A refined Bloch group and the third homology of SL_2 of a field

A REFINED BLOCH GR OUP AND THE THIRD HOMOLOGY OF SL 2 OF A FIELD KEVIN HUTCHINSON A bstra ct . W e use the properties of the reﬁned Bloch group to study the structure of H 3 (SL 2 ( F ) , Z ) for a ﬁeld F . W e compute this group up to 2-torsion when F is a local ﬁeld with ﬁnite residue ﬁeld of odd order , and we show that for any global ﬁeld F it is not ﬁnitely-generated. 1. I ntr oduction In [19], Chih-Han Sah quotes S. Lichtenbaum who mentions our lack of kno wledge of the precise structure of H 3 (SL 2 ( Q ) , Z ) as an example of the unsatisfactory state of our understand- ing of the homology of linear groups. This was nearly twenty ﬁ ve years ago, and to the au- thor’ s kno wledge the precise structure of this group is still unkno wn. (Observe, ho wev er, that H 3 (SL n ( Q ) , Z )  K ind 3 ( Q )  Z / 24 for all n ≥ 3 - by the results of [8], for example.) In this article, we study the structure of the third homology of SL 2 of ﬁelds by using the prop- erties of the r eﬁned Bloch gr oup of the ﬁeld, which was introduced in [7]. W e are particularly interested in understanding H 3 (SL 2 ( F ) , Z ) as a functor of F , and its possible relation to other functors in K -theory and algebraic geometry . What are no w referred to as Bloch gr oups of ﬁelds ﬁrst appeared in the work of S. Bloch in the late 1970s (see [2] for the lecture notes) as a way of constructing explicit maps - and, in particular , regulators - on K 3 of ﬁelds. In the 1980s, they were studied by Dupont and Sah (under the name scissors congruence gr oup ) because of their connection with 3-dimensional hyperbolic geometry ([4], [19]). This connection is still acti vely studied today: Bloch groups of number ﬁelds are tar gets for Bloch in variants of certain ﬁnite-v olume oriented hyperbolic 3- manifolds ([15], [16], [5]). These in v ariants are amenable to e xplicit calculation and are related in a known way to the Chern-Simons in variant. There are also intriguing connections between Bloch groups, conformal ﬁeld theories and e ven modular form theory ([22],[13]). The precise relationship between the Bloch group and K -theory was established via their mu- tual connection to the homology of linear groups. These connections were greatly clariﬁed and exploited in the work of Suslin ([21]): F or a ﬁeld F with at least 4 elements, the pre-Bloch group, P ( F ), of the ﬁeld is an abelian group with generators [ x ] , x ∈ F × \ { 1 } , subject to a family of 5-term relations. The Bloch group, B ( F ), is a subgroup of P ( F ) which also arises naturally as a quotient of H 3 (GL 2 ( F ) , Z ). Suslin has sho wn that this e xtends to a surjection from H 3 (GL 3 ( F ) , Z ) to B ( F ) ([21, section 3]). Of course, K 3 ( F ) admits a Hure wicz homomor- phism to H 3 (GL ( F ) , Z ) = H 3 (GL 3 ( F ) , Z ), and hence, by composition, a map to B ( F ). Using calculations of the homology of GL n ( F ) ( n = 2 , 3) as well as the homotopy theory of the plus construction, Suslin prov ed that there is a natural short exact sequence 0 → ] T or Z 1 ( µ F , µ F ) → K ind 3 ( F ) → B ( F ) → 0 Date : Nov ember 16, 2021. 1991 Mathematics Subject Classiﬁcation. 19G99, 20G10. K e y wor ds and phrases. K -theory , Group Homology . 1 2 KEVIN HUTCHINSON where ] T or Z 1 ( µ F , µ F ) is the unique nontri vial extension of T or Z 1 ( µ F , µ F ) by Z / 2 and K ind 3 ( F ) = Coker( K M 3 ( F ) → K 3 ( F )) . These results of Suslin were e xtended to ﬁnite ﬁelds (with at least 4 elements) in [7, section 7]. In a letter to Sah (see [19, section 4]), Suslin asked the question whether the composite H 3 (SL 2 ( F ) , Z ) → H 3 (SL ( F ) , Z ) → K ind 3 ( F ) induces an isomorphism H 3 (SL 2 ( F ) , Z ) F ×  K ind 3 ( F ) . The current state of knowledge on this question is that the map is surjectiv e ([8]) and, if we let A [ 1 2 ] denote A ⊗ Z [ 1 2 ] for any abelian group A , that the induced map H 3 (SL 2 ( F ) , Z [ 1 2 ]) F × → K ind 3 ( F )[ 1 2 ] is an isomorphism ([12]). The homology groups of the special linear groups SL n ( F ) are naturally modules ov er the group ring Z [ F × ] via the short exact sequences 1 / / SL n ( F ) / / GL n ( F ) det / / F × / / 1 . Since the scalar matrices a · I n are central and have determinant a n , it follows that ( F × ) n acts tri vially on H k (SL n ( F ) , Z ). In particular , the groups H k (SL 2 ( F ) , Z ) are modules for the group ring R F : = Z [ F × / ( F × ) 2 ]. When n > k (or n ≥ k when k is odd) in the groups H k (SL n ( F ) , Z ), we are in the range of stability (see [19], [8] and [9]) and this module structure is necessarily tri vial. But belo w the range of stability , the module structure appears to be nontrivial and interesting. For example, the unstable groups H 2 n (SL 2 n ( F ) , Z ) are modules ov er the Grothendieck-W itt ring of the ﬁeld (which is a quotient of R F ) and surject onto the e ven Milnor-W itt K -theory groups of the ﬁeld F ([9]). In [7] the author deﬁned a r eﬁned pr e-Bloch gr oup , RP ( F ), of a ﬁeld F and a subgroup, the r eﬁned Bloch gr oup , RB ( F ), which was sho wn to have the follo wing properties: (1) [7, Theorem 4.3] The group RB ( F ) is an R F -module and there is a natural surjectiv e homomorphism of R F -modules H 3 (SL 2 ( F ) , Z ) / / / / RB ( F ) (2) This induces a commutati ve diagram (of R F [ 1 2 ]-modules) with exact ro ws 0 / / T or Z 1 ( µ F , µ F )[ 1 2 ] / / =   H 3 (SL 2 ( F ) , Z [ 1 2 ]) / /     RB ( F )[ 1 2 ] / /     0 0 / / T or Z 1 ( µ F , µ F )[ 1 2 ] / / K ind 3 ( F )[ 1 2 ] / / B ( F )[ 1 2 ] / / 0 Here R F acts tri vially on the bottom row and furthermore (3) [7, Corollary 5.1] On taking F × -coin variants, the top row becomes isomorphic to the bottom ro w . In particular , R B ( F )[ 1 2 ] F ×  B ( F )[ 1 2 ] and I F RB ( F )[ 1 2 ]  I F H 3 (SL 2 ( F ) , Z [ 1 2 ]) = Ker(H 3 (SL 2 ( F ) , Z [ 1 2 ]) → K ind 3 ( F )[ 1 2 ]) where I F denotes the augmentation ideal of the group ring R F . Reﬁned Bloch group and third homology of SL 2 3 (4) [7, Lemma 5.2] The group I F RB ( F )[ 1 2 ] is also the k ernel of the stabilization homomor - phism H 3 (SL 2 ( F ) , Z [ 1 2 ]) → H 3 (SL 3 ( F ) , Z [ 1 2 ]) . The cokernel of this map is the third Milnor K -group K M 3 ( F )[ 1 2 ] and the image is isomorphic to K ind 3 ( F )[ 1 2 ] ([8, Theorem 4.7, Section 5]). The main result of the current article (Theorem 4.9 and its corollaries) tells us that giv en a v aluation v on the ﬁeld F with residue ﬁeld k , there are surjecti ve reduction or specialization homomorphisms RP ( F ) / / / / c RP ( k ) where c RP ( k ) is a certain quotient of RP ( k ). In particular , if a ∈ F × and v ( a ) is not a multiple of 2, there is a specialization homomorphism which induces a surjection e − a RB ( F )[ 1 2 ] / / / / b P ( k )[ 1 2 ] where e − a = 1 − h a i 2 ∈ I F [ 1 2 ] and h a i denotes the square class of a in R F and b P ( k ) is a certain quotient of the classical pre- Bloch group, P ( k ), of the residue ﬁeld. Using these results, we can prov e that I F RB ( F )[ 1 2 ] is large if F is a ﬁeld with many v aluations. In particular , it follo ws that H 3 (SL 2 ( F ) , Z ) is not ﬁnitely generated for any global ﬁeld F . By contrast, if F is a global ﬁeld then H 3 (SL 3 ( F ) , Z ) = K ind 3 ( F ) is well-kno wn to be ﬁnitely gen- erated. More precisely , if F is a global ﬁeld whose class group has odd order then there is a natural surjection I F RB ( F )[ 1 2 ] / / / / L v P ( k v )[ 1 2 ] where v runs through all the ﬁnite places of F and k v is the residue ﬁeld at v (see Corollary 5.2). Let n 0 denote the odd part of the nonzero integer n . By the results of [7], if F q is the ﬁnite ﬁeld with q elements, then P ( F q )[ 1 2 ] is ﬁnite cyclic of order ( q + 1) 0 . As another application, we also use the techniques de veloped to construct e xplicitly non-tri vial F 3 -vector spaces of kno wn dimension inside groups of the form H 3 (SL 2 ( O S ) , Z ) where O S is a ring of S -integers corresponding to a set S of primes of the ﬁeld F , and thus to gi ve lo wer bounds on the 3-ranks of such groups. Since there is still very little in the literature at present by way of explicit calculations of the homology or cohomology of S -arithmetic groups of this type, we hope that the techniques de veloped here will be useful in pro ving more general results of this type in the future. In particular , our results point to the importance of the O × S -module structure in calculations of this type. Finally , we use these specialization maps, together with the basic algebraic properties of the reﬁned Bloch groups, de veloped in section 3 belo w , to giv e a calculation of H 3 (SL 2 ( F ) , Z [ 1 2 ]) for local ﬁelds F with ﬁnite residue ﬁeld k of odd order (Theorem 6.19). In particular, for such ﬁelds (with some minor restrictions if Q 3 ⊂ F ) we hav e H 3 (SL 2 ( F ) , Z [ 1 2 ])  K ind 3 ( F )[ 1 2 ] ⊕ P ( k )[ 1 2 ] . Here the right-hand side is an R F module with F × acting tri vially on the ﬁrst factor while any uniformizer acts as − 1 on the second factor . 4 KEVIN HUTCHINSON T o return to our opening remarks, it follows from the results here that there is a natural surjec- tion H 3 (SL 2 ( Q ) , Z ) ⊗ Z [ 1 2 ] = H 3 (SL 2 ( Q ) , Z [ 1 2 ]) / / / / K ind 3 ( Q )[ 1 2 ] ⊕  L p P ( F p )[ 1 2 ]  . It is natural to ask whether this is an isomorphism and, furthermore, what adjustments, if any , need to be made to obtain a corresponding statement with integral coe ﬃ cients. Acknowledgements. The author thanks the anonymous referee for a v ery thorough reading of the original manuscript and for numerous detailed suggestions which hav e greatly improv ed the exposition. Any remaining obscurities are entirely the responsibility of the author . 2. R eview of B loch G r oups 2.1. Preliminaries and Notation. For a ﬁeld F , we let G F denote the multiplicati ve group, F × / ( F × ) 2 , of nonzero square classes of the ﬁeld. For x ∈ F × , we will let h x i ∈ G F denote the corresponding square class. Let R F denote the integral group ring Z [ G F ] of the group G F . W e will use the notation h h x i i for the basis elements, h x i − 1, of the augmentation ideal I F of R F . For any a ∈ F × , we will let p + a and p − a denote respectiv ely the elements 1 + h a i and 1 − h a i in R F . W e let e + a and e − a denote respecti vely the mutually orthogonal idempotents e + a : = p + a 2 = 1 + h a i 2 , e − a : = p − a 2 = 1 − h a i 2 ∈ R F [ 1 2 ] . (Of course, these operators depend only on the class of a in G F .) For an y abelian group A we will let A [ 1 2 ] denote A ⊗ Z [ 1 2 ]. For an integer n , we will let n 0 denote the odd part of n . Thus if A is a ﬁnite abelian group of order n , then A [ 1 2 ] is a ﬁnite abelian group of order n 0 . For an additi ve abelian group A , A / 2 denotes A ⊗ Z / 2. Howe ver , for multiplicativ e groups M we will use the notation M / M 2 . If n is a positiv e integer and A an abelian group, A ( n ) will denote the set { a ∈ A | na = 0 } . 2.2. The classical Bloch group. For a ﬁeld F , with at least 4 elements, the pre-Bloc h gr oup , P ( F ), is the group generated by the elements [ x ] , x ∈ F × \ { 1 } , subject to the relations R x , y : [ x ] −  y  +  y / x  − h (1 − x − 1 ) / (1 − y − 1 ) i +  (1 − x ) / (1 − y )  x , y . Let S 2 Z ( F × ) denote the group F × ⊗ Z F × < x ⊗ y + y ⊗ x | x , y ∈ F × > and denote by x ◦ y the image of x ⊗ y in S 2 Z ( F × ). The map λ : P ( F ) → S 2 Z ( F × ) , [ x ] 7→ ( 1 − x ) ◦ x is well-deﬁned, and the Bloch gr oup of F , B ( F ) ⊂ P ( F ), is deﬁned to be the kernel of λ . 