A Bloch-Wigner complex for SL_2

A BLOCH-WIGNER COMPLEX FOR SL 2 KEVIN HUTCHINSON A bstra ct . W e introduce a reﬁnement of the Bloch-W igner complex of a ﬁeld F . This reﬁnement is complex of modules over the multiplicati ve group of the ﬁeld. Instead of computing the K 2 ( F ) and K ind 3 ( F ) - as the classical Bloch-W igner complex does - it calculates the second and third integral homology of SL 2 ( F ). On passing to F × -coin v ariants we recover the classical Bloch- W igner complex. W e include the case of ﬁnite ﬁelds throughout the article. 1. I ntr oduction What is no w usually referred to the as the Bloch gr oup of a ﬁeld F arose ﬁrst in the work of S. Bloch as an explicitly-presented approximation to indecomposable K 3 of the ﬁeld which could be used to deﬁne a regulator map based on the dilogarithm (see the notes [3]). When F = C (and, more generally , when F × = ( F × ) 2 ) there is a natural identiﬁcation K ind 3 ( C ) = H 3 (SL 2 ( C ) , Z ), and this latter group is a natural target for in v ariants of hyperbolic 3-manifolds. It was because of this connection with hyperbolic geometry that Dupont and Sah ([6] and [22]) explored the properties of the Bloch group. In particular, they wrote down a proof of the so-called Bloch- W igner Theor em ([6], Theorem 4.10): The pre-Bloch group (or scissors congruence gr oup ) of the ﬁeld F is the group, P ( F ), with generators [ 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 . (These relations correspond to the 5-term functional equation satisﬁed by the classical dilog- arithm. See Zagier [27] for a beautiful exposition of these and related matters.) W e will let S 2 Z ( F × ) denote the antisymmetric product F × ⊗ Z F × < x ⊗ y + y ⊗ x | x , y ∈ F × > . Then there is a well-deﬁned group homomorphism λ : P ( F ) → S 2 Z ( F × ) , [ x ] 7→ (1 − x ) ⊗ x and the theorem of Bloch and W igner says that there is an exact sequence 0 / / µ C / / H 3 (SL 2 ( C ) , Z ) / / P ( C ) λ / / S 2 Z ( C ) / / H 2 (SL 2 ( C ) , Z ) / / 0 . The argument of Dupont and Sah works equally well for any algebraically closed ﬁeld and more generally for any quadratically closed ﬁeld (i.e. satisfying F × = ( F × ) 2 ). When the ﬁeld F is quadratically closed then the homology groups can be interpreted in terms of K -theory: H 3 (SL 2 ( F ) , Z ) = K ind 3 ( F ) and H 2 (SL 2 ( F ) , Z ) = K 2 ( F ) = K M 2 ( F ). Thus the homology groups of the Bloch-W igner complex P ( F ) λ / / S 2 Z ( F ) Date : Nov ember 26, 2024. 1991 Mathematics Subject Classiﬁcation. 19G99, 20G10. K e y wor ds and phrases. K -theory , Group Homology . 1 2 KEVIN HUTCHINSON are (essentially) the K -theory groups K ind 3 ( F ) and K M 2 ( F ). The group B ( F ) = Ker( λ ) is the Bloch gr oup of the F . Suslin showed that, interpreted in this way , the Bloch-W igner theorem extends to all (inﬁnite) ﬁelds. He proved (see [25], Theorem 5.2) that for any inﬁnite ﬁeld F there is a natural exact sequence 0 / / T or Z 1 ] ( µ F , µ F ) / / K ind 3 ( F ) / / P ( F ) λ / / S 2 Z ( F ) / / K M 2 ( F ) / / 0 . where T or Z 1 ] ( µ F , µ F ) is 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 ) = T or Z 1 ( µ F , µ F ) in characteristic 2. The purpose of the current article is to extend the original sequence of Bloch-W igner-Dupont- Sah in another direction: namely , to construct a complex, which coincides with the one abov e when F is quadratically closed, but which calculates in the general case - instead of K -theory - the homology groups H 3 (SL 2 ( F ) , Z ) and H 2 (SL 2 ( F ) , Z ). Our main goal is to understand better the structure of the unstable homology group H 3 (SL 2 ( F ) , Z ) and its relation to K ind 3 ( F ). T o put this project in context, we recall some of what is known about the relationship between the homology groups and the K -theory groups. In general, the group extension 1 → SL n ( F ) → GL n ( F ) → F × → 1 deﬁnes an action of F × on the homology groups H k (SL n ( F ) , Z ). Since the determinant of a scalar matrix is an n -th po wer , the subgroup ( F × ) n acts tri vially . In the particular , the groups H k (SL 2 ( F ) , Z ) are modules over the integral group ring R F : = Z [ F × / ( F × ) 2 ]. The natural map H 2 (SL 2 ( F ) , Z ) → K 2 ( F ) (via stabilization and an in verse Hurewicz map) is surjective and in- duces an isomorphism on F × -coin variants H 2 (SL 2 ( F ) , Z ) F ×  K 2 ( F ) . Ho we ver , the action of F × on H 2 (SL 2 ( F ) , Z ) is in general nontrivial. The action of R F factors through the Grothendieck-W itt ring GW( F ) of the ﬁeld, and the k ernel of the surjecti ve map H 2 (SL 2 ( F ) , Z ) → K 2 ( F ) is isomorphic, as a GW( F )-module, to I( F ) 3 , the third power of the fundamental ideal I( F ) of the augmented ring GW( F ). (See Suslin [24], Appendix for the details.) T o be more explicit H 2 (SL 2 ( F ) , Z ) can be e xpressed as a ﬁbre product H 2 (SL 2 ( F ) , Z )  I( F ) 2 × I( F ) 2 / I( F ) 3 K M 2 ( F ) . H 2 (SL 2 ( F ) , Z ) is of interest in its own right to K -theorists and geometers because it coincides with the second Milnor -W itt K -group, K MW 2 ( F ), of the ﬁeld F (see, for example, [18] or [19]). More generally , the calculation of the groups H n (SL n ( F ) , Z ), which are at the boundary of the homology stability range, in volves the Milnor -W itt K -groups K MW n ( F ) ([11]). The group H 3 (SL 2 ( F ) , Z ) is of interest, among other reasons, because it is strictly belo w the range of homology stability . Ho we ver there is, for any ﬁeld F , a natural homomorphism H 3 (SL 2 ( F ) , Z ) → K ind 3 ( F ) which induces a surjecti ve homomorphism (see [10]) H 3 (SL 2 ( F ) , Z ) F × / / / / K ind 3 ( F ) . Suslin has ask ed the question whether this is an isomorphism, and it is known (see Mirzaii [17]) that the kernel consists of - at worst - 2-primary torsion. In order to reﬁne the Bloch-W igner sequence to a sequence which captures the homology of SL 2 ( F ), it is necessary to b uild in the R F -module structures at each stage. Thus, in this article we Bloch-W igner-Suslin comple x 3 introduce ﬁrst the r eﬁned pr e-Bloch gr oup RP ( F ) of a ﬁeld F . This is the R F -module generated by symbols [ x ] , x ∈ F × \ { 1 } subject to the relations 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 where h x i denotes the class of x in F × / ( F × ) 2 . Similarly we introduce an R F -module RS 2 Z ( F × ) which has natural generators [ x , y ], x , y ∈ F × . The ‘reﬁned Bloch-W igner complex’ is then the complex of R F -modules RP ( F ) Λ / / RS 2 Z ( F ) , [ x ] 7→ [1 − x , x ] . On taking F × -coin variants this reduces to the classical Bloch-W igner complex. Our main result (Theorem 4.3) is that there is, for any ﬁeld F , a natural complex of R F -modules 0 / / T or Z 1 ( µ F , µ F ) / / H 3 (SL 2 ( F ) , Z ) / / RP ( F ) Λ / / RS 2 Z ( F × ) / / H 2 (SL 2 ( F ) , Z ) / / 0 which is exact at e very term e xcept possibly at the term H 3 (SL 2 ( F ) , Z ), where the homology of the complex is annihilated by 4. The r eﬁned Bloch gr oup of the ﬁeld F is the R F -module RB ( F ) : = Ker( Λ ). The main theorem tells us that it is a good approximation to the H 3 (SL 2 ( F ) , Z ). In particular , we show that - up to some 2-primary torsion - RB ( F ) F ×  B ( F ) and K er( RB ( F ) → B ( F ))  Ker(H 3 (SL 2 ( F ) , Z ) → K ind 3 ( F )) . In a separate article we will de velop further the algebraic properties of the reﬁned Bloch group (see [8], for example). In particular , when the ﬁeld F has a valuation with residue ﬁeld k , there are useful specialization homomorphisms from RB ( F ) to P ( k ). W e will use these maps to sho w that if F is a nondyadic local ﬁeld with (ﬁnite) residue ﬁeld k there is a natural isomorphism H 3 (SL 2 ( F ) , Z [1 / 2])   K ind 3 ( F ) ⊕ P ( k )  ⊗ Z [1 / 2] . Similarly , we will show (see [8] for details) that for an y global ﬁeld F the kernel K er(H 3 (SL 2 ( F ) , Z [1 / 2]) → K ind 3 ( F ) ⊗ Z [1 / 2]) maps homomorphically onto the inﬁnite direct sum ( ⊕ v P ( k v ) ) ⊗ Z [1 / 2] , the sum being ov er all ﬁnite places v of F . Thus the (reﬁned) Bloch groups of ﬁnite ﬁelds will play an important role in future applications. Because of this, and unlike most of the references above, throughout the paper we include the case of ﬁnite ﬁelds. At times, they require separate treatment and methods. For this reason we include a separate section - section 3- recalling the results we need on the homology of SL 2 ( F ) for ﬁnite ﬁelds F . In the last section of the paper we combine our main theorem with these homology calculations to gi ve a proof of Suslin’ s theorem in the case of ﬁnite ﬁelds and to make some useful calculations in Bloch groups of ﬁnite ﬁelds. Remark 1.1. Se veral authors (W . Neumann [20], W . Nahm, S. Goette and C. Zickert [7]) hav e introduced and studied an e xtended Bloc h gr oup , which is e xactly isomorphic to the K ind 3 ( F ) - at least for some ﬁelds F . This is a quite di ﬀ erent object from the r eﬁned Bloch gr oup introduced here, which e ﬀ ectiv ely bears the same relationship to H 3 (SL 2 ( F ) , Z ) as the classical Bloch group does to K ind 3 ( F ). 4 KEVIN HUTCHINSON 2. B loch G r oups and the B loch -W igner map In this section, we revie w the deﬁnition of the classical Bloch group and pre-Bloch group of a ﬁeld, and we deﬁne our basic objects of study in this article, the reﬁned Bloch group and reﬁned pre-Bloch group. 2.1. Some notation in this article. For a ﬁeld F , we let G F denote the multiplicativ e 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 a commutativ e ring A and an A -module M , we let T n A ( M ) denote the n -fold tensor product of M ov er A . W e let ∧ n A ( M ) denote the n -th exterior power of M ov er A ; i.e. the n -th term of the graded ring T n A ( M ) / I where I is the ideal generated by the elements m ⊗ m , m ∈ M . W e let Sym n A ( M ) denote the n -th symmetric po wer of M o ver A ; i.e. the n -th term of the graded ring T n A ( M ) / J where J is the ideal generated by the elements m ⊗ n − n ⊗ m , m , n ∈ M . For an y abelian group A we let A 0 denote A ⊗ Z [1 / 2]. 2.2. The classical Bloch group. Let F be a ﬁeld with at least 4 elements and let X n denote the set of all ordered n -tuples of distinct points of P 1 ( F ). PGL 2 ( F ), and hence also GL 2 ( F ), acts on P 1 ( F ) via fractional linear transformations. Thus these groups act on X n via a diagonal action. No w let A ( F ) be the cok ernel of the homomorphism of GL 2 ( F )-modules δ : Z X 5 → Z X 4 , ( x 1 , . . . , x 5 ) 7→ 5 X j = 1 ( − 1) j + 1 ( x 1 , . . . , ˆ x j , . . . , x 5 ) . Then the pr e-Bloch gr oup of F is the group P ( F ) : = A ( F ) GL 2 ( F ) = Coker( ¯ δ : ( Z X 5 ) GL 2 ( F ) → ( Z X 4 ) GL 2 ( F ) ) . No w the orbits of GL 2 ( F ) on X 4 are classiﬁed by the cross-ratio: i.e., in general, ( x 1 , . . . , x 4 ) is in the orbit of (0 , ∞ , 1 , x ) where x ∈ P 1 ( F ) \ {∞ , 0 , 1 } = F × \ { 1 } is the cross-ratio ( x 4 − x 1 )( x 3 − x 2 ) ( x 3 − x 1 )( x 4 − x 2 ) of x 1 , . . . , x 4 . Thus ( Z X 4 ) GL 2 ( F )  M x ∈ F × \{ 1 } Z · (0 , ∞ , 1 , x ) and, similarly , ( Z X 5 ) GL 2 ( F )  M x , y ∈ F × \{ 1 } x , y Z · (0 , ∞ , 1 , x , y ) . For x , y in F × \ { 1 } , ¯ δ (0 , ∞ , 1 , x , y ) = ( ∞ , 1 , x , y ) − (0 , 1 , x , y ) + (0 , ∞ , x , y ) − (0 , ∞ , 1 , y ) + (0 , ∞ , 1 , x ) =  0 , ∞ , 1 , 1 − x 1 − y  −  0 , ∞ , 1 , 1 − x − 1 1 − y − 1  +  0 , ∞ , 1 , y x  − (0 , ∞ , 1 , y ) + (0 , ∞ , 1 , x ) Thus, if we let [ x ] denote the class of the orbit of (0 , ∞ , 1 , x ) in P ( F ) then 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 )  Bloch-W igner-Suslin comple x 5 for 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 k ernel of λ . 