THE THIRD HOMOLOGY OF SL2(Q)

We calculate the structure of H3 ( SL2 (Q) ,Z [ 1 2 ]) . Let H3 (SL2 (Q) ,Z)0 denote the kernel of the (split) surjective homomorphism H3 (SL2 (Q) ,Z) → Kind 3 (Q). Each prime number p determines an operator 〈p〉 on H3 (SL2 (Q) ,Z) with square the identity. H3 ( SL2 (Q) ,Z [ 1 2 ]) 0 is the direct sum of the (−1)-eigenspaces of these operators. The (−1)-eigenspace of 〈p〉 is a cyclic group whose order is the odd part of p + 1. We explore some applications to the groups H3 ( SL2 ( Z [ 1 N ]) ,Z [ 1 2 ]) .


Introduction
Many years ago, in an article on the homology of Lie groups made discrete, Chi-Han Sah, quoting S. Lichtenbaum, cited our lack of any precise knowledge of the structure of H 3 (SL 2 (Q) , Z) as an example of the poor state of understanding of the homology of linear groups of general fields (see [12, pp 307-8]). Where such understanding does exist, even now, it tends often to come from connections with algebraic K-theory or Lie group theory where a bigger suite of mathematical tools is available. For example, we know the structure of H 3 (SL 3 (Q) , Z) because homology stability theorems tell us that it is isomorphic to H 3 (SL n (Q) , Z) for all larger n ( [8]) and this stable homology group is in turn isomorphic, via a Hurewicz homomorphism, to K 3 (Q)/{−1} · K 2 (Q) = K ind 3 (Q) (indecomposable K 3 ) by [14,Lemma 5.2], which is known to be cyclic of order 24 by the result of Lee and Szczarba ( [9]). For any field F, the natural map H 3 (SL 2 (F) , Z) → H 3 (SL 3 (F) , Z) K 3 (F)/{−1} · K 2 (F) → K ind 3 (F) can be shown to be surjective ( [8]). When F = C, or more generally when F is algebraically closed, it has long been known, thanks to the work of Sah and his co-authors, that this map is an isomorphism. When F is a number field, or a global function field, the map H 3 (SL 3 (F) , Z) → K ind 3 (F) is an isomorphism and the issue at stake therefore is the kernel of the stability homomorphism H 3 (SL 2 (F) , Z) → H 3 (SL 3 (F) , Z). One natural obstruction to the injectivity or surjectivity of the stability homomorphisms H • (SL n (F) , Z) → H • (SL n+1 (F) , Z) lies in the action of the multiplicative group F × : For any a ∈ F × conjugation on SL n (F) by a matrix M of determinant a induces an automorphism of H • (SL n (F) , Z) which depends only on a. In particular, a n = det(diag(a, . . . , a)) acts trivially. Since the stability homomorphism is a map of Z[F × ] modules, both a n and a n+1 act trivially on its image, and so the action of F × on the image of this map trivial. It follows that the stability homomorphism factors through the coinvariants of F × on H • (SL n (F) , Z) and has image lying in the invariants of F × on H • (SL n+1 (F) , Z). In particular, when F × acts nontrivially on H • (SL n (F) , Z), the stability homomorphism has a nontrivial kernel, since it contains I F H • (SL n (F) , Z), where I F denotes the augmentation ideal of the group ring Z[F × ].
For example, the calculations of Suslin in [13] tell us that for any infinite (or sufficiently large) field F the map H 2 (SL 2 (F) , Z) → H 2 (SL 3 (F) , Z) K 2 (F) is surjective with kernel I F H 2 (SL 2 (F) , Z) isomorphic to I(F) 3 where I(F) denotes the fundamental ideal in the Grothendieck-Witt ring of the field F. In the case F = Q, this kernel is isomorphic to the Z[Q × ]-module Z on which −1 acts by negation and all primes act trivially. B. Mirzaii has shown ( [10]) for infinite fields F that the kernel of the stability homomorphism H 3 (SL 2 (F) , Z) → H 3 (SL 3 (F) , Z) = K ind 3 (F), when tensored with Z 1 2 , is I F H 3 SL 2 (F) , Z 1 2 ; ie., it is again the case that the only obstruction to injective stability is the nontriviality of the action of the multiplicative group. He subsequently ( [11] ) generalised this result to rings with many units (including local rings with infinite residue fields).