2.3. The r eﬁned Bloch group. The reﬁned pr e-Bloch gr oup , R P ( F ), of a ﬁeld F which has at least 4 elements, is the R F -module with generators [ x ] , x ∈ F × subject to the relations [ 1 ] = 0 and S x , y : 0 = [ x ] − [ y ] + h x i  y / x  − D x − 1 − 1 E h (1 − x − 1 ) / (1 − y − 1 ) i + h 1 − x i  (1 − x ) / (1 − y )  , x , y , 1 Of course, from the deﬁnition it follows immediately that P ( F ) = ( RP ( F )) F × = H 0 ( F × , RP ( F )). Reﬁned Bloch group and third homology of SL 2 5 For an y ﬁeld F we deﬁne the R F -module RS 2 Z ( F × ) : = I 2 F × Sym 2 F 2 ( G F ) S 2 Z ( F × ) ⊂ I 2 F ⊕ S 2 Z ( F × ) where S 2 Z ( F × ) has the tri vial R F -module structure. As an R F -module, RS 2 Z ( F × ) is generated by the elements [ a , b ] : = ( h h a i i h h b i i , a ◦ b ) ∈ RS 2 Z ( F × ) . The r eﬁned Bloch-W igner homomorphism Λ to be the R F -module homomorphism Λ : RP ( F ) → RS 2 Z ( F ) , [ x ] 7→ [1 − x , x ] which is well-deﬁned by [7, proof of Theorem 4.3]. In view of the deﬁnition of RS 2 Z ( F × ), we can express Λ = ( λ 1 , λ 2 ) where λ 1 : RP ( F ) → I 2 F is the map [ x ] 7→ h h 1 − x i i h h x i i , and λ 2 is the composite RP ( F ) / / / / P ( F ) λ / / S 2 Z ( F × ) . Finally , we can deﬁne the r eﬁned Bloc h gr oup of the ﬁeld F (with at least 4 elements) to be the R F -module RB ( F ) : = K er( Λ : RP ( F ) → RS 2 Z ( F × )) . 2.4. The ﬁelds F 2 and F 3 . Throughout this paper it will be con venient for us to ha ve (reﬁned and classical) pre-Bloch and Bloch groups for the ﬁelds with 2 and 3 elements. F or this reason, we introduce the follo wing ad hoc deﬁnitions. P ( F 2 ) = RP ( F 2 ) = RB ( F 2 ) = B ( F 2 ) is simply an additiv e group of order 3 with distinguished generator , denoted C F 2 . RP ( F 3 ) is the cyclic R F 3 -module generated by the symbol [ − 1 ] and subject to the one relation 0 = 2 · ( [ − 1 ] + h − 1 i [ − 1 ] ) . The homomorphism Λ : RP ( F 3 ) → RS 2 Z ( F × 3 ) = I 2 F 3 = 2 · Z h h − 1 i i is the R F 3 -homomorphism sending [ − 1 ] to h h − 1 i i 2 = − 2 h h − 1 i i . Then RB ( F 3 ) = Ker( Λ ) is the submodule of order 2 generated by [ − 1 ] + h − 1 i [ − 1 ] . Furthermore, we let P ( F 3 ) = RP ( F 3 ) F × 3 . This is a cyclic Z -module of order 4 with generator [ − 1 ] . Let λ : P ( F 3 ) → S 2 Z ( F × 3 ) = Sym 2 F 2 ( F × 3 ) be the map [ − 1 ] 7→ − 1 ◦ − 1. Then B ( F 3 ) : = K er( λ ) is cyclc of order 2 with generator 2 [ − 1 ] and the natural map R B ( F 3 ) → B ( F 3 ) is an isomorphism. 2.5. The reﬁned Bloch Gr oup and H 3 (SL 2 ( F ) , Z ) . W e recall some results from [7]: The main result there is Theorem 2.1. Let F be an inﬁnite ﬁeld. Ther e is a natural complex 0 → T or Z 1 ( µ F , µ F ) → H 3 (SL 2 ( F ) , Z ) → RB ( F ) → 0 . which is exact everywher e except possibly at the middle term. The middle homology is annihi- lated by 4 . In particular , for any inﬁnite ﬁeld ther e is a natural short exact sequence 0 → T or Z 1 ( µ F , µ F )[ 1 2 ] → H 3 (SL 2 ( F ) , Z [ 1 2 ]) → RB ( F )[ 1 2 ] → 0 . 6 KEVIN HUTCHINSON The follo wing result is Corollary 5.1 in [7]: Lemma 2.2. Let F be an inﬁnite ﬁeld. Then the natural map RB ( F ) → B ( F ) is surjective and the induced map RB ( F ) F × → B ( F ) has a 2 -primary torsion kernel. No w for any ﬁeld F , let H 3 (SL 2 ( F ) , Z ) 0 : = K er(H 3 (SL 2 ( F ) , Z ) → K ind 3 ( F )) and RB 0 ( F ) : = K er( RB ( F ) → B ( F )) Lemma 2.3. [7, Lemma 5.2] Let F be an inﬁnite ﬁeld. Then (1) H 3 (SL 2 ( F ) , Z [ 1 2 ]) 0 = RB ( F )[ 1 2 ] 0 (2) H 3 (SL 2 ( F ) , Z [ 1 2 ]) 0 = I F H 3 (SL 2 ( F ) , Z [ 1 2 ]) and RB 0 ( F )[ 1 2 ] = I F RB ( F )[ 1 2 ] . (3) H 3 (SL 2 ( F ) , Z [ 1 2 ]) 0 = K er(H 3 (SL 2 ( F ) , Z [ 1 2 ]) → H 3 (SL 3 ( F ) , Z [ 1 2 ])) = K er(H 3 (SL 2 ( F ) , Z [ 1 2 ]) → H 3 (GL 2 ( F ) , Z [ 1 2 ])) . On the other hand, the corresponding results for ﬁnite ﬁelds are as follo ws (the results in [7] apply to ﬁelds with at least 4 elements, b ut it is straightforward to v erify that they e xtend to the ﬁelds F 2 and F 3 with the deﬁnitions supplied abov e): Lemma 2.4. [7, Lemma 7.1] . F or a ﬁnite ﬁeld k the natural map RP ( k ) → P ( k ) induces an isomorphism RB ( k )  B ( k ) . For a ﬁeld F , we let T or Z 1 ] ( µ F , µ F ) denote the unique nontrivial extension of T or Z 1 ( µ F , µ F ) by Z / 2 if the characteristic of F is not 2, and T or Z 1 ( µ F , µ F ) in characteristic 2. Theorem 2.5. [7, Corollary 7.5] Ther e is a natural short exact sequence 0 → T or Z 1 ] ( µ F q , µ F q ) → H 3 (SL 2 ( F q ) , Z [1 / p ]) → B ( F q ) → 0 for any ﬁnite ﬁeld F q of or der q = p f . Furthermor e, ther e is a natural isomorphism H 3 (SL 2 ( F q ) , Z [1 / p ])  K ind 3 ( F q ) . Lemma 2.6. [7, Section 5,7] B ( F q )  ( Z / ( q + 1) / 2 , q odd Z / ( q + 1) , q even and if K ⊂ SL 2 ( F q ) is a cyclic subgr oup of or der ( q + 1) 0 then the composite map Z / ( q + 1) 0  H 3 ( K , Z [ 1 2 ]) → H 3 (SL 2 ( F q ) , Z [ 1 2 ]) → B ( F q )[ 1 2 ] is an isomorphism. Corollary 2.7. F or any prime power q, P ( F q )[ 1 2 ] is cyclic of or der ( q + 1) 0 . Pr oof. S 2 Z ( F × q ) has order di viding 2, so the inclusion B ( F q ) → P ( F q ) induces an isomorphism B ( F q )[ 1 2 ]  P ( F q )[ 1 2 ].  Reﬁned Bloch group and third homology of SL 2 7 2.6. The map H 3 ( G , Z ) → RB ( F ) . If G is any subgroup of SL 2 ( F ), then by composing the map H 3 (SL 2 ( F ) , Z ) → RB ( F ) of Theorem 2.1 with the map H 3 ( G , Z ) → H 3 (SL 2 ( F ) , Z ) induced by the inclusion of G into SL 2 ( F ) we obtain a map H 3 ( G , Z ) → RB ( F ). In [7, Section 6] a recipe is gi ven for calculating this map for subgroups G of SL 2 ( F ) which don’ t act transiti vely on P 1 ( F ). W e recall this calculation in the case that G is a ﬁnite cyclic subgroup. First, giv en 4 distinct points x 0 , x 1 , x 2 , x 3 ∈ P 1 ( F ) we deﬁne the r eﬁned cr oss ratio cr ( x 0 , x 1 , x 2 , x 3 ) ∈ RP ( F ) by cr( x 0 , x 1 , x 2 , x 3 ) =                                              D ( x 2 − x 0 )( x 0 − x 1 ) x 2 − x 1 E h ( x 2 − x 1 )( x 3 − x 0 ) ( x 2 − x 0 )( x 3 − x 1 ) i , if x i , ∞ h x 1 − x 2 i h x 1 − x 2 x 1 − x 3 i , if x 0 = ∞ h x 2 − x 0 i h x 3 − x 0 x 2 − x 0 i , if x 1 = ∞ h x 0 − x 1 i h x 3 − x 0 x 3 − x 1 i , if x 2 = ∞ D ( x 2 − x 0 )( x 0 − x 1 ) x 2 − x 1 E h x 2 − x 1 x 2 − x 0 i , if x 3 = ∞ Lemma 2.8. [7, Section 6] Suppose that G is a ﬁnite cyclic subgr oup of SL 2 ( F ) of or der r with gener ator t . Let x ∈ P 1 ( F ) with trivial stabilizer G x = 1 , and let y ∈ P 1 ( F ) \ G · x. The composite Z / r  H 3 ( G , Z ) → RB ( F ) is given by the formula 1 7→ r − 1 X i = 0 cr( β x , y 3 (1 , t , t i + 1 , t i + 2 )) . wher e β x , y 3 (1 , t , t , t 2 ) = 0 β x , y 3 (1 , t , t i + 1 , t i + 2 ) = ( x , t ( x ) , t i + 1 ( x ) , t i + 2 ( x )) for 1 ≤ i ≤ r − 3 β x , y 3 (1 , t , t r − 1 , 1) = ( y , t ( x ) , t − 1 ( x ) , x ) − ( y , x , t ( x ) , t − 1 ( x )) β x , y 3 (1 , t , t r , t r + 1 ) = β x , y 3 (1 , t , 1 , t ) = ( 0 , y = t ( y ) ( y , t ( y ) , x , t ( x )) + ( y , t ( y ) , t ( x ) , x ) , y , t ( y ) Furthermor e, the r esulting map is independent of the particular choice of x and y. 3. S ome algebra in RP ( F ) In this section we study certain key elements and submodules of the reﬁned pre-Bloch group of a ﬁeld F . 3.1. The elements ψ i ( x ) and the modules K ( i ) F . W e recall the elements { x } : = [ x ] + h x − 1 i ∈ P ( F ) (for x ∈ F × ). A straightforward calculation - see Suslin [21] - shows that these symbols allow us to deﬁne a group homomorphism F × → P ( F ) , x 7→ { x } 8 KEVIN HUTCHINSON whose kernel contains ( F × ) 2 ; i.e. we have n x 2 o = 0 and { xy } = { x } + { y } for all x , y . In particular , these elements satisfy 2 { x } = 0 for all x . W e no w consider two liftings of these elements in RP ( F ): For x ∈ F × we let ψ 1 ( x ) : = [ x ] + h − 1 i h x − 1 i and ψ 2 ( x ) : = ( h 1 − x i  h x i [ x ] + h x − 1 i , x , 1 0 , x = 1 (If F = F 2 , we interpret this as ψ i ( 1 ) = 0 for i = 1 , 2. F or F = F 3 , we hav e ψ 1 ( − 1 ) = ψ 2 ( − 1 ) = [ − 1 ] + h − 1 i [ − 1 ] . ) The maps F × → RP ( F ) , x 7→ ψ i ( x ) are no longer homomorphisms, b ut are 1-cocycles for the action of F × : Lemma 3.1. Let F be a ﬁeld. F or i ∈ { 1 , 2 } , the map F × → RP ( F ) , x 7→ ψ i ( x ) is a 1 -cocycle; i.e. we have ψ i ( xy ) = h x i ψ i ( y ) + ψ i ( x ) for all x , y ∈ F × . Pr oof. The statement is trivial for F = F 2 or F 3 . W e can thus assume F has at least 4 elements. If x = 1 or y = 1, the required identities are clear . If x , 1 and y , x − 1 the relation 0 = S x , xy + h − 1 i S x − 1 , x − 1 y − 1 in RP ( F ) yields the identity ψ 1 ( x ) − ψ 1 ( xy ) + h x i ψ 1 ( y ) = 0 . Thus we must also prove that h x i ψ 1  x − 1  + ψ 1 ( x ) = 0 for all x , 1. Fix x , 1 and choose y < { 1 , x − 1 } (here we use that F has at least 4 elements). Then h y i ψ 1 ( x ) = ψ 1 ( xy ) − ψ 1 ( y ) = − h xy i ψ 1  x − 1  and multiplying by h y i gi ves the required identity . No w , for x , y ∈ F × , let Q ( x , y ) : = h x i " x y # + h y i  y x  ∈ RP ( F ) . Then Q ( x , y ) = h y i * x y + " x y # +  y x  ! = h y i * 1 − x y + ψ 2 x y ! = h y − x i ψ 2 x y ! . For a , b , 1, the relation 0 = S a , b + S b , a in RP ( F ) giv es the identity Q ( a − 1 − 1 , b − 1 − 1) = Q ( a − 1 , b − 1 ) + Q (1 − a , 1 − b ) . Thus D b − 1 − a − 1 E ψ 2 a − 1 − 1 b − 1 − 1 ! = D b − 1 − a − 1 E ψ 2 b a ! + h a − b i ψ 2 1 − a 1 − b ! and hence ψ 2 b − 1 − 1 a − 1 − 1 ! = ψ 2 b a ! + * b a + ψ 2 1 − a 1 − b ! . Reﬁned Bloch group and third homology of SL 2 9 No w if we ﬁx x , y , 1 with xy , 1, we can solve the equations x = b a , y = 1 − a 1 − b for a and b and prov e the required identity for ψ 2 ( ) .  Observe from the deﬁnitions that, for i = 1 , 2, ψ i  x − 1  = h − 1 i ψ i ( x ) for all x ∈ F × . In particular , h − 1 i ψ i ( − 1 ) = ψ i ( − 1 ) for i = 1 , 2. The properties enumerated in following proposition will be used often in the remainder of this section. Proposition 3.2. F or i ∈ { 1 , 2 } we have: (1) h h x i i ψ i ( y ) = h h y i i ψ i ( x ) for all x , y (2) ψ i  xy 2  = ψ i ( x ) + ψ i  y 2  for all x , y (3) h h x i i ψ i  y 2  = 0 for all x , y (4) 2 · ψ i ( − 1 ) = 0 for all i (5) ψ i  x 2  = − h h x i i ψ i ( − 1 ) for all x (6) 2 · ψ i  x 2  = 0 for all x and if − 1 is a squar e in F then ψ i  x 2  = 0 for all x. (7) h h x i i h h y i i ψ i ( − 1 ) = 0 for all x , y (8) h − 1 i h h x i i ψ i ( y ) = h h x i i ψ i ( y ) for all x , y (9) Let  ( F ) : = ( 1 , − 1 ∈ ( F × ) 2 2 , − 1 < ( F × ) 2 The map G F → RP ( F ) , h x i 7→  ( F ) ψ i ( x ) is a well-deﬁned 1 -cocycle. Pr oof. The identities ψ i ( 1 ) = 0 and ψ i  x − 1  = h − 1 i ψ i ( x ) follo w from the deﬁnition of ψ i ( ) . More generally , let M be an R F -module and let ψ : F × → M be a 1-cocycle satisfying ψ (1) = 0 and ψ ( x − 1 ) = h − 1 i ψ ( x ) for all x ∈ F × . (1) By the cocycle condition, for all x , y ∈ F × we hav e ψ ( xy ) = h x i ψ ( y ) + ψ ( x ) = h y i ψ ( x ) + ψ ( y ) . Rearranging this latter equality , we deduce: h h x i i ψ ( y ) = h h y i i ψ ( x ) for all x , y . (2) For all x , y we have ψ ( xy 2 ) = D y 2 E ψ ( x ) + ψ ( y 2 ) = ψ ( x ) + ψ ( y 2 ) since D y 2 E = 1 in R F . (3) From (1) together with the fact that D D y 2 E E = 0 for all y , we deduce: h h x i i ψ ( y 2 ) = D D y 2 E E ψ ( x ) = 0 . (4) W e hav e h − 1 i ψ ( − 1) = ψ ( − 1) and thus 0 = ψ (1) = ψ ( − 1 · − 1) = ψ ( − 1) + h − 1 i ψ ( − 1) = 2 ψ ( − 1) . (5) For all x we ha ve ψ ( x ) = ψ 1 x · x 2 ! = ψ 1 x ! + ψ ( x 2 ) = h − 1 i ψ ( x ) + ψ ( x 2 ) . Thus ψ ( x 2 ) = − h h − 1 i i ψ ( x ) = − h h x i i ψ ( − 1) . 10 KEVIN HUTCHINSON (6) The ﬁrst statement follo ws from (4) and (5). For the second, observe that for any x we hav e ψ ( x 2 ) = − h h − 1 i i ψ ( x ) and h h − 1 i i = 0 if − 1 is a square. (7) This statement follo ws from (3) and (5). (8) This is a restatement of (7); namely h h − 1 i i h h x i i ψ ( y ) = h h x i i h h y i i ψ ( − 1) = 0 . (9) By (6),  ( F ) ψ ( x 2 ) = 0 in M for all x and thus  ( F ) ψ ( xy 2 ) =  ( F ) ψ ( x ) for all x , y . Thus the proposed map is well-deﬁned (and is thus clearly a 1-cocycle).  