2.3. The reﬁned pre-Bloch group. Let F be a ﬁeld with at least 4 elements. The r eﬁned pr e-Bloch gr oup of F is the group RP ( F ) : = A ( F ) SL 2 ( F ) = Coker( ¯ δ : ( Z X 5 ) SL 2 ( F ) → ( Z X 4 ) SL 2 ( F ) ) . Since for any ﬁeld F we ha ve a short e xact sequence of groups 1 → PSL 2 ( F ) → PGL 2 ( F ) → G F → 1 it follo ws that if X is any PGL 2 ( F )-set, then SL 2 ( F ) \ X is a G F -set and ( Z X ) SL 2 ( F )  Z [SL 2 ( F ) \ X ] is an R F -module. The stabilizer in SL 2 ( F ) of (0 , ∞ ) is the subgroup T consisting of all diagonal matrices D ( a ) : = " a 0 0 a − 1 # For x ∈ P 1 ( F ), D ( a ) · x = a 2 x . Gi ven x , y in P 1 ( F ), let T x , y ∈ SL 2 ( F ) be the matrix T x , y =                                    " 1 − x 1 x − y − y x − y # x , y , ∞ " 1 − x 0 1 # y = ∞ " 0 − 1 1 − y # x = ∞ Then T x , y ( x ) = 0, T x , y ( y ) = ∞ , and, by the preceding remarks, if S ∈ SL 2 ( F ) satisﬁes S ( x ) = 0 and S ( y ) = ∞ , then S = D ( a ) · T x , y for some a ∈ F × . In particular , if A ∈ SL 2 ( F ), it follows that T A x , Ay = D ( a ) · T x , y · A − 1 for some a = a ( x , y , A ) ∈ F × . For x , y , z distinct points of P 1 ( F ), we deﬁne φ ( x , y , z ) : = T x , y ( z ) =              ( z − x )( x − y )( z − y ) − 1 , x , y , z , ∞ ( y − z ) − 1 , x = ∞ z − x , y = ∞ x − y , z = ∞ Thus φ ( x , y , z ) ∈ P 1 ( F ) \ {∞ , 0 } = F × , and φ (0 , ∞ , z ) = z for z ∈ F × . Furthermore, if A ∈ SL 2 ( F ), then φ ( A x , Ay , Az ) = T A x , Ay ( Az ) = D ( a ) · T x , y · A − 1 ( Az ) = a 2 φ ( x , y , z ) for some a ∈ F × . 6 KEVIN HUTCHINSON No w , for n ≥ 1, let Y n denote the set of ordered n -tuples of distinct points of F × . Y n is an F × -set via the diagonal action. Lemma 2.1. F or n ≥ 3 , the map Φ n : X n → Y n − 2 ( x 1 , x 2 , . . . , x n ) 7→ ( φ ( x 1 , x 2 , x 3 ) , φ ( x 1 , x 2 , x 4 ) , . . . , φ ( x 1 , x 2 , x n )) induces a bijection of G F -sets SL 2 ( F ) \ X n ← → ( F × ) 2 \ Y n − 2 . Pr oof. By the remarks abov e Φ n descends to a well-deﬁned map ¯ Φ n : SL 2 ( F ) \ X n → ( F × ) 2 \ Y n − 2 . Furthermore, the map Ψ n : Y n − 2 → X n ( y 1 , . . . , y n − 2 ) 7→ (0 , ∞ , y 1 , . . . , y n − 2 ) gi ves a set-theoretic section of Φ n which descends to an in verse of ¯ Φ n . Since, for any a ∈ F × , φ ( a x 1 , a x 2 , ay ) = ( a φ ( x 1 , x 2 , y ) , x 1 , ∞ a − 1 φ ( x 1 , x 2 , y ) , x 1 = ∞ it also follo ws that ¯ Φ n is a map of G F -sets.  Corollary 2.2. F or n ≥ 0 , let Z n denote the set of or der ed n-tuples, [ z 1 , . . . , z n ] , of distinct points of F × \ { 1 } . Then for all n ≥ 3 ther e is an isomorphism of R F -modules ( Z X n ) SL 2 ( F )  R F [ Z n − 3 ] . Pr oof. By Lemma 2.1 we ha ve R F -isomorphisms ( Z X n ) SL 2 ( F )  Z [SL 2 ( F ) \ X n ]  Z [( F × ) 2 \ Y n − 2 ] . Finally , we have an R F -isomorphism Z [( F × ) 2 \ Y n − 2 ]  R F [ Z n − 3 ] via the map ( y 1 , . . . , y n − 2 ) 7→ h y 1 i " y 2 y 1 , . . . , y n − 2 y 1 # .  It follo ws that the R F -isomorphism ( Z X n ) SL 2 ( F )  R F [ Z n − 3 ] is gi ven by ( x 1 , . . . , x n ) 7→ h φ ( x 1 , x 2 , x 3 ) i " φ ( x 1 , x 2 , x 4 ) φ ( x 1 , x 2 , x 3 ) , . . . , φ ( x 1 , x 2 , x n ) φ ( x 1 , x 2 , x 3 ) # . In particular , we have R F -isomorphisms ( Z X 3 ) SL 2 ( F )  R F , ( Z X 4 ) SL 2 ( F )  R F [ F × \ { 1 } ] , ( Z X 5 ) SL 2 ( F )  R F [ Z 2 ] Note that taking G F -coin variants of the terms in Corollary 2.2 we obtain Corollary 2.3. F or all n ≥ 3 there is an isomorphism of gr oups ( Z X n ) GL 2 ( F )  Z [ Z n − 3 ] . Bloch-W igner-Suslin comple x 7 In particular , for n = 4, the isomorphism ( Z X 4 ) GL 2 ( F )  Z [ F × \ { 1 } ] is giv en by ( x 1 , x 2 , x 3 , x 4 ) 7→ " φ ( x 1 , x 2 , x 4 ) φ ( x 1 , x 2 , x 3 ) # which is just the classical cross-ratio map. No w it follo ws from the calculations abov e that the map R F [ Z 2 ]  / / ( Z X 5 ) SL 2 ( F ) ¯ δ / / ( Z X 4 ) SL 2 ( F )  / / R F [ Z 1 ] is the R F -module homomorphism [ x , y ] 7→ [ 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 )  (since φ ( ∞ , 1 , a ) = (1 − a ) − 1 , φ (0 , 1 , a ) = ( a − 1 − 1) − 1 and φ (0 , ∞ , a ) = a .) Thus we ha ve: Lemma 2.4. The r eﬁned pr e-Bloch gr oup RP ( F ) is the R F -module with gener ators [ x ] , x ∈ F × subject to the r elations [ 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, by deﬁnition, we hav e P ( F ) = ( RP ( F )) F × = H 0 ( F × , RP ( F )). 2.4. The module RS 2 Z ( F × ) and the reﬁned Bloch gr oup of a ﬁeld. Lemma 2.5. Let G be an abelian gr oup. Ther e is a natural short exact sequence of Z [ G ] - modules 0 → I 3 G → I 2 G → Sym 2 Z ( G ) → 0 (wher e G acts trivially on the fourth term). Pr oof. In fact if R is an y commutati ve ring and I an ideal in R , then there is a natural e xact sequence Sym n + 1 R ( I ) η / / Sym n R ( I ) / / Sym n R / I ( I / I 2 ) / / 0 where η ( a 0 ∗ a 1 ∗ · · · ∗ a n ) = a 0 · ( a 1 ∗ · · · ∗ a n ) = ( a 0 a 1 ) ∗ · · · ∗ a n . In the particular case n = 2, R = Z [ G ], I = I G this gi ves an e xact sequence Sym 3 Z [ G ] ( I G ) → Sym 2 Z [ G ] ( I G ) → Sym 2 Z ( G ) → 0 since I G / I 2 G  G . No w there is a natural surjecti ve homomorphism of graded Z [ G ]-algebras Sym • Z [ G ] ( I G ) → I • G , a 1 ∗ · · · ∗ a n 7→ h h a 1 i i · · · h h a n i i . This is an isomorphism in dimension 2 (and,for trivial reasons, in dimensions 0 and 1). T o see this, apply the functor − ⊗ Z [ G ] I G to the short exact sequence 0 → I G → Z [ G ] → Z → 0 to obtain the exact sequence 0 → T or Z [ G ] 1 ( Z , I G ) → T 2 Z [ G ] ( I G ) → I 2 G → 0 . But T or Z [ G ] 1 ( Z , I G ) = H 1 ( G , I G )  H 2 ( G , Z )  ∧ 2 Z ( G ). A straightforward calculation no w shows that the map ∧ 2 Z ( G )  T or Z [ G ] 1 ( Z , I G ) → T 2 Z [ G ] ( I G ) sends g 1 ∧ g 2 to h h g 1 i i ⊗ h h g 2 i i − h h g 2 i i ⊗ h h g 1 i i . 8 KEVIN HUTCHINSON Finally , we observe that the image of the map Sym 3 Z [ G ] ( I G ) → Sym 2 Z [ G ] ( I G )  I 2 G is clearly I 3 G .  Remark 2.6. It is a straightforward matter to v erify that Sym • Z [ G ] ( I G ) has the follo wing presen- tation as a graded ring: It is generated in degree 1 by the elements h h g i i , g ∈ G , subject to the relations (N) h h 1 i i = 0 (R) h h g 1 i i ∗ h h g 2 g 3 i i + h h g 2 i i ∗ h h g 3 i i = h h g 1 g 2 i i ∗ h h g 3 i i + h h g 1 i i ∗ h h g 2 i i for all g 1 , g 2 , g 3 ∈ G . (S) h h g 1 i i ∗ h h g 2 i i = h h g 2 i i ∗ h h g 1 i i for all g 1 , g 2 ∈ G For abelian groups the surjecti ve homomorphism of graded rings α : Sym • Z [ G ] ( I G ) → I • G is not generally injecti ve in dimensions greater than 2. Ho we ver , the following is kno wn: If G is either torsion-free or c yclic then α is an isomorphism (see Bak and T ang [1]). On the other hand, if G is an elementary abelian 2-group, then the k ernel of α is the ideal generated by the degree 3 terms h h g 1 i i ∗ h h g 2 i i ∗ h h g 1 g 2 i i for g 1 , g 2 ∈ G (see Bak and V avilo v [2]). It is easy to see that these latter terms are nonzero (by considering their image in Sym 3 Z / 2 ( G ), for example). Applying Lemma 2.5 to the case G = G F gi ves: Corollary 2.7. Let F be a ﬁeld. Ther e is a natural e xact sequence of R F -modules 0 → I 3 F → I 2 F → Sym 2 F 2 ( G F ) → 0 . On the other hand, clearly there is also a natural homomorphism of additi ve groups S 2 Z ( F × ) / / / / S 2 Z ( F × ) ⊗ Z / 2  / / Sym 2 F 2 ( G F ) . 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. Gi ven a , b ∈ F × , we let [ a , b ] denote the element [ a , b ] : = ( h h a i i h h b i i , a ◦ b ) ∈ RS 2 Z ( F × ) . Lemma 2.8. Let F be a ﬁeld. (1) I F RS 2 Z ( F × )  I 3 F (2) RS 2 Z ( F × ) F ×  S 2 Z ( F × ) (3) RS 2 Z ( F × ) is gener ated as an R F -module by the elements [ a , b ] , a , b ∈ F × . Pr oof. (1) W e hav e an injecti ve homomorphism I 3 F → I 2 F ⊕ S 2 Z ( F × ) , x 7→ ( x , 0) . But, since x maps to 0 in Sym 2 F 2 ( G F ) = I 2 F / I 3 F , in fact I 3 F ⊂ RS 2 Z ( F × ). On the other hand, if a , b , c ∈ F × , then h h a i i [ b , c ] = ( h h a i i h h b i i h h c i i , 0). So I 3 F ⊂ I F RS 2 Z ( F × ). Con versely , if ( x , y ) ∈ RS 2 Z ( F × ) and a ∈ F × , then h h a i i ( x , y ) = ( h h a i i x , 0) ∈ I 3 F ; i.e. I F RS 2 Z ( F × ) ⊂ I 3 F . Bloch-W igner-Suslin comple x 9 (2) Suppose that ( x , y ) lies in the kernel of the surjecti ve R F -homomorphism RS 2 Z ( F × ) → S 2 Z ( F × ). Then y = 0 and thus x maps to 0 in Sym 2 F 2 ( G F ). By Corollary 2.7, x ∈ I 3 F = and hence ( x , y ) ∈ I F RS 2 Z ( F × ). Observe that it follo ws that there is a natural short exact sequence of R F -modules 0 → I 3 F → RS 2 Z ( F × ) → S 2 Z ( F × ) → 0 . (3) Let K ( F ) be the R F -submodule of RS 2 Z ( F × ) generated by the elements [ a , b ]. Since h h a i i [ b , c ] = h h a i i h h b i i h h c i i ∈ I 3 F ⊂ RS 2 Z ( F × ), it follows that I 3 F ⊂ K ( F ). On the other hand the homomorphism RS 2 Z ( F × ) → S 2 Z ( F × ) maps K ( F ) onto S 2 Z ( F × ), since the latter is generated by the elements a ◦ b . Thus K ( F ) = RS 2 Z ( F × ) as required.  Observe that the R F -module structure on RS 2 Z ( F × ) is gi ven by the formula h b i [ a , c ] = [ ab , c ] − [ b , c ] = [ a , bc ] − [ a , b ] . W e deﬁne 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 ] . In vie w 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 × ) . It is a tedious calculation to v erify directly λ 1 is a well-deﬁned homomorphism of R F -modules. Ho we ver , we will see belo w that λ 1 arises naturally as a di ﬀ erential in a spectral sequence. Recall that the homology groups H k (SL 2 ( F ) , Z ) are naturally R F -modules for all k . Theorem 2.9. F or any ﬁeld F with at least 10 elements, ther e is a natural surjective R F -module homomorphism RS 2 Z ( F × ) → H 2 (SL 2 ( F ) , Z ) inducing an isomorphism Coker( Λ )  H 2 (SL 2 ( F ) , Z ) . Pr oof. Suppose ﬁrst that F is ﬁnite. Then H 2 (SL 2 ( F ) , Z ) = 0. The statement of the theorem amounts to the surjecti vity of Λ . For a ﬁnite ﬁeld, since F × is cyclic, S 2 Z ( F × ) = Sym 2 Z / 2 ( G F ) and thus RS 2 Z ( F × ) = I 2 F . Recall that the Grothendieck-W itt ring of the ﬁeld F is the ring GW( F ) = R F / J F , where J F is the ideal generated by the elements h h 1 − x i i h h x i i . Thus Coker( Λ ) is I( F ) 2 , where I( F ) is the fundamental ideal I F / J F in GW( F ). Howe ver , it is well-kno wn that I( F ) 2 = 0 for any ﬁnite ﬁeld F (see, for example, [16], section III.5). Thus we can suppose that F is inﬁnite. In this case, the symplectic case of the theorem of Mat- sumoto and Moore ([13]), gi ves a presentation of the group H 2 (SL 2 ( F ) , Z ). It has the follo wing form: The generators are symbols h a 1 , a 2 i , a i ∈ F × , subject to the relations: (i) h a 1 , a 2 i = 0 if a i = 1 for some i (ii) h a 1 , a 2 i = D a − 1 2 , a 1 E (iii) D a 1 , a 2 a 0 2 E + D a 2 , a 0 2 E = D a 1 a 2 , a 0 2 E + h a 1 , a 2 i (i v) h a 1 , a 2 i = h a 1 , − a 1 a 2 i (v) h