b B E x ( F ) (cid:12)(cid:12) α E y = Ψ xy ∗ ( α E x ) o , where the product is over primitive roots of unity. Remark . If F is not free, we can still define b B ( F ) as an inverse limit, but in thegeneral case, the primitive extensions do not form a directed set, and we do not seehow to establish the desired connection with the classical Bloch group. Proposition 4.7.
There is a natural action of
Aut( F ) on b B ( F ) , where each auto-morphism acts by an isomorphism.Proof. This follows directly from (4.1). (cid:3)
Proposition 4.8.
There is an exact sequence f µ F χ / / b B ( F ) π / / B ( F ) / / . Proof.
This is an easy consequence of Corollary 3.30. (cid:3)
Note that the action of
Aut( F ) on f µ F is through the quadratic character.4.1. Embeddings in C and the regulator. Recall that b B ( C ) is the extendedBloch group associated to the extension of C ∗ given by the exponential map.Let σ be an embedding of F in C . By Lemma 4.3, each primitive extension E x admits a covering b σ x : E x → C of σ . Lemma 4.9.
The induced map b σ x ∗ : b P E x ( F ) → b P ( C ) restricts to an injection E x / Z → C / πi Z .Proof. Clearly, b σ x : E x → C is injective. Since b σ x must take to πik , where k isrelatively prime to | µ F | , the result follows from Lemma 3.18. (cid:3) Corollary 4.10.
Let F be a free field admitting an embedding in C . The map χ : E/ Z → b P E ( F ) is injective for all primitive extensions E .Proof. We may assume that E = E x . It is enough to prove that b σ x ∗ ◦ χ : E x / 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 / πi Z → b P ( C ) . By Corollary 4.10 (and Remark 3.15), this is a compositionof injective maps, hence injective. (cid:3) Since b σ x and b σ y ◦ Ψ xy both cover σ , the induced maps satisfy 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 of Lemma 4.9 andCorollary 4.10. HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 15 Proposition 4.11.
Let F be a free field admitting an embedding σ in C . The map σ ∗ : b B ( F ) → b B ( C ) restricts to an injection f µ F → µ C . Furthermore, the sequence (4.4) / / f µ F χ / / b B ( F ) π / / B ( F ) / / . is exact. (cid:3) The regulator.
A number field F of type [ r , r ] has r real embeddings and r pairs of complex embeddings. The regulator map (2.4) is equivariant with respectto the action on b B ( C ) by complex conjugation given by [ z ; p, q ] [¯ z ; − p, − q ] . Wetherefore obtain a regulator(4.5) b B ( F ) → ( R / π Z ) r ⊕ ( C / π Z ) r . Example 4.12.
Let x be a root of p ( x ) = x − x + 2 x − x + 1 , and considerthe number field F = Q ( x ) . One can check that µ F has order and is generatedby w = x + x . Let E be the primitive extension corresponding to w and let ˜ w bethe generator of E µ F . Let u = − x − x + 1 and let v = x − x + 1 . The relations − u = u w , v = w u − , − v = u − w, are easily verified, and it follows that α = [ u ] + 2[ v ] ∈ B ( F ) . Let ˜ u be a lift of u ,and consider the lift(4.6) ˜ α = (˜ u, u + 4 ˜ w ) + 2( − u + 3 ˜ w, − u + ˜ w ) − χ (˜ u ) ∈ b P E ( F ) of α . One easily checks that b ν (˜ α ) = 0 , so ˜ α is in b B E ( F ) . One can check, e.g. usingLemma 3.16, that ˜ α is independent of the particular choice of ˜ u .Let z be the root of p given by z = − . . . . + i . . . . , and let σ denote thecorresponding embedding. Then σ ( w ) = exp( − πi/ . Let b σ : E → C be a coveringof σ . Letting log denote the principal branch of logarithm, we may assume that b σ ( ˜ w ) = − πi/ , b σ (˜ u ) = log( σ ( u )) = − . . . . − i . . . . . Using Lemma 3.18, we see that b σ ∗ takes χ (˜ u ) to − χ (log( σ ( u ))) . We now have σ ∗ ( α ) = ( − . . . . − i . . . . , − . . . . − i . . . . )+ 2(0 . − i . . . . , . . . . + i . . . . )+ 3 χ ( − . . . . − i . . . . ) . Using (2.4) or (2.6), (and Remark 3.15), we obtain R ( σ ∗ ( α )) = − . . . . − i . . . . ∈ C / π Z . Remark . Examples like the above can be produced in abundance using com-puter software like PARI/GP.5.
The other version of the extended Bloch group
As mentioned in the introduction there are two versions, b B ( C ) SL and b B ( C ) PSL , ofthe extended Bloch group. They are isomorphic to H (SL(2 , C )) and H (PSL(2 , C )) ,respectively. In this section we define the algebraic version of b B ( C ) PSL , and dis-cuss its relationship with hyperbolic geometry. We stress that this version is onlydefined when the extension E of F is –primitive. Let F be a field and let E be a –primitive extension of F ∗ by Z . By Corol-lary 3.26, ∈ Z ⊂ E is uniquely two-divisible, so ∈ E is well defined. Considerthe set of odd flattenings (5.1) F E = (cid:8) ( e, f ) ∈ E × E (cid:12)(cid:12) ± π ( e ) ± π ( f ) = 1 ∈ F (cid:9) , Since knowledge of z and − z up to a sign determines z , we have a map π : F E → F \ { , } . Define FT E as in Definition 3.2, and define b P E ( F ) PSL to be the abeliangroup generated by F E subject to the relations X i =0 ( − i ( e i , f i ) = 0 for (cid:0) ( e , f ) , . . . ( e , f ) (cid:1) ∈ FT E ( e + 12 , f + 12 ) + ( e, f ) = ( e + 12 , f ) + ( e, f + 12 ) . The second relation is the analog of the transfer relation ; see Goette–Zickert [5] orNeumann [12, Proposition 7.2]. The extended Bloch group b B E ( F ) PSL is defined asin Definition 4.5. Note that π : F E → F \ { , } induces maps from the extendedgroups b P E ( F ) PSL and b B E ( F ) PSL to the classical ones.For a ∈ E \ Z let χ ( a ) = ( e, f + 1 / − ( e, f ) , where ( e, f ) is any flattening of π ( a ) . The analog of Lemma 3.6 holds, proving independence of f , and indepen-dence of e follows from the transfer relation. As in Corollary 3.10, χ extends toa homomorphism χ : E → b P E ( F ) PSL , and a computation as in (3.10) shows that χ (1) = 2 χ ( ) = 0 . Lemma 5.1.