No w , for i ∈ { 1 , 2 } , let K ( i ) F denote the R F -submodule of RP ( F ) generated by the set { ψ i ( x ) | x ∈ F × } . Lemma 3.3. Let F be a ﬁeld. Then for i ∈ { 1 , 2 } λ 1  K ( i ) F  = p + − 1 ( I F ) ⊂ I 2 F and K er( λ 1 | K ( i ) F ) is annihilated by 4 . Pr oof. W e use the identities h h a i i h h b i i = h h ab i i − h h a i i − h h b i i , h − 1 i h h a i i = h h − a i i − h h − 1 i i , D D ab 2 E E = h h a i i in I F . Thus λ 1 ( ψ 1 ( x ) ) = λ 1 ( [ x ] ) + h − 1 i λ 1 ( h x − 1 i ) = h h x i i h h 1 − x i i + h − 1 i h h x i i h h x ( x − 1) i i = h h x (1 − x ) i i − h h x i i − h h 1 − x i i + h − 1 i ( h h x − 1 i i − h h x i i − h h x ( x − 1) i i ) = h h x (1 − x ) i i − h h x i i − h h 1 − x i i + h h 1 − x i i − h h − x i i − h h x (1 − x ) i i + h h − 1 i i = h h − 1 i i − h h x i i − h h − x i i = h h − x i i · h h x i i = − p + − 1 · h h x i i Thus λ 1 ( K (1) F ) = p + − 1 ( I F ). For x , 1 we hav e ψ 2 ( x ) = h x (1 − x ) i [ x ] + h 1 − x i h x − 1 i and thus λ 1 ( ψ 2 ( x ) ) = h x (1 − x ) i h h x i i h h 1 − x i i + h 1 − x i h h x i i h h x ( x − 1) i i = h x (1 − x ) i ( h h x (1 − x ) i i − h h x i i − h h 1 − x i i ) + h 1 − x i ( h h x − 1 i i − h h x ( x − 1) i i − h h x i i ) = − h h 1 − x i i − h h x i i + h h x (1 − x ) i i + h h − 1 i i − h h − x i i − h h x (1 − x ) i i + h h 1 − x i i = h h − 1 i i − h h x i i − h h − x i i = h h x i i · h h − x i i = − p + − 1 · h h x i i . Thus λ 1 ( K (2) F ) = p + − 1 ( I F ) also. For the second statement, recall that for any group G and any Z [ G ]-module M a 1-cocycle ρ : G → M giv es rise to Z [ G ]-homomorphism I G → M , deﬁned by g − 1 7→ ρ ( g ). Thus, for i ∈ { 1 , 2 } , we ha ve a well-deﬁned R F -homomorphism I F → RP ( F ) , h h x i i 7→ 2 ψ i ( x ) . Combining this with the inclusion p + − 1 ( I F ) → I F we obtain an R F -module homomorphism µ : p + − 1 ( I F ) → K ( i ) F sending p + − 1 h h x i i = h h x i i + h − 1 i h h x i i to 2 ψ i ( x ) + h − 1 i 2 ψ i ( x ) . Ho wev er , Reﬁned Bloch group and third homology of SL 2 11 for any x ∈ F × we hav e 2 h h − 1 i i ψ i ( x ) = 2 h h x i i ψ i ( − 1 ) = 0 by Proposition 3.2 (4), so that 2 h − 1 i ψ i ( x ) = 2 ψ i ( x ) and thus 2 ψ i ( x ) + h − 1 i 2 ψ i ( x ) = 4 ψ i ( x ) . It follo ws that µ ◦  λ 1 | K ( i ) F  is just multiplication by 4, and the result is prov ed.  Remark 3.4. Since p + − 1 I F is a free abelian group, it follows that, as an abelian group, K ( i ) F decomposes as a direct sum A ⊕  K ( i ) F  tors where A is a free abelian group and 4 annihilates  K ( i ) F  tors = K er( λ 1 | K ( i ) F ). Furthermore, if h h − 1 i i ψ i ( x ) = 0 for all x (for example, if − 1 ∈ ( F × ) 2 ) then ψ i  x 2  = 0 for all x and the map G F → RP ( F ) , h x i 7→ ψ i ( x ) is already a well-deﬁned 1-cocycle. The above arguments then sho w that  K ( i ) F  tors = K er( λ 1 | K ( i ) F ) is annihilated by 2. 3.2. The constants C F and D F . In the classical pre-Bloch group P ( F ) the expression [ x ] + [ 1 − x ] is kno wn to be independent of x ∈ F \ { 0 , 1 } (see, for example, [21, Section 1]). Further- more, this constant has order di viding 6. W e consider no w an analogous constant in R P ( F ). Let F be a ﬁeld with at least 4 elements. For x ∈ F × , x , 1 we let ˜ C ( x ) : = [ x ] + h − 1 i [ 1 − x ] and C ( x ) = ˜ C ( x ) + h h 1 − x i i ψ 1 ( x ) . Lemma 3.5. Let F be a ﬁeld with at least 4 elements. Then C ( x ) is constant; i.e. for all x , y ∈ F \ { 0 , 1 } we have C ( x ) = C ( y ) in RP ( F ) . Pr oof. In RP ( F ) we hav e 0 = S x , y + h − 1 i S 1 − x , 1 − y = [ x ] −  y  + h x i  y x  − D x − 1 − 1 E " x − 1 − 1 y − 1 − 1 # + h 1 − x i " 1 − x 1 − y # + h − 1 i [ 1 − x ] − h − 1 i  1 − y  + h x − 1 i " 1 − y 1 − x # − D 1 − x − 1 E " y − 1 − 1 x − 1 − 1 # + h − x i " x y # = ˜ C ( x ) − ˜ C ( y ) + h x i ψ 1  y x  − D 1 − x − 1 E ψ 1 1 − y − 1 1 − x − 1 ! + h x − 1 i ψ 1 y − 1 x − 1 ! =  ˜ C ( x ) − ψ 1 ( x ) + ψ 1  1 − x − 1  − ψ 1 ( x − 1 )  −  ˜ C ( y ) − ψ 1 ( y ) + ψ 1  1 − y − 1  − ψ 1 ( y − 1 )  (using the cocycle property of ψ 1 ( ) to obtain the last line). Furthermore ψ 1  1 − x − 1  − ψ 1 ( x − 1 ) − ψ 1 ( x ) = ψ 1  ( x − 1) x − 1  − ψ 1 ( x − 1 ) − ψ 1 ( x ) = h x − 1 i ψ 1  x − 1  − ψ 1 ( x ) = h 1 − x i ψ 1 ( x ) − ψ 1 ( x ) = h h 1 − x i i ψ 1 ( x ) .  Deﬁnition 3.6. Thus, for a giv en ﬁeld F with at least 4 elements, we will denote by C F the common v alue of the expressions C ( x ) for x ∈ F \ { 0 , 1 } . Furthermore, we let C F 2 denote the distinguished generator of RP ( F 2 ) = R B ( F 2 ) and we let C F 3 : = ψ 1 ( − 1 ) = (1 + h − 1 i ) [ − 1 ] ∈ RB ( F 3 ). For any ﬁeld F , we let Φ ( T ) denote the polynomial T 2 − T + 1 ∈ F [ T ]. Observe that if F has characteristic 3 then − 1 is a root of Φ ( T ). In any other characteristic, a root of Φ ( T ) will be of the form − ζ , where ζ is a primiti ve cube root of unity . Corollary 3.7. Let F be a ﬁeld. 12 KEVIN HUTCHINSON (1) h − 1 i C F = C F (2) If Φ ( T ) has a r oot in F , then C F = ψ 1 ( − 1 ) . (3) F or any ﬁeld F , C F ∈ RB ( F ) ; i.e Λ ( C F ) = 0 . Pr oof. (1) For x < { 0 , 1 } we have C F = C (1 − x ) = h − 1 i C ( x ) = h − 1 i C F (since h − 1 i h h 1 − x i i ψ 1 ( x ) = h h 1 − x i i ψ 1 ( x ) by Proposition 3.2 (8)). (2) F 2 contains no root of Φ ( T ). If F = F 3 , then − 1 is a root of Φ ( T ), but the result is true by deﬁnition. So we can assume that F has at least 4 elements. Let x be a root of Φ ( T ). Then 1 − x = x − 1 and thus C F = C ( x ) = [ x ] + h − 1 i [ 1 − x ] + h h 1 − x i i ψ 1 ( x ) = ψ 1 ( x ) + D D x − 1 E E ψ 1 ( x ) = ψ 1 ( x ) + h h x i i ψ 1 ( x ) = h x i ψ 1 ( x ) . No w x 3 = − 1, so that x = − 1 · x 4 and hence h x i = h − 1 i . Furthermore, using Proposition 3.2 (2) and (6), it follo ws that = ψ 1 ( x ) = ψ 1  − 1 · x 4  = ψ 1 ( − 1 ) + ψ 1  x 4  = ψ 1 ( − 1 ) + 2 ψ 1  x 2  = ψ 1 ( − 1 ) Thus C F = h − 1 i ψ 1 ( − 1 ) = ψ 1 ( − 1 ) as claimed. (3) Fix x ∈ F × \ { 1 } . Recall that λ 1 ( ψ 1 ( x ) ) = − p + − 1 h h x i i (see the proof of Lemma 3.3). Thus λ 1 ( C F ) = λ 1 ( [ x ] ) + h − 1 i λ 1 ( [ 1 − x ] ) − h h 1 − x i i λ 1 ( ψ 1 ( x ) ) = h h 1 − x i i h h x i i + h − 1 i h h x i i h h 1 − x i i − h h 1 − x i i p + − 1 h h x i i = p + − 1 h h 1 − x i i h h x i i − p + − 1 h h 1 − x i i h h x i i = 0 . On the other hand, recalling that F × acts tri vially on S 2 Z ( F × ), we hav e λ 2 ( C F ) = λ 2 ( [ x ] ) + h − 1 i λ 2 ( [ 1 − x ] ) − h h 1 − x i i λ 2 ( ψ 1 ( x ) ) = (1 − x ) ◦ x + x ◦ (1 − x ) = 0 in S 2 Z ( F × ) proving the result.  Deﬁnition 3.8. For an y ﬁeld F , we deﬁne the element D F ∈ RB ( F ) by D F : = 2 C F . Observe that D F = (1 + h − 1 i ) C F by Corollary 3.7 (1). Lemma 3.9. F or any x ∈ F \ { 0 , 1 } we have D F = [ x ] + h − 1 i " 1 1 − x − 1 # − ψ 1 1 1 − x ! Pr oof. D F = 2 C F = C ( x ) + C 1 1 − x ! = [ x ] + h − 1 i [ 1 − x ] + h h 1 − x i i ψ 1 ( x ) + " 1 1 − x # + h − 1 i " 1 1 − x − 1 # + D D 1 − x − 1 E E ψ 1 1 1 − x ! . Ho wev er , h − 1 i [ 1 − x ] + " 1 1 − x # = ψ 1 1 1 − x ! Reﬁned Bloch group and third homology of SL 2 13 and h h 1 − x i i ψ 1 ( x ) = * * 1 1 − x + + ψ 1 ( x ) = h h x i i ψ 1 1 1 − x ! . Thus D F = [ x ] + h − 1 i " 1 1 − x − 1 # +  1 + h h x i i + D D 1 − x − 1 E E ψ 1 1 1 − x ! = [ x ] + h − 1 i " 1 1 − x − 1 # +  h x i + D 1 − x − 1 E − 1  ψ 1 1 1 − x ! . So it remains to prov e that h x i ψ 1 1 1 − x ! = − D 1 − x − 1 E ψ 1 1 1 − x ! for any x , 0 , 1. No w , by the cocycle property , h x i ψ 1 1 1 − x ! = ψ 1  x 1 − x  − ψ 1 ( x ) . Ho wev er , for any a ∈ F × , we have ψ 1 ( a ) = h − 1 i ψ 1 ( − a ) − ψ 1 ( − 1 ) , and thus for any a , b ∈ F × we hav e ψ 1 ( a ) − ψ 1 ( b ) = h − 1 i ψ 1 ( − a ) − h − 1 i ψ 1 ( − b ) = ψ 1  − a − 1  − ψ 1  − b − 1  . It follo ws that h x i ψ 1 1 1 − x ! = ψ 1 − 1 − x x ! − ψ 1  − x − 1  = ψ 1  1 − x − 1  − ψ 1  − x − 1  . On the other hand, using the cocycle property again, D 1 − x − 1 E ψ 1 1 1 − x ! = ψ 1 1 − x − 1 1 − x ! − ψ 1  1 − x − 1  = ψ 1  − x − 1  − ψ 1  1 − x − 1  completing the proof.  Lemma 3.10. F or any ﬁeld F we have 3 D F = 6 C F = 0 . Pr oof. Fix x , 0 , 1. Then 3 D F = D 1 x ! + D 1 1 − x − 1 ! + D (1 − x ) = " 1 x # + h − 1 i " 1 1 − x # − " 1 1 − x − 1 # − h − 1 i h 1 − x − 1 i + " 1 1 − x − 1 # + h − 1 i [ x ] − [ 1 − x ] − h − 1 i " 1 1 − x # + [ 1 − x ] + h − 1 i h 1 − x − 1 i − " 1 x # − h − 1 i [ x ] = 0 .  14 KEVIN HUTCHINSON Remark 3.11. Examples show that this is best possible. Under the natural map RP ( F ) → P ( F ) the image of C F is the constant C : = [ x ] + [ 1 − x ] ∈ B ( F ). It can be shown that this element has order 6 for example when F = R ([21, Section 1]) or when F is a ﬁnite ﬁeld with q elements and q ≡ − 1 (mod 12) ([7, Lemma 7.11]). Theorem 3.12. Let F be a ﬁeld. Then (1) F or all x ∈ F × , h h x i i D F = ψ 1 ( x ) − ψ 2 ( x ) . (2) Let E be the ﬁeld extension obtained fr om F by adjoining a r oot of Φ ( T ) . Then h h x i i D F = 0 if ± x ∈ N E / F ( E × ) ⊂ F × . Pr oof. W e consider ﬁrst the case of a ﬁnite ﬁeld F . The results in the ﬁnal section of [7] sho w that the natural map RP ( F ) → P ( F ) induces an isomorphism R B ( F )  B ( F ). No w D F ∈ RB ( F ) and thus h h x i i D F = 0 for all x . Similarly for all x , ψ 1 ( x ) − ψ 2 ( x ) ∈ RB ( F ) and this maps to { x } − { x } = 0 in B ( F ). Thus, we can assume without loss that F is an inﬁnite ﬁeld. Let t : = " − 1 1 − 1 0 # ∈ SL 2 ( F ) . So t 2 = t − 1 = " 0 − 1 1 − 1 # and for x ∈ P 1 ( F ) we hav e t ( x ) = 1 − x − 1 and t − 1 ( x ) = (1 − x ) − 1 . Thus t ( x ) = x if and only if Φ ( x ) = 0. W e no w choose x ∈ F × \ { 1 } with t ( x ) , x (i.e. x not a root of Φ ( T )) and y ∈ F × \ { 1 } satisfying t ( y ) , y and y < { x , t ( x ) , t − 1 ( x ) } . By Lemma 2.8, the natural composite map Z / 3 = H 3 ( h t i , Z ) → H 3 (SL 2 ( F ) , Z ) → RB ( F ) is gi ven by the formula 1 7→ 2 X i = 0 cr( β x , y 3 (1 , t , t i + 1 , t i + 2 )) : = C ( x , y ) . Thus, using Lemma 2.8 again, we ha ve C ( x , y ) = cr( β x , y 3 (1 , t , t − 1 , 1)) + cr( β x , y 3 (1 , t , 1 , t ) = cr( y , t ( x ) , t − 1 ( x ) , x ) − cr( y , x , t ( x ) , t − 1 ( x )) + cr( y , t ( y ) , x , t ( x )) + cr( y , t ( y ) , t ( x ) , x ) = * ( t − 1 ( x ) − y )( y − t ( x )) t − 1 ( x ) − t ( x ) + " ( t − 1 ( x ) − t ( x ))( x − y ) ( t − 1 ( x ) − y )( x − t ( x )) # − * ( t ( x ) − y )( y − x ) t ( x ) − x + " ( t ( x ) − x )( t − 1 ( x ) − y ) ( t ( x ) − y )( t − 1 ( x ) − x ) # + * ( x − y )( y − t ( y )) x − t ( y ) + " ( x − t ( y ))( t ( x ) − y ) ( x − y )( t ( x ) − t ( y )) # + * ( t ( x ) − y )( y − t ( y )) t ( x ) − t ( y ) + " ( t ( x ) − t ( y ))( x − y ) ( t ( x ) − y )( x − t ( y )) # No w let a = a ( x , y ) = ( x − y )( t ( x ) − y ) t ( x ) − x . Reﬁned Bloch group and third homology of SL 2 15 Multiplying C ( x , y ) by the square class h a i gi ves h a i C ( x , y ) = * ( t − 1 ( x ) − t ( x ))( x − y ) ( t − 1 ( x ) − y )( x − t ( x )) + " ( t − 1 ( x ) − t ( x ))( x − y ) ( t − 1 ( x ) − y )( x − t ( x )) # − h − 1 i " ( t ( x ) − x )( t − 1 ( x ) − y ) ( t ( x ) − y )( t − 1 ( x ) − x ) # + * ( y − t ( y ))( x − t ( x ) ( x − t ( y ))( y − t ( x )) + " ( x − t ( y ))( t ( x ) − y ) ( x − y )( t ( x ) − t ( y )) # + * ( y − t ( y ))( t ( x ) − x ) ( x − y )( t ( x ) − t ( y ) + " ( t ( x ) − t ( y ))( x − y ) ( t ( x ) − y )( x − t ( y )) # No w let s = s ( x , y ) : = ( t − 1 ( x ) − t ( x ))( x − y ) ( t − 1 ( x ) − y )( x − t ( x )) and u = u ( x , y ) : = ( x − y )( t ( x ) − t ( y )) ( x − t ( y ))( t ( x ) − y ) . Thus h a i C ( x , y ) = h s i [ s ] − h − 1 i " 1 1 − s # + h 1 − u i h u − 1 i + D u − 1 − 1 E [ u ] = h s i [ s ] − h − 1 i " 1 1 − s # + ψ 2 ( u ) . No w t − 1 ( x ) − t ( x ) = x 2 − x + 1 x ( x − 1) , t − 1 ( x ) − y = 1 − y + xy 1 − x , x − t ( x ) = − x 2 − x + 1 x so that s = x − y 1 − y + xy . On the other hand, u = ( x − y ) 2 ( xy − y + 1)( xy − x + 1) = s · s s − 1 = s 1 − s − 1 . Hence C ( x , y ) = h a i E where E = E ( x , y ) = h s i [ s ] − h − 1 i " 1 1 − s # + ψ 2  s 1 − s − 1  . No w , from the deﬁnition of ψ 2 ( s ) and the identity h − 1 i ψ 2 ( a ) = ψ 2  a − 1  , we hav e h s − 1 i ψ 2  s − 1  = h 1 − s i ψ 2 ( s ) = h s i [ s ] + h s − 1 i . Furthermore D F = D  s − 1  = h s − 1 i + h − 1 i " 1 1 − s # − ψ 1 1 1 − s − 1 ! . It follo ws that E = h s − 1 i ψ 2  s − 1  − D F − ψ 1 1 1 − s − 1 ! + ψ 2  s 1 − s − 1  . But h s − 1 i =  s 1 − s − 1  . 16 KEVIN HUTCHINSON Thus, using the cocycle property of ψ 2 ( ) we get E =  s 1 − s − 1  ψ 2  s − 1  + ψ 2  s 1 − s − 1  − D F − ψ 1 1 1 − s − 1 ! = ψ 2 1 1 − s − 1 ! − ψ 1 1 1 − s − 1 ! − D F . No w let r : = 1 1 − s − 1 = x − y x − 1 − xy and observe that a = ( x − y )( t ( x ) − y ) t ( x ) − x = ( x − y )( x − 1 − xy ) x − 1 − x 2 = ( x − y )( x − 1 − xy ) − Φ ( x ) and hence h a i = h − Φ ( x ) r i . Thus C = C ( x , y ) = h − Φ ( x ) r i ( ψ 2 ( r ) − ψ 1 ( r ) − D F ) ∈ RB ( F ) (1) has order di viding 3 and is independent of the choice of x and y . For the remainder of the proof, we will suppose that x satisﬁes ( x + 1) Φ ( x ) , 0. Since for any r ∈ F × \ { x − 1 , x / ( x − 1) } we can solve for y y = r x − r − x r x − 1 ∈ F × then clearly r can assume an y nonzero v alue other than x − 1 and x / ( x − 1) by appropriate choice of y . In particular , taking r = − 1 and multiplying both sides of the equation by 4 we see that C = − h Φ ( x ) i D F since 4 ψ i ( − 1 ) = 0 while 4 C = C and 4 D F = D F . It follo ws that h Φ ( x ) i D F is independent of x . Using the identity Φ ( x 1 ) Φ ( x 2 ) = ( x 1 + x 2 − 1) 2 Φ x 1 x 2 − 1 x 1 + x 2 − 1 ! it follo ws that h Φ ( x 1 ) i h Φ ( x 2 ) i = * Φ x 1 x 2 − 1 x 1 + x 2 − 1 !