a 1 , a 2 i = h a 1 , (1 − a 1 ) a 2 i 10 KEVIN HUTCHINSON Furthermore, Suslin has shown ([24], appendix) that for an inﬁnite ﬁeld F , there is an isomor- phism of R F -modules H 2 (SL 2 ( F ) , Z )  I( F ) 2 × I( F ) 2 / I( F ) 3 K M 2 ( F ) , h a , b i ↔ ( h h a i i h h b i i , { a , b } ) . No w we hav e a map of diagrams of R F -modules S 2 Z ( F × )   K M 2 ( F )   I 2 F / / Sym 2 Z ( G F ) / / I( F ) 2 / / I( F ) 2 / I( F ) 3 which induces a map of pullbacks RS 2 Z ( F × ) → I( F ) 2 × I( F ) 2 / I( F ) 3 K M 2 ( F )  H 2 (SL 2 ( F ) , Z ) sending the elements [ a , b ] to the elements h a , b i . This map is surjectiv e, since the elements h a , b i generate H 2 (SL 2 ( F ) , Z ), and the image of Λ is contained in its kernel since { 1 − x , x } = 0 in K M 2 ( F ) and h h 1 − x i i h h x i i = 0 in I( F ) 2 . T o complete the proof of the theorem we must show that there is an R F -homomorphism H 2 (SL 2 ( F ) , Z ) → Coker( Λ ) sending h a , b i to [ a , b ] (mod Im( Λ )); i.e. we must show that the elements [ a , b ] ∈ RS 2 Z ( F × ) satisfy the Matsumoto-Moore relations modulo the image of Λ . No w the elements [ a , b ] are easily seen to satisfy relations (i), (ii) and (iii). On the other hand, since [ a , 1 − a ] ≡ 0 (mod Im( Λ )) for all a ∈ F × , and since Λ is an R F -homomorphism, it follo ws that 0 ≡ h b i [ a , 1 − a ] ≡ [ a , (1 − a ) b ] − [ a , b ] (mod Im( Λ )) for all a , b ∈ F × . No w , for any a ∈ F × , Λ ( [ a ] + h − 1 i h a − 1 i ) = [ − a , a ], since λ 1 ( [ a ] + h − 1 i h a − 1 i ) = h h 1 − a i i h h a i i + h − 1 i h h a ( a − 1) i i h h a i i = ( h h (1 − a ) a i i − h h 1 − a i i − h h a i i ) + h − 1 i ( h h a − 1 i i − h h a ( a − 1) i i − h h a i i ) = h h (1 − a ) a i i − h h 1 − a i i − h h a i i + h h 1 − a i i − h h a (1 − a ) i i − h h − a i i + h h − 1 i i = h h − 1 i i − h h a i i − h h − a i i = h h a i i h h − a i i and λ 2 ( [ a ] + h − 1 i h a − 1 i ) = (1 − a ) ◦ a + (1 − a − 1 ) ◦ a − 1 = (1 − a ) ◦ a − 1 − a − a ! ◦ a = ( − a ) ◦ a . Thus [ a , − a ] ≡ 0 (mod Im( Λ )) for all a ∈ F × and hence 0 ≡ h b i [ a , − a ] ≡ [ a , − ab ] − [ a , b ] (mod Im( Λ )) for all a , b ∈ F × . Thus relations (iv) and (v) also hold in Coker( Λ ), and the theorem is prov en.  Remark 2.10. The restriction to ﬁelds with at least 10 elements is to rule out the exceptional cases of the ﬁeld with 4 and the ﬁeld with 9 elements for which H 2 (SL 2 ( F ) , Z ) = Z / p is nonzero (see Lemma 3.14 belo w) Finally , we can deﬁne the r eﬁned Bloch gr oup of the ﬁeld F (with at least 4 elements) to be the R F -module RB ( F ) : = Ker( Λ : R P ( F ) → RS 2 Z ( F × )) . Thus we hav e: Bloch-W igner-Suslin comple x 11 Corollary 2.11. F or any ﬁeld F (with at least 10 elements) ther e is an exact sequence of R F - modules 0 → RB ( F ) → RP ( F ) → RS 2 Z ( F × ) → H 2 (SL 2 ( F ) , Z ) → 0 . For future reference, we mak e the follo wing observ ation: Lemma 2.12. Let F be a ﬁnite ﬁeld with at least 4 elements. Then the natural map RB ( F ) → B ( F ) induces an isomorphism RB ( F ) F ×  B ( F ) . Pr oof. W e can assume F has odd characteristic, since otherwise G F = { 1 } and RP ( F ) = P ( F ), RB ( F ) = B ( F ). Thus if a is a generator of F × , G : = G F is cyclic of order 2 generated by the class h a i . Thus, I F = Z · h h a i i is inﬁnite cyclic, and I n F = 2 n − 1 I F for all n ≥ 1 (since h h a i i 2 = − 2 h h a i i ). Since F × is cyclic, S 2 Z ( F × ) = Sym 2 Z ( G F ) and thus RS 2 Z ( F × ) = I 2 F . The fact that I( F ) 2 = 0 thus amounts to the statement that the map Λ : RP ( F ) → I 2 F is surjecti ve. Thus taking G -coin v ariants of the short exact sequence 0 → RB ( F ) → RP ( F ) → I 2 F → 0 gi ves the e xact sequence H 1 ( G , I 2 F ) / / RB ( F ) F × / / P ( F ) λ / / Sym 2 Z ( G F ) / / 0 . Ho we ver , I 2 F  Z and the generator h a i of the cyclic group G acts as − 1 (since h a i h h a i i = − h h a i i ). Thus H 1 ( G , I 2 F ) = 0 and RB ( F ) F × = B ( F ) as required.  Remark 2.13. W e will sho w below that for a ﬁnite ﬁeld F the action of F × on RB ( F ) is tri vial. It will thus follo w that RB ( F ) = B ( F ) when F is ﬁnite. In general the action of F × on RB ( F ) is nontri vial (for example, if F is a local or global ﬁeld). Ho we ver , we will see below that for any ﬁeld F the natural map RB ( F ) → B ( F ) is always surjecti ve and that the induced surjectiv e map R B ( F ) F × → B ( F ) has a kernel annihilated by a po wer of 2. 3. T he homology of SL 2 of finite fields In this section p is a prime number , q = p f for some f ≥ 1, and F q denotes the ﬁnite ﬁeld with q elements. Recall that the group SL 2 ( F q ) has order q ( q 2 − 1) = q ( q − 1)( q + 1). W e re vie w - for want of an explicit reference - some of the main facts about the integral homomlogy of SL 2 ( F q ). W e recall the relev ant results we will use (for details, see [4], III.9, III.10): Let G be a ﬁnite group, ` a prime number and H a Sylo w ` -subgroup of G . For any g ∈ G , conjugation by g induces a homomorphism H k ( H , Z ) → H k ( g H g − 1 , Z ) , z 7→ g · z . For g ∈ G , we say that z ∈ H k ( H , Z ) is g-in variant if res H H ∩ gH g − 1 z = res gH g − 1 H ∩ gH g − 1 g · z . Let in v G H k ( H , Z ) : = { z ∈ H k ( H , Z ) | z is g -in variant for all g ∈ G } . Then for k ≥ 1, the corestriction homomorphism cor G H : H k ( H , Z ) → H k ( G , Z ) induces an isomorphism in v G H k ( H , Z )  H k ( G , Z ) ( ` ) = H k ( G , Z ( ` ) ) . 12 KEVIN HUTCHINSON No w , for ` , p , the ` -Sylow subgroups of G = SL 2 ( F q ) are cyclic or generalised quaternion, and hence the (co)homology is ` -periodic. This means that there is a number d = d ( ` ) ≥ 2 such that H k ( G , Z ( ` ) )  H k + d ( G , Z ( ` ) ) for all k ≥ 1 and that this happens if and only if H d − 1 ( G , Z ( ` ) )  Z ( ` ) / | G | . In this case, d = d ( ` ) is called the ` -period of G . W e recall also the follo wing useful results of Swan [26]: Theorem 3.1. [Swan [26] , Theor ems 1 and 2] (1) Suppose that ` is odd and the the ` -Sylow subgr oup of G is cyclic. Let H be a ` - Sylow subgr oup of G and let Φ ` be the gr oup of automorphisms of H induced by inner automorphisms of G . Then the ` -period of G is 2 · | Φ ` | . (2) If the 2 -Sylow subgr oup of G is cyclic, the 2 -period is 2 . If the 2 -Sylow subgr oup of G is gener alised quaternion then the 2 -period is 4 . Furthermore, we recall the well-kno wn calculation Lemma 3.2. H 1 (SL 2 ( F q ) , Z ) = ( Z / p , q = 2 , 3 0 , otherwise Corollary 3.3. Suppose p is odd. Then the 2 -period of SL 2 ( F q ) is 4 and for k ≥ 1 H k (SL 2 ( F q ) , Z (2) ) = ( Z (2) / q ( q 2 − 1) , k ≡ 3 (mod 4) 0 , otherwise Pr oof. When p , 2, the 2-Sylow subgroups of SL 2 ( F q ) are generalized quaternion groups. So Swan’ s theorem tells us that the 2-period is 4, and hence that H k (SL 2 ( F q ) , Z (2) ) = Z (2) / q ( q 2 − 1) whene ver k ≡ 3 (mod 3). On the other hand, the ev en-dimensional integral homology of the (generalized) quaternion groups are zero. Finally , by Lemma 3.2, H k (SL 2 ( F q ) , Z (2) ) = 0 for k ≡ 1 (mod 4).  No w we consider the ` -Sylo w subgroups for odd ` dividing q − 1. W e let T denote the diagonal subgroup { D ( x ) | x ∈ F × } of SL 2 ( F ). T is c yclic of order q − 1. Since the order of SL 2 ( F ) is q ( q 2 − 1), it follows that for any odd prime ` dividing q − 1, T ( ` ) is a Sylo w ` -subgroup of SL 2 ( F ). Lemma 3.4. Let ` be an odd prime dividing q − 1 . The ` -period of SL 2 ( F q ) is 4 and for k ≥ 1 H k (SL 2 ( F q ) , Z ( ` ) ) = ( Z ( ` ) / q ( q 2 − 1) , k ≡ 3 (mod 4) 0 , otherwise Pr oof. Fix ` odd dividing q − 1, and let D ( a ) ∈ T ( ` ) . So a 2 , 1. No w suppose that A = " x y z w # normalises T ( ` ) . Then since " x y z w # " a 0 0 a − 1 # " w − y − z x # = " a x w − a − 1 yz ( a − 1 − a ) xy ( a − a − 1 ) zw a − 1 xw − ayz # ∈ T it follo ws that xy = zw = 0. Thus A belongs to the group of order 2( q − 1) generated by T and w = " 0 1 − 1 0 # . Bloch-W igner-Suslin comple x 13 Conjugation by w acts as − 1 (i.e in version) on T ( ` ) . Hence Φ ` = h − 1 i (in the notation of Theorem 3.1 abov e). So the ` -period is 2 | Φ ` | = 4. The rest follows as in the case ` = 2.  Next, we deal with the case ` | q + 1 ( ` odd). Let E / F q be a quadratic extension of ﬁelds.(So E = F q 2 .) Then there is an embedding of groups E × → Aut F q ( E ) , a 7→ µ a , and Aut F q ( E )  GL 2 ( F q ) on choosing an F q -basis of E . The composite E × / / Aut F q ( E ) det / / F × q is just the norm map N E / F q . Thus, if we let K = Ker( N E / F q : E × → F × ), we obtain an embedding K → SL 2 ( F q ) on choosing an F q -basis of E . Since the norm map is surjecti ve, it follo ws that K is cyclic of order q + 1. Thus for any odd ` dividing q + 1, K ( ` ) is a Sylo w ` -subgroup of SL 2 ( F q ). Lemma 3.5. Let ` be an odd prime dividing q + 1 . The ` -period of SL 2 ( F q ) is 4 and for k ≥ 1 H k (SL 2 ( F q ) , Z ( ` ) ) = ( Z ( ` ) / q ( q 2 − 1) , k ≡ 3 (mod 4) 0 , otherwise Pr oof. Let µ : E × → GL 2 ( F q ) be the embedding described above. If σ ∈ Gal( E / F q ) then σ ∈ Aut F q ( E ) and thus, gi ven the choice of basis, is represented by an element ˜ σ ∈ GL 2 ( F q ). Then Γ : = µ ( E × ) · h ˜ σ i ⊂ GL 2 ( F q ) is a semidirect product in which the ˜ σ by conjugation on µ ( E × ) corresponds to the Galois action of σ on E × . Fix an odd prime ` dividing q + 1. So µ ( K ( ` ) ) is an ` -Sylow subgroup of s pl 2 F q . W e will show that the normalizer µ ( K ( ` ) ) in SL 2 ( F q ) is Γ ∩ SL 2 ( F ). Since the elements of µ ( E × ) act trivially by conjugation on µ ( K ) and since ˜ σ acts by in version on µ ( K ) the result follows from this as in the case ` | q − 1. W e must distinguish two cases: Case (i): p , 2. Let E = F q ( θ ) where θ 2 = α ∈ F q (and thus α is not square in F q ). T ake the basis { 1 , θ } of E . Then the associated embedding µ : K → SL 2 ( F q ) is gi ven by a + b θ 7→ " a b α b a # . Suppose that A = " x y z w # ∈ SL 2 ( F q ) normalizes µ ( K ( ` ) ). Let λ = a + b θ ∈ K ( ` ) . Since ` does not di vide q − 1, λ < F q (so that b , 0). Then from A µ ( λ ) A − 1 = " a + b ( yw − xz α ) b ( x 2 α − y 2 ) b ( w 2 − z 2 α ) a − b ( yw − xz α ) # ∈ µ ( K ) we obtain the conditions yw − xz α = 0 x 2 α − y 2 = ( w 2 − z 2 α ) α (since 2 b , 0). No w we ﬁx x and z . Eliminating w from these equations gi ves the quartic ( y 2 − x 2 α )( y 2 − z 2 α 2 ) = 0 14 KEVIN HUTCHINSON in y . Since α is not square in F , this leads to the two solutions ( y , w ) = ( z α, x ) and ( − z α, − x ). The ﬁrst of these gi ves A = " x z α z x # ∈ µ ( E × ) and the second gi ves A = " x − z α z − x # = " x z α z x # " 1 0 0 − 1 # = " x z α z x # ˜ σ ∈ Γ . So the normalizer of µ ( K ( ` ) ) is contained in Γ as required. Case (ii): p = 2 W e write E = F q ( θ ) where θ satisﬁes θ 2 + θ = α and σ ( θ ) = 1 + θ . Again we choose the basis { 1 , θ } of E . The embedding µ : E × → GL 2 ( F q ) then has the form a + b θ 7→ " a b α b a + b # and we hav e ˜ σ = " 1 1 0 1 # . Suppose again that A = " x y z w # ∈ SL 2 ( F ) normalizes µ ( K ( ` ) ) and that λ = a + b θ ∈ K ( ` ) (so that b , 0 as abov e). Then from A µ ( λ ) A − 1 = " a ( x w + yz ) + b ( xz α + yz + yw ) b ( x 2 α + y 2 + xy ) b ( w 2 + z 2 α + zw ) a ( x w + zy ) + b ( xz α + xw + yw ) # ∈ µ ( E × ) we get the conditions: xw + yz = z 2 α + w 2 + zw x 2 α + xy + y 2 = ( z 2 α + w 2 + zw ) α. If we ﬁx x and z , then the four solutions of this pair of binary quadratic equations is ( y , w ) = ( z α, 0) , ( z α, x + z ) , ( x + z α, x ) and ( x + z α, z ). The ﬁrst and last of these gi ve singular matrices. The second gi ves A = " x z α z x + z # = µ ( x + z θ ) ∈ µ ( E × ) . The third solution gi ves A = " x x + z α z x # = " x z α z x + z # · ˜ σ ∈ Γ . Thus, again, the normalizer of µ ( K ( ` ) ) is contained in Γ and the lemma is prov en.  