There is a commutative diagram of exact sequences. (5.2) E/ Z χ / / (cid:15) (cid:15) b P E ( F ) p (cid:15) (cid:15) / / P ( F ) / / E/ Z χ / / b P E ( F ) PSL / / P ( F ) / / , Proof.
Exactness of the bottom sequence is proved as in Theorem 3.13. For com-mutativity of the left diagram, note that in b P E ( F ) PSL , we have(5.3) ( e, f + 1) − ( e, f ) = ( e, f + 1) − ( e, f + 12 )+ ( e, f + 12 ) − ( e, f ) = 2 χ ( e ) = χ (2 e ) . This proves the result. (cid:3)
It follows from Proposition 3.29 that we have an exact sequence(5.4) µ F χ / / b B E ( F ) PSL / / B ( F ) / / If F is a free field, we 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 . Inparticular, χ is injective, and it follows from Lemma 5.1 that there is an exactsequence(5.5) / / Z / Z / / b B ( F ) p / / b B ( F ) PSL / / Z / Z / / . HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 17 Remark . One can concretely determine if an element α ∈ b B E ( F ) PSL lifts: Pickany 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 from Lemma 5.1 that α lifts if and only if x is a squarein F ∗ .5.1. Parabolic representations.
The result below generalizes the main result inZickert [21].
Theorem 5.3.
Let M be a tame manifold, and let F be a free field. A parabolicrepresentation ρ : π ( M ) → PSL(2 , F ) defines a fundamental class [ ρ ] ∈ b B ( F ) PSL .If σ : F → C is an embedding, σ ∗ ([ ρ ]) = [ σ ◦ ρ ] ∈ b B ( C ) PSL .Proof.
In Zickert [21] we defined a homomorphism H (PSL(2 , C ) , P ) → b B ( C ) PSL ,where P = { ( ∗ ) } . This map can be defined over any free field by replacing thelogarithms in Zickert [21, (3.6)] by a lift of E → F ∗ . By Zickert [21, Theorem 5.13] adecorated parabolic representation ρ : π ( M ) → PSL(2 , F ) of M has a fundamentalclass H (PSL(2 , F ) , P ) , and as in Zickert [21, Theorem 6.10], the image of thefundamental class in b B ( F ) PSL is independent of the decoration. This proves the firststatement. The second statement is an immediate consequence of the definition. (cid:3)
Remark . We stress that b B ( F ) PSL is not isomorphic to H (PSL(2 , F )) in general.6. Ideal cochains and flattenings of –cycles Fix a field F and a primitive extension E of F ∗ by Z . By a simplex we will alwaysmean a standard simplex together with a fixed vertex ordering. Unless otherwisespecified, a simplex means a –simplex. Definition 6.1. An (algebraic) flattening of a simplex ∆ is an association of analgebraic flattening ( e, f ) ∈ b F E to ∆ . If ( e, f ) is a flattening of z ∈ F \ { , } , werefer to z as the cross-ratio of the flattened simplex. Remark . Definition 6.1 is a generalization of even flattenings , i.e. flattenings [ z ; p, q ] , with p and q even. Neumann [12] also allows odd values of p and q .We will associate elements in E to edges of a flattened simplex as indicated inFigure 1. We will refer to these elements as log-parameters . Definition 6.3. An (ordered, oriented) –cycle is a space K obtained from acollection of simplices by gluing together pairs of faces using simplicial attachingmaps preserving the vertex orderings. If all faces have been glued, we say that K is closed . We assume that the manifold K with boundary (and corners) obtainedby removing disjoint regular neighborhoods of the –cells is oriented. If ∆ i is asimplex in K , we let ε i be a sign encoding whether or not the orientation of ∆ i coming from the vertex ordering agrees with the orientation of K . Remark . Neumann [12] only considers closed –cycles. With our definition, asingle simplex is a –cycle.The definition below is the analog of Neumann [12, Definition 4.4], which thereader may consult for further details. Definition 6.5.
Let K be a closed –cycle. A flattening of K is a choice of flat-tening of each simplex of K such that the total log-parameter (summed according to the sign conventions of Neumann [12, Definition 4.3]) around each edge is zero.If the total log-parameter along any normal curve in the star of each zero-cell iszero, it is called a strong flattening . Remark . We do not need any conditions on the parity. This is because theparity condition is automatically satisfied for even flattenings. The proof of thisfact is identical to the proof of Neumann [12, Proposition 5.3].If K is a –cycle, and G is an abelian group, we let C ( K ; G ) denote the setof cellular –cochains in K with values in G . A cochain c ∈ C ( K ; G ) naturallyassociates to each edge of each simplex of K an element in G . Edges that areidentified in K acquire the same labeling. If ∆ is a simplex of K , we let c ij (∆) denote the labeling of the edge joining the vertices i and j in ∆ , see Figure 2. e e − f − f − e + f − e + f Figure 1.
Associat-ing log-parameters toedges of a flattenedsimplex. c c c c c c Figure 2.
Edgelabelings arisingfrom a cochain.
Definition 6.7.
Let K be a –cycle. A cochain c ∈ C ( K ; F ∗ ) is called an idealcochain if for each simplex ∆ i in K , there is an element z i ∈ F ∗ \ { , } , such thatthe associated labeling of edges satisfies(6.1) c i c i c i c i = z i , c i c i c i c i = 1 − z i . An ideal cochain thus associates cross-ratios to each simplex.
Remark . Not all –cycles admit ideal cochains. The fact that –cycles admittingideal cochains exist follows from Remark 6.16 below.We wish to prove that each lift ˜ c ∈ C ( K ; E ) of an ideal cochain c determinesan 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 isindependent of the choice of lift. In other words, an ideal cochain on a closed –cycle determines an element in b B E ( F ) .Let I n be the free abelian group on cochains ˜ c ∈ C (∆ n ; E ) on an n –simplex ∆ n , whose restriction to each –dimensional face is the lift of an ideal cochain. Theusual boundary map induces boundary maps ∂ : I n → I n − , making I ∗ into a chaincomplex. A lift ˜ c of an ideal cochain c on K determines an element in I given by P ε i ˜ c i . We may thus regard lifts of ideal cochains as elements in I . Note that if K is closed, ˜ c is a cycle, i.e. ∂ ˜ c = 0 ∈ I . HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 19 Consider the maps b σ : I → Z [ b F E ] and µ : I → ∧ ( E ) defined on generators by b σ (˜ c ) = (˜ c + ˜ c − ˜ c − ˜ c , ˜ c + ˜ c − ˜ c − ˜ c ) , (6.2) µ (˜ c ) = − ˜ c ∧ ˜ c + ˜ c ∧ ˜ c − ˜ c ∧ ˜ c + ˜ c ∧ ˜ c . (6.3) Lemma 6.9.