+ ( = h Φ ( z ) i , say) . Thus, from h Φ ( x 2 ) i D F = h Φ ( z ) i D F we deduce, on multiplying by h Φ ( x 1 ) i , that h Φ ( z ) i D F = h Φ ( x 1 ) i h Φ ( z ) i D F and thus that h Φ ( x 1 ) i D F = D F . Since we also hav e h h − 1 i i D F = 0 (by Corollary 3.7 (1)), it follo ws more generally that D D ± Φ ( x ) z 2 E E D F = 0 for any z . Note that if E , F then a norm from E will be an element of the form Φ ( a ) b 2 for some a , b ∈ F × . Thus, statement (2) of the theorem follows immediately . No w choose r = b 2 in formula (1). This gives C = h Φ ( x ) i ( ψ 2  b 2  − ψ 1  b 2  ) − D F . Reﬁned Bloch group and third homology of SL 2 17 Multiplying both sides by 4 again sho ws that C = − D F . Using this, formula (1) says that − D F = h − Φ ( x ) r i ( ψ 2 ( r ) − ψ 1 ( r ) − D F ) or , equiv alently , h − Φ ( x ) r i D F = D F + ψ 1 ( r ) − ψ 2 ( r ) and hence h h − Φ ( x ) r i i D F = ψ 1 ( r ) − ψ 2 ( r ) for all r . Since h h − Φ ( x ) i i D F = 0 for all x and since h h − Φ ( x ) r i i = h h − Φ ( x ) i i h h r i i + h h − Φ ( x ) i i + h h r i i it follo ws that h h r i i D F = ψ 1 ( r ) − ψ 2 ( r ) for all r ∈ F × , proving statement (1) of the theorem.  Remark 3.13. W e will see belo w that statement (2) is in general best possible. For example if F is a local or global ﬁeld not containing a primiti ve cube root of unity , ζ 3 , then often we have h h x i i D F , 0 (and hence ψ 1 ( x ) , ψ 2 ( x ) ) when x is not a norm from E = F ( ζ 3 ). Remark 3.14. Observe that the element t chosen in the last proof lies in the image of SL 2 ( Z ) → SL 2 ( F ). No w it is well known that SL 2 ( Z ) can be expressed as an amalg amated product C 4 ∗ C 2 C 6 . Here C 4 is cyclic of order 4 with generator a and C 6 is cyclic of order 6 with generator b with a = " 0 1 − 1 0 # and b = " 0 − 1 1 − 1 # . Thus b 2 = t is our matrix of order 3. A straightforward direct calculation, using this decompo- sition (see, for example, [10, Theorem 4.1.1]), sho ws that H 3 (SL 2 ( Z ) , Z ) is cyclic of order 12 and that the inclusion G : = h t i → SL 2 ( Z ) induces an isomorphism H 3 ( G , Z )  H 3 (SL 2 ( Z ) , Z ) (3) . It follo ws that for any ﬁeld F , the image of the natural map H 3 (SL 2 ( Z ) , Z ) (3) → RB ( F ) (3) is the cyclic subgroup generated by − D F . 4. V alu a tions and S pecializa tion In this section, we prove the existence of specialization or reduction maps from the reﬁned pre-Bloch group of a ﬁeld to the reﬁned pre-Bloch group of the residue ﬁeld of a valuation. W e use these specialization maps to obtain lo wer bounds for the groups I F RB ( F ) for ﬁelds with v aluations. 4.1. Some preliminary deﬁnitions. In this subsection we deﬁne certain quotients of the clas- sical and reﬁned Bloch and pre-Bloch groups which we will use in the remainder of the paper . For an y ﬁeld F we will let g RP ( F ) : = RP ( F ) / K (1) F . Observe that for any x we hav e λ 1 ( ψ 1 ( x ) ) = h h − x i i · h h x i i ∈ I 2 F , by the proof of Lemma 3.3, and λ 2 ( ψ 1 ( x ) ) = (1 − x ) ◦ x + (1 − x − 1 ) ◦ x − 1 = (1 − x ) ◦ x − 1 − x − x ! ◦ x = (1 − x ) ◦ x − (1 − x ) ◦ x + ( − x ) ◦ x = ( − x ) ◦ x ∈ S 2 Z ( F × ) . 18 KEVIN HUTCHINSON It follows that Λ ( ψ 1 ( x ) ) = [ − x , x ] ∈ RS 2 Z ( F × ) so that Λ ( K (1) F ) is the R F -submodule of RS 2 Z ( F × ) generated by the symbols [ − x , x ]. Thus we set ] RS 2 Z ( F × ) : = RS 2 Z ( F × ) / Λ ( K (1) F ) and let e Λ : g RP ( F ) → ] RS 2 Z ( F × ) be the resulting R F -homomorphism induced by Λ . Finally we let g RB ( F ) : = K er( e Λ ). Lemma 4.1. F or any ﬁeld F the natural map RP ( F ) → g RP ( F ) induces a surjection R B ( F ) → g RB ( F ) whose kernel is annihilated by 4 . In particular , RB ( F )[ 1 2 ]  g RB ( F )[ 1 2 ] . Pr oof. The surjectivity of the map is clear from the deﬁnitions. On the other hand, the kernel of the map RB ( F ) → g RB ( F ) is RB ( F ) ∩ K (1) F which is contained in K er( λ 1 | K (1) F ) and which in turn is annihilated by 4 by Lemma 3.3.  For ﬁnite ﬁelds, the results of [7] allo w us to be more precise: Lemma 4.2. Let F q be a ﬁnite ﬁeld with q elements. (1) If q is even or if q ≡ 1 (mod 4) then B ( F q ) = RB ( F q ) = g RB ( F q ) . This gr oup is cyclic of or der ( q + 1) when q is even and ( q + 1) / 2 when q ≡ 1 (mod 4) . (2) If q ≡ 3 (mod 4) then g RB ( F q )  B ( F q ) / h{ − 1 }i is cyclic of or der ( q + 1) / 4 . Pr oof. The cases q = 2 or q = 3 are immediate. When q ≥ 4, by [7, Lemma 7.1], the natural map RP ( F q ) → P ( F q ) induces an isomorphism RB ( F q )  B ( F q ) and for all x the image of ψ 1 ( x ) ∈ R B ( F q ) is { x } ∈ B ( F q ), which has order di visible by 2. If q is ev en or if q ≡ 1 (mod 4) then B ( F q ) is cyclic of order q + 1 or ( q + 1) / 2 respecti vely , and hence is of odd order . On the other hand, if q ≡ 3 (mod 4) then [7, Lemma 7.8] sho ws that K (1) F q = h ψ 1 ( − 1 ) i is c yclic of order 2 and is contained in RB ( F q )  B ( F q ).  W e let K F denote the R F -submodule K (1) F + K (2) F of RP ( F ). Lemma 4.3. F or any ﬁeld F we have K F = K (1) F + I F C F and K F  K (1) F ⊕ I F D F . Pr oof. By Theorem 3.12, for any ﬁeld F we ha ve an identity of R F -submodules of RP ( F ) K F = K (1) F + K (2) F = K (1) F + I F D F . On the other hand since D F ≡ " 1 x # + h − 1 i " 1 1 − x # (mod K (1) F ) C F ≡ [ x ] + h − 1 i [ 1 − x ] (mod K (1) F ) and h x − 1 i ≡ − h − 1 i [ x ] (mod K (1) F ) , Reﬁned Bloch group and third homology of SL 2 19 it follo ws that C F ≡ − h − 1 i D F ≡ − D F (mod K (1) F ) (using Corollary 3.7 (1)) and thus K (1) F + I F C F = K (1) F + I F D F . Finally , suppose that a ∈ K (1) F ∩ I F D F . Then 3 a = 0 since 3 D F = 0 and 4 a = 0 since a ∈ K (1) F ∩ RB ( F ), so that a = 0. Thus K (1) F ∩ I F D F = 0.  For an y ﬁeld F we set c RP ( F ) : = RP ( F ) / K F = g RP ( F ) / I F D F and d RB ( F ) = g RB ( F ) / I F D F . Corollary 4.4. F or any ﬁeld F ther e is natural short e xact sequence 0 → I F D F → g RB ( F ) → d RB ( F ) → 0 . If F is a ﬁnite ﬁeld, by Lemma 2.4, RB ( F )  B ( F ) and hence the action of G F on RB ( F ) is tri vial and thus we hav e Corollary 4.5. F or any ﬁnite ﬁeld F , we have d RB ( F ) = g RB ( F ) . W e will use the follo wing identities repeatedly below: Lemma 4.6. F or any ﬁeld F and any x ∈ F × , we have h x i [ x ] = h − 1 i [ x ] = − h x − 1 i in c RP ( F ) . Pr oof. Since ψ 1 ( x ) = 0 in c RP ( F ), it follows that h − 1 i [ x ] = − h x − 1 i . Since ψ 2 ( x ) = 0, we have h x i [ x ] = − h x − 1 i also.  W e will also need to consider the corresponding quotient of the classical pre-Bloch group: Let S F denote the (2-torsion) subgroup of P ( F ) generated by the elements { x } , and let b P ( F ) : = P ( F ) S F Thus, the natural map RP ( F ) → P ( F ) induces an ismorphism c RP ( F ) F ×  b P ( F ). Recall that D F denotes the R F -module R F D F . Finally , we let RP ( F ) : = RP ( F ) K (1) F + D F = g RP ( F ) D F = c RP ( F ) Z · D F and RB ( F ) : = g RB ( F ) D F . In section 6 belo w , we will also need the corresponding classical version: P ( k ) : = P ( k ) Z D k + S k = b P ( k ) Z D k . Note that D F = I F D F + Z D F and that the sum K (1) F + D F is direct (by the ar gument of Lemma 4.3). Thus we have 20 KEVIN HUTCHINSON Lemma 4.7. F or any ﬁeld F ther e are short e xact sequences 0 → D F → g RB ( F ) → R B ( F ) → 0 and 0 → Z D F → d RB ( F ) → R B ( F ) → 0 . 4.2. V aluations. Now let Γ be an ordered (additiv e) abelian group and let v : F × → Γ be a v aluation, which we will always assume to be surjective . As usual, let O = O v = { 0 } ∪ { x ∈ F × | v ( x ) ≥ 0 } be the v aluation ring, let M = M v = { 0 } ∪ { x ∈ F × | v ( x ) > 0 } be the maximal ideal, let k = k v = O / M be the residue ﬁeld and U = U v = { x ∈ F × | v ( x ) = 0 } = O \ M . Also let f v be the group homomorphism U → k × , u 7→ ¯ u : = u + M , and let U 1 = U 1 , v = K er( f v ) = 1 + M . Since Γ is a torsion-free group we hav e a short exact sequence of F 2 -vector spaces 1 → U / U 2 → G F → Γ / 2 → 0 . (2) W e hav e homomorphisms of commutativ e rings Z [ U / U 2 ]   / /     R F R k Thus, if M is any R k module M F = M F , v : = R F ⊗ Z [ U / U 2 ] M is an R F -module. W e will require the follo wing result sev eral times in the next section: Lemma 4.8. Let F be a ﬁeld with valuation v : F × → Γ and corr esponding r esidue ﬁeld k . Let a , b ∈ F and suppose that v ( a ) ≡ v ( b ) (mod 2 Γ ) . Then h a i ⊗ C k = h b i ⊗ C k in c RP ( k ) F . Pr oof. W e hav e v ( a ) = v ( b ) + 2 γ for some γ ∈ Γ . Choose c ∈ F × with v ( c ) = γ . Then u = a / ( bc 2 ) ∈ U . So h a i − h b i = h b i h h u i i in R F . Thus h a i ⊗ C k − h b i ⊗ C k = h a i ⊗ h h ¯ u i i C k = 0 since h h ¯ u i i C k ∈ I k C k and hence represents 0 in c RP ( k ).  4.3. The specialization homomorphisms. Theorem 4.9. Let F be a ﬁeld with valuation v and corr esponding r esidue ﬁeld k . Then ther e is a surjective R F -module homomorphism S v : RP ( F ) → c RP ( k ) F [ a ] 7→          1 ⊗ [ ¯ a ] , v ( a ) = 0 1 ⊗ C k , v ( a ) > 0 − (1 ⊗ C k ) , v ( a ) < 0 Reﬁned Bloch group and third homology of SL 2 21 Pr oof. Let Z 1 denote the set of symbols of the form [ x ] , x , 1 and let T : R F [ Z 1 ] → c RP ( k ) F be the unique R F -homomorphism gi ven by [ a ] 7→          1 ⊗ [ ¯ a ] , v ( a ) = 0 1 ⊗ C k , v ( a ) > 0 − (1 ⊗ C k ) , v ( a ) < 0 W e must prov e that T ( S x , y ) = 0 for all x , y ∈ F × \ { 1 } . Through the remainder of this proof we will adopt the following notation: Giv en x , y ∈ F × \ { 1 } , we let u = y x and w = 1 − x 1 − y . Note that 1 − x − 1 1 − y − 1 = y x · x − 1 y − 1 = uw . Thus, with this notation, S x , y becomes [ x ] −  y  + h x i [ u ] − D x − 1 − 1 E [ uw ] + h 1 − x i [ w ] . W e di vide the proof into sev eral cases: Case (i): v ( x ) , v ( y ) , 0 Subcase (a): v ( x ) = v ( y ) > 0. Then 1 − x , 1 − y ∈ U 1 and hence w ∈ U 1 , so that ¯ w = 1 and uw = ¯ u . Thus T ( S x , y ) = 1 ⊗ C k − 1 ⊗ C k + h x i ⊗ [ ¯ u ] − D x − 1 − 1 E ⊗ [ ¯ u ] . Ho wev er , x − 1 − 1 = x − 1 (1 − x ), so that D x − 1 − 1 E ⊗ ¯ u = h x i ⊗ h 1 − ¯ x i [ ¯ u ] = h x i ⊗ [ ¯ u ] , and thus T ( S x , y ) = 0 as required. Subcase (b): v ( x ) = v ( y ) < 0. Then u ∈ U and uw ∈ U 1 so that ¯ w = ¯ u − 1 . Thus T ( S x , y ) = − 1 ⊗ C k + 1 ⊗ C k + h x i ⊗ [ ¯ u ] + h 1 − x i ⊗ h ¯ u − 1 i . But 1 − x = − x (1 − x − 1 ) and 1 − x − 1 ∈ U 1 , so that the last term is h − x i ⊗ h ¯ u − 1 i and hence T ( S x , y ) = h x i ⊗ ψ 1 ( ¯ u ) = 0 in c RP ( k ) F . Subcase (c): v ( x ) > v ( y ) > 0. Then w ∈ U 1 and v ( u ) , v ( uw ) < 0. So T ( S x , y ) = 1 ⊗ C k − 1 ⊗ C k − h x i ⊗ C k + D x − 1 − 1 E ⊗ C k . But since x − 1 − 1 = x − 1 (1 − x ) and 1 − x ∈ U 1 it follows that D x − 1 − 1 E ⊗ C k = h x i ⊗ C k and hence T ( S x , y ) = 0. Subcase (d): v ( x ) > 0 > v ( y ). Then v ( u ) = v ( y ) − v ( x ) < 0, v ( w ) = − v (1 − y ) = − v ( y ( y − 1 − 1)) = − v ( y ) > 0 and v ( uw ) = v ( u ) + v ( w ) = − v ( x ) < 0. So T ( S x , y ) = 1 ⊗ C k + 1 ⊗ C k − h x i ⊗ C k + D x − 1 − 1 E ⊗ C k + h 1 − x i ⊗ C k . But 1 − x ∈ U 1 and D x − 1 − 1 E = h x i h 1 − x i . So this gi ves T ( S x , y ) = 1 ⊗ 3 C k . Ho wev er , 3 C k = − 3 D k = 0 in c RP ( k ). Subcase (e): 0 > v ( x ) > v ( y ). Then v ( u ) = v ( y ) − v ( x ) < 0 and uw ∈ U 1 , so that v ( w ) = − v ( u ) > 0 and ¯ w = ¯ u − 1 . Thus T ( S x , y ) = − (1 ⊗ C k ) + 1 ⊗ C k − h x i ⊗ C k + h 1 − x i ⊗ C k . 