Pulling together the statements of Corollary 3.3 and Lemmas 3.4 and 3.5 we hav e Corollary 3.6. F or all k ≥ 1 H k (SL 2 ( F q ) , Z [1 / p ]) = ( Z / ( q 2 − 1) , k ≡ 3 (mod 4) 0 , otherwise Furthermore, we can deduce the follo wing: Bloch-W igner-Suslin comple x 15 Corollary 3.7. Let H be any subgr oup of SL 2 ( F q ) of or der not divisible by p. If k ≡ 3 (mod 4) then H k ( H , Z ) = Z / | H | and the cor estriction map H k ( H , Z ) → H k (SL 2 ( F q ) , Z ) is injective. Pr oof. It is clearly su ﬃ cient to consider the case where H is an ` -group for some prime ` , p . In particular , H is cyclic or generalised quaternion and H k ( H , Z ) = Z / | H | . Now H is contained in an ` -Sylow subgroup, L say , of SL 2 ( F q ). It is a straightforward calculation to show that whenever L is a cyclic ` -group or generalized quaternion 2-group and whenev er H is a subgroup that the corestriction map H k ( H , Z ) → H k ( L , Z ) is injectiv e (for k ≡ 3 (mod 4)). On the other hand, the results abov e sho w that the corestriction map H k ( L , Z ) → H k (SL 2 ( F ) , Z ) is injectiv e.  W e will also need the follo wing result in the next section: Lemma 3.8. F or all k, the natural action of F × q on H k (SL 2 ( F q ) , Z [1 / p ]) is trivial. Pr oof. By Corollary 3.6, we can assume k ≡ 3 (mod 4). By Corollary 3.7, the corestriction map H k (SL 2 ( F q ) , Z [1 / p ]) → H k (SL 2 ( F q 2 ) , Z [1 / p ]) is injectiv e. But for any a ∈ F × q , a ∈ ( F × q 2 ) 2 and thus h a i = 1 in G F q 2 , so that h a i acts trivially on H k (SL 2 ( F q 2 ) , Z [1 / p ]).  Corollary 3.9. F or any ﬁnite ﬁeld F q ther e is a natural isomorphism H 3 (SL 2 ( F q ) , Z [1 / p ])  K ind 3 ( F q ) = K 3 ( F q ) . Pr oof. For an y ﬁeld F , the Hure wicz map induces an isomorphism K 3 ( F ) / ( − 1 · K 2 ( F ))  H 3 (SL ( F ) , Z ) (see Suslin, [25], Corollary 5.2). Thus, for any ﬁeld F there is a natural composite homomor- phism H 3 (SL 2 ( F ) , Z ) → H 3 (SL ( F ) , Z ) → K ind 3 ( F ) . When F is algebraically closed, this map is an isomorphism (Sah, [22]). For a ﬁnite ﬁeld F q we hav e K 3 ( F q ) = K ind 3 ( F q ) = Z / ( q 2 − 1) = K ind 3 ( F q ) ⊗ Z [1 / p ] and the functorial maps K 3 ( F q ) → K 3 ( F q n ) are injecti ve (Quillen, [21]) Thus, if we let ¯ F q denote an algebraic closure of F q we hav e a commutative diagram H 3 (SL 2 ( F q ) , Z [1 / p ]) / /  _   K ind 3 ( F q )  _   H 3 (SL ( ¯ F q ) , Z [1 / p ])  / / K ind 3 ( ¯ F q ) in which the vertical arrows are injectiv e, from which it follows that the top horizontal map is injecti ve, and hence an isomorphism of ﬁnite abelian groups of equal order .  In the remainder of this section, we will calculate, for completeness, the p -Sylow subgroups of H k (SL 2 ( F q ) , Z ) for k ≤ 3. Of course, for g ∈ G and z ∈ H k ( H , Z ) the condition res H H ∩ gH g − 1 z = res gH g − 1 H ∩ gH g − 1 g · z 16 KEVIN HUTCHINSON is trivially satisﬁed if H ∩ g H g − 1 = { 1 } . Thus, in order to determine in v G H k ( H , Z ) for an ` - Sylo w subgroup H , it is enough to consider only the set Conj( G , H ) of those elements g for which H ∩ gH g − 1 , { 1 } . A Sylo w p -subgroup of SL 2 ( F q ) is the group of unipotents U : = (" 1 a 0 1 # | a ∈ F q )  F q . W e ﬁrst determine the set Conj(SL 2 ( F q ) , U ) of those A ∈ SL 2 ( F q ) for which AU A − 1 ∩ U , { 1 } . Let A = " x y z w # ∈ SL 2 ( F q ) . Then A " 1 a 0 1 # A − 1 = " 1 − a xz a x 2 − az 2 1 + a xz # . Thus A ∈ Conj(SL 2 ( F q ) , U ) if and only if z = 0 and in this case A = " x y 0 x − 1 # ∈ B where B is the subgroup of upper triangular matrices in SL 2 ( F q ). Corollary 3.10. (1) F or all k ≥ 1 , we have H k ( B , Z )  H k ( T , Z ) ⊕ H k ( B , Z ) ( p ) . (2) The inclusion B → SL 2 ( F q ) induces an isomorphism H k ( B , Z ) ( p )  H k (SL 2 ( F q ) , Z ) ( p )  H 0  ( F × q ) 2 , H k ( F q , Z )  . Pr oof. (1) The result follows from the Hochschild-Serre spectral sequence associated to the split short exact sequence 1 → U → B → T → 1 together with the fact that ( | T | , | B | ) = 1. (2) The Sylow p -subgroup U of B is also a Sylow p -subgroup of SL 2 ( F q ) and the calcula- tions abov e sho w that Conj(SL 2 ( F q ) , U ) = Conj( B , U ) = B . Thus H k ( B , Z ) ( p ) = in v B H k ( U , Z ) = inv SL 2 ( F ) H k ( U , Z ) = H k (SL 2 ( F ) , Z ) ( p ) . The ﬁnal isomorphism deriv es from the fact that U  F q is normal in B with quotient T  F × q and with these identiﬁcations a ∈ F × q acts by conjug ation on U = F q as multipli- cation by a 2 .  Lemma 3.11. SL 2 ( F p ) is p periodic with p-period d = d ( p ) given by d ( p ) = ( 2 , p = 2 p − 1 , p , 2 Furthermor e, for k ≥ 1 H k (SL 2 ( F p ) , Z ) ( p ) = ( Z / p , k ≡ − 1 (mod d ( p )) 0 , otherwise Bloch-W igner-Suslin comple x 17 Pr oof. The ﬁrst statement follo ws from Swan’ s Theorem, since Φ p  ( F × p ) 2 . For the second statement, we can suppose p is odd and let x ∈ ( F × p ) 2 of order ( p − 1) / 2. Then multiplication by x on F p = Z / p induces multiplication by x k + 1 on H 2 k + 1 ( F p , Z ) = Z / p . The statement no w follo ws easily , since Z / p is in variant only if x k + 1 = 1.  When q = p f > p , of course the integral homology is no longer p -periodic. Ho we ver , we will calculate H k (SL 2 ( F q ) , Z ) ( p ) for k ≤ 3. The follo wing is a minor v ariation on [23], Lemma 1.: Lemma 3.12. Let m , n ≥ 1 . Suppose that ( p − 1) f > mn. Then ther e exists a ∈ F × q such that for any (not necessarily distinct) φ 1 , . . . , φ n ∈ Gal( F q / F p ) we have Q n i = 1 φ ( a m ) , 1 . Pr oof. Let a be a generator of F × q . Suppose, for the sake of contradiction that there exist φ 1 , . . . , φ n ∈ Gal( F q / F p ) with Q i φ i ( a m ) = 1. Since Gal( F q / F p ) is generated by x 7→ x p , for each i ∈ { 1 , . . . , n } there exists k i < f such that φ i ( x ) = x p k i . Hence a P n i = 1 m p k i = 1 and thus P n i = 1 m p k i ≡ 0 (mod p f − 1). F or 0 ≤ t ≤ f − 1 let s t ≥ 0 be the number of i for which k i = t . Thus f − 1 X t = 0 m s t p t ≡ 0 (mod p f − 1) . Let k t = m s t for t < f . Then P k t ≥ ( p − 1) f . For if some k t ≥ p , then replacing k t by k t − p and (ordering the t cyclically) k t + 1 by k t + 1 + 1 we get a new system of k t satisfying the same congruence b ut ha ving a smaller sum. By iterating this operation we arri ve at a collection k 0 1 , . . . , k 0 f − 1 satisfying the congruence and also k 0 t < p for all t and P k 0 t ≤ P k t . But then the inequalities p f − 1 ≤ f − 1 X t = 1 k 0 t p t ≤ ( p − 1)(1 + p + · · · + p f − 1 ) = p f − 1 imply that k 0 t = p − 1 for all t and hence P k t ≥ P k 0 t = ( p − 1) f . But this gi ves the contradiction ( p − 1) f ≤ P k t = m P s t = mn , proving the lemma.  Corollary 3.13. Suppose that ( p − 1) f > mn. Let F × q act on F q by multiplication. Then H 0  ( F × q ) m , T n Z ( F q )  = H 0  ( F × q ) m , ∧ n Z ( F q )  = 0 . Pr oof. By Lemma 3.12, there exists a ∈ F × q such that if b = a m then Q n i = 1 φ i ( b ) , 1 for all φ 1 , . . . , φ n ∈ Gal( F q / F p ). Let µ b denote the F p -linear endomorphism x 7→ b x of F q . Let φ be the F p -linear endomorphism µ b ⊗ · · · ⊗ µ b of T n F p ( F q ). Then b acts as φ on T n F p ( F ). Thus b ﬁxes a nonzero element of this space if and only if 1 is not an eigen v alue of φ . The eigen values of φ are products of the form λ 1 · · · λ n where λ 1 , . . . , λ n are (not necessarily distinct) eigen v alues of µ b . But the eigen v alues of µ b are precisely the v alues of the elements of Gal( F q / F p ) at b . This proves the result. The same ar gument applies to ∧ n Z ( F q ) = ∧ n F p ( F q ), since it is a quotient module of T n F p ( F q ) for the action of F × q .  Lemma 3.14. H k (SL 2 ( F q ) , Z ) ( p ) = Z / p if ( k , q ) ∈ { (1 , 2) , (1 , 3) , (2 , 4) , (2 , 9) , (3 , 2) , (3 , 3) , (3 , 4) , (3 , 5) , (3 , 8) , (3 , 9) , (3 , 27) } . and H k (SL 2 ( F q ) , Z ) ( p ) = 0 for any other value of ( k , q ) with k ≤ 3 . 18 KEVIN HUTCHINSON Pr oof. The cases k = 1 or q = p are already co vered by Lemmas 3.2 and 3.11. So we can suppose that f , k ≥ 2. W e recall that H 2 ( F q , Z ) = ∧ 2 Z ( F q ) and H 3 ( F q , Z )   ∧ 3 Z ( F q )  ⊕  F q ⊗ Z F q  σ where the second term denotes the subgroup ﬁxed by the twist operator σ (sending x ⊗ y to y ⊗ x ). By Corollary 3.13, H 0  ( F × q ) 2 , ∧ n Z ( F q )  = 0 unless p = 2 and f ≤ n or p > 2 and ( p − 1) f ≤ 2 n . Of course, ∧ n Z ( F q ) = 0 if n > f . Thus when n = 2 we need only consider the cases f = 2 and p = 2 or 3; i.e. q = 4 or 9. W e observe also ∧ f Z ( F q )  F p and that x ∈ F × q acts on this module as multiplication by N F q / F p ( x ). Thus H 0  F × 4 , ∧ 2 Z ( F 4 )  = Z / 2 since the norm map F × 4 → F × 2 is necessarily tri vial. Similarly H 0  ( F × 9 ) 2 , ∧ 2 Z ( F 9 )  = Z / 3 since ( F × 9 ) 2 is the kernel of the norm map to F × 3 . This dispenses with the case k = 2. No w suppose k = 3. By the remarks abov e, H 0  ( F × q ) 2 , ∧ 3 Z ( F q )  = 0 unless q = 8 or 27. In both of these cases we obtain H 0  ( F × q ) 2 , ∧ 3 Z ( F q )  = Z / p as in the case n = 2. Again, by Corollary 3.13, H 0  ( F × q ) 2 , ( F q ⊗ F q ) σ  = 0 unless q = 4 or 9. W e consider these cases indi vidually: q = 4: Let F 4 = F 2 ( a ) where a 2 = 1 + a . Then  T 2 F 2 ( F )  σ is a 3-dimensional F 2 -space with basis e 1 : = 1 ⊗ 1, e 2 : = a ⊗ a and e 3 : = 1 ⊗ a + a ⊗ 1. If φ is the map induced by multiplication by a , then it has a 1-dimensional 1-eigenspace with basis e 3 . Thus H 0 ( F × , H 3 (SL 2 ( F ) , Z )) ( p )  Z / 2 . q = 9: W e let F 9 = F 3 ( i ) where i 2 = − 1 and λ : = 1 − i is an element of order 8. Then (T 2 F 3 ( F 9 )) σ is the 3-dimensional subspace of T 2 F 3 ( F 9 ) with basis e 1 = 1 ⊗ 1, e 2 = i ⊗ i , and e 3 = 1 ⊗ i + i ⊗ 1. Then conjugation by D ( λ ) induces φ : = µ λ 2 ⊗ µ λ 2 = µ i ⊗ µ i on this space. Since φ ( e 1 ) = e 2 , φ ( e 2 ) = e 1 and φ ( e 3 ) = 2 e 3 , it easily follo ws that H 0  ( F × 9 ) 2 , ( F 9 ⊗ F 9 ) σ  is the 1-dimensional subspace with basis e : = e 1 + e 2 . This complete the proof of the lemma.  4. T he third homology of SL 2 ( F ) 4.1. Statement of the main theorem. W e recall two standard subgroups of SL 2 ( F ): T : = ( D ( a ) = " a 0 0 a − 1 # | a ∈ F × ) B : = (" a b 0 a − 1 # | a ∈ F × , b ∈ F ) Lemma 4.1. If F is an inﬁnite ﬁeld, then the inclusion T → B induces homology isomorphisms H k ( T , Z )  H k ( B , Z ) for all k ≥ 0 . Pr oof. This is, for example, a special case of Lemma 9 in [9].  