The main theorem of this article (Theorem 3.1) describes the structure of I Q H 3 SL 2 (Q) , Z 1 2 as a Z[Q × ]-module. −1 ∈ Q × acts trivially, but each prime acts nontrivially. Since the the squares of rational numbers act trivially, each prime induces a decomposition into (+1)− and (−1)-eigenspace. The theorem states that this module is the direct sum over all primes of these (−1)-eigenspaces. The (−1)-eigenspace of the prime p is isomorphic, via a natural residue homomorphism S p , to P(F p ), the scissors congruence group of the field F p . It follows that as an abelian group The main tool we use is the description of H 3 SL 2 (F) , Z 1 2 in terms of refined scissors congruence groups. The scissors congruence group P(F) of a field F was introduced by Dupont and Sah in [2]. It is an abelian group defined by a presentation in terms of generators and relations and it was shown by the authors to be closely related to K ind 3 (F) = H 3 (SL 2 (F) , Z) when F is algebraically closed. Soon afterwards Suslin proved ( [14,Theorem 5.2]) that the connection to K ind 3 (F) persists for all infinite fields F (see Theorem 2.4 below). However, to derive an analagous result for H 3 (SL 2 (F) , Z) for general fields it is necessary to factor in the action of the multiplicative group of the field. The refined scissors congruence RP(F) of the field F -introduced in [4] -is defined by generators and relations analagously to the scissors congruence group but as a module over Z[F × ] and not merely an abelian group. It can then be shown to bear approximately the same relation to H 3 (SL 2 (F) , Z) as P(F) has to K ind 3 (F). (For a precise statement, see Theorem 2.5 below.) Using some later results of the author about erfined scissors congruence groups, our starting point in this article is essentially a presentation of I Q H 3 SL 2 (Q) , Z 1 2 as a module of the group ring Z[Q × /(Q × ) 2 ] as well as the existence of module homomorphisms S p : I Q H 3 (SL 2 (Q) , Z) → P(F p ) (where the target is a module via a · x = (−1) v p (a) x for a ∈ Q × ), one for each prime p. Remark 1.1. In our main theorem, we prove that the module homomorphism I Q H 3 (SL 2 (Q) , Z) → p P(F p ) induced by the maps S p , ranging over all primes p, becomes an isomorphism after tensoring with Z 1 2 . It is natural to ask whether the original homomorphism is an isomorphism over Z. I do not know. Our methods of proof and 2-torsion ambiguities in existing results require us to work over Z 1 2 . However, it is not hard to show even over Z that the cokernel of this map is annihilated by 4 (see the argument in Lemma 4.7 below). Remark 1.2. It is to be expected that some version of the main result should hold for general number fields and even global fields. In order to arrive at such a result it would appear necessary first to determine whether the action of the (square classes of) the global units is trivial on the groups H 3 (SL 2 (F) , Z). There is some mild evidence suggesting that this is so: (i) for any field the square class −1 acts trivially and (ii) for local fields with finite residue field, the units act trivially. We hope to examine these questions elsewhere.
1.1. Layout of the article. In section 2 we review some of the relevant known results about scissors congruence groups and the third homology of SL 2 of fields. Section 3 contains the proof of the main theorem (Theorem 3.1). We begin by recalling an elementary character-theoretic 'local-global' principle for proving that homomorphisms of Z[Q × /(Q × ) 2 ]-modules isomorphisms. The rest is straightforward manipulation of identities in the refined scissors congruence group. In section 4 we look at further applications to SL 2 (A) for subrings A of Q. The key result in this section is Theorem 4.1. This result guarantees, for quite general commutative rings A, the existence of elements of H 3 (SL 2 (A) , Z) satisfying certain conditions. We use it first to show that H 3 SL 2 Z 1 2 , Z surjects onto K ind 3 (Q) and to deduce the module structure of this homology group. We then use it to characterise the image of H 3 In section 5, we describe some further applications of the main theorem; for example, the calculation of H 3 SL 2 Q[t, t −1 ] , Z 1 2 and an explicit description of a basis for the F 3 -vector space elements of order dividing 3 in H 3 (SL 2 (Q) , Z). In section 6 we give the details of the technical proof of Theorem 4.1, which was deferred from Section 4.