There is a commutative diagram (6.4) I ∂ / / b σ (cid:15) (cid:15) I ∂ / / b σ (cid:15) (cid:15) I µ (cid:15) (cid:15) Z [ c FT E ] b ρ / / Z [ b F E ] b ν / / ∧ ( E ) . Proof.
Let ˜ c ∈ I be a generator, and suppose b σ ◦ ∂ (˜ c ) = P ( − i ( e i , f i ) . We mustprove that the flattenings ( e i , f i ) satisfy the five equations of Definition 3.2. Wecheck the first equation, and leave the verification of the four others to the reader.By (6.2) we have e = ˜ c + ˜ c − ˜ c − ˜ c e = ˜ c + ˜ c − ˜ c − ˜ c e = ˜ c + ˜ c − ˜ c − ˜ c , and it follows that e = e − e .Letting ˜ c ∈ I be a generator, we see that b ν ◦ b σ (˜ c ) and µ ◦ ∂ (˜ c ) are both a sum of terms of the form ˜ c ij ∧ ˜ c kl with { i, j } 6 = { k, l } and two-torsion terms summingto ˜ c ∧ ˜ c + ˜ c ∧ ˜ c (the two other terms cancel out). The two-torsion terms thusmatch up, and one easily checks that the other terms match up as well, provingcommutativity of the right square. (cid:3) Corollary 6.10. If K is closed and ˜ c is a lift of an ideal cochain on K , b σ (˜ c ) is in b B E ( F ) . (cid:3) Proposition 6.11.
Let K be a closed –cycle. The set of flattenings coming froma lift of an ideal cochain is a strong flattening of K .Proof. The proof is identical to the proof of Zickert [21, Theorem 6.5], so we omitsome details. Consider a curve α in the star of a –cell as shown in Figure 3. When α passes through a simplex, it picks up a log-parameter, which is a signed sumof four terms. The signs are shown in the figure. If α is a closed curve, it is notdifficult to see that all terms must cancel out. (cid:3) α + − + − + − + − + −− + − + − + − + − + Figure 3.
A normal curve in the star of a –cell. Each edge and eachvertex corresponds to a –cell in K . Lemma 6.12.
Let K be a –cycle and let c be an ideal cochain on K . Let e bean interior –cell of K , and let α e ∈ C ( K ; Z ) be the cochain taking e to and allother –cells to . For every lift ˜ c of c , we have (6.5) b σ (˜ c + α e ) = b σ (˜ c ) ∈ b P E ( F ) . Proof.
The map b σ associates flattenings to the simplices of K , and using (6.2) onechecks that the flattenings coming from ˜ c and ˜ c + α differ by Neumann’s cyclerelation [12, Section 6] about e (or rather the obvious generalization of this relationto algebraic flattenings). Neumann’s proof that the cycle relation is a consequenceof the lifted five term relation carries over to the algebraic setup word by word. (cid:3) Corollary 6.13. If K is a closed –cycle, b σ (˜ c ) = b σ (˜ c + α ) ∈ b B E ( F ) for any α ∈ C ( K ; Z ) . Hence, an ideal cochain c on K determines an element in b σ ( c ) ∈ b B E ( F ) . (cid:3) The action of Z ( K ; Z / Z ) on ideal cochains. Let K be a closed –cycle,and suppose that the characteristic of F is not . The group Z ( K ; Z / Z ) of cellular –cocycles on K acts on the set of ideal cochains by multiplication. Note that theaction does not change the cross-ratios. A cochain α ∈ Z ( K ; Z / Z ) determines amap Bα : K → B ( Z / Z ) = R P ∞ , and we wish to prove that the elements in b B E ( F ) associated to ideal cochains c and αc differ by a two-torsion element which is zero if and only if Bα ∗ ([ K ]) is zero in H ( R P ∞ ) = Z / Z .The homology of a group G is the homology of the complex B ∗ ( G ) where B n ( G ) is generated by symbols h g | . . . | g n i with g i ∈ G . Such tuples are in one-one corre-spondence with G –cocycles on ∆ n ; a cocycle is uniquely given by its values on theedges between vertices i and i + 1 . Under this correspondence, the boundary mapsare induced by the standard ones. Given a cocycle α ∈ Z ( K ; G ) the restriction of α to ∆ i determines a tuple h g i | g i | g i i and by Zickert [21, Proposition 5.7] we have Bα ∗ ([ K ]) = X ε i h g i | g i | g i i . Let α ∈ Z (∆ ; Z / Z ) and let c ∈ C (∆ ; F ∗ ) be an ideal cochain. Let c ′ = αc ,and pick a lift ˜ c of c . Then ˜ c endows ∆ with a flattening b σ (˜ c ) given by ( e, f ) = (˜ c + ˜ c − ˜ c − ˜ c , ˜ c + ˜ c − ˜ c − ˜ c ) . Let w ∈ E be the sum of the log-parameters at the edges where α ij = − . Oneeasily checks that w is always (uniquely) two-divisible, e.g. if α = h− | | − i , w = 2 e + 2( − e + f ) = 2 f . Let ˜ c ′ be the lift of c ′ defined by(6.6) ˜ c ′ ij = ˜ c ij + ( if α ij = − otherwise. Lemma 6.14.
Let δ ∈ Z ⊂ E be if α = h− | − | − i and otherwise. We have (6.7) b σ (˜ c ′ ) − b σ (˜ c ) = χ ( 12 w ) + χ ( δ ) ∈ b P E ( F ) . Proof.
This is done case by case using Lemma 3.16. If e.g. α = h | − | i , w = − e and we have b σ (˜ c ′ ) − b σ (˜ c ) = ( e, f − − ( e, f ) = χ ( − e ) = χ ( 12 w ) + χ ( δ ) . HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 21 The other seven cases are similar and left to the reader. (cid:3)
Theorem 6.15.
Let α ∈ Z ( K ; Z / Z ) . For any ideal cochain c , b σ ( αc ) − b σ ( c ) istwo-torsion in b B E ( F ) , which is trivial if and only if Bα ∗ ([ K ]) ∈ H ( R P ∞ ) = Z / Z is trivial.Proof. Let ˜ c be a lift of c and let ˜ c ′ be the lift of αc defined by (6.6). For eachsimplex ∆ i of K , we have elements w i and δ i as above. Since the flattening of K defined by ˜ c is strong, P ε i w i = 0 ∈ E , and since E has no two-torsion, Lemma 6.14implies that b σ (˜ c ′ ) − b σ (˜ c ) = X ε i χ ( δ i ) . One easily checks that the isomorphism H ( B Z / Z ) ∼ = Z / Z is induced by the map B ( Z / Z ) → Z / Z taking h− | − | − i to and all other generators to . Thisproves the result. (cid:3) Ideal cochains and homology of linear groups.