22 KEVIN HUTCHINSON No w 1 − x = − x (1 − x − 1 ) and 1 − x − 1 ∈ U 1 , so the last term is h x i ⊗ h − 1 i C k = h x i ⊗ C k by Corollary 3.7 (1). This gives T ( S x , y ) = 0 as required. Subcases (f),(g),(h): The corresponding calculations when v ( y ) > v ( x ) are almost identical. Case (ii): x , y ∈ U 1 . Subcase (a): v (1 − x ) , v (1 − y ). Then u ∈ U 1 and v ( w ) = v ( uw ) , 0. So T ( S x , y ) = ± D x − 1 − 1 E ⊗ C k − h 1 − x i ⊗ C k  = 0 since D x − 1 − 1 E = D x − 1 E h 1 − x i and x − 1 ∈ U 1 . Subcase (b): v (1 − x ) = v (1 − y ). Then u ∈ U 1 and w , uw ∈ U with ¯ uw = ¯ w . So T ( S x , y ) = − D x − 1 − 1 E ⊗ [ ¯ w ] + h 1 − x i ⊗ [ ¯ w ] which is 0 by the same argument as the pre vious (sub)case. Case (iii): x ∈ U 1 , v ( y ) , 0. Then v ( u ) = v ( y ). Observe that v (1 − y ) = min( v (1) , v ( y )) = min(0 , v ( y )) ≤ 0. Of course, v (1 − x ) > 0. Thus v ( w ) = v (1 − x ) − v (1 − y ) > 0 and v ( uw ) = v ( u ) + v ( w ) = v (1 − x ) + v ( y ) − v (1 − y ) > 0 since v ( y ) − v (1 − y ) ≥ 0. So T ( S x , y ) = ± (1 ⊗ C k − 1 ⊗ C k ) − D x − 1 − 1 E ⊗ C k + h 1 − x i ⊗ C k = 0 since x ∈ U 1 and thus D x − 1 − 1 E = h 1 − x i . Case (i v): v ( x ) , 0, y ∈ U 1 Arguing as in the last case, v ( w ) , v ( uw ) < 0 in this case. Subcase (a): v ( x ) > 0 Then v ( u ) = − v ( x ) < 0. So T ( S x , y ) = 1 ⊗ C k − h x i ⊗ C k + D x − 1 − 1 E ⊗ C k − h 1 − x i ⊗ C k = 0 using Lemma 4.8 together with the identities v (1 − x ) = 0 = v (1) and v ( x − 1 − 1) = − v ( x ). Subcase (b): v ( x ) < 0. Then v ( u ) > 0 and T ( S x , y ) = − 1 ⊗ C k + h x i ⊗ C k + D x − 1 − 1 E ⊗ C k − h 1 − x i ⊗ C k . This vanishes by Lemma 4.8 together with the identities v (1 − x ) = v ( x ) and v ( x − 1 − 1) = 0 = v (1). Case (v): x ∈ U \ U 1 and v ( y ) , 0. Subcase (a): v ( y ) > 0. Then v ( u ) = v ( y ) > 0. Since 1 − x ∈ U and 1 − y ∈ U 1 , it follows that v ( w ) = 0, ¯ w = 1 − ¯ x in k and v ( uw ) > 0. Thus T ( S x , y ) = 1 ⊗ [ ¯ x ] − 1 ⊗ C k + h x i ⊗ C k − D x − 1 − 1 E ⊗ C k + h 1 − x i ⊗ [ 1 − ¯ x ] = 1 ⊗ ( [ ¯ x ] + h 1 − ¯ x i [ 1 − ¯ x ] − C k ) using Lemma 4.8 to eliminate terms. Now h 1 − ¯ x i [ 1 − ¯ x ] = h − 1 i [ 1 − ¯ x ] in c RP ( k ) by Lemma 4.6, and hence [ ¯ x ] + h 1 − ¯ x i [ 1 − ¯ x ] = C k in c RP ( k ). Thus we conclude that T ( S x , y ) = 0 as required. Reﬁned Bloch group and third homology of SL 2 23 Subcase (b): v ( y ) < 0 W e hav e v ( u ) = v ( y ) < 0 and v ( w ) = − v (1 − y ) = − v ( y ) > 0. Thus v ( uw ) = v ( u ) + v ( w ) = v ( y ) − v ( y ) = 0. Furthermore, since 1 − y − 1 ∈ U 1 , ¯ uw = 1 − ¯ x − 1 . Thus (using Lemma 4.8 again) S ( T x , y ) = 1 ⊗ [ ¯ x ] + 1 ⊗ [ k ] − h x i ⊗ C k − D x − 1 − 1 E ⊗ h 1 − ¯ x − 1 i + h 1 − x i ⊗ C k = 1 ⊗  C k + [ ¯ x ] − D ¯ x − 1 − 1 E h 1 − ¯ x − 1 i = 1 ⊗  C k + [ ¯ x ] − D ¯ x − 1 − 1 E h 1 − ¯ x − 1 i . No w D ¯ x − 1 − 1 E h 1 − ¯ x − 1 i = h 1 − ¯ x − 1 i in c RP ( k ) by Lemma 4.6, and [ ¯ x ] = − h − 1 i h ¯ x − 1 i since ψ 1 ( ¯ x ) = 0. Thus [ ¯ x ] − D ¯ x − 1 − 1 E h 1 − ¯ x − 1 i = − h − 1 i ( h ¯ x − 1 i + h − 1 i h 1 − ¯ x − 1 i ) = − h − 1 i C k = − C k in c RP ( k ) and again T ( S x , y ) = 0 as required. Case (vi): y ∈ U \ U 1 and v ( x ) , 0 Subcase (a): v ( x ) > 0 we have v ( u ) = − v ( x ) < 0, v ( w ) = 0 and 1 − x ∈ U 1 so that ¯ w = (1 − ¯ y ) − 1 . Finally v ( uw ) = v ( u ) < 0. Thus T ( S x , y ) = 1 ⊗ C k − 1 ⊗  ¯ y  − h x i ⊗ C k + D x − 1 − 1 E ⊗ C k + h 1 − x i ⊗ " 1 1 − ¯ y # = 1 ⊗ C k −  ¯ y  + " 1 1 − ¯ y #! using Lemma 4.8 again (after observing that v ( x − 1 − 1) = − v ( x ) when v ( x ) > 0). Ho wev er " 1 1 − ¯ y # = − h − 1 i  1 − ¯ y  in c RP ( k ), and hence T ( S x , y ) = 1 ⊗ ( C k − C k ) = 0. Subcase (b): v ( x ) < 0 W e have v ( u ) = − v ( x ) > 0 and v ( w ) = v (1 − x ) = v ( x ) < 0 and v ( uw ) = v ( u ) + v ( w ) = 0. Furthermore, since 1 − x − 1 ∈ U 1 , ¯ uw = 1 / (1 − ¯ y − 1 ). Thus T ( S x , y ) = − (1 ⊗ C k ) − 1 ⊗  ¯ y  + h x i ⊗ C k − D x − 1 − 1 E ⊗ " 1 1 − ¯ y − 1 # − h 1 − x i ⊗ C k = 1 ⊗ − C k −  ¯ y  − h − 1 i " 1 1 − ¯ y − 1 #! using Lemma 4.8 and the fact that D x − 1 − 1 E = h − 1 i D 1 − x − 1 E and 1 − x − 1 ∈ U 1 . But in c RP ( k ) we ha ve, by Lemma 4.6, −  ¯ y  − h − 1 i " 1 1 − ¯ y − 1 # = h − 1 i h ¯ y − 1 i + h 1 − ¯ y − 1 i = C k . Case (vii): x ∈ U \ U 1 and y ∈ U 1 . 24 KEVIN HUTCHINSON W e hav e v ( u ) = 0 and ¯ u = ¯ x − 1 . Furthermore, v ( w ) = − v (1 − y ) < 0 and thus v ( uw ) = v ( w ) < 0 also. Thus T ( S x , y ) = 1 ⊗ [ ¯ x ] + h x i ⊗ h ¯ x − 1 i + D x − 1 − 1 E ⊗ C k − h 1 − x i ⊗ C k = 1 ⊗  [ ¯ x ] + h ¯ x i h ¯ x − 1 i (by Lemma 4.8) = 1 ⊗ h ¯ x i h 1 − ¯ x i ψ 2 ( ¯ x ) = 0 . Case (viii): x ∈ U 1 and y ∈ U \ U 1 W e hav e u ∈ U and ¯ u = ¯ y , v ( w ) = v (1 − x ) > 0 and v ( uw ) = v ( w ) > 0. So T ( S x , y ) = − (1 ⊗  ¯ y  ) + h x i ⊗  ¯ y  − D x − 1 − 1 E ⊗ C k + h 1 − x i ⊗ C k = 0 by Lemma 4.8 and the fact that x ∈ U 1 . Case (ix): x , y ∈ U \ U 1 In this case ¯ x , ¯ y ∈ k × \ { 1 } and T ( S x , y ) = 1 ⊗ S ¯ x , ¯ y = 1 ⊗ 0 .  Gi ven a valuation v on a ﬁeld F and an element a ∈ F × with v ( a ) , 0, we will let  v ( a ) denote sign( v ( a )) ∈ {± 1 } . Suppose that ρ : R F → R k is a ring homomorphism satisfying ρ ( h u i ) = h ¯ u i for any u ∈ U . Then by composing S v with the surjecti ve R F -module homomorphism c RP ( k ) F → c RP ( k ) , x ⊗ y 7→ ρ ( x ) y . we obtain: Corollary 4.10. Let ρ : R F → R k be any ring homomorphism satisfying ρ ( h u i ) = h ¯ u i if u ∈ U . Then ρ induces a R F -module structur e on R P ( k ) and ther e is a surjective homomorphism of R F -modules S = S ρ : RP ( F ) → c RP ( k ) determined by [ a ] 7→ ( [ ¯ a ] , a ∈ U v  v ( a ) C k , a < U v No w , if we choose a splitting, j : Γ / 2 → G F of the sequence (2), we obtain a ring isomorphism R F  Z [ U / U 2 × j ( Γ / 2)]  Z [ U / U 2 ] ⊗ Z Z [ Γ / 2] . On the other hand, any character χ : Γ / 2 → µ 2 = { 1 , − 1 } induces a ring homomorphism Z [ Γ / 2] → Z . Thus, gi ven a splitting j and a character χ , we obtain a ring homomorphism ρ = ρ j ,χ : R F → R k , h u · j ( γ ) i 7→ χ ( γ ) h ¯ u i and a corresponding surjecti ve specialization map S = S j ,χ : RP ( F ) → c RP ( k ) which is also an R F -homomorphism. Example 4.11. For example, if v is a discrete v aluation, then Γ = Z and Γ / 2 = Z / 2. Any choice, π , of uniformizing parameter , determines a splitting Z / 2 → G F , 1 7→ h π i . There are two characters,  , on Z / 2, namely 1 and − 1. Thus we get ring homomorphisms ρ π, : R F → R k , h u π r i 7→  r h ¯ u i . and corresponding specialization homomorphisms S π, : RP ( F ) → c RP ( k ) . Reﬁned Bloch group and third homology of SL 2 25 In general, the specialization homomorphisms S j ,χ do not restrict to homomorphisms RB ( F ) → d RB ( k ) of Bloch groups. T o see this, we begin with the follo wing observation: Lemma 4.12. F or a ﬁeld F and x ∈ F × we have p + − 1 h h x i i g RP ( F ) ⊂ g RB ( F ) . Pr oof. Since G F acts tri vially on S 2 Z ( F × ) we hav e h h x i i S 2 Z ( F × ) = 0 and thus e Λ ( h h x i i p + − 1 [ a ] ) = ( h h x i i h h 1 − a i i p + − 1 h h a i i , 0) = ( h h x i i h h 1 − a i i h h − a i i h h a i i , 0) = h h x i i h h 1 − a i i [ − a , a ] = 0 ∈ ] RS 2 Z ( F × ) .  Corollary 4.13. Let F be a ﬁeld with valuation v : F × → Γ , with corr esponding r esidue ﬁeld k. Let j : Γ / 2 → G F be a splitting, and let χ be a nontri vial char acter on Γ / 2 . Then the image of S j ,χ | RB ( F ) : RB ( F ) → c RP ( k ) contains 2p + − 1 c RP ( k ) . Pr oof. First observe that S j ,χ induces a homomorphism g RP ( F ) → c RP ( k ) since S j ,χ ( ψ 1 ( x ) ) = ( ψ 1 ( ¯ x ) , v ( x ) = 0 0 , v ( x ) , 0 Suppose that γ ∈ Γ with χ ( γ ) = − 1 and let π : = j ( γ ). Since c RP ( k ) is an R F -module via the homomorphism ρ j ,χ , h π i acts on c RP ( k ) as multiplication by − 1. Thus h h π i i c RP ( k ) = 2 c RP ( k ) and hence S j ,χ induces a surjecti ve homomorphism h h π i i p + − 1 g RP ( F ) → 2p + − 1 c RP ( k ) .  Remark 4.14. As just noted, the maps S j ,χ : R P ( F ) → c RP ( k ) descend to maps g RP ( F ) → c RP ( k ). Howe ver , they do not usually induce maps c RP ( F ) → c RP ( k ). In fact, below we will use the homomorphisms S j ,χ to detect the non-tri viality of I F C F for global and (some) local ﬁelds. Corollary 4.15. Let F be a ﬁeld with valuation v : F × → Γ and quadratically closed r esidue ﬁeld k . Let j : Γ / 2 → G F be a splitting, and let χ : Γ / 2 → µ 2 be a nontrivial character . Then S j ,χ induces a surjective map RB ( F ) / / / / RP ( k ) . Pr oof. Since k is quadratically closed, RP ( k ) = P ( k ) and ψ i ( x ) = { x } = 0 for all x for i = 1 , 2. Thus c RP ( k ) = P ( k ) and p + − 1 c RP ( k ) = p + − 1 P ( k ) = 2 P ( k ). Ho wev er, since e very element of k × is a square, P ( k ) is 2-divisible ([4, Lemma 5.8])and thus 2p + − 1 RP ( k ) = 4 P ( k ) = P ( k ) = RP ( k ).  26 KEVIN HUTCHINSON Corollary 4.16. Let F be a ﬁeld with valuation v : F × → Γ and corresponding r esidue ﬁeld k . Let j : Γ / 2 → G F be a splitting, and let χ : Γ / 2 → µ 2 be a nontri vial char acter . Choose γ ∈ Γ with χ ( γ ) = − 1 and let π = j ( γ ) . Then the map S j ,χ induces a surjective homomorphism of R F [ 1 2 ] -modules e − π e + − 1  RB ( F )[ 1 2 ]  → e + − 1  c RP ( k )[ 1 2 ]  In particular , if − 1 ∈ ( F × ) 2 , then ther e is a surjective homomorphism e − π  RB ( F )[ 1 2 ]  → c RP ( k )[ 1 2 ] Pr oof. Since e − π e + − 1  g RP ( F )[ 1 2 ]  ⊂ g RB ( F )[ 1 2 ] by Lemma 4.12, it follo ws that e − π e + − 1  g RP ( F )[ 1 2 ]  = e − π e + − 1  g RB ( F )[ 1 2 ]   e − π e + − 1  RB ( F )[ 1 2 ]  (using Lemma 4.1)  Remark 4.17. In general, the specialization maps S j ,χ depend on the choice of splitting j and character χ . T o spell this out: If γ ∈ Γ then the R F -module structure on c RP ( k ) associated to S j ,χ requires that h j ( γ ) i acts as multiplication by χ ( γ ). If j 0 is another splitting, then we will hav e j ( γ ) = j 0 ( γ ) · u for some u ∈ U , and the action of h j ( γ ) i with respect to the module-structure associated to the pair ( j 0 , γ ) will be multipication by h ¯ u i χ ( γ ). Note that h ¯ u i ∈ R k . W e can free ourselves of the dependence on the choice of splitting j (or the choice of uni- formizer π , in the case of a discrete valuation) by composing S j ,χ with the natural surjectiv e map c RP ( k ) → b P ( k ) . Since R k acts tri vially on P ( k ), we get well-deﬁned surjecti ve specialization maps S χ : RP ( F ) → b P ( k ) which depend only on the choice of character χ : Γ / 2 → µ 2 . It is important to note that although b P ( k ) is a tri vial R k -module, the R F -module structure in- duced from a character χ is not usually trivial. Thus, if χ is not the trivial character and if π ∈ G F satisﬁes χ ( v ( π )) = − 1, then π acts as multiplication by − 1 on b P ( k ) and S χ induces a surjecti ve homomorphism e − π  RB ( F )[ 1 2 ]  → b P ( k )[ 1 2 ] 5. A pplica tions : global fields From the e xistence of these specialization maps it follows that if F is ﬁeld with many valua- tions, then H 3 (SL 2 ( F ) , Z ) 0 must be large. For e xample: Theorem 5.1. Let F be a ﬁeld and let V be a family of discr ete values on F satisfying (1) F or any x ∈ F × , v ( x ) = 0 for all but ﬁnitely many v ∈ V (2) The map F × → ⊕ v ∈V Z / 2 , a 7→ { v ( a ) } v is surjective. Reﬁned Bloch group and third homology of SL 2 27 Then ther e is a natural surjective homomorphism H 3 (SL 2 ( F ) , Z [ 1 2 ]) 0 → M v ∈V b P ( k v )[ 1 2 ] . Pr oof. W e hav e H 3 (SL 2 ( F ) , Z [ 1 2 ]) 0 = RB ( F ) 0 [ 1 2 ] = I F RB ( F )[ 1 2 ] = X a ∈ G F e − a RB ( F )[ 1 2 ] by Lemma 2.3. W e denote by S v , − 1 the specialization map RP ( F ) → b P ( k v ) corresponding to the character  = − 1 (see Remark 4.17). Note that if a ∈ F × with v ( a ) ≡ 0 (mod 2), then for an y y ∈ RP ( F ), S v (e − a y ) = e − a S v ( y ) = 0 since h a i acts trivially on P ( k v ). Thus, for any a ∈ F × , y ∈ RP ( F ), we hav e S v , − 1 (e − a y ) = 0 for almost all v . It follows that for any x ∈ I F RB ( F )[ 1 2 ], S v , − 1 ( x ) = 0 for almost all v ∈ V . Thus the maps S v , − 1 induce a well-deﬁned R F [ 1 2 ]-homomorphism S : RB ( F )[ 1 2 ] 0 → ⊕ v b P ( k v )[ 1 2 ] , x 7→ { S v , − 1 ( x ) } v . Finally , let y = { y v } v ∈ ⊕ v b P ( k v )[ 1 2 ]. Let T = { v ∈ V | y v , 0 } . F or each v ∈ T , choose x v ∈ RB ( F )[ 1 2 ] with S v , − 1 ( x v ) = y v . For each v ∈ T , choose π v ∈ F × satisfying w ( π v ) ≡ δ w , v (mod 2) for all w ∈ V . Then y = S        X v ∈ T e − π v · x v        .  