Bloch-W igner-Suslin comple x 19 Remark 4.2. For ﬁnite ﬁelds, the calculations of section 3 show that this result f ails in general. Thus, when F is ﬁnite of characteristic p , we have H k ( B , Z ) = H k ( T , Z ) ⊕ H k ( B , Z ) ( p ) = H k ( T , Z ) ⊕ H 0  ( F × ) 2 , H k ( F , Z )  and we tabulate the ﬁnite number of ﬁelds such that H k ( B , Z ) ( p ) , 0 for k ≤ 3. The rest of this section will to de voted to the proof of Theorem 4.3. Let F be a ﬁeld with at least 4 elements. (1) If F is inﬁnite, ther e is a natur al 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 annihilated by 4 . (2) If F is ﬁnite of odd characteristic, ther e is a complex 0 → H 3 ( B , Z ) → H 3 (SL 2 ( F ) , Z ) → RB ( F ) → 0 which is e xact except possible at the middle term, wher e the homology has or der at most 2 . (3) If F is ﬁnite of characteristic 2 , ther e is an exact sequence 0 → H 3 ( B , Z ) → H 3 (SL 2 ( F ) , Z ) → RB ( F ) → 0 . 4.2. Preliminaries. Let G be a group and let P be a (left) G -set. W e can use the action of G on P to study the homology of G in the follo wing way . Let X n be the set consisting of ordered n -tuples of distinct points of P . W e let G act on X n via a diagonal action. Let C n = C n ( P ) = ( Z [ X n ] , n ≥ 1 Z , n = 0 Let d = d n : C n → C n − 1 be the Z [ G ]-module homomorphism determined by d n ( x 1 , . . . , x n ) = n X i = 1 ( − 1) i + 1 ( x 1 , . . . , ˆ x i , . . . , x n ) for n ≥ 2 and d 1 ( x ) = 1 for all x ∈ P = X 1 . Then C • = ( C n , d n ) is a complex of Z [ G ]-modules. It is almost acyclic: Lemma 4.4. Suppose that the set P has car dinality c. Then H n ( C ) = 0 if n , c. In particular , C • is acyclic if P is inﬁnite. Pr oof. If S ⊂ P , we will let D n ( S ) denote the subgroup of C n generated by those n -tuples in which all elements of S occur . Thus D n ( S ) = 0 if S has more than n elements, and D n ( S 1 ∪ S 2 ) = D n ( S 1 ) ∩ D n ( S 2 ). For x ∈ P we deﬁne Z [ G ]-homomorphisms S x : C n → C n + 1 by S x ( x 1 , . . . , x n ) = ( ( x , x 1 , . . . , x n ) , if x < { x 1 , . . . , x n } 0 , otherwise Thus, if ( x 1 , . . . , x n ) ∈ X n and x < { x 1 , . . . , x n } then d S x ( x 1 , . . . , x n ) = d ( x , x 1 , . . . , x n ) = ( x 1 , . . . , x n ) − S x d ( x 1 , . . . , x n ) On the other hand, if x = x j for some j , then S x ( d ( x 1 , . . . , x n )) = ( − 1) j + 1 ( x j , x 1 , . . . , ˆ x j , . . . , x n ) 20 KEVIN HUTCHINSON and thus 0 = d ( S x ( x 1 , . . . , x n ) = ( x 1 , . . . , x n ) − S x ( d ( x 1 , . . . , x n )) − n ( x 1 , . . . , x n ) + ( − 1) j ( x j , x 1 , . . . , x n ) o Either way , whether x belongs to { x 1 , . . . , x n } or not, we hav e d S x ( x 1 , . . . , x n ) = ( x 1 , . . . , x n ) − S x ( d ( x 1 , . . . , x n )) + w where w ∈ D n ( { x } ). Furthermore, if ( x 1 , . . . , x n ) ∈ D n ( S ), then w ∈ D n ( S ∪ { x } ). No w suppose that x 1 , . . . , x n + 1 are n + 1 distinct elements of P . Let z ∈ C n be a cycle. Then ( d S x 1 − Id ) z = S x 1 ( d z ) + z 1 = z 1 where z 1 is a cycle and z 1 ∈ D n ( { x 1 } ). Thus ( d S x 2 − Id )( z 1 ) = z 2 where z 2 is a cycle in D n ( { x 1 , x 2 } ). Repeating the process, we get ( d S x n + 1 − Id )( dS x n − Id ) · · · ( d S x 1 − Id )( z ) ∈ D n ( { x 1 , . . . , x n + 1 } ) = 0 . This has the form d y + ( − 1) n + 1 ( z ) = 0 and thus z = d (( − 1) n y ) is a boundary , as required.  Remark 4.5. If P is ﬁnite of size c ≥ 2, then it is easy to see that H c ( C ) , 0; in fact, a straightforward Euler characteristic calculation sho ws that it is a free abelian group of rank c ! 1 2! − 1 3! + · · · + ( − 1) c 1 c ! ! . Let no w L • = L • ( P ) be the comple x deﬁned by L n : = ( C n + 1 , n ≥ 0 0 , n < 0 If P is inﬁnite, Lemma 4.4 sho ws that L • is weakly equi valent to the Z (considered as a complex concentrated in dimension 0) and more generally , if P has cardinality c then H n ( L ) = 0 for n , 0 , c − 1 and H 0 ( L )  Z . Lemma 4.6. Let L • be a complex of Z [ G ] -modules and suppose that H n ( L ) = 0 for 1 ≤ n ≤ k. Then H n ( G , L • ) = H n ( G , H 0 ( L )) for 0 ≤ n ≤ k . Pr oof. Recall (see, for example, [4], VII.5) that H n ( G , L • ) is by deﬁnition the homology of the total complex B • ⊗ Z [ G ] L • . where B • is a (right) projecti ve resolution of Z o ver Z [ G ]. This is the total complex of a bounded double complex and there are thus two ﬁltrations and two associated spectral sequences con ver ging to H n ( G , L • ). The ﬁrst takes the form E 2 p , q = H p ( G , H q ( L )) = ⇒ H p + q ( G , L • ) , with di ﬀ erentials d r : E r p , q → E r p − r , q + r − 1 . By our assumptions, E 2 p , q = 0 for 1 ≤ q ≤ k . In particular , all higher di ﬀ erentials leaving E r p , 0 , 0 ≤ p ≤ k are 0, so that E ∞ p , 0 = E 2 p , 0 = H p ( G , H 0 ( L )) for p ≤ k , and there are no other nonzero terms in (total) dimension at most k .  In particular , if P has cardinality c , then H n ( G , L • ( P )) = H n ( G , Z ) for n ≤ c − 2. The second spectral sequence for H n ( G , L • ) has the form E 1 p , q = H p ( G , L q ) = ⇒ H p + q ( G , L • ) , d r : E r p , q → E r p + r − 1 , q − r . Bloch-W igner-Suslin comple x 21 The map d 1 : E 1 p , q = H p ( G , L q ) → H p ( G , L q − 1 ) = E 1 p , q − 1 is just the map induced by d q : L q → L q − 1 . Thus, by Lemma 4.6, when L • = L • ( P ) for a G -set P of cardinality c , we hav e a spectral sequence with E 1 p , q = H p ( G , L q ) whose abutment in dimensions less than c − 2 is H n ( G , Z ). W e no w apply this set-up to the particular case G = SL 2 ( F ) and P = P 1 ( F ) (the resulting spectral sequence has been studied elsewhere; for example in [14]). If F has q elements, then P 1 ( F ) has q + 1 elements and thus we hav e a spectral sequence which abuts to H k (SL 2 ( F ) , Z ) for k ≤ q − 1. Since we wish to use the spectral sequence to calculate H 3 (SL 2 ( F ) , Z ), we will require that q ≥ 4; i.e. in the spectral sequence arguments belo w F is a ﬁeld with at least 4 elements. 4.3. Module structure on the spectral sequence. Our spectral sequence has a natural graded module structure (in a sense to be detailed) ov er the Pontryagin ring H • ( µ 2 , Z ) which facilitates the calculation of some higher di ﬀ erentials. More generally , we hav e the follo wing situation: Let G be a group and let H be a subgroup of the centre, Z ( G ), of G . The integral homology of G is a graded module for the Pontryagin ring H • ( H , Z ) of the abelian group H (see, for e xample, Bro wn [4], Chapter V): Let B • (respecti vely B 0 • ) be a right projecti ve resolution of Z over Z [ G ] (respecti vely Z [ H ]). Then B 0 ⊗ B is a projecti ve resolution of Z o ver Z [ H × G ]. Let τ : B 0 ⊗ Z B → B be a map an augmentation-preserving chain map compatible with the group homomorphism H × G → G , ( h , g ) 7→ h · g . Then the induced composite map ( B 0 ⊗ Z [ H ] Z ) ⊗ ( B ⊗ Z [ G ] Z ) → ( B 0 ⊗ B ) ⊗ Z [ H × G ] Z → B ⊗ Z [ G ] Z induces the required homomorphisms H k ( H , Z ) ⊗ H p ( G , Z ) → H k + p ( G , Z ) which deﬁne the module structure. No w suppose that C • is a complex of Z [ G ]-modules which is weakly equiv alent to Z , consid- ered as comple x concentrated in dimension 0. Then, as noted, we ha ve a spectral sequence abutting to H • ( G , Z ) associated to the double complex D p , q = B • ⊗ Z [ G ] C • . If we further assume that H acts tri vially on the complex C • , then, using τ , we obtain (replacing Z by C • abov e) a map of double complex es ( B 0 • ⊗ Z [ H ] Z ) ⊗ ( B • ⊗ Z [ G ] C • ) → ( B 0 • ⊗ B • ) ⊗ Z [ H × G ] C • → B • ⊗ Z [ G ] C • which induces, for all r ≥ 1 maps H k ( H , Z ) ⊗ E r p , q → E r k + p , q such that the diagrams H k ( H , Z ) ⊗ E r p , q / / ( − 1) k ⊗ d r   E r k + p , q d r   H k ( H , Z ) ⊗ E r p + r − 1 , q − r / / E r k + p + r − 1 , q − r commute; i.e. we have d r ( α · z ) = ( − 1) k α · d r ( z ) for α ∈ H k ( H , Z ), z ∈ E r p , q . 22 KEVIN HUTCHINSON 4.4. The E 1 -page of the spectral sequence. Let X n denote the set of ordered n -tuples of dis- tinct points of P 1 ( F ) and L n = Z X n + 1 . Thus there is a spectral sequence of the form E 1 p , q = H p (SL 2 ( F ) , L q ) = ⇒ H p + q (SL 2 ( F ) , Z ) deri ved from the double comple x E 0 • , • = B • ⊗ Z [SL 2 ( F )] L • where B • is the standard (right) bar resolution of Z ov er SL 2 ( F ), tensored with Z . Let δ : F × → GL 2 ( F ) be the map a 7→ diag( a , 1) (a splitting of the determinant map). Let F × act on E 0 p , q by a ·  [ g 1 | · · · | g p ] ⊗ ( x 0 , . . . , x q )  = [ δ ( a ) g 1 δ ( a ) − 1 | · · · | δ ( a ) g p δ ( a ) − 1 ] ⊗ δ ( a ) · ( x 0 , . . . , x q ) . This action mak es E 0 • , • into a double complex of R F -modules and the induced actions on E 1 p , q = H p (SL 2 ( F ) , L q ) are the natural actions deri ved from the GL 2 ( F )-action on L q and the short exact sequence 1 → SL 2 ( F ) → GL 2 ( F ) → F × → 1. The E 1 -page of the spectral sequence is easily calculated using Shapiro’ s Lemma since the SL 2 ( F )-modules L n are permutation modules, and hence induced modules. Thus, SL 2 ( F ) acts transiti vely on X 1 = P 1 ( F ) and the stabilizer of ( ∞ ) is B . Thus L 0 = Z [ X 1 ]  Z [ B \ SL 2 ( F )]  Ind Z [SL 2 ( F )] Z [ B ] Z so that E 1 p , 0 = H p (SL 2 ( F ) , L 0 )  H p ( B , Z ) by Shapiro’ s Lemma. Similarly , SL 2 ( F ) acts transiti vely on X 2 and the stabilizer of (0 , ∞ ) is T . So E 1 p , 1 = H p (SL 2 ( F ) , L 1 )  H p ( T , Z ) . For n ≥ 3 the stabilizer of an element ( x 1 , . . . , x n ) in SL 2 ( F ) is µ 2 ( F ) = Z (SL 2 ( F )). Using Corollary 2.2 abov e, it follows that for q ≥ 2, and when the characteristic of F is not 2, we ha ve E 1 p , q = R F [ Z q − 2 ] ⊗ H p ( µ 2 , Z )           R F [ Z q − 2 ] , p = 0 0 , p > 0 ev en R F [ Z q − 2 ] ⊗ Z / 2 , p > 0 odd where Z n is the set of ordered n -tuples [ z 1 , . . . , z n ] of distinct points of P 1 ( F ) \ {∞ , 0 , 1 } . When the characteristic is 2, of course, µ 2 ( F ) = { 1 } and E 1 p , q = 0 whene ver p ≥ 1 and q ≥ 2. Note also that for q ≥ 2, the module structure mentioned above is reﬂected in the tensor product decomposition of the terms; if α ∈ H p ( µ 2 , Z ) and z ∈ E 1 p , q = R F [ Z q − 2 ], then α · z = z ⊗ α ∈ R F [ Z q − 2 ] ⊗ H k ( µ 2 , Z ) = E 1 p , q . Bloch-W igner-Suslin comple x 23 Thus our E 1 -page has the form . . . . . . . . . . . . R F [ Z 2 ] d 1   R F [ Z 2 ] ⊗ µ 2 d 1   . . . . . . . . . R F [ Z 1 ] d 1   R F [ Z 1 ] ⊗ µ 2 d 1   0 R F [ Z 1 ] ⊗ Z / 2 d 1   . . . R F d 1   R F ⊗ µ 2 d 1   0 R F ⊗ Z / 2 d 1   . . . Z d 1   H 1 ( T , Z ) d 1   H 2 ( T , Z ) d 1   H 3 ( T , Z ) d 1   . . . Z H 1 ( B , Z ) H 2 ( B , Z ) H 3 ( B , Z ) . . . when the characteristic of F is not 2. 