In this section we definea canonical map b λ : H (SL(2 , F )) → b B E ( F ) . This is purely algebraic and followsDupont–Zickert [2]. We assume that F is infinite.Let C ∗ ( F ) be the chain complex generated in dimension n by ( n + 1) –tuplesof vectors in F \ { } in general position, together with the usual boundary map.Letting p : F \ { } → P ( F ) be the canonical projection, a simple computation(as in Dupont–Zickert [2, Section 3.1]) shows that(6.8) z = det( v , v ) det( v , v )det( v , v ) det( v , v ) , − z = det( v , v ) det( v , v )det( v , v ) det( v , v ) , where z is the cross-ratio of the tuple ( p ( v ) , p ( v ) , p ( v ) , p ( v )) . It follows thatthere is a chain map(6.9) Γ : C ∗ ( F ) → I ∗ , Γ( v , . . . , v n ) ij = log det( v i , v j ) . Here log denotes a fixed section of π : E → F ∗ . We refer to it as a logarithm . Let b λ = b σ ◦ Γ . Then b λ is SL(2 , F ) –invariant, and by Lemma 6.9 it induces a map H ( C ∗ ( F ) SL(2 ,F ) ) → b B E ( F ) .Recall that the homology of a group G is the homology of ( F ∗ ) G = F ∗ ⊗ Z [ G ] Z , where F ∗ is any free resolution of Z by G -modules. One such resolution isthe 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 assume (see e.g. Dupont–Zickert [2, Section 3.2]) that all tuples ( g , . . . g n ) are in general position in thesense that ( g v, . . . , g n v ) ∈ C n ( F ) for some fixed v = 0 ∈ F (the particular choiceis inessential). It follows that λ canonically extends to a map(6.10) b λ : H (SL(2 , F )) → b B E ( F ) , ( g , g , g , g ) b σ ◦ Γ( g v, g v, g v, g v ) . Remark . Note that for each α ∈ H (SL(2 , F )) , b λ ( α ) is induced by an idealcochain on a –cycle. The fact that b λ is independent of the choice of logarithmfollows from Corollary 6.13. Remark . The map π ◦ b λ : C ( F ) → P ( F ) is GL(2 , F ) –invariant, and it followsthat there is an induced map H (GL(2 , F )) → B ( F ) . This map factors through H ( C ∗ ( P ( F )) GL(2 ,F ) ) , and thus agrees with that of Suslin [18]. The extended Bloch group and algebraic K –theory In this section, we prove our main result Theorem 1.1. We do this in three steps.The first and most difficult step is to extend the map b λ from Section 6.2 to a map b λ : H (GL(3 , F )) → b B E ( F ) . Once this has been done, we obtain a map(7.1) K ( F ) H / / H (GL( F )) ∼ = / / H (GL(3 , F )) b λ / / b B E ( F ) , where H is the Hurewicz map and the middle isomorphism is Suslin’s stabilityresult Theorem 2.7. In the first step, we only require that F be infinite and that E be primitive. The second step is to prove that this map takes the image of K M ( F ) to . To do this, we need to assume that F is a number field, or more generally,a free field. The third and final step is to show that b λ induces a map between thediagrams (1.1) and (4.4). The result then follows from the five-lemma.7.1. Step one: Extension of b λ to H (GL(3 , F )) . We start by constructing b λ on H (SL(2 , F )) . We assume that F is infinite and that E is primitive. Muchof the construction draws inspiration from Igusa [7], Fock–Goncharov [3], and pri-vate discussions with Dylan Thurston and Stavros Garoufalidis. In Garoufalidis–Thurston–Zickert [4] we generalize to SL( n, F ) and discuss some of the underlyinggeometric ideas motivating the construction.In Section 6.2 we associated an ideal cochain to a quadruple of vectors in F .We now generalize this to tuples of vectors in F . Let v = ( v , . . . , v n ) be a tupleof vectors in F in general position, and let w ∈ F be in general position withrespect to the v i ’s. For each i ∈ { , . . . , n + 1 } we have a cochain ˜ c iw ∈ C (∆ n ; E ) given by(7.2) ˜ c iw ( v ) jk = 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, as in Section 6.2, log is a fixed section of π : E → F ∗ . Lemma 7.1.
Each ˜ c iw ( v ) is in I n , and for each restriction to a –dimensional faceof ∆ n , the cross-ratio is independent of i .Proof. We may assume that v = ( v , v , v , v ) . Let c iw ( v ) be the projection of ˜ c iw ( v ) to C (∆ ; F ∗ ) , and let p : F \ { w } → P ( F / h w i ) denote the map induced byprojection. By applying a linear transformation if necessary, we may assume that w = (1 , , , and identify F / h w i with F . It now follows from (6.8) that the cross-ratio z of the tuple ( p ( v ) , p ( v ) , p ( v ) , p ( v )) of elements in P ( F / h w i ) ≈ P ( F ) satisfies z = det( w, v , v ) det( w, v , v )det( w, v , v ) det( w, v , v ) , − z = det( w, v , v ) det( w, v , v )det( w, v , v ) det( w, v , v ) . It follows that c w ( v ) is an ideal cochain with cross-ratio z . Since the expressions c iw ( v ) c iw ( v ) c iw ( v ) c iw ( v ) , c iw ( v ) c iw ( v ) c iw ( v ) c iw ( v ) are independent of i , it follows that the same is true for all the c iw ( v ) ’s. (cid:3) HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 23 If v = ( v , v , v , v ) , we let ( v , v , v , v ) iw denote the flattening b σ (˜ c iw ( v )) . Thelog-parameters are given by(7.3) e = ˜ c iw ( v ) + ˜ c iw ( v ) − ˜ c iw ( v ) − ˜ c iw ( v ) f = ˜ c iw ( v ) + ˜ c iw ( v ) − ˜ c iw ( v ) − ˜ c iw ( v ) . Lemma 7.2.