Since S k is a 2-torsion group, we hav e b P ( k )[ 1 2 ] = P ( k )[ 1 2 ] for a ﬁnite ﬁeld k , and we deduce: Corollary 5.2. Let F be a global ﬁeld and let S be a ﬁnite set of places of F . Let O S = { a ∈ F × | v ( a ) ≥ 0 for all v < S } ∪ { 0 } . Suppose that Cl( O S ) / 2 = 0 . Let V be the set of all ﬁnite places of F not in S . Then there is a surjective homomorphism of R F -modules H 3 (SL 2 ( F ) , Z [ 1 2 ]) 0 → M v ∈V P ( k v )[ 1 2 ] . Her e, if π v is a uniformizer for v, then the squar e class of π v acts as − 1 on the factor P ( k v )[ 1 2 ] on the right. Remark 5.3. Note that it follows that the groups H 3 (SL 2 ( F ) , Z ) are not ﬁnitely generated, for any global ﬁeld F . By contrast, when F is a global ﬁeld, the groups H 3 (SL 3 ( F ) , Z )  K ind 3 ( F ) are well-kno wn to be ﬁnitely generated. Furthermore the stabilization map H 3 (SL 2 ( F ) , Z ) → H 3 (SL 3 ( F ) , Z ) is kno wn to be surjectiv e in this case, by the results of [8]. Remark 5.4. Observe that in the case F = Q the short exact sequence 0 → H 3 (SL 2 ( Q ) , Z [ 1 2 ]) 0 → H 3 (SL 2 ( Q ) , Z [ 1 2 ]) → K ind 3 ( Q )[ 1 2 ] → 0 is split, since the composite H 3 (SL 2 ( Z ) , Z [ 1 2 ]) → H 3 (SL 2 ( Q ) , Z [ 1 2 ]) → K ind 3 ( Q )[ 1 2 ] = B ( Q )[ 1 2 ] = Z · D Q 28 KEVIN HUTCHINSON is an isomorphism (of groups of order 3). Thus there is a natural surjective homomorphism H 3 (SL 2 ( Q ) , Z [ 1 2 ]) / / / / K ind 3 ( Q )[ 1 2 ] ⊕  L p P ( F p )[ 1 2 ]  When k is algebraically closed, b P ( k ) = P ( k ) = RP ( k ) and P ( k ) is uniquely divisible – i.e. a Q -vector space – by the results of Suslin ([21]). Corollary 5.5. Let k be an algebraically closed ﬁeld and let X be an algebraic variety over k , with rational function ﬁeld k ( X ) . Suppose that X is r e gular in codimension 1 and that Cl( X ) / 2 = 0 . Let V be the set of codimension 1 subvarieties. The theor em says that ther e is a surjective homomorphism of R k ( X ) -modules H 3 (SL 2 ( k ( X )) , Q ) 0 → ⊕ v ∈ V P ( k ) Remark 5.6. Note that this sho ws that H 3 (SL 2 ( F ) , Z ) 0 may hav e a large torsion-free part. For an y global ﬁeld F we ha ve H 3 (SL 2 ( F ) , Z ) = lim − − → S H 3 (SL 2 ( O S ) , Z ) (where S v aries ov er all ﬁnite sets of places). It is natural to inquire to what extent the results described above for ﬁelds may extend to rings of S -integers O S (at least when S is large). For example, it follo ws from the equation just quoted that the square classes of O × S must act nontri vially on the group H 3 (SL 2 ( O S ) , Z ) for lar ge S . As noted in the introduction, there are very fe w explicit calculations of the homology or coho- mology of S -arithmetic groups such as SL 2 ( O S ), in spite of their great interest in geometry and number theory . (Over number ﬁelds, the homology groups are kno wn to be ﬁnitely-generated ([3]), and calculation of homology and cohomology are essentially equiv alent by the uni versal coe ﬃ cient theorem.) For example, the cohomology of the groups SL 2 ( Z ) and SL 2 ( Z [1 / p ]) ( p a prime) are known (see [1]) and there has been extensi ve work on the calculation of the integral cohomology of the groups SL 2 ( O ) where O is the ring of integers in an imaginary quadratic ﬁeld (see [18], [17],[20]). Little is known, on the other hand, about the integral (co)homology of SL 2 ( Z [1 / N ]) when N has at least 2 prime factors (see [6] and [11] for some partial results). Here we giv e an example of how the module structure and specialization maps dev eloped above can allo w us to construct nontrivial homology classes in the groups H 3 (SL 2 ( O S ) , Z ): Let t : = " − 1 1 − 1 0 # ∈ SL 2 ( Z ) and let G be the cyclic group of order 3 generated by t . For any ring A , we will let D : = D A denote the image of the generator of H 3 ( G , Z ) in H 3 (SL 2 ( A ) , Z ) (see Remark 3.14 abo ve). Thus if A is a subring of the ﬁeld F , D A maps to − D F under the map H 3 (SL 2 ( A ) , Z ) → RB ( F ). Theorem 5.7. Let F be a number ﬁeld. Let S be a nonempty set of ﬁnite primes of F such that Cl S ( F ) / 2 = 0 , wher e Cl S ( F ) denotes the subgr oup of Cl( O F ) gener ated by the primes of S . Then ther e is a natural surjective map of F 3 [ O × S / 2] -modules H 3 (SL 2 ( O S ) , Z ) (3) → ⊕ v ∈ S P ( k v ) (3) . Pr oof. For v ∈ S , let R v denote the composite H 3 (SL 2 ( O S ) , Z ) / / H 3 (SL 2 ( F ) , Z ) / / RB ( F ) S v , − 1 / / b P ( k v ) . Reﬁned Bloch group and third homology of SL 2 29 Observe that the associated action of O × S on b P ( k v ) is decribed as follows: For a ∈ O × S , h a i acts as multiplication by ( − 1) v ( a ) . By [7], section 7, b P ( k ) (3) = P ( k ) (3) is generated by D k = 2 C k for any ﬁnite ﬁeld k . Our hypothesis on Cl S ( F ) guarantees that the map O × S → ⊕ v ∈ S Z / 2 is surjectiv e, and thus for each w ∈ S , there exists u w ∈ O × S satisfying v ( u w ) ≡ δ v , w (mod 2) for all v ∈ S . Finally , for each w ∈ S , let D w , S : = e − u w · D O S ∈ H 3 (SL 2 ( O S ) , Z ) (3) (see Remark 3.14) and hence R v ( D w , S ) = − e − u w · D k v = − δ v , w D k v .  . Corollary 5.8. Let F be a number ﬁeld not containing ζ 3 . Let S be a ﬁnite set of primes of F for which Cl S ( F ) / 2 = 0 . Let r ( S ) = | { v ∈ S | | k v | ≡ − 1 (mod 3) } | . Then 3 -rank ( H 3 (SL 2 ( O S ) , Z ) ) ≥ 1 + r ( S ) . Pr oof. The 3-rank of ⊕ v ∈ S P ( k v ) (3) is r ( S ) since P ( k v ) (3) = B ( k v ) (3) , and B ( k v ) is cyclic of order ( | k v | + 1) / 2 or | k v | + 1 by the results of [7]. On the other hand, letting u w be as in the proof of Theorem 5.7, the element D + : =        Y w ∈ S e + u w        D O S ∈ H 3 (SL 2 ( O S ) , Z ) (3) lies in the kernel of the map ⊕ v R v : H 3 (SL 2 ( O S ) , Z ) (3) → ⊕ v ∈ S B ( k v ) (3) Ho wev er , for any place v of the ﬁeld F , the image of D + in RB ( F ) maps to − D k v under the specialization map S v , 1 : R B ( F ) → b P ( k v ) corresponding to the tri vial character χ = 1. If we choose v for which 3 di vides | k v | + 1 then D k v has order 3. It follows that D + has order 3.  6. T he third homology of SL 2 of local fields In this section, we use the properties of the reﬁned Bloch group to calculate H 3 (SL 2 ( F ) , Z ) up to 2-torsion when F is a local ﬁeld – i.e. is complete with respect to discrete v alue – with ﬁnite residue ﬁeld of odd characteristic (Theorem 6.19). 6.1. Preliminary results. W e will begin by recalling some of the rele vant facts about structure of the multiplicati ve group of a local ﬁeld. Let F be a ﬁeld with discrete v alue v and residue ﬁeld k . Let π be a uniformizer . Then the homomorphism v : F × → Z splits and we hav e F × = U · π Z  U × Z . Thus G F  U / U 2 × Z / 2 . 30 KEVIN HUTCHINSON No w suppose that F is complete with respect to v and that 2 ∈ U (i.e. 2 , 0 in k ). Then Hensel’ s Lemma ([14, Chapter II, 4.6]) implies that, for any u ∈ U , if ¯ u ∈ ( k × ) 2 then u ∈ U 2 . In particular , in this case U 1 = U 2 1 and U / U 2  k × / ( k × ) 2 = G k . It follo ws that if F is complete with respect to v and if k is ﬁnite of odd order , then G F  G k × Z / 2 has order 4 with generators h π i and h u i where u is any nonsquare unit. Suppose that L / F is an extension of local ﬁelds. Let π be a uniformizer of F . Then there is an induced extension of residue ﬁelds k L / k F and [ L : F ] = v L ( π )[ k L : k F ] for any uniformizer π of F ([14, Chapter II, 6.8]). An extension L / F of local ﬁelds is said to be unramiﬁed if a uniformizer π of F is also a uniformizer in L . For example, if ζ n is a primiti ve n -th root of unity in some separable closure of F and if ( n , char( k )) = 1, then F ( ζ n ) / F is an unramiﬁed extension ([14, Chapter II, Proposition 7.12]). Furthermore, Hensel’ s Lemma implies that, when ( n , char( k )) = 1, ζ n ∈ F if and only if k contains a primitiv e n th root of unity . Local class ﬁeld theory giv es us valuable information about norm groups N L / F ( L × ) of ﬁnite algebraic extensions L / F of local ﬁelds: (1) If L / F is a ﬁnite abelian extension of local ﬁelds of degree d , then F × / N L / F ( L × ) is a group of order d ([14, Chapter 5, Theorem 1.3]). (2) If L / F is a ﬁnite unramiﬁed extension of local ﬁelds, then N L / F ( U L ) = U F ([14, Chapter V , Corollary 1.2]). (3) The theory of the Hilbert symbol tells us that if ( n , char( k )) = 1 and if ζ n ∈ F , then u ∈ U F is a norm from F ( n √ b ) if and only if ¯ u v ( b ) | k × | n = 1 in k × . (See [14, Chapter V , 3.2 (iii) and 3.4].) As abov e, E denotes the ﬁeld obtained from F by adjoining a root of Φ ( T ) = T 2 − T + 1. Lemma 6.1. Let F be a local ﬁeld with ﬁnite r esidue ﬁeld k of or der q = p f . Suppose that Q 3 1 F . Then (1) h h a i i D F = 0 if and only if a ∈ h − 1 i · N E / F ( E × ) . (2) F or any uniformizer , π , of F the specialization map S π, − 1 : RP ( F ) → b P ( k ) induces an isomorphism of R F -modules I F · D F  Z · D k = P ( k ) (3) . Pr oof. W e hav e already seen that if a ∈ h − 1 i · N E / F ( E × ) then h h a i i D F = 0 (Theorem 3.12). In other words, the action of R F on D F factors through the group ring Z [ F × / h − 1 i · N E / F ( E × )]. If char ( F ) = 3, then E = F and hence D F = 0 by Corollary 3.7 (2). Similarly , D k = 0. So the result holds tri vially . So we may assume that char( F ) , char( k ) , 3. In this case E = F ( √ − 3) = F ( ζ 3 ). By the facts just quoted, E / F is unramiﬁed and e very unit is a norm. In particular , − 1 ∈ N E / F ( E × ). If ζ 3 ∈ F , then h h a i i D F = 0 for all a , and also D k = 0. The result holds again for tri vial reasons. Otherwise, ζ 3 < F and F × / N E / F ( E × ) is cyclic of order 2 generated by the class of a uniformizer π . It follows that I F D F = Z h h π i i D F Reﬁned Bloch group and third homology of SL 2 31 has order 1 or 3. Howe ver , since ζ 3 < k , q . 1 (mod 3) so 3 | q + 1 and D k has order 3. S π, − 1 ( D F ) = D k , 0 and since π acts as − 1 on the right, S π, − 1 ( h h π i i D F ) = − 2 D k = D k , 0, so that the statements of the lemma follo w immediately .  Lemma 6.2. Suppose that Q 3 ⊂ F . Then − 1 ∈ N E / F ( E × ) if and only if [ F : Q 3 ] is even. Pr oof. By (3) above (with n = 2 and u = − 3), − 1 ∈ N E / F ( E × ) ⇐ ⇒ ( − 1) v (3) 3 f − 1 2 = 1 ⇐ ⇒ v (3) 3 f − 1 2 is e ven ⇐ ⇒ v (3) f is even . Ho wev er , by the result on the degree of local ﬁeld extensions quoted before Lemma 6.1 above, [ F : Q 3 ] = v (3) f .  Corollary 6.3. If [ F : Q 3 ] is odd then I F D F = 0 = Z · D k . Pr oof. Under these hypotheses, − 1 < N E / F ( E × ) and thus F × = h − 1 i · N E / F ( E × ) (since [ F × : N E / F ( E × )] = 2). Thus I F D F = 0 by Theorem 3.12. But char( k ) = 3 and hence D k = 0 also.  Remark 6.4. Of course, if Q 3 ⊂ F and − 1 ∈ N E / F ( E × ) then our results do not rule out the pos- sibility that I F D F has order 3 (rather than 1), but this module, if nontrivial, cannot be detected by S π, − 1 (since D k = 0 in this case). The follo wing lemma will be central to our computations below: Lemma 6.5. In any ﬁeld F we have e − b [ a ] = 0 in c RP ( F )[ 1 2 ] whenever a , b ∈ F × with a ≡ − b (mod ( F × ) 2 ) . Pr oof. Let a ∈ F × , a , 1. Then h a i [ a ] = h − 1 i [ a ] in c RP ( F ) by Lemma 4.6. Thus h − a i [ a ] = [ a ] and hence h h − a i i [ a ] = 0. It follo ws that e − − a [ a ] = 0 in c RP ( F )[ 1 2 ]. Of course, e − x only depends on the square class of x , so the result follows.  For the remainder of this section we let F be a local ﬁeld complete with respect to the discrete v aluation v , with residue ﬁeld k of order q = p f , with p odd . Furthermore, for simplicity , we will suppose that if Q 3 ⊂ F then [ F : Q 3 ] is odd. Recall that G F has order 4: if π is a uniformizing parameter and if u is a nonsquare unit G F = { 1 , h π i , h u i , h u π i } . (If q ≡ 3 (mod 4), we can take u = − 1.) W e let c G F denote the group of characters Hom ( G F , µ 2 ) = Hom ( G F , {± 1 } ) of G F . Giv en χ ∈ c G F , we hav e the associated idempotent e χ = 1 4 X g ∈ G F χ ( g ) h g i . Observe that if χ is a nontrivial character on G F , then e χ = Y a ∈ χ − 1 ( − 1) e − a ∈ R F [ 1 2 ] . 