4.5. The E 2 -page. By the calculations of section 2 abov e the di ﬀ erential d 1 : E 1 0 , 4 = R F [ Z 2 ] → R F [ Z 1 ] = E 1 0 , 3 is gi ven by [ x , y ] 7→ S x , y = [ x ] −  y  + h x i  y x  − D x − 1 − 1 E " 1 − x − 1 1 − y − 1 # + h 1 − x i " 1 − x 1 − y # . Thus E 1 0 , 3 / Im( d 1 ) : = RP ( F ). On the other hand, for x ∈ Z 1 we hav e d 1 ( [ x ] ) = d ((0 , ∞ , 1 , x )) = ( ∞ , 1 , x ) − (0 , 1 , x ) + (0 , ∞ , x ) − (0 , ∞ , 1) which corresponds to the element h φ ( ∞ , 1 , x ) i − h φ (0 , 1 , x ) i + h φ (0 , ∞ , x ) i − h φ (0 , ∞ , 1) i = h 1 − x i − h x (1 − x ) i + h x i − h 1 i = − h h x i i h h x ( x − 1) i i = − h h 1 − x i i h h x i i = − λ 1 ( [ x ] ) ∈ R F = E 1 0 , 2 . Thus E 2 0 , 3 = RP 1 ( F ) : = Ker( λ 1 : RP ( F ) → I 2 F ). Using the module structure, the map d 1 : E 1 1 , 3 = R F [ Z 1 ] ⊗ µ 2 → R F ⊗ µ 2 = E 1 1 , 2 is the map [ x ] ⊗ z 7→ − λ 1 ([ x ]) ⊗ z . Thus E 1 1 , 2 Im( d 1 ) = R F J F ⊗ µ 2 = GW( F ) ⊗ µ 2 Similarly , the map d 1 : E 1 0 , 2 = R F → E 1 0 , 1 = Z is easily seen to be the natural augmentation homomorphism sending h x i ∈ G F to 1, and hence the di ﬀ erential d 1 : E 1 1 , 2 = R F ⊗ µ 2 → F ×  H 1 ( T , Z ) = E 1 1 , 1 24 KEVIN HUTCHINSON sends h x i ⊗ z to z (for all x ∈ F × , z ∈ µ 2 ⊂ F × ). It follows that E 2 1 , 2 = I( F ) ⊗ µ 2 . Similarly , we obtain that E 2 1 , 3 = RP 1 ( F ) ⊗ µ 2 (keeping in mind that all the groups R F [ Z i ] are Z -free). No w let w : = " 0 − 1 1 0 # ∈ SL 2 ( F ) . Then w ( ∞ ) = 0 and w (0) = ∞ . It follo ws easily that the di ﬀ erential d 1 : E 1 p , 1 = H p ( T , Z ) → H p ( B , Z ) = E 1 p , 0 is the composite H p ( T , Z ) w p − 1 / / H p ( T , Z ) / / H p ( B , Z ) where w p : H p ( T , Z ) → H p ( T , Z ) is the map induced by conjugation by w . Ho wever , conjugat- ing by w is just the in version map on T  F × . For future con venience, we will deﬁne A i ( F ) = ( 0 , F is inﬁnite H i ( B , Z ) ( p ) = H i (SL 2 ( F ) , Z ) ( p ) , F is ﬁnite of characteristic p . Thus d 1 = w 1 − 1 : E 1 1 , 1 = F × → E 1 1 , 0 = H 1 ( B , Z ) = F × ⊕ A 1 ( F ) is the map x 7→ x − 2 . It follows that E 2 1 , 0 = G F ⊕ A 1 ( F ). Furthermore, w 2 is the identity map on H 2 ( T , Z ) = F × ∧ F × and hence d 1 : E 1 2 , 1 → E 1 2 , 0 is the zero map. So E 2 2 , 0 = E 3 2 , 0 = H 2 ( B , Z ) = ∧ 2 ( F × ) ⊕ A 1 ( F ). Recall that E 1 3 , 1 = H 3 ( T , Z )  H 3 ( F × , Z )  ∧ 3 Z ( F × ) ⊕ T or Z 1 ( µ F , µ F ) and E 1 3 , 0 = H 3 ( B , Z )  H 3 ( T , Z ) ⊕ A 3 ( F ) . The the map d 1 : E 1 3 , 1 → E 1 3 , 0 restricts to the identity on the factors ∧ 2 ( F × ) and to the zero map on T or Z 1 ( µ F , µ F ). It follows that E 2 3 , 0 = T or Z 1 ( µ F , µ F ) ⊕ A 3 ( F ). Thus the rele v ant part of the E 2 -page has the form RP 1 ( F ) d 2   RP 1 ( F ) ⊗ µ 2 d 2   0 . . . I( F ) d 2   I( F ) ⊗ µ 2 d 2   0 . . . 0 0 ∧ 2 Z ( F × ) . . . Z G F ⊕ A 1 ( F ) ∧ 2 Z ( F ) ⊕ A 2 ( F ) T or Z 1 ( µ F , µ F ) ⊕ A 3 ( F ) Bloch-W igner-Suslin comple x 25 4.6. The E 3 -page. W e begin by observing that since the edge homomorphisms A i ( F ) → H i (SL 2 ( F ) , Z ) are necessarily injecti ve, it follows that the base terms E r p , 0 always factor in the form G r p , 0 ⊕ A p ( F ) and that any di ﬀ erential d r with target E r p , 0 has image contained in G r p , 0 . No w the di ﬀ erential d 2 : E 2 0 , 2 = I( F ) → G F ⊂ E 2 2 , 0 has been calculated by Mazzoleni ([14], Lemma 5): it sends h h x i i to h x i (for x , 1). If h x i ⊗ − 1 ∈ I( F ) ⊗ µ 2 = E 2 1 , 2 , it follows - using the module structure on the spectral sequence - that d 2 ( h x i ⊗ − 1) = − d 2 ( h x i ) · − 1 = x ∧ − 1 ∈ F × ∧ F × ⊂ E 2 2 , 0 . (Here we use the fact that under the identiﬁcation F × ∧ F × = H 2 ( F × , Z ), the wedge product corresponds to the Pontryagin product on homology). Thus E 3 2 , 0 = SE 2 Z ( F × ) ⊕ A 2 ( F ) where we set SE 2 Z ( F × ) : = F × ∧ F × F × ∧ µ 2 . If F is a ﬁnite ﬁeld F × ∧ F × = 0 and E 3 1 , 2 = E ∞ 1 , 2 is a quotient of I( F ) ⊗ µ 2  µ 2 ( F ). Thus E ∞ 1 , 2 has order at most 2 if F is ﬁnite of odd characteristic, and is 0 if F is ﬁnite of characteristic 2. In any case, for an y ﬁeld F , the term E 3 1 , 2 = E ∞ 1 , 2 is annihilated by 2. Of course, the di ﬀ erential d 2 : E 2 3 , 0 = RP 1 ( F ) → E 2 1 , 1 = 0 is necessarily the zero map, and it follo ws, using the module structure again, that the di ﬀ erential d 2 : E 2 3 , 1 = RP 1 ( F ) ⊗ µ 2 → ∧ 2 ( F × ) = E 2 2 , 2 is also the zero map. Thus the rele v ant part of the E 3 -page has the form E 3 0 , 4 d 3   . . . . . . . . . RP 1 ( F ) d 3   . . . . . . . . . I( F ) 2 E 3 1 , 2 . . . . . . 0 0 ∧ 2 Z ( F × ) . . . Z A 1 ( F ) SE 2 Z ( F ) ⊕ A 2 ( F ) T or Z 1 ( µ F , µ F ) ⊕ A 3 ( F ) 4.7. The E 4 -page. W e begin by calculating the di ﬀ erential d 3 : E 3 0 , 3 = RP 1 ( F ) → SE 2 Z ( F × ) ⊂ E 3 2 , 0 . Let E ( G L ) 1 p , q = H p (GL 2 ( F ) , L q ) = ⇒ H p + q (GL 2 ( F ) , Z ) be the spectral sequence deri ved form the the action of GL 2 ( F ) on the complex L • and con- ver ging to the integral homology of GL 2 ( F ). (This spectral has been studied in [9].) Then the inclusion SL 2 ( F ) → GL 2 ( F ) induces a map of spectral sequences E r p , q → E ( G L ) r p , q . Now 26 KEVIN HUTCHINSON E ( G L ) 3 0 , 3 = P ( F ) and E ( G L ) 3 2 , 0 = H 2 ( ˜ T , Z ) / ( w 2 − 1), where ˜ T ⊂ GL 2 ( F ) is the subgroup con- sisting of all diagonal matrices. There is a split-exact sequence ([9], Lemma 4) 0 / / S 2 Z ( F × ) / / H 2 ( ˜ T , Z ) / ( w 2 − 1) det / / H 2 ( F × , Z ) / / 0 No w the image of d 3 : E ( G L ) 3 0 , 3 = P ( F ) → E ( G L ) 3 2 , 0 factors through the term S 2 Z ( F × ) and is gi ven by the formula P ( F ) → S 2 Z ( F × ) , [ x ] 7→ (1 − x ) ⊗ x . (See [9], p190, and allow for the fact that the term [ x ] ∈ P ( F ) in this paper corresponds to [ 1 / x ] there.) W e observe that, for an y ﬁeld F there is a natural injectiv e homomorphism F × ∧ F × F × ∧ µ 2 = SE 2 Z ( F × ) → S 2 Z ( F × ) , a ∧ b 7→ 2( a ◦ b ) . It is easily seen that the inclusion T → ˜ T induces the map F × ∧ F × = H 2 ( F × , Z )  H 2 ( T , Z ) → S 2 Z ( F × ) ⊂ H 2 ( ˜ T , Z ) / ( w 2 − 1) a ∧ b 7→ 2( a ⊗ b ) . and thus induces the injection SE 2 Z ( F × ) → S 2 Z ( F × ). Putting all of this together we get the commutati ve diagram RP 1 ( F ) d 3 / /   λ 2 % % SE 2 Z ( F × )  _   P ( F ) λ / / S 2 Z ( F × ) from which it follo ws that E ∞ 0 , 3 = E 4 0 , 3 = K er( d 3 ) = K er( λ 2 : RP 1 ( F ) → S 2 Z ( F × )) = RB ( F ) . Finally , we will show that 2 · E ∞ 2 , 1 = 2 · E 4 2 , 1 = 0. In order to this we will need a tec hnical lemma : Let G be a group and L • a comple x of Z [ G ]-modules concentrated in non-negati ve dimensions. Let L • ( m ) denote the truncated complex L k ( m ) : = ( L k , k ≥ m 0 , k < m (So L • = L • (0).) Consider the spectral sequences E 1 ( m ) p , q = H p ( G , L q ( m )) = ⇒ H p + q ( G , L • ( m )) . If m 0 ≥ m , the natural map of complex es L ( m 0 ) → L ( m ) induces a map of spectral sequences E r ( m 0 ) p , q → E r ( m ) p , q compatible with the map on abutments H p + q ( G , L • ( m 0 )) → H p + q ( G , L • ( m )). Note that E r ( m ) p , q = 0 for q < m and thus E r ( m ) 0 , q = E ∞ ( m ) 0 , q for r > q − m . Similarly , the if m 0 ≥ m then E r ( m 0 ) p , q = E r ( m ) p , q as long as r ≤ q − m 0 + 1. If A is a Z [ G ]-module, we let A [ m ] denote the module A considered as a comple x concentrated in dimension m . Observe, in particular , that the di ﬀ erential d : L m + 1 → L m induces a map of complex es φ : L • ( m + 1) → L m [ m + 1]. Our technical lemma then states: Bloch-W igner-Suslin comple x 27 Lemma 4.7. The following diagr am commutes for any r ≥ 1 , m ≥ 0 H r + m ( G , L • ( m + 1)) φ / /     H r + m ( G , L m [ m + 1]) =   E ∞ ( m + 1) 0 , r + m =   H r − 1 ( G , L m ) =   E r ( m + 1) 0 , r + m =   E 1 ( m ) r − 1 , m     E r ( m ) 0 , r + m d r / / E r ( m ) r − 1 , m Pr oof. This is a tedious but straightforward veriﬁcation from the deﬁnitions (it is clearly enough to consider the case m = 0).  Applying this to the group G = SL 2 ( F ) and the complex L • = L • ( P 1 ( F )) in the case r = 3 and m = 1 gi ves a commutati ve diagram H 4 (SL 2 ( F ) , L • (2)) φ / / =   H 4 (SL 2 ( F ) , L 1 [2]) =   E 3 (1) 0 , 4 =   H 2 (SL 2 ( F ) , L 1 ) =   E 3 0 , 4 d 3 / / E 3 2 , 1 No w let W k = K er( L k → L k − 1 ), so that (in su ﬃ ciently low dimensions) we hav e short exact sequences 0 → W k → L k → W k − 1 → 0 by Lemma 4.4. The map d : L 2 → W 1 induces a map of complexes L • (2) → W 1 [2] which is a weak equi v alence in lo w dimensions. In particular, it induces an isomorphism H 4 (SL 2 ( F ) , L • (2))  H 4 (SL 2 ( F ) , W 1 [2]) = H 2 (SL 2 ( F ) , W 1 ) . Putting these facts together gi ves us: Corollary 4.8. Ther e is a commutative diagr am H 2 (SL 2 ( F ) , W 1 ) / /    H 2 (SL 2 ( F ) , L 1 )    E 3 0 , 4 d 3 / / E 3 2 , 1 wher e the top horizontal map is induced by the inclusion W 1 → L 1 . Thus from this corollary and the long exact homology sequence of the e xact sequence 0 → W 1 → L 1 → W 0 → 0 it follo ws that the image of d 3 : E 3 0 , 4 → E 3 2 , 1 = H 2 ( T , Z ) is equal to the kernel of the map, µ say , F × ∧ F ×  H 2 ( T , Z ) = H 2 (SL 2 ( F ) , L 1 ) → H 2 (SL 2 ( F ) , W 0 ) . 28 KEVIN HUTCHINSON W e will sho w that the map 2 · µ is zero: Let B • → Z be a projectiv e resolution of Z as a Z [SL 2 ( F )]-module. So H • ( T , Z ) is the homology of the complex B • ⊗ Z [ T ] Z , and the composite H • ( T , Z ) → H • (SL 2 ( F ) , L 1 ) → H • (SL 2 ( F ) , W 0 ) is described on the le vel of chains B • ⊗ Z [ T ] Z → B • ⊗ Z [SL 2 ( F )] L 1 → B • ⊗ Z [SL 2 ( F )] W 0 by γ ⊗ 1 7→ γ ⊗ ( ∞ , 0) 7→ γ ⊗ ((0) − ( ∞ )) . Recall that w = " 0 − 1 1 0 # acts on T by conjug ation, and the action of w on the homology of T is described on the le vel of chains by γ ⊗ 1 7→ γ · w − 1 ⊗ 1 . Thus, if z ∈ H 2 ( T , Z ) is represented by γ ⊗ 1, then w 2 · z is represented by γ · w − 1 ⊗ 1. Thus µ ( w 2 · z ) = µ ( γ · w − 1 ⊗ 1) = γ · w − 1 ⊗ ((0) − ( ∞ )) = γ ⊗ w − 1 · ((0) − ( ∞ )) = γ ⊗ (( ∞ ) − (0)) = − µ ( z ) . Since, as observ ed abo ve, w 2 is the identity map, we ha ve 2 µ ( z ) = 0 as required. It follo ws that 2 · E ∞ 2 , 1 = 0. 4.8. The calculation of H 3 (SL 2 ( F ) , Z ) . Now the map T or Z 1 ( µ F , µ F ) → H 3 (SL 2 ( F ) , Z ) is injec- ti ve, since, for e xample, the composite T or Z 1 ( µ F , µ F ) → H 3 (SL 2 ( F ) , Z ) → H 3 (GL 2 ( F ) , Z ) is injecti ve when F is inﬁnite by the results of Suslin ([25]), while for ﬁnite ﬁelds we hav e sho wn that the map T or Z 1 ( µ F , µ F ) = H 3 ( T , Z ) → H 3 (SL 2 ( F ) , Z ) is injecti ve in section 3 abo ve. It thus follo ws that E ∞ 3 , 0 = T or Z 1 ( µ F , µ F ) ⊕ A 3 ( F ) = ( T or Z 1 ( µ F , µ F ) , F inﬁnite H 3 ( B , Z ) , F ﬁnite No w , by the computations above, 2 · E ∞ 1 , 2 = 2 · E ∞ 2 , 1 for any ﬁeld F . Furthermore, clearly E ∞ 2 , 1 = 0 for any ﬁnite ﬁeld F since E 1 1 , 2  F × ∧ F × = 0 in this case. W e ha ve also seen that E ∞ 1 , 2 has order at most 2 for any ﬁnite ﬁeld and that this term is already 0 for ﬁnite ﬁelds of characteristic 2. Thus the con vergence of the spectral sequence gi ves us a complex 0 → E ∞ 3 , 0 → H 3 (SL 2 ( F ) , Z ) → E ∞ 0 , 3 → 0 . which is exact except possible at the middle term. If we denote the middle homology group by H ( F ), then it admits a short exact sequence 0 → E ∞ 1 , 2 → H ( F ) → E ∞ 2 , 1 → 0 . This completes the proof of Theorem 4.3. Bloch-W igner-Suslin comple x 29 5. T he refined B loch gr oup , the classical B loch group and indecomposable K 3 Recall that for any ﬁeld F there is a natural homomorphism H 3 (SL 2 ( F ) , Z ) → K ind 3 ( F ) which factors as follo ws: H 3 (SL 2 ( F ) , Z ) / / H 3 (SL ( F ) , Z ) K 3 ( F ) / ( {− 1 } · K 2 ( F ))  o o / / / / K ind 3 ( F ) . No w , for any inﬁnite ﬁeld F this map is surjectiv e (see [10]), and the induced homomorphism H 3 (SL 2 ( F ) , Z ) F × / / / / K ind 3 ( F ) has a 2-primary torsion kernel (see Mirzaii [17]). Suslin, [25], has sho wn that for any inﬁnite ﬁeld F there is a natural short exact sequence 0 → T or Z 1 ] ( µ F , µ F ) → K ind 3 ( F ) → B ( F ) → 0 where T or Z 1 ] ( µ F , µ F ) denotes the unique nontrivial extension of T or Z 1 ( µ F , µ F ) by Z / 2 if the char- acteristic of F is not 2, and denotes T or Z 1 ( µ F , µ F ) in characteristic 2. (W e will sho w that this result extends to ﬁnite ﬁelds in section 7 belo w .) Corollary 5.1. 