The following formulas hold in b P E ( F ) (note the superscripts). ( v , v , v , v ) w − ( v , v , v , v ) w +( v , v , v , v ) w − ( v , v , v , v ) w +( v , v , v , v ) w = 0( v , v , v , v ) w − ( v , v , v , v ) w +( v , v , v , v ) w − ( v , v , v , v ) w +( v , v , v , v ) w = 0( v , v , v , v ) w − ( v , v , v , v ) w +( v , v , v , v ) w − ( v , v , v , v ) w +( v , v , v , v ) w = 0( v , v , v , v ) w − ( v , v , v , v ) w +( v , v , v , v ) w − ( v , v , v , v ) w +( v , v , v , v ) w = 0( v , v , v , v ) w − ( v , v , v , v ) w +( v , v , v , v ) w − ( v , v , v , v ) w +( v , v , v , v ) w = 0 . We will refer to the left hand sides of the five equations above as boundaries , anddenote them by ∂ i ( v , v , v , v , v ) w , i ∈ { , . . . , } .Proof. We will show that each of the boundaries corresponds to a lifted five termrelation. To do this we must prove that the flattenings satisfy the five equationsof Definition 3.2. Suppose we wish to verify that f = f − f for the boundary ∂ . The relevant terms involved in this are ( v , v , v , v ) w , ( v , v , v , v ) w and( v , v , v , v ) w . If we denote their flattenings by ( e , f ) , ( e , f ) and ( e , f ) , itfollows from (7.3) and (7.2) that f = log( v , v , w ) + log( w, v , v ) − log( v , w, v ) − log( v , w, v ) f = log( v , v , w ) + log( w, v , v ) − log( v , w, v ) − log( v , w, v ) f = log( v , w, v ) + log( w, v , v ) − log( v , w, v ) − log( w, v , v ) , where log( u, v, w ) denotes log(det( u, v, w )) . Hence, f = f − f as desired. Theverification of the other formulas are similar and are thus left to the reader. (cid:3) Lemma 7.3.
We have (7.4) ( v , v , v , v ) v − ( v , v , v , v ) v + ( v , v , v , v ) v − ( v , v , v , v ) v + ( v , v , v , v ) v = 0 ∈ b P E ( F ) . We will denote the left hand side by ∂ ( v , v , v , v , v ) .Proof. As in the proof of Lemma 7.2, we can verify that the flattenings satisfy thefive equations in Definition 3.2. We leave this to the reader. (cid:3)
If F is an ordered basis of F k , we let F i denote the i th basis vector. A set S of ordered bases is in general position if any set of k basis vectors from S islinearly independent. Let CF ∗ be the chain complex generated in dimension n by tuples ( F , . . . , F n ) of ordered bases of F in general position, together withthe usual boundary map. Left multiplication makes CF ∗ into a chain complex offree GL(3 , F ) –modules. Since F is assumed to be infinite, it is easy to see that CF ∗ is acyclic. Hence, the complexes ( CF ∗ ) SL(3 ,F ) and ( CF ∗ ) GL(3 ,F ) compute thehomology groups H ∗ (SL(3 , F )) and H ∗ (GL(3 , F )) , respectively. Consider the
SL(3 , F ) –invariant map b λ : CF → b P E ( F ) given by sending a gen-erator ( F , F , F , F to(7.5) ( F , F , F , F ) F + ( F , F , F , F ) F + ( F , F , F , F ) F + ( F , F , F , F ) F . We will often abbreviate the notation by omitting the F, and writing a subscriptF i as i , e.g. we abbreviate ( F , F , F , F ) F to (0 , , , ) .To help the reader visualize the arguments that follow, we give a geometric wayof viewing the map b λ : A generator of CF can be thought of as a simplex ∆ togetherwith an association of an ordered basis F i to each vertex. We may think of each ofthe four terms in (7.5) as a standard simplex endowed with an ideal cochain. Wemark ∆ with two points on each edge and a point on each face as shown in Figure 4.We will refer to these points as edge points and face points respectively. Each pointis given uniquely by a tuple β = ( x , x , x , x ) with x + x + x + x = 3 , wherethe coordinate x i measures the “distance” to the face opposite vertex i . For eachsuch β , let β i be the ordered set { F i , . . . , F i x i } and let S β = β ∪ β ∪ β ∪ β .Note that S β always has exactly elements. Hence, det( S β ) is well defined andgives a labeling of each marked point of ∆ . As an example, the edge point, closestto vertex , between vertices and is labeled by det( F , F , F ) .We can think of ∆ as a union of four simplices ∆ i , where ∆ i is the simplexspanned by the i th vertex of ∆ and the marked points with x i = 0 . We thinkof the ∆ i ’s as being disjoint. The labelings of the marked points in ∆ gives riseto cochains on ∆ i , and using (7.2), one can check that these are exactly the idealcochains of the terms in (7.5). F0 F1 F2 F3 c c c c c c c c c c c c c c c c c c c c c c c c Figure 4.
The ideal cochains on the simplices ∆ i arising from a labelingof marked points in ∆ . The dashed lines mark the bottom of ∆ . Remark . An ordered basis determines an affine flag, and one easily checks that b λ only depends on the underlying affine flags.If τ ∈ ( CF ) SL(3 ,F ) is a cycle, we can represent τ by a –cycle K together witha labeling of the marked points in each of the simplices of K . Identified pointsacquire the same labeling. From the geometric description of the map b λ , it followsthat we can represent b λ ( τ ) by a –cycle C with boundary together with an idealcochain on C . Note that C is homeomorphic to the disjoint union of the cones onthe links of the –cells of K . HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 25 Lemma 7.5.
The restriction of b λ to cycles in ( CF ) SL(3 ,F ) is independent of thechoice of logarithm. In fact, we may choose different logarithms for each of themarked points as long as we use the same logarithm for identified points.Proof. For edge points this follows from Lemma 6.12. A face point occurs in exactly of the simplices ∆ i . Consider the face point opposite vertex , and supposethe flattenings of ∆ , ∆ and ∆ are ( e, f ) , ( e ′ , f ′ ) and ( e ′′ , f ′′ ) . If we add tothe logarithm of the face point, it follows from (7.3) that the flattenings become ( e, f + 1) , ( e ′ − , f ′ − and ( e ′′ + 1 , f ′′ ) . By Lemma 3.16, this changes the elementin 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 , , ) − (0 , , ) − (1 , , ) − (cid:0) (0 , , ) + (1 , , ) − (0 , , ) − (1 , , ) (cid:1) + (0 , , ) + (2 , , ) − (0 , , ) − (1 , , ) − (cid:0) (0 , , ) + (2 , , ) − (0 , , ) − (1 . , ) (cid:1) = (cid:0) (1 , , ) − (1 , , ) (cid:1) + (cid:0) (1 , , ) − (1 , , ) (cid:1) + (cid:0) (2 , , ) − (2 , , ) (cid:1) , where ( i j , k l , m n ) denotes log det( F i j , F k l , F m n ) . Note that each term is a logarithmof one of the six edge points on the face opposite vertex . By a similar calculationone can check that this holds in general, i.e. the change in the b P E ( F ) element whenadding to the logarithm of a face point, is a signed sum of logarithms of the edgepoints on that face (plus χ (1) ). The signs are shown in Figure 5.
13 2 −− − + ++ 30 2 −− − + ++ 10 3 −− − + ++ 12 0 −− − + ++
Figure 5.