32 KEVIN HUTCHINSON If M is an R F [ 1 2 ]-module and if χ ∈ c G F , then M χ : = e χ ( M ) is a submodule and g · m = χ ( g ) m for all g ∈ G F . Observe that the functor M → M χ is an exact functor on the category of R F [ 1 2 ]-modules. For the rest of this section we ﬁx the following: Let π be a uniformizing parameter for F and let u be a ﬁxed nonsquare unit, which we take to be − 1 in the case q ≡ 3 (mod 4). Clearly a nontri vial character in c G F is determined by χ − 1 ( − 1). W e label the four characters as follo ws: χ 1 is the tri vial character , χ is the character with χ − 1 ( − 1) = { h π i , h u π i } , ψ is the character with ψ − 1 ( − 1) = { h u π i , h u i } and ψ 0 is the remaining character . Note that for any R F [ 1 2 ]-module M , we hav e a decomposition M = M χ 1 ⊕ M χ ⊕ M ψ ⊕ M ψ 0 where M χ 1 = M G F  M G F = H 0 ( F × , M ) and I F M = M χ ⊕ M ψ ⊕ M ψ 0 . Since T or Z 1 ( µ F , µ F ) is a tri vial G F -module, from the exact sequence 0 → T or Z 1 ( µ F , µ F )[ 1 2 ] → H 3 (SL 2 ( F ) , Z [ 1 2 ]) → RB ( F )[ 1 2 ] → 0 it follo ws that H 3 (SL 2 ( F ) , Z [ 1 2 ]) ρ = RB ( F )[ 1 2 ] ρ for any ρ , χ 1 . On the other hand, H 3 (SL 2 ( F ) , Z [ 1 2 ]) χ 1 = H 0 ( F × , H 3 (SL 2 ( F ) , Z [ 1 2 ])) = K ind 3 ( F )[ 1 2 ] . Thus we hav e a R F [ 1 2 ]-module decomposition H 3 (SL 2 ( F ) , Z [ 1 2 ])  K ind 3 ( F )[ 1 2 ] ⊕ RB ( F )[ 1 2 ] χ ⊕ RB ( F )[ 1 2 ] ψ ⊕ RB ( F )[ 1 2 ] ψ 0 . Our goal in the remainder of this section is to sho w that the last two factors on the right are zero and that the second factor is isomorphic, via S π, − 1 , to P ( k )[ 1 2 ]. In order to do this, we make a couple of reductions: Lemma 6.6. Let F be as stated. Then (1) ( D F ) ψ = 0 = ( D F ) ψ 0 . (2) ( D F ) χ = I F D F . Pr oof. Note that I F D F = ( D F ) χ ⊕ ( D F ) ψ ⊕ ( D F ) ψ 0 . So we must prov e that the last two factors are 0. Recall that e ψ = e − u e − u π and e ψ 0 = e − u e − π . Now e − u = − h h u i i / 2 and our conditions guarantee that u ∈ h − 1 i N E / F ( E × ) so that h h u i i D F = 0 by Theorem 3.12.  In vie w of Corollary 4.4 and Lemma 4.7 we hav e: Corollary 6.7. RB ( F )[ 1 2 ] ψ = RB ( F )[ 1 2 ] ψ and RB ( F )[ 1 2 ] ψ 0 = RB ( F )[ 1 2 ] ψ 0 . Observe that S π, − 1 induces a well-deﬁned surjecti ve homomorphism ¯ S π, − 1 : RP ( F ) → P ( k ) . Reﬁned Bloch group and third homology of SL 2 33 Lemma 6.8. The homomorphism S π, − 1 : RB ( F )[ 1 2 ] χ → b P ( k )[ 1 2 ] is an isomorphism if and only if the homomorphism ¯ S π, − 1 : RB ( F )[ 1 2 ] χ → P ( k )[ 1 2 ] is an isomorphism. Pr oof. Using Lemma 6.6 (2), we hav e a commutativ e diagram of R F -modules with exact ro ws 0 / / I F D F / /    RB ( F )[ 1 2 ] χ / / S π, − 1   RB ( F )[ 1 2 ] χ / / ¯ S π, − 1   0 0 / / Z D k / / b P ( k )[ 1 2 ] / / P ( k )[ 1 2 ] / / 0 in which the left vertical arro w is an isomorphism by Lemma 6.1 (2).  6.2. S 3 -dynamics. Again, in this section, F denotes a ﬁeld complete with respect to a discrete v alue v with ﬁnite residue ﬁeld k of odd order q . For an y ﬁeld L , the transformations σ , τ : P 1 ( L ) → P 1 ( L ) , σ ( x ) = x − 1 and τ ( x ) = 1 − x determine an action of the symmetric group on 3 letters S 3 . { 0 , 1 , ∞} is an orbit for this action, and hence S 3 acts on P 1 ( L ) \ { 0 , 1 , ∞} = F × \ { 1 } . Lemma 6.9. If α ∈ R F and if α [ x ] = 0 in RP ( F ) . Then α  γ ( x )  = 0 for all γ ∈ S 3 . Pr oof. In RP ( F ) we hav e D F = 0 and ψ 1 ( x ) = 0 for all x , so that h x − 1 i = [ 1 − x ] = − h − 1 i [ x ] . It follows that if α [ x ] = 0 for some α ∈ R F , then α [ σ ( x ) ] = α [ τ ( x ) ] = 0 also. The statement follo ws since σ and τ generate S 3  Remark 6.10. W e will apply this below in the case where ρ is a character of G F and α = e ρ ∈ R F [ 1 2 ] Let ¯ R = ( k × ) 2 and let ¯ N = k × \ ( k × ) 2 . Let ¯ R 1 = { a ∈ ¯ R \ { 1 }| τ ( a ) ∈ ¯ R } , ¯ R − 1 = { a ∈ ¯ R | τ ( a ) ∈ ¯ N } . Let ¯ N 1 = { a ∈ ¯ N | τ ( a ) ∈ ¯ R } , ¯ N − 1 = { a ∈ ¯ N | τ ( a ) ∈ ¯ N } . Thus we hav e a partition k × = { 1 } ∪ ¯ R 1 ∪ ¯ R − 1 ∪ ¯ N 1 ∪ ¯ N − 1 Let U be the group of units of F and let U 1 be the kernel of the reduction map U → k × . Then there is a corresponding partition U = R ∪ N = U 1 ∪ R 1 ∪ R − 1 ∪ N 1 ∪ N − 1 where R : = U 2 = { a ∈ U | ¯ a ∈ ¯ R } , N = U \ R , R 1 : = { a ∈ U | ¯ a ∈ ¯ R 1 } = { a ∈ R | τ ( a ) ∈ R } , R − 1 : = { a ∈ U | ¯ a ∈ ¯ R − 1 } , N 1 : = { a ∈ U | ¯ a ∈ ¯ N 1 } and N − 1 : = { a ∈ U | ¯ a ∈ ¯ N − 1 } . 34 KEVIN HUTCHINSON Lemma 6.11. (1) If q ≡ 1 (mod 4) then ¯ R 1 = σ ( ¯ R 1 ) = τ ( ¯ R 1 ) σ ( ¯ N 1 ) = ¯ N − 1 and σ ( ¯ R − 1 ) = ¯ R − 1 Furthermor e | ¯ R − 1 | = | ¯ N − 1 | = | ¯ N 1 | = q − 1 4 and | ¯ R 1 | = q − 5 4 . (2) If q ≡ 3 (mod 4) then ¯ N − 1 = σ ( ¯ N − 1 ) = τ ( ¯ N − 1 ) σ ( ¯ R 1 ) = ¯ R − 1 and σ ( ¯ N 1 ) = ¯ N 1 Furthermor e | ¯ R − 1 | = | ¯ N 1 | = | ¯ R 1 | = q − 3 4 and | ¯ N − 1 | = q + 1 4 . Pr oof. (1) Clearly τ ( ¯ R 1 ) = ¯ R 1 by deﬁnition. Furthermore, if a ∈ ¯ R 1 then 1 − 1 a = ( − 1) · (1 − a ) · 1 a ∈ ¯ R and thus a − 1 = σ ( a ) ∈ ¯ R 1 . It follo ws that σ ( ¯ R − 1 ) = ¯ R − 1 since σ ( ¯ R ) = ¯ R . Similarly , if a ∈ ¯ N 1 then a − 1 ∈ ¯ N 1 − a ∈ ¯ R and thus 1 − 1 a = ( − 1) · (1 − a ) · 1 a ∈ N so that σ ( ¯ N 1 ) = ¯ N − 1 . The second statement follo ws by simple counting since | ¯ N 1 | + | ¯ N − 1 | = | ¯ N | = ( q − 1) / 2 . (2) The argument is similar e xcept that this time a − 1 = ( − 1) · (1 − a ) ∈ ( ¯ N , 1 − a ∈ ¯ R ¯ R , 1 − a ∈ ¯ N since − 1 ∈ ¯ N .  T aking in verse images in U ⊂ F , and noting that U \ U 1 is closed under the action of S 3 , we obtain : Corollary 6.12. (1) If q ≡ 1 (mod 4) then R 1 = σ ( R 1 ) = τ ( R 1 ) σ ( N 1 ) = N − 1 and σ ( R − 1 ) = R − 1 (2) If q ≡ 3 (mod 4) then N − 1 = σ ( N − 1 ) = τ ( N − 1 ) σ ( R 1 ) = R − 1 and σ ( N 1 ) = N 1 W e will need the follo wing elementary result below: Lemma 6.13. Let k be a ﬁnite ﬁeld of odd order , and let n ∈ ¯ N . Then ther e exist r 1 , r 2 ∈ ¯ R such that n = r 1 + r 2 . Reﬁned Bloch group and third homology of SL 2 35 Pr oof. If q ≡ 1 (mod 4), there exists r ∈ ¯ R with r n ∈ ¯ N 1 . Thus 1 − r n = s ∈ ¯ R and n = 1 r +  − s r  = r 1 + r 2 . If q ≡ 3 (mod 4) then there exists r ∈ ¯ R with r n ∈ ¯ N − 1 . Then 1 − nr = m ∈ ¯ N and thus n = 1 r +  − m r  = r 1 + r 2 .  6.3. The main result. Let F be a local ﬁeld with discrete value v and ﬁnite residue ﬁeld k of odd order . Throughout π will denote a choice of uniformizer . Let M ( F ) denote the R F [ 1 2 ]-module e − − 1 I F [ 1 2 ]. Lemma 6.14. (1) If ρ , χ 1 then ther e is a natural short exact sequence of R F [ 1 2 ] -modules 0 / / RB ( F )[ 1 2 ] ρ / / RP ( F )[ 1 2 ] ρ ¯ λ 1 / / M ( F ) ρ / / 0 . (2) If q ≡ 1 (mod 4) and if ρ , χ 1 then RB ( F )[ 1 2 ] ρ = RP ( F )[ 1 2 ] ρ . (3) If q ≡ 3 (mod 4) then RB ( F )[ 1 2 ] χ = RP ( F )[ 1 2 ] χ . Pr oof. There is an exact sequence 0 / / RB ( F ) / / RP ( F ) Λ / / ] RS 2 Z ( F × ) The kernel of the surjection RS 2 Z ( F × ) → I 2 F is a trivial R F -module. Furthermore, since I F / I 2 F  G F is a 2-torsion group, we have I 2 F [ 1 2 ] = I F [ 1 2 ]. Putting these facts together we obtain an isomorphism of R F -modules RS 2 Z ( F × )[ 1 2 ] ρ  I F [ 1 2 ] ρ for all ρ , χ 1 . This in turn induces an isomorphism ] RS 2 Z ( F × )[ 1 2 ] ρ        I F [ 1 2 ] hh h x i i h h − x i ii       ρ =       I F [ 1 2 ] p + − 1 I F [ 1 2 ]       ρ =       I F [ 1 2 ] e + − 1 I F [ 1 2 ]       ρ  e − − 1 I F [ 1 2 ] ρ = M ( F ) ρ and the homomorphism RP ( F )[ 1 2 ] ρ → ] RS 2 Z ( F × )[ 1 2 ] ρ  M ( F ) ρ is induced by the map ¯ λ 1 : RP ( F ) → M ( F ) [ x ] 7→ e − − 1 λ 1 ( [ x ] ) = e − − 1 h h x i i h h 1 − x i i . T o see that this homomorphism is surjectiv e, observe ﬁrst that if q ≡ 1 (mod 4), then h − 1 i = 1 and hence e − − 1 = 0 in R F [ 1 2 ]. Thus M ( F ) = 0 in this case, and there is nothing to pro ve. This also implies statement (2) of the lemma. So we can suppose that q ≡ 3 (mod 4). Then, as a Z [ 1 2 ]-module, I F [ 1 2 ] is free of rank 3 with basis e − − 1 , e − π and e − − π . Then e − − 1 ( I F [ 1 2 ]) is a free Z [ 1 2 ]-module of rank 2 with basis e − − 1 and e − π e − − 1 (since, for example, e − − 1 e − − π = − (e − − 1 + e − π e − − 1 ).) As an R F [ 1 2 ]-module, it is thus generated by e − − 1 . 36 KEVIN HUTCHINSON No w let n ∈ N − 1 . Then 1 − n ∈ N − 1 and λ 1 1 4 [ n ] ! = 1 4 h h n i i h h 1 − n i i = 1 4 h h − 1 i i h h − 1 i i = (e − − 1 ) 2 = e − − 1 so that the map is surjecti ve. Finally , observe also that if q ≡ 3 (mod 4) then e χ e − − 1 = 0 so that M ( F ) χ = 0 and statement (3) also follo ws.  W e will use the following notation in the remainder of this section: Giv en a character , ρ of G F and a ∈ F × , we will denote e ρ ( [ a ] ) ∈ RP ( F )[ 1 2 ] by [ a ] ρ . In the each of next few lemmas, the strategy of proof is largely the same. The proof in volv es sho wing that [ a ] ρ = 0 for a belonging to some subset of F × . W e begin with Lemma 6.5 to show that [ a ] ρ = 0 for a belonging to one or more square classes, and then use Lemma 6.9 to deduce the v anishing of [ a ] ρ for a larger set of a . In order to follo w the ar guments, it may be helpful for the reader to keep the follo wing decom- positions in mind: Recall that there are four square classes, or cosets modulo ( F × ) 2 , in the group F × . Thus F × = ( F × ) 2 ∪ u · ( F × ) 2 ∪ π · ( F × ) 2 ∪ u π · ( F × ) 2 and ( F × ) 2 = [ n ∈ Z R π 2 n = R ∪ [ n , 0 R π 2 n = U 1 ∪ R 1 ∪ R − 1 ∪ [ n , 0 R π 2 n . u · ( F × ) 2 = [ n ∈ Z N π 2 n = N ∪ [ n , 0 N π 2 n = N 1 ∪ N − 1 ∪ [ n , 0 N π 2 n . π · ( F × ) 2 = [ n ∈ Z R π 2 n − 1 . u π · ( F × ) 2 = [ n ∈ Z N π 2 n − 1 . U 1 = [ m ≥ 1 (1 − U π m ) = [ n ≥ 1 (1 − U π 2 n ) ∪ [ n ≥ 1 (1 − U π 2 n − 1 ) . W e will repeatedly use the ke y fact that U 1 = U 2 1 and thus 1 − a is a square whenev er v ( a ) > 0. Lemma 6.15. Let ρ be a char acter of G F . Then the following are equivalent: (1) [ a ] ρ = 0 in RP ( F ) for all a ∈ U 1 (2) [ a ] ρ = 0 in RP ( F ) for all a satisfying v ( a ) , 0 . Pr oof. Suppose ﬁrst that [ a ] ρ = 0 for all a ∈ U 1 . Let b ∈ F × with v ( b ) > 0. Then c : = τ ( b ) = 1 − b ∈ U 1 . Thus [ c ] ρ = 0 by assumption. It follo ws that [ b ] ρ = [ τ ( c ) ] ρ = 0 by Lemma 6.9 with α = e ρ . On the other hand, if b ∈ F × with v ( b ) < 0, then v ( b − 1 ) > 0 and thus h b − 1 i ρ = 0. But then [ b ] ρ = h σ ( b − 1 ) i ρ = 0 by Lemma 6.9 again. Con versely , suppose that [ a ] ρ = 0 whene ver v ( a ) , 0 and that b ∈ U 1 . Then v (1 − b ) > 0. Thus [ 1 − b ] ρ = 0 and hence [ b ] ρ = [ τ (1 − b ) ] ρ = 0.  Lemma 6.16. The R F [ 1 2 ] -module homomorphism S π, − 1 : RP ( F )[ 1 2 ] χ → b P ( k )[ 1 2 ] is an isomorphism. Reﬁned Bloch group and third homology of SL 2 37 Pr oof. By Lemma 6.8 it is enough to prove that the map ¯ S π, − 1 : RP ( F )[ 1 2 ] χ → P ( k )[ 1 2 ] is an isomorphism. T o prov e the result we will sho w that there is a well-deﬁned R F [ 1 2 ]-module homomorphism T : P ( k )[ 1 2 ] → RP ( F )[ 1 2 ] χ , T ( [ ¯ a ] ) = [ a ] χ (where a ∈ U maps to ¯ a ∈ k × ) which is in verse to ¯ S π, − 1 . T o sho w that T is well-deﬁned we must show that [ au ] χ = [ a ] χ whene ver a ∈ U and u ∈ U 1 . Since clearly ¯ S π, − 1 ◦ T = Id d RB ( k ) [ 1 2 ] , it is then only necessary to show that T is surjectiv e: we must show that [ a ] χ = 0 whenev er v ( a ) , 0 (since if v ( a ) = 0 then [ a ] χ = T ( [ ¯ a ] ) by deﬁnition of T , and thus [ a ] χ lies in the image of T for all a ∈ F × ). Note, to begin with, that since e χ = e − π e − u π , we hav e [ a ] χ = 0 if a ≡ − π, − u π (mod ( F × ) 2 ) by Lemma 6.5. This is clearly equi valent to [ a ] χ = 0 if a lies in one of the two square classes π · ( F × ) 2 and u π · ( F × ) 2 . Thus [ a ] χ = 0 if a ∈ U π 2 n − 1 for any n ∈ Z . Thus for any n ≥ 1, [ a ] χ = 0 if a ∈ τ ( U π 2 n − 1 ) = 1 − U π 2 n − 1 ⊂ U 1 . Consider now the case a = w π 2 n , w ∈ U and n ≥ 1. By considering the relation  S 1 /π, a /π  χ , we see that 0 = " 1 π # χ −  a π  χ + h π i [ a ] χ − h π − 1 i " − w π 2 n − 1 1 − π 1 − w π 2 n − 1 # χ + h − π i " 1 π · 1 − π 1 − w π 2 n − 1 # χ = h π i [ a ] χ since all terms other than the third belong to the square classes π · ( F × ) 2 or u π · ( F × ) 2 . Thus we deduce that [ a ] χ = 0 if a ∈ U π 2 n for an y n ≥ 1. It follo ws that [ a ] χ = 0 whenev er v ( a ) > 0. By considering σ ( a ) = 1 / a it follows that [ a ] χ = 0 whenev er v ( a ) , 0. From Lemma 6.15 it follo ws that [ a ] χ = 0 whene ver a ∈ U 1 also. Finally , let a ∈ U \ U 1 and w ∈ U 1 . Then 0 =  S a , aw  χ = [ a ] χ − [ aw ] χ + h a i [ w ] χ − D a − 1 − 1 E " w · 1 − a 1 − aw # χ + h 1 − a i " 1 − a 1 − aw # χ which gi ves [ a ] χ = [ aw ] χ since the last three terms all lie in U 1 . This proves the Lemma.  