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 tor sion kernel. Pr oof. Combining the preceding remarks with Theorem 4.3 giv es the commutati ve diagram (deﬁning K ) 0 / / K   / / H 3 (SL 2 ( F ) , Z )   / / RB ( F ) / /   0 0 / / T or Z 1 ] ( µ F , µ F ) / / K ind 3 ( F ) / / B ( F ) / / 0 from which the ﬁrst statement follows. T aking F × -coin variants of the top row and noting that the natural map K → T or Z 1 ] ( µ F , µ F ) has cokernel annihilated by 4, the second statement also follo ws.  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 ( F ) 0 : = K er( RB ( F ) → B ( F )) Lemma 5.2. Let F be an inﬁnite ﬁeld. Then (1) H 3 (SL 2 ( F ) , Z 0 ) 0 = RB ( F ) 0 0 (2) H 3 (SL 2 ( F ) , Z 0 ) 0 = I F H 3 (SL 2 ( F ) , Z 0 ) and RB ( F ) 0 0 = I F RB ( F ) 0 . (3) H 3 (SL 2 ( F ) , Z 0 ) 0 = K er(H 3 (SL 2 ( F ) , Z 0 ) → H 3 (SL 3 ( F ) , Z 0 )) = K er(H 3 (SL 2 ( F ) , Z 0 ) → H 3 (GL 2 ( F ) , Z 0 )) Pr oof. (1) This follows from applying − ⊗ Z [1 / 2] to the diagram in the proof of Corollary 5.1 and noting that K 0 = T or Z 1 ( µ F , µ F ) 0 = T or Z 1 ] ( µ F , µ F ) 0 . (2) By Corollary 5.1 again, we ha ve RB ( F ) 0 F × = B ( F ) 0 and by the result of Mirzaii men- tioned abov e we ha ve H 3 (SL 2 ( F ) , Z 0 ) F × = K ind 3 ( F ) 0 . Of course, for any F × -module M , we hav e I F · M = Ker( M → M F × ). 30 KEVIN HUTCHINSON (3) For the ﬁrst equality , observe ﬁrst that the stabilization map H 3 (SL 2 ( F ) , Z ) → H 3 (SL 3 ( F ) , Z ) factors through H 3 (SL 2 ( F ) , Z ) F × , since, for example, ( F × ) 2 acts tri vially on H 3 (SL 2 ( F ) , Z ) while ( F × ) 3 acts trivially on H 3 (SL 3 ( F ) , Z ). From the isomorphism H 3 (SL 2 ( F ) , Z 0 ) F ×  K ind 3 ( F ) 0 it thus follo ws that K er(H 3 (SL 2 ( F ) , Z 0 ) → H 3 (SL 3 ( F ) , Z 0 )) ⊂ H 3 (SL 2 ( F ) , Z 0 ) 0 . On the other hand, the natural map H 3 (SL 2 ( F ) , Z ) → K ind 3 ( F ) f actors through H 3 (SL 3 ( F ) , Z ), giving us the re v erse inclusion. Furthermore, by [17], the map H 3 (GL 2 ( F ) , Z 0 ) → H 3 (GL 3 ( F ) , Z 0 ) = H 3 (GL ( F ) , Z 0 ) is injectiv e, while the map H 3 (SL 3 ( F ) , Z ) → H 3 (GL 3 ( F ) , Z ) is always injectiv e (by the stability results in [10]). This implies the second equality .  Corollary 5.3. Let F be a ﬁeld of char acteristic other than 2 . Let ¯ F be an algebr aic closur e of F and let ˜ F be the smallest quadratically closed subﬁeld of ¯ F containing F . Then H 3 (SL 2 ( F ) , Z 0 ) 0 = K er(H 3 (SL 2 ( F ) , Z 0 ) → H 3 (SL 2 ( ˜ F ) , Z 0 )) . Pr oof. This follo ws from the fact that the natural map H 3 (SL 2 ( E ) , Z ) → K ind 3 ( E ) is an isomorphism, when E is quadratically closed ([22]), together with the f act that K ind 3 ( F ) sat- siﬁes Galois descent for ﬁnite Galois extensions of degree relativ ely prime to the characteristic of the ﬁeld (Le vine [12], Merkurje v and Suslin [15]).  Remark 5.4. In [10], it is sho wn that, for any inﬁnite ﬁeld F , H 3 (SL 3 ( F ) , Z ) = H 3 (SL ( F ) , Z ) = K 3 ( F ) {− 1 } · K 2 ( F ) . Thus, it follows that H 3 (SL 3 ( F ) , Z 0 )  K 3 ( F ) 0 (since {− 1 } · K 2 ( F ) ⊂ K 3 ( F ) is clearly killed by 2). Again, in [10] it is sho wn that the cok ernel of H 3 (SL 2 ( F ) , Z ) → H 3 (SL 3 ( F ) , Z ) is 2 · K M 3 ( F ), while the image of this map is isomorphic to K ind 3 ( F ). 6. T he map H 3 ( G , Z ) → RB ( F ) for subgr oups G of SL 2 ( F ) 6.1. Preliminary Remarks. Let G be a group and let P be a left G -set with at least 5 elements. As in section 4, let L • be the complex of Z [ G ]-modules deﬁned by L n is the free abelian group on ( n + 1)-tuples of distinct points of P , and let d n : L n → L n − 1 be the simplicial boundary map. Thus we hav e a spectral sequence E 1 p , q = H p ( G , L q ) = ⇒ H p + q ( G , L • ) and H n ( G , L • ) = H n ( G , Z ) for n ≤ 3. Thus we hav e edge homomorphisms H n ( G , Z ) / / / / E ∞ 0 , n   / / E 2 0 , n = H n ( ( L • ) G ) These edge homomorphisms can be constructed as follows: Let F • be a (left) projectiv e resolu- tion of Z as a Z [ G ]-module. Let β : F • → L • be an augmentation-preserving map of complex es Bloch-W igner-Suslin comple x 31 of Z [ G ]-modules. Then β is determined uniquely up to chain homotopy (see, for example, [4] I.7.4). There is an induced map of complexes Z ⊗ Z [ G ] F • = ( F • ) G ( β ) G / / ( L • ) G and, hence, on taking homology , induced maps H n ( G , Z ) = H n ( ( F • ) G ) → H n ( ( L • ) G ) (which are independent of the particular chain map β ). 6.2. Construction of β . W e will now let F • = F • ( G ) be the homogeneous (left) standard resolution of Z over Z [ G ]. Thus F n is the free Z -module on ( n + 1)-tuples ( g 0 , . . . , g n ) of elements of G and d n : F n → F n − 1 is again the standard simplicial boundary map. G acts diagonally on the left on F n . So F n is a free left Z [ G ]-module with basis consisting of the elements of the form (1 , g 1 , . . . , g n ). No w , suppose that x ∈ P and that the orbit of x , G · x , is not all of P (so that G does not act transiti vely on P ). Fix y ∈ P \ G · x . In dimensions less than or equal to 3, we will use x and y to construct a chain map β = β x , y : F • → L • . (It follows, of course, that the resulting maps on homology are independent of the choice of x and y ). In dimension 0, we set β x , y 0 ( g ) = g ( x ) ∈ P . In dimension 1, we deﬁne β x , y 1 ( g 0 , g 1 ) = ( ( g 0 ( x ) , g 1 ( x )) , if g 0 ( x ) , g 1 ( x ) 0 , if g 0 ( x ) = g 1 ( x ) In dimension 2, we deﬁne β x , y 2 ( g 0 , g 1 , g 2 ) =          ( g 0 ( x ) , g 1 ( x ) , g 2 ( x )) , if g 0 ( x ) , g 1 ( x ) , g 2 ( x ) are distinct 0 , if g i ( x ) = g i + 1 ( x ) for i ∈ { 0 , 1 } ( g 0 ( y ) , g 0 ( x ) , g 1 ( x )) + ( g 0 ( y ) , g 1 ( x ) , g 0 ( x )) , if g 0 ( x ) = g 2 ( x ) , g 1 ( x ) In dimension 3, we deﬁne β x , y 3 ( g 0 , g 1 , g 2 , g 3 ) =                                                        ( g 0 ( x ) , g 1 ( x ) , g 2 ( x ) , g 3 ( x )) , if g 0 ( x ) , . . . , g 3 ( x ) are distinct 0 , if g i ( x ) = g i + 1 ( x ) for i ∈ { 0 , 1 , 2 } 0 , if g 0 ( x ) = g 2 ( x ) and g 1 ( x ) = g 3 ( x ) and g 0 ( y ) = g 1 ( y ) ( g 0 ( y ) , g 1 ( y ) , g 0 ( x ) , g 1 ( x )) + ( g 0 ( y ) , g 1 ( y ) , g 1 ( x ) , g 0 ( x )) if g 0 ( x ) = g 2 ( x ) and g 1 ( x ) = g 3 ( x ) and g 0 ( y ) , g 1 ( y ) ( g 0 ( y ) , g 0 ( x ) , g 1 ( x ) , g 3 ( x )) + ( g 0 ( y ) , g 1 ( x ) , g 0 ( x ) , g 3 ( x )) , if g 0 ( x ) = g 2 ( x ) , and g 0 ( x ) , g 1 ( x ) , g 3 ( x ) are distinct ( g 0 ( x ) , g 1 ( y ) , g 1 ( x ) , g 2 ( x )) + ( g 0 ( x ) , g 1 ( y ) , g 2 ( x ) , g 1 ( x )) , if g 1 ( x ) = g 3 ( x ) , and g 0 ( x ) , g 1 ( x ) , g 2 ( x ) are distinct ( g 0 ( y ) , g 1 ( x ) , g 2 ( x ) , g 0 ( x )) − ( g 0 ( y ) , g 0 ( x ) , g 1 ( x ) , g 2 ( x )) , if g 0 ( x ) = g 3 ( x ) and g 0 ( x ) , g 1 ( x ) , g 2 ( x ) are distinct It can be directly veriﬁed that these giv e a well-deﬁned augmentation-preserving chain map in dimensions less than or equal to 3. 32 KEVIN HUTCHINSON 6.3. The reﬁned cr oss ratio map. W e specialize no w to the case where G is a subgroup of SL 2 ( F ) for some ﬁeld F and P = P 1 ( F ). From the calculations of sections 2 and 4, we have H 3  ( L • ( P 1 ( F ))) SL 2 ( F )   RP 1 ( F ) ⊂ RP ( F ) and the isomorphism is induced by the map ( L 3 ( P 1 ( F ))) SL 2 ( F ) → RP ( F ) , ( x 0 , x 1 , x 2 , x 3 ) 7→ h φ ( x 0 , x 1 , x 2 ) i " φ ( x 0 , x 1 , x 3 ) φ ( x 0 , x 1 , x 2 ) # W e will call this map the r eﬁned cr oss ratio and will denote it by cr. Thus, if x 0 , . . . , x 3 are distinct points of P 1 ( F ), we ha ve 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 = ∞ Putting all of this together , we conclude: If G is a subgroup of SL 2 ( F ), then the composite homomorphism H 3 ( G , Z ) → H 3 (SL 2 ( F ) , Z ) → RP ( F ) can be calculated on the le vel of chains as ( F 3 ) G β / / L 3 ( P 1 ( F )) G / / L 3 ( P 1 ( F )) SL 2 ( F ) cr / / RP ( F ) 6.4. Finite cyclic subgroups of SL 2 ( F ) . W e begin with the follo wing observation: Let G be a group and let F • be the standard (left) homogeneous resolution of Z as a Z [ G ]-module. (For con venience below , we will work modulo degenerate simplices; i.e., we set ( g 0 , . . . , g n ) = 0 if g i = g i + 1 for some i ≤ n − 1.) The augmented resolution · · · → F n → · · · → F 0 → Z = F − 1 → 0 is contractible as a complex of abelian groups via the homotopy h n : F n → F n + 1 sending ( g 0 , . . . , g n ) to (1 , g 0 , . . . , g n ). Thus if C • is any complex of free Z [ G ]-modules, with C 0 = Z [ G ] and C 1 → C 0 → Z the zero map, we can recursi vely construct an augmentation preserving chain map of Z [ G ]-complexes α • : C • → F • as follo ws: W e let α 0 = Id Z [ G ] , and if e 1 , . . . , e s is a basis of C n + 1 , we set α n + 1 ( e i ) = h n ( α n ( d e i )) . No w t ∈ SL 2 ( F ) hav e ﬁnite order r and let G = h t i be the cyclic group generated by t , and let C • be the standard 2-periodic resolution of Z by free Z [ G ]-modules: · · · t − 1 / / Z [ G ] N / / Z [ G ] t − 1 / / Z [ G ] / / / / Z where N = 1 + t + · · · + t r − 1 ∈ Z [ G ]. Bloch-W igner-Suslin comple x 33 Applying the recipe above to this situation giv es a chain map of complexes of Z [ G ]-modules C • → F • which is gi ven in dimension 3 by the formula Z [ G ] = C 3 → F 3 , 1 7→ r − 1 X i = 0 (1 , t , t i + 1 , t i + 2 ) . Finally , if we choose x ∈ P 1 ( F ) and if we choose y ∈ P 1 ( F ) \ G · x , then it follows that composite Z / n  H 3 ( G , Z ) → RP ( F ) is gi ven by the formula 1 7→ r − 1 X i = 0 cr( β x , y 3 (1 , t , t i + 1 , t i + 2 )) . Furthermore, the uniqueness up to chain homotopy of the chain map β guarantees us that the resulting map is independent of the particular choice of x and y . If we suppose that G x = { 1 } , then, from the deﬁnition of β x , y 3 abov e, we ha ve: 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 ) For example, if G ∞ = G ∩ B = { 1 } and if r ≥ 3, then we can take x = ∞ and y , t i ( ∞ ) for i = 0 , . . . , n − 1. Suppose also that y , t ( y ). Then, using the formulae for cr gi v en abo ve, we see that the map Z / n = H 3 ( G ) → RP ( F ) is gi v en by 1 7→        r − 3 X i = 1 D t ( ∞ ) − t i + 1 ( ∞ ) E " t ( ∞ ) − t i + 1 ( ∞ ) t ( ∞ ) − t i + 2 ( ∞ ) #        + * ( t − 1 ( ∞ ) − y )( y − t ( ∞ )) t − 1 ( ∞ ) − t ( ∞ )) + " t − 1 ( ∞ ) − t ( ∞ ) t − 1 ( ∞ ) − y # − h t ( ∞ ) − y i " t − 1 ( ∞ ) − y t ( ∞ ) − y # + h y − t ( y ) i " t ( ∞ ) − y t ( ∞ ) − t ( y ) # + * ( y − t ( y ))( t ( ∞ ) − y ) t ( ∞ ) − t ( y ) + " t ( ∞ ) − t ( y ) t ( ∞ ) − y # . 6.5. Third homology of generalised quaternion groups. Let t be an e ven integer and let Q = Q 4 t = h x , y | x t = y 2 , xy x = y i . Again, let F • be the standard (nondegenerate) resolution of Z over Z [ Q ]. Let C • be the 4-periodic resolution of Z ov er Z [ Q ] (see Cartan-Eilenberg [5], XII.7). W e can use the recipe abov e to construct an augmentation-preserving chain map α • : C • → F • . In particular , C 3 = Z [ Q ] and we obtain α 3 (1) =        t − 1 X i = 1 (1 , x , x i + 1 , x i + 2 )        − (1 , x , xy , xy 2 ) − (1 , xy , y 2 , xy 2 ) − (1 , xy , y , y 2 ) . No w H 3 ( Q , Z )  Z / 4 t and thus the cycle on the right represents a generator of this cyclic group. If q = p f with p an odd prime, then the 2-Sylow subgroups of SL 2 ( F q ) are generalised quaternion and we will use the maps β 3 and cr as abo ve to calculate the image H 3 ( Q , Z ) → RB ( F q ). 