Change in the b P E ( F ) element when adding to the loga-rithm of a face point. There is a contribution for each edge point on thegiven face. In a cycle, each face point lies in exactly two simplices, and since the face pairingspreserve orderings, it follows from Figure 5 that the changes in the element in b P E ( F ) , resulting from adding to the logarithm, appear with opposite signs. (cid:3) Lemma 7.6. b λ takes boundaries in CF to ∈ b P E ( F ) .Proof. Using (7.5), we see that b λ ( ∂ ( F , . . . , F ∈ b P E ( F ) equals + (1 , , , ) − (0 , , , ) + (0 , , , ) − (0 , , , ) + (0 , , , ) + (1 , , , ) − (0 , , , ) + (0 , , , ) − (0 , , , ) + (0 , , , ) + (1 , , , ) − (0 , , , ) + (0 , , , ) − (0 , , , ) + (0 , , , ) + (1 , , , ) − (0 , , , ) + (0 , , , ) − (0 , , , ) + (0 , , , ) . Using Lemma 7.2, this simplifies to − (1 , , , ) + (0 , , , ) − (0 , , , ) + (0 , , , ) − (0 , , , ) , which by Lemma 7.3 is ∈ b P E ( F ) . (cid:3) We thus obtain an induced map b λ : H (SL(3 , F )) → b P E ( F ) . Lemma 7.7.
The image of b λ is in b B E ( F ) .Proof. Consider the sequence of maps J n : CF n → I n given by(7.7) ( F , . . . , F n ) n X i =0 ( F , . . . , F i , . . . , F n ) i F i . Note that J is not a chain map. By definition, b λ : CF → b P E ( F ) is equal to b σ ◦ J ,where b σ : I → b P E ( F ) is the map given by (6.2). Consider the diagram(7.8) CF J / / ∂ (cid:15) (cid:15) I b σ / / ∂ (cid:15) (cid:15) b P E ( F ) b ν (cid:15) (cid:15) CF J / / I µ / / ∧ ( E ) . By Lemma 6.9 the right square is commutative. Using the usual notational abbre-viations, i.e. omitting the F, and shortening subscripts to i , a direct computationshows that(7.9) δ := ( ∂J − J ∂ )( F , F , F , F
3) =(1 , , ) − (0 , , ) + (0 , , ) − (0 , , ) . One easily checks that µ takes δ to ∈ ∧ ( E ) , and the result follows. (cid:3) Lemma 7.8.
The restriction of b λ to H (SL(2 , F )) agrees with the map from theprevious section.Proof. We consider F as a subspace of F using the inclusion ( x, y ) (0 , x, y ) .Let p : F → F be the natural projection, and let D ∗ be the subcomplex of CF ∗ consisting of tuples ( F , . . . , F n ) such that ( p F , . . . , p F n ) ∈ C n ( F ) . Note that D ∗ is an acyclic SL(2 , F ) –complex, where SL(2 , F ) is regarded as a subgroup of SL(3 , F ) in the natural way. Consider the GL(2 , F ) –equivariant map(7.10) Ψ : D ∗ → C ∗ ( F ) , ( F , . . . , F n ) ( p F , . . . , p F n ) . Let b τ denote the map C ( F ) → b P E ( F ) from Section 6.2. We wish to prove that b τ ◦ Ψ and b λ differ by a coboundary. Note that (0 , , , ) w = b τ ◦ Ψ( F , F , F , F .By definition, b λ takes ( F , F , F , F ∈ D to (0 , , , ) + (0 , , , ) + (0 , , , ) + (0 , , , ) . We may subtract boundaries without effecting the image in b P E ( F ) , and after sub-tracting ∂ ( w, , , , ) + ∂ ( w, , , , ) + ∂ ( w, , , , ) + ∂ ( w, , , , ) , HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 27 the remaining terms become ( w, , , ) − ( w, , , ) + ( w, , , ) − ( w, , , ) ( w, , , ) − ( w, , , ) + ( w, , , ) − ( w, , , ) ( w, , , ) − ( w, , , ) + ( w, , , ) − ( w, , , ) ( w, , , ) − ( w, , , ) + ( w, , , ) − ( w, , , ) . By Lemma 7.3 the diagonal terms sum to (0 , , , ) w = b τ ◦ Ψ( F , F , F , F .If we define φ : D → b P E ( F ) by φ ( F , F , F
2) = ( w, , , ) + ( w, , , ) + ( w, , , ) , the remaing terms are easily seen to equal φ ◦ ∂ ( F , F , F , F . Hence, b τ ◦ Ψ and b λ differ by a coboundary as desired. (cid:3) Remark . As in Remark 6.17, the map π ◦ b λ : CF → P ( F ) is GL(3 , F ) invariant,and induces a map H (GL(3 , F )) → B ( F ) , which factors through the homology ofthe complex P CF ∗ of projective bases of F . The proof of Lemma 7.8 shows thatthe map H (GL(3 , F )) → B ( F ) agrees with the map in Remark 6.17, and that themap H (GL(2 , F )) → B ( F ) lifts to b B E ( F ) via H (SL(3 , F )) and the stabilizationmap GL(2 , F ) → SL(3 , F ) .7.1.1. Extension to H (GL( n, F )) . In Garoufalidis–Thurston–Zickert [4] we con-struct maps H (SL( n, F )) → b B E ( F ) commuting with the stabilization maps. Thesemaps are induced by an SL( n, F ) –invariant map b λ : CF n → b P E ( F ) , where CF n ∗ isthe complex of ordered bases of F n (or affine flags, c.f. Remark 7.4). Remark 7.9generalizes, i.e. the maps(7.11) H (GL( n, F )) → H (SL( n + 1 , F )) → b B E ( F ) commute with stabilization. Hence, by (7.1), we obtain a map K ( F ) → b B E ( F ) .If F is free (and infinite), b λ commutes with the maps Ψ xy from Proposition 4.3, so b λ induces a map K ( F ) → b B ( F ) . The map b λ commutes with Galois actions, andrespects the maps induced by embeddings in C . This implies that the regulators(1.2) and (4.5) agree.7.2. Step two: K M ( F ) maps to zero. From now on, we assume that F is a freefield admitting an embedding in C . This is used in Proposition 7.11 but not inLemma 7.10. Lemma 7.10.
The composition H (GL(3 , F )) b λ / / b B E ( F ) π / / B ( F ) agrees with the map constructed by Suslin [18, Section 3] .Proof. By a result of Suslin [17], H (GL(3 , F )) is generated by H (GL(2 , F )) and H ( T ) . By Remark 6.17 the two maps on H (GL(2 , F )) . By Remark 7.9 the map π ◦ b λ factors through the complex P CF ∗ of projective bases. Since T acts triviallyon projective bases, P CF → Z has a T –equivariant section, and it follows that π ◦ b λ is on H ( T ) . By Suslin [18, Proposition 3.1], this also holds for Suslin’smap. Hence, the two maps agree. (cid:3) Proposition 7.11.
