THE THIRD HOMOLOGY OF THE SPECIAL LINEAR GROUP OF A FIELD

KEVIN HUTCHINSON, LIQUN TAO

Abstract. We prove that for any infinite field F, the map H_{3}(SL_{n}(F), ℤ) → H_{3}(SL_{n+1}(F), ℤ) is an isomorphism for all n ≥ 3. When n = 2 the cokernel of this map is naturally isomorphic to 2 · K_{3}^{M}(F), where K_{n}^{M}(F) is the nth Milnor K-group of F . We deduce that the natural homomorphism from H_{3}(SL_{2}(F), ℤ) to the indecomposable K_{3} of F, K_{3}(F)^{ind}, is surjective for any infinite field F.

1. Introduction

A well-known result of A. A. Suslin ([Sus84]) relates the Milnor K-theory of a field, F, to the question of homology stability for the general linear group of the field. He shows that the maps H_{p}(GL_{n}(F), ℤ) → H_{p}(GL_{n+1}(F), ℤ) are isomorphisms for n ≥ p and that, when n = p − 1, the cokernel is naturally isomorphic to K_{p}^{M}(F). In fact, if we let H_{p}(F) := H_{p}(GL_{p}(F), ℤ)/H_{p}(GL_{p−1}(F), ℤ), his arguments show that there is an isomorphism of graded rings H_{•}(F) ≅ K_{•}^{M}(F) (where the multiplication on the first term comes from direct sum of matrices and cross product on homology). In particular, the non-negatively graded ring H_{•}(F) is generated in dimension 1. Recent work of Barge and Morel ([BM99]) suggests that Milnor-Witt K-theory may play a somewhat analogous role for the homology of the special linear group. The Milnor-Witt K-theory of F is a ℤ-graded ring K_{•}^{MW}(F) surjecting naturally onto Milnor Ktheory. It arises as a ring of operations in stable motivic homotopy theory. (For a definition see section 2 below, and for more details see [Mor04a, Mor04b, Mor06].) Let SH_{p}(F) := H_{p}(SL_{p}(F), ℤ)/H_{p}(SL_{p−1}(F), ℤ) for p ≥ 1, and let SH_{0}(F) = ℤ [F^{×}] for convenience. Barge and Morel construct a map of graded algebras SH_{•}(F) → K_{•}^{MW}(F) for which the square

\xymatrix{SH_{•}(F) \ar[r] \ar[d] & K_{•}^{MW}(F) \ar[d] \\ H_{•}(F) \ar[r] & K_{•}^{M}(F)}

commutes. A result of Suslin ([Sus87]) implies that the map H_{2}(SL_{2}(F), ℤ) = SH_{2}(F) → K_{2}^{MW}(F) is an isomorphism. Since positive-dimensional Milnor-Witt K-theory is generated in dimension 1, it follows that the map of Barge and Morel is surjective in even dimensions greater than or equal to 2. They ask the question whether it is in fact an isomorphism in even dimensions.

Date: August 7, 2008.

1991 Mathematics Subject Classification. 19G99, 20G10.

Key words and phrases. K-theory, Special Linear Group, Group Homology .