1.2.
Notation. For a commutative unital ring A, A × denotes the group of units of A.
For any abelian group A, we denote A ⊗ Z 1 n by A 1 n . For any prime p, A (p) denotes the group of p-primary torsion elements in A. If q is a prime power, F q will denote the finite field with q elements. Given an abelian group G we let S 2 Z (G) denote the group and, for x, y ∈ G, we denote by x • y the image of x ⊗ y in S 2 Z (G). For any rational prime p, v p : Q × → Z denotes the corresponding discrete valuation, determined by a = p v p (a) · (m/n) with m, n not divisible by p. For a field F, we let R F denote the group ring Z[F × /(F × ) 2 ] of the group of square classes of F and we let I F denote the augmentation ideal of R F . If x ∈ F × , we denote the corresponding square-class, considered as an element of R F , by x . The generators x − 1 of I F will be denoted x .

Refined scissors congruence groups and H 3 (SL 2 (F) , Z)
In this section we review some of the relevant known facts about the third homology of SL 2 of fields and its description in terms of refined scissors congruence groups.
2.1. Indecomposable K 3 . For any field F there is a natural surjective homomorphism When F is quadratically closed (i.e. when F × /(F × ) 2 = 1) this map is an isomorphism. However, in general, the group extension It can be shown that the map (1) is a homomorphism of R F -modules (where F × /(F × ) 2 acts trivially on K ind 3 (F) and induces an isomorphism (see [10,Proposition 6.4]), but -as our calculations in [3] show -the action of F × /(F × ) 2 on H 3 (SL 2 (F) , Z) is in general non-trivial. Let H 3 (SL 2 (F) , Z) 0 denote the kernel of the surjective homomorphism H 3 (SL 2 (F) , Z) → K ind 3 (F). This is an R F -submodule of H 3 (SL 2 (F) , Z). Note that the isomorphism (2) implies that is split as a map of Z-modules. In fact, K ind 3 (F) is a finitely generated abelian group and it is enough to show that there is a torsion subgroup of H 3 (SL 2 (F) , Z) mapping isomorphically to the (cyclic) torsion subgroup of K ind 3 (F). But this latter statement follows from the explicit calculations of C. Zickert in [15,Section 8]. It follows that, as an abelian group, H 3 (SL 2 (F) , Z) K ind 3 (F) ⊕ H 3 (SL 2 (F) , Z) 0 for any number field F. However, there is no such decomposition of H 3 (SL 2 (F) , Z) as an R F -module. For details, see Remark 2.11 below.
2.2. Scissors Congruence Groups. For a field F, with at least 4 elements, the scissors congruence group (also called the pre-Bloch group), P(F), is the group generated by the elements [x], x ∈ F × , subject to the relations x is well-defined, and the Bloch group of F, B(F) ⊂ P(F), is defined to be the kernel of λ. For the fields with 2 and 3 elements the following definitions allow us to include these fields in the statements of some of our results: P(F 2 ) = B(F 2 ) is a cyclic group of order 3 with generator denoted C F 2 . We let [1] := 0 in P(F 2 ). Lemma 2.2. If q is a prime power then B(F q ) is cyclic of order (q + 1)/2 when q is odd and q + 1 when q is even.
The following corollary is particularly relevant to this article: Corollary 2.3. If q is a prime power then P(F q ) 1 2 is cyclic of order (q + 1) odd .
The Bloch group is closely related to the indecomposable K 3 of the field F: (See Suslin [14] for infinite fields and [4] for finite fields.) 2.3. The refined scissors congruence group. For a field F with at least 4 elements, Of course, from the definition it follows immediately that It can be shown that Λ is well-defined.
The refined scissors congruence group of F is the R F -module RP 1 (F) := Ker(λ 1 ).