34 KEVIN HUTCHINSON 7. B loch gr oups of finite fields In this section we use the calculations of sections 3 and section 6 as well as Theorem 4.3 to gi ve an e xplicit description of the Bloch groups of ﬁnite ﬁelds. W e begin by observing: Lemma 7.1. F or a ﬁnite ﬁeld F (with at least 4 elements) the natural map RP ( F ) → P ( F ) induces an isomorphism RB ( F )  B ( F ) . Pr oof. By Lemma 2.12 we kno w that RB ( F ) F ×  B ( F ). Ho we ver , by Lemma 3.8, F × acts tri vially on H 3 (SL 2 ( F ) , Z [1 / p ]) (where p is the characteristic of F ). By Theorem 4.3, RB ( F ) is naturally a quotient of the R F -module H 3 (SL 2 ( F ) , Z [1 / p ]), and thus F × acts tri vially on RB ( F ).  Remark 7.2. On the other hand, for a ﬁnite ﬁeld F the map RP ( F ) → P ( F ) is not an isomor - phism if the characteristic is odd. W e have a commutati ve diagram with exact ro ws 0 / / RB ( F ) / /    RP ( F ) / /     I 2 F / /     0 0 / / B ( F ) / / P ( F ) / / Sym 2 F 2 ( G F ) / / 0 from which we deri ve the short e xact sequence 0 → I 3 F → RP ( F ) → P ( F ) → 0 . If the characteristic is odd, then I 3 F  Z with a nontrivial R F -structure; any nonsquare element of F × acts as multiplication by − 1. For an y ﬁeld F and for x ∈ F × we deﬁne the element { x } : = [ x ] + h x − 1 i ∈ P ( F ). The following lemma (due to Suslin, [25]) is easily veriﬁed: Lemma 7.3. (1) F or any ﬁeld F (with at least 4 elements) ther e is a well-deﬁned gr oup homomorphism G F → P ( F ) , h x i 7→ { x } . In particular , { x } = 0 if x ∈ ( F × ) 2 , and 2 · { y } = 0 for all y ∈ F × . (2) F or x , 1 let C F ( x ) = [ x ] + [ 1 − x ] ∈ B ( F ) Then C F ( x ) = C F ( y ) for all x , y ∈ F × \ { 1 } , and 3 C F = { − 1 } . For a ﬁnite ﬁeld F q of characteristic 2, Theorem 4.3 tells us that RB ( F q ) = B ( F q ) = P ( F q ) is cyclic of order q + 1. If F q has odd characteristic, then Theorem 4.3 tells us that B ( F q ) is cyclic of order q + 1 or ( q + 1) / 2. In fact, it is alw ays the latter: Lemma 7.4. Let F q be a ﬁnite ﬁeld of odd characteristic. Then RB ( F q ) = B ( F q ) is cyclic of or der ( q + 1) / 2 . Pr oof. As abov e, we let w = " 0 1 − 1 0 # ∈ SL 2 ( F q ) . Suppose ﬁrst that q ≡ 1 (mod 4). Then F q contains an element i satisfying i 2 = − 1. Bloch-W igner-Suslin comple x 35 Let 2 t be the exact po wer of 2 dividing q − 1. Then a 2-Sylow subgroup of SL 2 ( F q ) is the generalised quaternion group Q generated by w and { D ( z ) | z ∈ µ 2 t } . A typical element of Q has the form g = D ( z ) w e with e ∈ { 0 , 1 } . Clearly , we must prove that the composite H 3 ( Q , Z ) → H 3 (SL 2 ( F q ) , Z ) → P ( F q ) is the zero map. W e will calculate using the standard (homogeneous) resolution of Q . No w suppose that g k = D ( z k ) w e ( k ) ∈ Q ,0 ≤ k ≤ 3. W e will show that cr( β ∞ , i 3 ( g 0 , g 1 , g 2 , g 3 )) = 0 in P ( F q ) . Since Q · ∞ = { 0 , ∞} , it follo ws that either two successi ve terms of ( g 0 ( ∞ ) , g 1 ( ∞ ) , g 2 ( ∞ ) , g 3 ( ∞ )) are equal (in which case β ∞ , i 3 ( g 0 , g 1 , g 2 , g 3 ) = 0 or this term has one of the forms ( ∞ , 0 ∞ , 0) or (0 , ∞ , 0 , ∞ ) . Either way , since w · i = i , we must have β ∞ , i 3 ( g 0 , g 1 , g 2 , g 3 ) = ( z 2 0 i , z 2 1 i , ∞ , 0) + ( z 2 0 i , z 2 1 i , 0 , ∞ ) and applying cr to this and taking the image in P ( F q ) gi ves the element       z 1 z 0 ! 2       +       z 0 z 1 ! 2       =        z 1 z 0 ! 2        = 0 by Lemma 7.3. On the other hand, if q ≡ − 1 (mod 4), we let G be the cyclic subgroup of SL 2 ( F q ) generated by w . Then it will be enough to show that the composite Z / 4  H 3 ( G , Z ) → H 3 (SL 2 ( F q ) , Z ) → RB ( F ) is the zero map. For the map H 3 ( G , Z ) → H 3 (SL 2 ( F q ) , Z ) is injecti ve (by Corollary 3.7) and thus will follo w that RB ( F q ) has order ( q + 1) / 2 in this case also. Using the formulae of section 6, 1 ∈ Z / 4 maps to the cycle represented by (1 , w , w , w 2 ) + (1 , w , w 2 , w 3 ) + (1 , w , w 3 , 1) + (1 , w , 1 , w ) in the standard resolution for G . Applying β ∞ , y 3 to this gi ves the term 2 · [( y , w · y , ∞ , 0) + ( y , w · y , 0 , ∞ )] . Finally , applying cr to this and taking the image in P ( F q ) gi ves the element 2 · ( w ( y ) y ) = 2 ( − 1 y 2 ) ∈ P ( F q ) which is zero by Lemma 7.3 again.  W e can thus extend the main result of [25] to the case of ﬁnite ﬁelds: Corollary 7.5. F or any ﬁnite ﬁeld F ther e is a natural short exact sequence 0 → T or Z 1 ] ( µ F , µ F ) → K ind 3 ( F ) → B ( F ) → 0 Pr oof. By Corollary 3.6, Theorem 4.3 and Lemma 7.4 we ha ve a short e xact sequence 0 → T or Z 1 ] ( µ F q , µ F q ) → H 3 (SL 2 ( F q ) , Z [1 / p ]) → R B ( F q ) → 0 for any ﬁnite ﬁeld F q of order q = p f . Howe ver , by Corollary 3.9 and Lemma 7.1 we have natural isomorphisms H 3 (SL 2 ( F q ) , Z [1 / p ])  K ind 3 ( F q ) and RB ( F q )  B ( F q ) . 36 KEVIN HUTCHINSON  Corollary 7.6. If q ≡ 1 (mod 4) then P ( F q ) is cyclic of or der q + 1 . Pr oof. In this case ( q + 1) / 2 is odd. Since Sym 2 F 2 ( G F ) has order 2, the statement follows from Lemma 7.4 and the short exact sequence 0 → B ( F q ) → P ( F q ) → Sym 2 F 2 ( G F ) → 0 .  In fact we can use the methods of the last section to write down a formula for a generator of this cyclic group: Fix a nonsquare element a ∈ F × q . Thus F q 2 = F q ( √ a ) and we hav e an associated embedding µ : F × q 2 → GL 2 ( F q ) , x + y √ a 7→ " x yb y x # . No w let θ ∈ F q 2 hav e order r : = ( q + 1) / 2. Then t = µ ( θ ) ∈ SL 2 ( F q ). Let G = h t i ⊂ SL 2 ( F q ). The results abov e guarantee that the composite homomorphism H 3 ( G , Z ) → H 3 (SL 2 ( F q ) , Z ) → B ( F q ) is an isomorphism. Since B ⊂ SL 2 ( F q ) has order q ( q − 1), it follows that G ∩ B = { 1 } . It follo ws that the orbit G · ∞ has size ( q + 1) / 2. Choosing any y ∈ P 1 ( F q ) \ G · ∞ we obtain a generator        r − 3 X i = 1 " t ( ∞ ) − t i + 1 ( ∞ ) t ( ∞ ) − t i + 2 ( ∞ ) #        + " t − 1 ( ∞ ) − t ( ∞ ) t − 1 ( ∞ ) − y # − " t − 1 ( ∞ ) − y t ( ∞ ) − y # + " t ( ∞ ) − y t ( ∞ ) − t ( y ) # + " t ( ∞ ) − t ( y ) t ( ∞ ) − y # of B ( F q ). The last four terms can be simpliﬁed: Observe that if θ = w + z √ a then θ − 1 = w − z √ a , since θ has norm 1. Thus t ( ∞ ) = w / z , t − 1 ( ∞ ) = − w / z and t ( y ) = ( wy + az ) / ( zy + w ). If we let A = ( t − 1 ( ∞ ) − y ) / ( t ( ∞ ) − y ), then we ha ve " t − 1 ( ∞ ) − t ( ∞ ) t − 1 ( ∞ ) − y # − " t − 1 ( ∞ ) − y t ( ∞ ) − y # + " t ( ∞ ) − y t ( ∞ ) − t ( y ) # + " t ( ∞ ) − t ( y ) t ( ∞ ) − y # = C F q ( A − 1 ) − { A } + ( t ( ∞ ) − t ( y ) t ( ∞ ) − y ) = C F q + ( t − 1 ( ∞ ) − y t ( ∞ ) − t ( y ) ) = C F q + { − 1 } = C F q So when q ≡ 1 (mod 4) a generator of B ( F q ) is        r − 3 X i = 1 " t ( ∞ ) − t i + 1 ( ∞ ) t ( ∞ ) − t i + 2 ( ∞ ) #        + C F q . On the other hand, note that, since λ ( { a } ) = (1 − a ) ◦ a + (1 − a − 1 ) ◦ a − 1 = a ◦ a ∈ Sym 2 F 2 ( G F ) it follo ws that { a } ∈ P ( F q ) has order 2 and thus H θ : =        r − 3 X i = 1 " t ( ∞ ) − t i + 1 ( ∞ ) t ( ∞ ) − t i + 2 ( ∞ ) #        + C F q + { a } is a generator of the cyclic group P ( F q ). Bloch-W igner-Suslin comple x 37 Remark 7.7. If we let R θ be the corresponding isomorphism R θ : Z / ( q + 1) → P ( F q ) , m 7→ m H θ then the in verse map L θ : P ( F q ) → Z / ( q + 1) is a ‘uni versal discrete dilogarithm’ on F q in the following sense: If A is an (additi ve) abelian group and if L : F × q → A is an y map of sets satisfying L (1) = 0 and L ( x ) − L ( y ) + L  y x  − L 1 − x − 1 1 − y − 1 ! + L 1 − x 1 − y ! = 0 for all x , y ∈ F q \ { 0 , 1 } then there is a unique homomorphism τ : Z / ( q + 1) → A such that L ( x ) = τ ( L θ ( [ x ] )) for all x ∈ F × q . When q ≡ − 1 (mod 4), we can similarly obtain a formula for a generator of B ( F q ), but in this case we must compute a (more complicated) homomorphism H 3 ( Q , Z ) → B ( F q ) where Q is a generalised quaternion subgroup of SL 2 ( F q ). As an example of a related calculation we prov e Lemma 7.8. Suppose that q ≡ − 1 (mod 4) . Then { − 1 } has or der 2 in B ( F q ) . Pr oof. The calculations abov e allo w us to conclude that SL 2 ( F q ) contains a quaternion subgroup Q of order 8 with the property that the composite map Z / 8  H 3 ( Q , Z ) → H 3 (SL 2 ( F q ) , Z ) → B ( F q ) has image of order 2. Now we can take generators x and y of Q satisfying x = w = " 0 1 − 1 0 # , y 2 = x 2 = − I , xy x = y . By the calculations of the last section, a generator of H 3 ( Q , Z ) is represented by the cycle (1 , x , x 2 , x 3 ) − (1 , x , xy , x y 2 ) − (1 , xy , y 2 , xy 2 ) − (1 , xy , y , y 2 ) . Let a : = y · ∞ . Then x · ∞ = 0 and ( xy ) · ∞ = x · a = − a − 1 . Choose z ∈ P 1 ( F q ) \ {∞ , 0 , a , − 1 / a } . Applying β ∞ , z 3 to this cycle gi ves the element [ ( z , − 1 / z , ∞ , 0 ) + ( z , − 1 / z , 0 , ∞ ) ] − [ ( ∞ , − 1 / z , 0 , a ) + ( ∞ , − 1 / z , a , 0 ) ] − [ ( z , ∞ , a , 0 ) + ( z , a , ∞ , 0) ] − [ ( z , a , − 1 / a , ∞ ) − ( z , ∞ , a , − 1 / a ) ] . Applying cr to this gi ves the element X : = n − z 2 o − { 1 + az } −  z z − a  −  z a  − " 1 + a 2 1 + az # + " 1 + az a ( z − a ) # ∈ B ( F q ) . W e easily verify that −  z z − a  −  z a  = C F q −  z z − a  −  z a  and − " 1 + a 2 1 + az # + " 1 + az a ( z − a ) # = ( 1 + az a ( z − a ) − C F q from which it follo ws that X = { − 1 } .  Corollary 7.9. If q ≡ 3 (mod 8) , then P ( F q )  Z / ( q + 1) . Pr oof. Since { − 1 } = 2 [ − 1 ] in P ( F q ), the computation just completed sho ws that [ − 1 ] has order 4 in P ( F q ) when q ≡ 3 (mod 4). On the other hand, if q ≡ 3 (mod 8), then the 2-Sylow subgroup of P ( F q ) has order 4.  38 KEVIN HUTCHINSON Remark 7.10. On the other hand, if q ≡ 7 (mod 8) then 2 is a square in F × q and hence [ − 1 ] ∈ B ( F q ) has order 4. In particular , if q ≡ 7 (mod 16), then [ − 1 ] generates the 2-Sylow subgroup of B ( F q ). Finally , for any prime po wer q let t = " 0 1 − 1 − 1 # ∈ SL 2 ( F q ) of order 3 and let G = h t i ⊂ SL 2 ( F q ). Then the composite Z / 3 = H 3 ( G , Z ) → H 3 (SL 2 ( F q ) , Z ) → B ( F q ) sends 1 to C F q + { − 1 } = 4 C F q . Thus, this element has order 3 if 3 divides q + 1 and (of course) has order 1 otherwise. In view of Lemma 7.8 we deduce Lemma 7.11. The or der of C F q ∈ B ( F q ) is gcd(6 , ( q + 1) / 2) . R eferences [1] Anthony Bak and Guoping T ang . Solution to the presentation problem for powers of the augmentation ideal of torsion free and torsion abelian groups. Adv . Math. , 189(1):1–37, 2004. [2] Anthony Bak and Nikolai V avilov . Presenting powers of augmentation ideals and Pﬁster forms. K -Theory , 20(4):299–309, 2000. Special issues dedicated to Daniel Quillen on the occasion of his sixtieth birthday , Part IV . [3] Spencer J. Bloch. Higher r e gulators, algebraic K -theory , and zeta functions of elliptic curves , volume 11 of CRM Monograph Series . 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A Bloch-Wigner complex for SL_2

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