Lemma 6.17. Let F be a local ﬁeld whose r esidue ﬁeld k has or der q with q ≡ 1 (mod 4) . Then RP ( F )[ 1 2 ] ψ = RP ( F )[ 1 2 ] ψ 0 = 0 . Pr oof. W e treat the case of ψ . Clearly the case of ψ 0 is identical since it only in volv es a switch in our choice of uniformizer . Since ψ = e − u e − π and since − 1 ∈ ( F × ) 2 we hav e [ a ] ψ = 0 if a ≡ u , π (mod ( F × ) 2 ) by Lemma 6.5. Thus [ a ] ψ = 0 if a ∈ N π 2 n ∪ R π 2 n − 1 for any n ∈ Z . W e must prov e that [ a ] χ = 0 also for a ∈ ( F × ) 2 ∪ u π · ( F × ) 2 . 38 KEVIN HUTCHINSON T aking the S 3 -action into account as in the last lemma, it follo ws that [ a ] ψ = 0 if a ∈ 1 − R π 2 n − 1 ∪ 1 − N π 2 n for any n ≥ 1 . Furthermore, if a ∈ R − 1 then τ ( a ) ∈ u · ( F × ) 2 . Thus [ τ ( a ) ] ψ = 0 and hence [ a ] ψ = 0. Consider no w a = r π 2 n with r ∈ R and n ≥ 1. Then we have 0 =  π a  ψ − [ π ] ψ + h π i [ a ] ψ −  a − π 1 − π  ψ + h π i " 1 a · a − π 1 − π # ψ which forces h π i [ a ] ψ = 0 = [ a ] ψ since the other four terms all belong to the square class π · ( F × ) 2 . By considering σ ( a ) = a − 1 , we deduce that [ b ] ψ = 0 for all b ∈ R π 2 n and all n , 0. Next, we consider a = n π 2 m − 1 with n ∈ N and m ≥ 1. By Lemma 6.13, we can write n = r + s with r , s ∈ R . Let x = r π 2 m − 1 . Observe that w : = 1 − x 1 − a = 1 − r π 2 m − 1 1 − n π 2 m − 1 = (1 − r π 2 m − 1 )(1 + n π 2 m − 1 + · · · ) = 1 + ( n − r ) π 2 m − 1 + · · · = 1 + st π 2 m − 1 with t ∈ U 1 . Thus w ∈ 1 − R π 2 n − 1 since s ∈ R . Thus we hav e 0 = [ x ] ψ − [ a ] ψ + h π i  n r  ψ − h π i  n r · w  ψ + [ w ] ψ = − [ a ] ψ since x ∈ π · ( F × ) 2 , n r , nw r ∈ N ⊂ u · ( F × ) 2 and, as noted abov e, w ∈ 1 − R π 2 m − 1 . By considering σ ( a ) also, we deduce that [ b ] ψ = 0 for all b ∈ N π 2 m − 1 for any m ∈ Z . Thus we hav e shown that [ a ] ψ = 0 whenev er v ( a ) , 0 and whenev er a ∈ R − 1 . By Lemma 6.15 it follo ws that [ a ] ψ = 0 whene ver a ∈ U 1 . It remains to prov e that [ a ] ψ = 0 whene ver a ∈ R 1 . Let a ∈ R 1 . Observe that { a s | s ∈ R − 1 } ∩ R − 1 , ∅ since | ¯ R − 1 | = 1 + | ¯ R 1 | by Lemma 6.11 (1). It follo ws that we can write a = t / s with s , t ∈ R − 1 . Observe that (1 − s ) / (1 − t ) ∈ R − 1 also: Since 1 − s , 1 − t ∈ N we hav e (1 − s ) / (1 − t ) ∈ R . Since a = s / t ∈ R 1 , 1 − s t = t − s t ∈ R and thus s − t , t − s ∈ R . Hence 1 − 1 − s 1 − t = s − t 1 − t ∈ N so that (1 − s ) / (1 − t ) ∈ R − 1 . Applying the same argument to s − 1 , t − 1 sho ws that (1 − s − 1 ) / (1 − t − 1 ) ∈ R − 1 also. Thus 0 = [ s ] ψ − [ t ] ψ + [ a ] ψ − h u i " 1 − s − 1 1 − t − 1 # ψ + h u i " 1 − s 1 − t # ψ = [ a ] ψ proving the lemma.  Lemma 6.18. Let F be a local ﬁeld with r esidue ﬁeld k of or der q and q ≡ 3 (mod 4) . Then RB ( F )[ 1 2 ] ψ = RB ( F )[ 1 2 ] ψ 0 = 0 . Reﬁned Bloch group and third homology of SL 2 39 Pr oof. By Lemma 6.14 it is enough to show that λ 1 induces isomorphisms of R F [ 1 2 ]-modules RP ( F )[ 1 2 ] ψ  M ( F ) ψ and RP ( F )[ 1 2 ] ψ 0  M ( F ) ψ 0 . As remarked in the proof of Lemma 6.17, it is enough to treat the case of the character ψ . Observe that M ( F ) ψ is a free Z [ 1 2 ]-module of rank 1 with generator e ψ = e − − 1 e − π . Thus, if L is any R F [ 1 2 ]-module and if a ∈ L , there is a unique R F [ 1 2 ]-module homomorphism M ( F ) ψ → L sending e ψ to e ψ ( a ). Thus we ﬁx n ∈ N − 1 and let Φ : M ( F ) ψ → RP ( F )[ 1 2 ] ψ be the unique R F [ 1 2 ]-module homomor- phism sending e ψ to 1 4 [ n ] ψ . By our pre vious remarks, Φ is well-deﬁned and clearly ¯ λ 1 ◦ Φ = Id M ( F ) ψ . So it only remains to sho w that Φ is surjectiv e. T o do this we sho w that [ a ] ψ = 0 in RP ( F )[ 1 2 ] ψ whene ver a < N − 1 and that [ n 1 ] ψ = [ n 2 ] ψ for any n 1 , n 2 ∈ N − 1 . No w [ a ] ψ = 0 if a ∈ ( F × ) 2 ∪ − π · ( F × ) 2 . In particular, [ a ] ψ = 0 if a ∈ U 1 , since U 1 ⊂ ( F × ) 2 . By Lemma 6.15 it follows that [ a ] ψ = 0 if v ( a ) , 0. Thus [ a ] = 0 if a ∈ π · ( F × ) 2 and [ a ] = 0 if a ∈ − 1 · ( F × ) 2 and v ( a ) , 0. All of this sho ws that [ a ] ψ = 0 for any a < N since N = { a ∈ − 1 · ( F × ) 2 | v ( a ) = 0 } . Ho wev er , if a ∈ N 1 , then b = τ ( a ) ∈ ( F × ) 2 . Thus [ b ] ψ = 0 and hence [ a ] ψ = [ τ ( b ) ] ψ = 0. So we hav e shown that [ a ] ψ = 0 for all a < N − 1 . Finally , let n 1 , n 2 ∈ N − 1 . Then in R P ( F )[ 1 2 ] ψ we hav e 0 = [ n 1 ] ψ − [ n 2 ] ψ + h − 1 i " n 2 n 1 # ψ − h − 1 i " 1 − n 1 − 1 1 − n 2 − 1 # ψ + " 1 − n 1 1 − n 2 # ψ = [ n 1 ] ψ − [ n 2 ] ψ since clearly , from the deﬁnition of N − 1 , n 2 n 1 , 1 − n 1 1 − n 2 , 1 − n 1 − 1 1 − n 2 − 1 ∈ R . This prov es the lemma.  Putting all of these results together giv es the follo wing calculation of the third homology of SL 2 ( F ): Theorem 6.19. Let F be a local ﬁeld with residue ﬁeld k of odd order . If Q 3 ⊂ F , suppose that [ F : Q 3 ] is odd. Then ther e is an isomorphism of R F [ 1 2 ] -modules H 3 (SL 2 ( F ) , Z [ 1 2 ])  K ind 3 ( F )[ 1 2 ] ⊕ P ( k )[ 1 2 ] in which F × acts trivially on the ﬁrst factor , while (the square class of) any uniformizer acts as multiplication by − 1 on the second factor . 40 KEVIN HUTCHINSON Pr oof. As observed in section 6.1 abov e, there is a R F [ 1 2 ]-module decomposition H 3 (SL 2 ( F ) , Z [ 1 2 ])  K ind 3 ( F )[ 1 2 ] ⊕ RB ( F )[ 1 2 ] χ ⊕ RB ( F )[ 1 2 ] ψ ⊕ RB ( F )[ 1 2 ] ψ 0 . No w RB ( F )[ 1 2 ] χ  b P ( k )[ 1 2 ] = P ( k )[ 1 2 ] by Lemma 6.8, Lemma 6.14 and Lemma 6.16. Finally , we have RB ( F )[ 1 2 ] ψ = RB ( F )[ 1 2 ] ψ 0 = 0 by Corollary 6.7, Lemma 6.17 and Lemma 6.18.  Remark 6.20. If Q 3 ⊂ F and if [ F : Q 3 ] is ev en, then the arguments abov e show that there is a surjecti ve homomorphism of R F [ 1 2 ]-modules H 3 (SL 2 ( F ) , Z [ 1 2 ]) / / / / K ind 3 ( F )[ 1 2 ] ⊕ P ( k )[ 1 2 ] whose kernel has order 1 or 3. Remark 6.21. It is not di ﬃ cult to write down an explicit homology class which generates H 3 (SL 2 ( F ) , Z [ 1 2 ]) 0  P ( F q )[ 1 2 ]  B ( F q )[ 1 2 ] for a local ﬁeld F with residue ﬁeld F q of odd order . Let r = ( q + 1) 0 . Let a ∈ U \ U 2 and observe that the residue ﬁeld of F ( √ a ) is F q ( √ ¯ a ) = F q 2 . Let z be an element of F q ( √ ¯ a ) × of order r . By Hensel’ s Lemma, there exists ζ ∈ F ( √ a ) × of order r with the property that ¯ ζ = z ; i.e. ζ is a primiti ve r -th root of unity . Thus, if ζ = α + β √ a , then z = ¯ α + ¯ β √ ¯ a . No w , for any quadratic ﬁeld extension K / L with K = L ( √ b ) for some b ∈ L × , there is an injecti ve group homomorphism κ : K × → GL 2 ( L ) , x + y √ b 7→ " x by y x # with the property that det( κ ( x )) = N K / L ( x ) for all x ∈ K × . Let t = κ ( ζ ) = " α a β β α # ∈ SL 2 ( F ) . Let Γ t be a generator of H 3 ( h t i , Z [ 1 2 ])  Z / r . Then the image, γ , of Γ t under the map H 3 ( h t i , Z [ 1 2 ]) → H 3 (SL 2 ( F ) , Z [ 1 2 ]) is represented by the cycle n − 1 X i = 0 1 ⊗ (1 , t , t i , t i + 1 ) . Let ¯ t ∈ SL 2 ( F q ) be the matrix obtained by reducing the entries of t . Then ¯ t = κ ( z ) has order r by Lemma 2.6. Thus the class γ maps to a generator of B ( F q )[ 1 2 ] under the specialization homomorphism S π, − 1 since the diagram Reﬁned Bloch group and third homology of SL 2 41 H 3 ( h t i , Z [ 1 2 ]) / /    H 3 (SL 2 ( F ) , Z [ 1 2 ]) / / RB ( F )[ 1 2 ] S π, − 1   H 3 ( h ¯ t i , Z [ 1 2 ]) / / H 3 (SL 2 ( F q ) , Z [ 1 2 ]) / / B ( F q )[ 1 2 ] commutes, because the composite map on the bottom row is an isomorphism (see Lemma 2.6 abov e). Let π be a uniformizing parameter and let t π = " π 0 0 1 # " α a β β α # " 1 /π 0 0 1 # = " α π a β β/π α # . Then the homology class e − π ( γ ) = 1 2 n − 1 X i = 0 1 ⊗  (1 , t , t i , t i + 1 ) − (1 , t π , t i π , t i + 1 π )  is a generator of H 3 (SL 2 ( F ) , Z [ 1 2 ]) 0 = e − π H 3 (SL 2 ( F ) , Z [ 1 2 ]) by the arguments of this section. Remark 6.22. The same techniques as used abo ve apply also to case where the residue ﬁeld has characteristic 2. Howe ver , the number of square classes is of the form 2 k , k ≥ 3 and the condition U 1 = U 2 1 is not satisﬁed. The ‘ S 3 -dynamics’ – i.e. the manner in which the S 3 orbits intersect with the square classes in F × – become more complicated as k grows. The author has conﬁrmed, for example, that H 3 (SL 2 ( Q 2 ) , Z [ 1 2 ])  K ind 3 ( Q 2 )[ 1 2 ] ⊕ P ( F 2 ) as expected, b ut the calculations are long and unedifying. R eferences [1] Alejandro Adem and Nadim Na ﬀ ah. On the cohomology of SL 2 ( Z [1 / p ]). In Geometry and cohomology in gr oup theory (Durham, 1994) , volume 252 of London Math. Soc. Lecture Note Ser . , pages 1–9. Cambridge Univ . Press, Cambridge, 1998. [2] Spencer J. Bloch. Higher re gulators, algebraic K -theory , and zeta functions of elliptic curves , volume 11 of CRM Monograph Series . American Mathematical Society , Providence, RI, 2000. [3] A. Borel and J.-P . Serre. Cohomologie d’immeubles et de groupes S -arithm ´ etiques. T opology , 15(3):211–232, 1976. [4] Johan L. Dupont and Chih Han Sah. Scissors congruences. II. J. Pur e Appl. Algebra , 25(2):159–195, 1982. [5] Sebastian Goette and Christian K. Zickert. The extended Bloch group and the Cheeger -Chern-Simons class. Geom. T opol. , 11:1623–1635, 2007. [6] Sabine Hesselmann. Zur T orsion der K ohomologie S -arithmetischer Gruppen . Bonner Mathematische Schriften [Bonn Mathematical Publications], 257. Univ ersit ¨ at Bonn Mathematisches Institut, Bonn, 1993. [7] Ke vin Hutchinson. A Bloch-W igner complex for SL 2 . T o appear in Journal of K -theory . [8] Ke vin Hutchinson and Liqun T ao. The third homology of the special linear group of a ﬁeld. J . Pur e Appl. Algebra , 213:1665–1680, 2009. [9] Ke vin Hutchinson and Liqun T ao. Homology stability for the special linear group of a ﬁeld and Milnor-Witt K -theory . Doc. Math. , (Extra V ol.):267–315, 2010. [10] Ke vin P . Knudson. Homology of linear gr oups , volume 193 of Pro gr ess in Mathematics . Birkh ¨ auser V erlag, Basel, 2001. [11] Stefan K ¨ uhnlein. On torsion in the cohomology of PSL 2 ( Z [1 / N ]). J. Number Theory , 99(2):284–297, 2003. [12] B. Mirzaii. Third homology of general linear groups. J. Alg ebra , 320(5):1851–1877, 2008. [13] W erner Nahm. Conformal ﬁeld theory and torsion elements of the Bloch group. In F rontier s in number theory , physics, and geometry . II , pages 67–132. Springer , Berlin, 2007. 42 KEVIN HUTCHINSON [14] J ¨ urgen Neukirch. Alg ebraic number theory , volume 322 of Grundlehr en der Mathematisc hen W issenschaften [Fundamental Principles of Mathematical Sciences] . Springer-V erlag, Berlin, 1999. T ranslated from the 1992 German original and with a note by Norbert Schappacher , With a fore word by G. Harder . [15] W alter D. Neumann and Jun Y ang. Bloch inv ariants of hyperbolic 3-manifolds. Duke Math. J. , 96(1):29–59, 1999. [16] W alter D. Neumann and Don Zagier . V olumes of hyperbolic three-manifolds. T opology , 24(3):307–332, 1985. [17] Alexander D. Rahm. Homology and K -theory of the Bianchi groups. C. R. Math. Acad. Sci. P aris , 349(11- 12):615–619, 2011. [18] Alexander D. Rahm and Mathias Fuchs. The inte gral homology of PSL 2 of imaginary quadratic integers with nontrivial class group. J . Pure Appl. Alg ebra , 215(6):1443–1472, 2011. [19] Chih-Han Sah. Homology of classical Lie groups made discrete. III. J. Pure Appl. Algebra , 56(3):269–312, 1989. [20] Mehmet Haluk S ¸ eng ¨ un. On the integral cohomology of Bianchi groups. Exp. Math. , 20(4):487–505, 2011. [21] A. A. Suslin. K 3 of a ﬁeld, and the Bloch group. T rudy Mat. Inst. Steklo v . , 183:180–199, 229, 1990. Translated in Proc. Steklov Inst. Math. 1991 , no. 4, 217–239, Galois theory , rings, algebraic groups and their applications (Russian). [22] Don Zagier . The dilogarithm function. In F r ontiers in number theory , physics, and geometry . II , pages 3–65. Springer , Berlin, 2007. S chool of M a thema tical S ciences , U niversity C ollege D ublin E-mail addr ess : kevin.hutchinson@ucd.ie

A refined Bloch group and the third homology of SL_2 of a field

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