The refined Bloch group of the field F (with at least 4 elements) to be the R F -module is simply an additive group of order 3 with distinguished generator, denoted is the cyclic R F 3 -module generated by the symbol [−1] and subject to the one relation is then cyclic of order 4 generated by the symbol [−1]. The symbol [1] continues to denote 0 in RP(F 2 ) and RP(F 3 ). We recall some results from [4]: The main result there is Theorem 2.5. Let F be any field. There is a natural complex which is exact everywhere except possibly at the middle term. The middle homology is annihilated by 4. In particular, for any field there is a natural short exact sequence In particular, {x} = 0 in P(F) 1 2 . There are two natural liftings of these elements to RP(F): given x ∈ F × we define

Scissors congruence groups and
. In general, the elements ψ i (x) have infinite order however.
We define RP(F) to be RP(F) modulo the submodule generated by the elements ψ 1 (x), x ∈ F × . Likewise, P(F) is the group P(F) modulo the subgroup generated by the elements {x}, x ∈ F × . Note that since the elements {x} are annihilated by 2, we have P(F) 1 2 = P(F) 1 2 . For any field there is natural homomorphism of R F -modules H 3 (SL 2 (F) , Z) → RP(F) and we have ([6, Corollary 2.8,Corollary 4.4]): Theorem 2.6. For any field F, the map H 3 (SL 2 (F) , Z) → RP(F) induces an isomorphism of R F -modules Note that it follows that the square class −1 acts trivially on H 3 SL 2 (F) , Z 1 2 . To simplify the right-hand side we define the module RP + (F) to be RP(F) modulo the submodule generated by the elements ( The theorem then says that the map H 3 (SL 2 (F) , Z) → RP(F) induces an isomorphism 1 2 .
Note that the natural map RP + (F) → P(F) induces an isomorphism RP + (F) F × P(F). Furthermore, the results of [4,Section 7] immediately imply that k × acts trivially on RP + (k) when k is a finite field. Thus RP + (k) = P(k) for a finite field k.
We will also use the same symbol to denote the image of this element in P( We review some of the fundamental properties of the element C F ∈ RP + (F) (for proofs see [3, Section 3.2]).
Proposition 2.7. The element C F ∈ RP + (F) has the following properties: It will be convenient below to introduce the following additional notation in RP + (F): [0] := C F and [∞] := −C F .
With this notation, we then have and ψ 1 (x) = 0 for all x ∈ P 1 (F).
2.6. Discrete valuations and the specialization homomorphism. Suppose that v : F × → Z is a discrete valuation on the field F with residue field k = k(v). We give P(k) an R F -module structure via the requirement for all a ∈ F × , x ∈ k × and denote the resulting R F -module by P(k) {v}. Then there is a natural R F -module homomorphism (see [3,Section 4.3] and [6, Section 5]) Observe that this definition makes sense when k(v) = F 2 or F 3 . It is clear from the definition that the maps S v factor through the quotient RP + (F) of RP(F).
In this context we have the following result ([3, Theorem 5.1]): Theorem 2.8. Let F be a field and let V be a family of discrete valuations on F satisfying Then the maps {S v } v∈V induce a natural surjective homomorphism Taking F = Q and V = Primes, the set of all primes, we obtain a surjective homomorphism of R Q -modules Remark 2.9. Since −1 ∈ R Q acts trivially on both of the modules in (3), this is a map of The collection of maps {S v } v∈V as above induces an R F -module homomorphism with target the product -rather than the direct sum -of the scissors congruence groups: However, when we restrict to H 3 (SL 2 (F) , Z) 0 and tensor with Z 1 2 the image lies in the direct sum instead, in view of the isomorphism H 3 SL 2 (F) , Z 1 2 0 1 2 and the fact that Denote (also) by C Q ∈ H 3 (SL 2 (Q) , Z) the image of 1 ∈ Z/3 = H 3 ( t , Z) under the map induced by the inclusion t → SL 2 (Q). Then C Q ∈ H 3 (SL 2 (Q) , Z) maps to C Q ∈ RP + (Q) ([3, Remark 3.14]). Note that S p (C Q ) = C F p ∈ P(F p ) for all primes p. Furthermore, C F p 0 precisely when p ≡ 2 mod 3 (ie., precisely when 3|p + 1), by [4,Lemma 7.11].