The composition K M ( F ) → K ( F ) → b B ( F ) is .Proof. Let σ : F → C be an embedding and let σ ∗ : b B ( F ) → b B ( C ) be the inducedmap. By Lemma 7.10, the image of K M ( F ) in b B ( F ) is in f µ F . By Proposition 4.11, σ ∗ maps f µ F injectively to µ C , and since the regulator R is injective on µ C , it isenough to prove that the composition K M ( F ) / / f µ F / / µ C R / / C / π Z is zero. Since this factors through K M ( C ) , the result follows from Theorem 2.8 andTheorem 2.9. (cid:3) Step three: A five lemma argument.Lemma 7.12.
The exact sequences (1.1) and (4.4) fit together in a diagram / / f µ F / / K ind3 ( F ) / / b λ (cid:15) (cid:15) B ( F ) / / / / f µ F / / b B ( F ) / / B ( F ) / / . Proof.
Commutativity of the right square follows from Lemma 7.10. To provecommutativity of the left square, we proceed as in the proof of Lemma 7.11. Since σ ∗ is injective on f µ F , it is enough to prove the corresponding result with F replacedby C . The result now follows from Theorem 2.8 and Theorem 2.9. (cid:3) The theorem below summarizes our results.
Theorem 7.13.
Let F be a free field admitting an embedding in C . There is anatural isomorphism b λ : K ind3 ( F ) ∼ = b B ( F ) commuting with Galois actions. (cid:3) If F ⊂ E is a field extension, the natural map K ind3 ( F ) → K ind3 ( E ) is an inclu-sion. Furthermore, if F ⊂ E is Galois, we have K ind3 ( E ) Gal(
E,F ) = K ind3 ( F ) . Thisproperty is called Galois descent. We refer to Merkurjev–Suslin [10] for proofs. Corollary 7.14.
For any free subfield F of C , the map b B ( F ) → b B ( C ) induced byinclusion is injective. (cid:3) Corollary 7.15.
The extended Bloch group of a number field satisfies Galois de-scent. (cid:3) Torsion in the extended Bloch group
In this section we give a concrete description of the torsion in b B ( F ) . We startby reviewing some elementary properties of homology of cyclic groups. Proposition 8.1.
Let G be a cyclic group of order n generated by an element g ∈ G . The homology group H ( G ) is cyclic of order n and is generated by thecycle n X k =1 h g | g k | g i . We may thus identify G with H ( G ) . (cid:3) HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 29 We refer to Parry–Sah [15, Proposition 3.25] for an algebraic proof, and to Neu-mann [12] for a geometric proof using the lens space L ( n, .Let p be a prime number. As explained in the introduction, K ind ( F ) p has order p ν p for p odd and p ν p for p = 2 , where ν p = max { ν | ξ p ν + ξ − p ν ∈ F } . To constructthe torsion in b B ( F ) , it is thus enough to exhibit elements in b B ( F ) p of order p ν p for p odd and p ν p for p = 2 .Let n = p ν p , and let x be a primitive n th root of unity. Consider the matrices(8.1) g = (cid:18) x + x − −
11 0 (cid:19) , µ = (cid:18) x x − (cid:19) , X = (cid:18) x x (cid:19) . Note that g ∈ SL(2 , F ) and that g = XµX − . Hence, g generates a cyclic subgroupof SL(2 , F ) of order n . Let [ g ] ∈ H (SL(2 , F )) denote the homology class of thecycle P nk =1 h g | g k | g i . Lemma 8.2.
The element b λ ([ g ]) ∈ b B ( F ) has order n = p ν p .Proof. It follows from Proposition 8.1 that b λ ([ g ]) has order at most n . If we fix anembedding of F ( x ) in C , we can view 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 order atleast n . Hence, b λ ([ g ]) has order n . (cid:3) Corollary 8.3.
For p odd, b B ( F ) p is generated by b λ ([ g ]) . (cid:3) Explicit computations.
We now give an explicit expression for b λ ([ g ]) . As-sume for now that p is odd.For any h and h in SL(2 , F ) , there is a homogeneous representative of [ g ] ofthe form(8.2) n X i =1 ( h , gh , g k h , g k +1 h ) , see e.g. the example in Neumann [12, Section 12]. Using (6.10), we see that b λ takesa term ( h , gh , g k h , g k +1 h ) to a flattening ( e k , f k ) , with(8.3) e k = log(det( v , g k +1 v )) + log(det( v , g k − v )) − v , g k v )) f k = log(det( v , gv )) + log(det( v , gv )) − v , g k v )) , where v = h (cid:0) (cid:1) and v = h (cid:0) (cid:1) . It follows that b λ ([ g ]) = P ni =1 ( e k , f k ) ∈ b B ( F ) .Since the cycles (8.2) all represent [ g ] , we may choose v and v as we please(as long as the vectors v , gv , g k v , g k +1 v in F are in general position) withouteffecting the element in b B ( F ) . If we let v = (cid:0) − (cid:1) and v = (cid:0) (cid:1) , we have(8.4) det( v , g k v ) = det( v , Xµ k X − v )= ( x −
1) det( X − v , µ k X − v )= 1 x − (cid:0)(cid:0) x +1 − x − (cid:1) , µ k (cid:0) x − x − (cid:1)(cid:1) = det (cid:0)(cid:0) − (cid:1) , (cid:0) x k x − k (cid:1)(cid:0) (cid:1)(cid:1) = x k + x − k . Letting z k denote the corresponding cross-ratio of ( e k , f k ) , it follows from (8.3) that z k = ( x k +1 + x − k − )( x k − + x − k +1 )( x k + x − k ) . Since p is assumed to be odd, z k ∈ F \ { , } . This proves Theorem 1.3 for p odd.Suppose p = 2 . By computations similar to (8.4) using v = (cid:0) (cid:1) and v = (cid:0) − (cid:1) we obtain(8.5) det( v , g k v ) = x k − x − k +1 x − , det( v , gv ) = 1 , det( v , gv ) = 2+ x + x − . We wish to prove that b λ ([ g ]) ∈ b B ( F ) is –divisible.Let c k = det( v , g k v ) and let ˜ c k = log( c k ) . Also, let a = 2 + x + x − and let ˜ a = log( a ) . By (8.5), we see that c k = c n − k +1 and c k = − c k + n/ . By (8.3), ( e k , f k ) = (˜ c k +1 + ˜ c k − − c k , ˜ a − c k ) . We may choose different logarithms for each k without effecting the element b λ ([ g ]) = P ni =1 ( e k , f k ) . We will choose them such that ˜ c k − ˜ c n − k +1 and ˜ c k + n/ − ˜ c k areindependent of k and such that c k = 2˜ c k + n/ . With these particular choices, it iseasy to see that b λ ([ g ]) is –divisible. Indeed, b λ ([ g ]) = 2 Q , where(8.6) Q = n/ X i =1 ( e k , f k ) ∈ b P ( F ) . We now only need to prove that Q is in b B ( F ) . This follows from the computation b ν ( Q ) = X n/ k =1 (˜ c k +1 + ˜ c k − − c k ) ∧ (˜ a − c k )= X n/ k =1 c k ∧ (˜ c k +1 + ˜ c k − ) − X n/ k =1 c k ∧ ˜ a + X n/ k =1 (˜ c k +1 + ˜ c k − ) ∧ ˜ a = X n/ k =1 (2˜ c k ∧ ˜ c k +1 − c k − ∧ ˜ c k ) + X n/ k =1 (cid:0) (˜ c k +1 − ˜ c k ) − (˜ c k − ˜ c k − ) (cid:1) ∧ ˜ a = 2˜ c n/ ∧ ˜ c n/ − c ∧ ˜ c + (˜ c n/ − ˜ c ) ∧ ˜ a − (˜ c n/ − ˜ c ) ∧ ˜ a = 0 ∈ ∧ ( E ) . Since z k = c k +1 c k − c k = ( x k +1 − x − k )( x k − − x − k +2 )( x k − x − k +1 ) ∈ F, this proves Theorem 1.3 for p = 2 .We give some examples below. The computational details are left to the reader. Example 8.4.