Of course, since SL_{1}(F) = {1}, SH_{1}(F) = 0, and there is no possiblity that the map is an isomorphism in odd dimensions. Furthermore, Barge and Morel make the point that their map is also a map of graded ℤ [F^{×}]-modules, but that (F^{×})^{2} acts trivially on K_{•}^{MW}(F) while (F^{×})^{n} acts trivially on SH_{n}(F). It follows that, in odd dimensions, the image of their homomorphism must be contained in the F^{×}-invariants, K_{n}^{MW}(F)^{F^{×}}, of Milnor-Witt K-theory. Now 2K_{n}^{M}(F) occurs naturally as a subgroup of K_{n}^{MW}(F)^{F^{×}}, and these groups are very often equal (the obstruction to equality in general being a small subgroup of the torsion subgroup of the Witt ring of the field). The question thus arises whether the Barge-Morel homomorphism induces an isomorphism SH_{n}(F) ≅ 2K_{n}^{M}(F) in odd dimensions greater than or equal to 3. This paper is intended primarily as a first step towards answering the question of Barge and Morel. The authors believe that it will be useful to answer the case n = 3 in order to attack the higher-dimensional - even or odd - cases. Our main result is that the map SH_{3}(F) → K_{3}^{MW}(F) induces an isomorphism SH_{3}(F) ≅ 2K_{3}^{M}(F). In the process, we also prove the homology stability result that the maps H_{3}(SL_{n}(F), ℤ) → H_{3}(SL_{n+1}(F), ℤ) are isomorphisms for all n ≥ 3. Chih-Han Sah, in [Sah89], states (without proving it) that these maps are isomorphisms for all n ≥ 4. So our contribution here is to extend the known stability range by 1. Observe that it easily follows from our main result that, for all odd n greater than or equal to 3, the image of the Barge-Morel map contains 2K_{n}^{M}(F) (since in Milnor-Witt K-theory, (2K_{r}^{M}(F)) · K_{s}^{MW}(F) = 2K_{r+s}^{M}(F)). As another application, we deduce that for any infinite field F the natural map H_{0}(F^{×}, H_{3}(SL_{2}(F), ℤ)) → K_{3}(F)^{ind} ([Sus90, Sah89, EV98, Mir07]) is surjective. Suslin has asked the question whether this map is always an isomorphism. The homological techniques we use in this paper are standard in this area and go back to the earliest results on homology stability for linear groups. The new ingredient is our use of the Milnor conjecture and Milnor-Witt K-theory to prove Theorem 3.2 below. The discussion above suggests that the appearance of Milnor-Witt K-theory in this context is not merely incidental.

2. Notation and Background Results

Third Homology of SL 3

2.1. Group Rings and Grothendieck-Witt Rings. For a group G, we let ℤ [G] denote the corresponding integral groupring. It has an additive ℤ-basis consisting of the elements ⟨g⟩, g ∈ G, with the natural multiplication. (We use this notation in order to distinguish the elements ⟨1 − a⟩ from 1 − ⟨a⟩ when G is the multiplicative group, F^{×}, of a field F.) There is an augmentation homomorphism ϵ : ℤ [G] → ℤ, ⟨g⟩ ↦ 1, whose kernel is the augmentation ideal I_{G}, generated by the elements ⟨⟨g⟩⟩ := ⟨g⟩ − 1. The Grothendieck-Witt ring of a field F is the Grothendieck group, GW(F), of the set of isometry classes of nondgenerate symmetric bilinear forms under orthogonal sum. Tensor product of forms induces a natural multiplication on the group. As an abstract ring, this can be described as the quotient of the ring ℤ [F^{×}/(F^{×})^{2}] by the ideal generated by the elements ⟨⟨a⟩⟩ · ⟨⟨1 − a⟩⟩, a ≠ 0, 1. (This is just a mild reformulation of the presentation given in Lam, [Lam05], Chapter II, Theorem 4.1.) Here, the induced ring homomorphism ℤ [F^{×}] → ℤ [F^{×}/(F^{×})^{2}] → GW(F), sends ⟨a⟩ to the class of the 1-dimensional form with matrix [a]. This class is (also) denoted ⟨a⟩. GW(F) is again an augmented ring and the augmentation ideal, I(F), - also called the fundamental ideal - is generated by Pfister 1-forms, ⟨⟨a⟩⟩. It follows that the n-th power, I^{n}(F), of this ideal is generated by Pfister n-forms ⟨⟨a_{1}, …, a_{n}⟩⟩ := ⟨⟨a_{1}⟩⟩ ⋯ ⟨⟨a_{n}⟩⟩. We will use below the following result on the powers of the fundamental ideal.