In particular, the image of C Q under the map {S p } p lies in the product, but not the direct sum, of the scissors congruence groups of the residue fields.
Remark 2.11. Now Suslin's map gives a canonical isomorphism K ind 3 (Q) (3) B(Q) (3) = Z/3·C Q and we can let C Q also denote the corresponding element of K ind 3 (Q). Recall that R Q acts trivially on K ind 3 (Q). Suppose that there were an R Q -module splitting j : . Then we would have j(C Q ) = C Q +h for some h ∈ H 3 (SL 2 (Q) , Z) 0 . We must have R Q acts trivially on j(C Q ) and hence p j(C Q ) = p (C Q + h) = 0 for all primes p. However, we can choose a prime p such that p h = 0 in H 3 SL 2 (Q) , Z 1 2 0 and p ≡ 2 (mod 3). Then S p ( p (C Q + h)) = p C F p = −2C F p 0, giving us a contradiction. So no such R Q -splitting j can exist.

The Main Theorem
In this section we prove is an isomorphism of R + Q -modules We will use the following character-theoretic principles: Let G be an abelian group satisfying g 2 = 1 for all g ∈ G. Let R denote the group ring Z 1 2 [G]. For a character χ ∈ G := Hom(G, µ 2 ), let R χ be the ideal of R generated by the elements {g − χ(g) | g ∈ G}. In other words R χ is the kernel of the ring homomorphism ρ(χ) : R → Z sending g to χ(g) for any g ∈ G. We let R χ denote the associated R-algebra structure on Z; ie.
If M is an R-module, we let M χ = R χ M and we let Thus M χ is the largest quotient module of M with the property that g · m = χ(g) · m for all g ∈ G.
In particular, if χ = χ 0 , the trivial character, then R χ 0 is the augmentation ideal I G , M χ 0 = I G M and M χ 0 = M G . We will need the following results ([6, Section 3]) Proposition 3.2.
(1) For any χ ∈ G, M → M χ is an exact functor on the category of R-modules.

Recall now that R
As a multiplicative F 2 -space, the set of all primes form a (number-theoretically) natural basis of Q + /Q 2 + . Thus the space of characters Q + /Q 2 + is naturally parametrised by the subsets of the set Primes of positive prime numbers: if S ⊂ Primes then the corresponding character χ S is defined by for all p ∈ Primes or, equivalently, for all x ∈ Q × . Conversely, the character χ corresponds to the subset The following lemma is immediate from the definition of the R Q -module structure on P(F p ) {p}.
It thus follows from Proposition 3.2 that to prove Theorem 3.1 it is enough to prove that S p induces an isomorphism for any prime p, while whenever supp(χ) contains at least two distinct primes. These two statements are proved in Corollaries 3.13 and 3.11 below. Then Corollary 3.6. Let F be a field. Let χ ∈ F × /(F × ) 2 . Suppose that a ∈ F × satisfies χ(a) = 1 and for all a ∈ P 1 (F).
We divide further into sub-cases according to the value of χ(1 − y): for any a ∈ F.
for all t ∈ Z and a ∈ Q.
If q ≡ −1 (mod 8) we can take = 3q. Then χ( ) = −1 (since q 3). Furthermore we have − 1 ≡ 4 (mod 8) and This implies χ( − 1) = χ(1 − ) = 1 and we can conclude as before. Finally, if q ≡ 1 (mod 8) we take = −3q and argue as in the previous case. Proof. We will show that [a] χ = 0 for all a ∈ Q. By Proposition 3.10 we have [a] χ = [a + t] χ for all t ∈ Z, a ∈ Q × . It follows that [a] χ = [1] χ = 0 if a ∈ Z. Thus also [1/a] χ = 0 for all a ∈ Z \ {0}. Note that it is enough to prove [a] χ = 0 for all a > 0 (if necessary replacing a by a + t with t ∈ Z large). So let a = r/s with 0 < r, s ∈ Z. We proceed by induction on h := min(r, s). The case h = 1 has already been proved. Suppose now that n ≥ 1 and the statement is known for h ≤ n. Consider the case h = n + 1. Replacing a by 1/a if necessary we can suppose s < r and s = n + 1. Then there exists t ∈ Z such that 0 < r := r − ts < s. So [a] χ = [a − t] χ = r /s χ where now h = r ≤ n and we are done.