For any number field F , which does not contain a rd root of unity,the element −
2] + [ ] ∈ B ( F ) has order . Example 8.5.
Let F = Q ( √ . Doing the above computations, we obtain that Q = [ √ −
1; 0 ,
0] + [ √ −
1; 0 , −
2] + [ −√ −
1; 0 ,
0] + [ −√ − − , − ∈ b B ( F ) . It follows that the element β = 2[ √ −
1] + 2[ −√ − ∈ B ( F ) has order andgenerates B ( F ) . Note that β is not –divisible. Applying the regulator (2.4), weget R ( Q ) = π / ∈ C / π , which has order
16 = 2 ν +1 as expected. HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 31 Remark . The order of the torsion in K ind ( F ) is always divible by . A partic-ular generator of the –torsion in b B ( F ) is given by the element ( e, f ) + ( f, e ) fromLemma 3.11. We omit the proof of this.9. Hyperbolic –manifolds Let M be a complete, oriented, hyperbolic –manifold with finite volume, andlet K and k denote the trace field and invariant trace field of M . By a result ofGoncharov [6, Theorem 1.1], M defines an element [ M ] in K ind3 ( Q ) ⊗ Q , which equalsthe Bloch invariant of M (see e.g. Neumann–Yang [14]) under the isomorphism K ind3 ( Q ) ⊗ Q ∼ = B ( Q ) . Recall that a spin structure on M is equivalent to a lift of the geometric rep-resentation to SL(2 , C ) , and that the set of spin structures is an affine space over H ( M ; Z / Z ) . We thank Walter Neumann for assistance with the proof of theresult below. Theorem 9.1.
Suppose M is closed. A spin structure ρ on M determines afundamental class [ M ρ ] in K ind3 ( K ) lifting the Bloch invariant. For each α ∈ H ( M ; Z / Z ) , the element [ M ρ ] − [ M αρ ] is two-torsion, which is trivial if and onlyif the induced map Bα ∗ : H ( M ) → H ( B ( Z / Z )) = Z / Z is trivial. In particular, M ρ ] is independent of ρ . Moreover, M ρ ] is in K ind3 ( k ) .Proof. By Reid–Maclachlan [8, Corollary 3.2.4], we may assume that ρ has imagein SL(2 , K ( λ )) , where λ is an algebraic element of degree at most over K . Endow M with the structure of a closed –cycle, and fix an SL(2 , K ( λ )) –cocycle α on M representing the fundamental class [ ρ ] of ρ in H (SL(2 , K ( λ ))) ; see e.g. Zickert [21,Section 5]. Let [ M ρ ] = b λ ([ ρ ]) . Then [ M ρ ] , is given by the ideal cochain c on M defined by α using (6.10) and (6.9). If λ has degree , [ M ρ ] is obviously in b B ( K ) .If λ has degree , the non-trivial element in Gal( K ( λ ) , K ) preserves traces of ρ ( K is the trace field), and therefore takes ρ to a representation which is conjugate over C . It follows that the image of [ M ρ ] in b B ( C ) is invariant under Gal( K ( λ ) , K ) , soby the Corollaries 7.14 and 7.15, [ M ρ ] is in b B ( K ) .The second statement follows from Theorem 6.15, so we now only need to provethat M ρ ] is in b B ( k ) . By Neumann–Reid [13, Theorem 2.1], K is Galois over k .Let σ ∈ Gal(
K, k ) . Since k is the field of squares of traces of ρ , it follows that σρ asa representation in PSL(2 , C ) is conjugate to the geometric representation. After aconjugation (which does not change the fundamental class), we may thus assumethat ρ and σρ are equal as representations in PSL(2 , C ) . Hence, c and σ ( c ) differby a Z / Z –cocycle, so by Theorem 6.15, σ ([ M ρ ]) = 2[ M ρ ] ∈ b B ( C ) . As above, thisimplies that M ρ ] is in b B ( k ) . (cid:3) Cusped manifolds. If M has cusps, Reid–Maclachlan [8] shows that thegeometric representation has image in PSL(2 , K ) . It thus follows from Theorem 5.3that M has a fundamental class [ M ] ∈ b B ( K ) PSL . Neumann–Yang [14] show thatthe Bloch invariant of M is always in B ( k ) , but they define B ( k ) as the kernel of z z ∧ (1 − z ) . With our definition, only M ] is in B ( k ) . An explicit examplewhith [ M ] / ∈ B ( k ) is given by the manifold m in the SnapPea census. Similarly,only M ] is in b B ( k ) PSL . Using remark 5.2 one checks that M ] always lifts to b B ( k ) , and by Lemma 5.1, M ] lifts canonically. We do not believe that a canonicallift of M ] is possible, so this result is likely to be optimal.9.1.1. Knot complements. If M is a knot complement, Reid–Machlachlan [8, Corol-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 / Z –valued knot invariant, which by Remark 5.2 is explicitly computable.For example, [ M ] lifts for the figure knot complement and the knot comple-ment, but not for the knot complement. Since the significance of this invariantis unclear at this moment, we spare the reader for the computations. References [1] Johan L. Dupont and Chih Han Sah. Scissors congruences. II.
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HE EXTENDED BLOCH GROUP AND ALGEBRAIC K –THEORY 33 Department of Mathematics, University of California, Berkeley, CA 94720-3840,USA
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