Finally, we fix a (positive) prime number p. Let U p := {a ∈ Q × | v p (a) = 0} and let U 1,p := {a ∈ U | a = r/s with r ≡ s (mod p)}. Corollary 3.13. For any prime p , the homomorphism S p : RP + (Q) 1 2 → P(F p ) {p} 1 2 induces an isomorphism of R Q -modules Proof. By Lemma 3.12, Lemma 3.12. It follows that for all u ∈ U p \ U 1,p and w ∈ U 1,p we have since the last three terms of the first line line in U 1,p . Thus, for u ∈ U p , the element [u] χ p ∈ RP + (Q) 1 2 χ p depends only onū := u (mod p) ∈ F p . It follows that there is a well-defined R Q -module homomorphism for which S p • T p is the identity map: when p ≥ 3, T p is the map Thus T A is naturally a subgroup of B A . Proof. We defer the technical proof to Section 6 below.
The next two lemmas show that the hypothesis of Theorem 4.1 holds for many subrings of Q. Proof. By considering the Hochschild-Serre spectral sequence associated to the group extension it is enough to establish the vanishing of the groups H 0 (T A , H 2 (U A , Z)) and H 1 (T A , H 1 (U A , Z)). Furthermore, since A is a colimit of infinite cyclic groups, H 2 (A, Z) = 0. We need only prove the vanishing of H 1 (A × , A) where u ∈ A × T A acts on A U A as multiplication by u 2 . However, the pair of maps ( conjugation by u, u·) : (A × , A) → (A × , A) induces the identity on the groups H i (A × , A). Taking u = 2 ∈ A × , we deduce that multiplication by 4 = 2 2 is the identity map on H 1 (A × , A) and hence that this group is annihilated by 3. But 3 ∈ A × by hypothesis and so acts invertibly on H 1 (A × , A). Thus H 1 (A × , A) = 0 as required.
Proof. As in the preceding lemma, is is enough to show that H := H 1 (A × , A) = 0. Since 2 ∈ A × , we again deduce that H is annihilated by 3.
For a subring A of Q we denote by H 3 (SL 2 (A) , Z) 0 the kernel of the map H 3 (SL 2 (A) , Z) → K ind 3 (Q). Proposition 4.5. There is a split short exact sequence of R is a cyclic group of order 3 on which the square class 2 acts as multiplication by −1.
Example 4.9. For t ≥ 1, let N t := p 1 · · · p t where 2 = p 1 < · · · < p t are the first t prime numbers in order of size. By Corollary 4.8 the composite homomorphism  In view of Example 4.9 we deduce Corollary 4.14. For t ≥ 4, as above let N t := p 1 · · · p t where 2 = p 1 < · · · < p t are the first t prime numbers in order of size. Then H 3 SL 2 Z 1 and is isomorphic to Remark 4.15. The requirement 2, 3, 5, 7 ∈ S in Proposition 4.11 arise from the fact that the maps H 3 SL 2 Z p , Z 1 2 → K ind 3 (Q p ) 1 2 are proved to be isomorphisms (in [7]) only for p ≥ 11. This restriction is an artefact of the method of proof and these maps may well be isomorphisms for some or all of the remaining primes.
Now there is a natural map of complexes of Z[SL 2 (A)]-module L • → Z[0], where Z[0] denotes the module Z concentrated in degree 0 and the map sends each u ∈ X 1 to 1. For a field F with at least 4 elements (and for more general classes of rings with sufficiently many units) this is a weak equivalence in degrees ≤ 3, and so induces isomorphisms in homology H 3 (SL 2 (F) , L • ) H 3 (SL 2 (F) , Z). For a general commutative ring A this last statement is no longer usually true.