Lemma 2.1 ([HT08], Cor 2.16). For all n ≥ 0, I^{n+2}(F) is generated, as an additive group, by Pfister forms ⟨⟨a_{0}, …, a_{n+1}⟩⟩ subject to the following relations:

(a) ⟨⟨a_{0}, …, a_{n+1}⟩⟩ = 0 if a_{n+1} ∈ (F^{×})^{2}

(b) ⟨⟨a_{0}, …, a_{i}, a_{i+1}, …, a_{n+1}⟩⟩ = ⟨⟨a_{0}, …, a_{i+1}, a_{i}, …, a_{n+1}⟩⟩

(c) ⟨⟨a_{0}, …, a_{n}, a_{n+1}a′_{n+1}⟩⟩ + ⟨⟨a_{0}, …, a_{n−1}, a_{n+1}, a′_{n+1}⟩⟩ = ⟨⟨a_{0}, …, a_{n}a_{n+1}, a′_{n+1}⟩⟩ + ⟨⟨a_{0}, …, a_{n}, a_{n+1}⟩⟩

(d) ⟨⟨a_{0}, …, a_{n}, a_{n+1}⟩⟩ = ⟨⟨a_{0}, …, a_{n}, (1 − a_{n})a_{n+1}⟩⟩

Now let h := ⟨1⟩ + ⟨−1⟩ = ⟨⟨−1⟩⟩ + 2 ∈ GW(F). Then h · I(F) = 0, and the Witt ring of F is the ring

W (F) := \frac{GW(F)}{⟨h⟩} = \frac{GW(F)}{h · ℤ}.

Since h ↦ 2 under the augmentation, there is a natural ring homomorphism W (F) → ℤ/2. The fundamental ideal I(F) of GW(F) maps isomorphically to the kernel of this ring homomorphism under the map GW(F) → W (F), and we also let I(F) denote this ideal. For n ≤ 0, we define I^{n}(F) := W (F). The graded additive group I^{•}(F) = {I^{n}(F)}_{n∈ℤ} is given the structure of a commutative graded ring using the natural graded multiplication induced from the multiplication on W (F). In particular, if we let η ∈ I^{−1}(F) be the element corresponding to 1 ∈ W (F), then multiplication by η : I^{n+1}(F) → I^{n}(F) is just the natural inclusion.

2.2. Milnor K-theory and Milnor-Witt K-theory. The Milnor ring of a field F (see [Mil70]) is the graded ring K_{•}^{M}(F) with the following presentation: Generators: {a} , a ∈ F^{×}, in dimension 1. Relations:

(a) {ab} = {a} + {b} for all a, b ∈ F^{×}.

(b) {a} · {1 − a} = 0 for all a ∈ F^{×} \ {1}.

The product {a_{1}} ⋯ {a_{n}} in K_{n}^{M}(F) is also written {a_{1}, …, a_{n}}. So K_{0}^{M}(F) = ℤ and K_{1}^{M}(F) is an additive group isomorphic to F^{×}. We let k_{•}^{M}(F) denote the graded ring K_{•}^{M}(F)/2 and let i^{n}(F) := I^{n}(F)/I^{n+1}(F), so that i^{•}(F) is a non-negatively graded ring. In the 1990s, Voevodsky and his collaborators proved a fundamental and deep theorem - originally conjectured by Milnor ([Mil71]) - relating Milnor K-theory to quadratic form theory:

Theorem 2.2 ([OVV07]). There is a natural isomorphism of graded rings k_{•}^{M}(F) ≅ i^{•}(F) sending {a} to ⟨⟨a⟩⟩. In particular for all n ≥ 1 we have a natural identification of k_{n}^{M}(F) and i^{n}(F) under which the symbol {a_{1}, …, a_{n}} corresponds to the class of the form ⟨⟨a_{1}, …, a_{n}⟩⟩.

The Milnor-Witt K-theory of a field is the graded ring K_{•}^{MW}(F) with the following presentation (due to F. Morel and M. Hopkins, see [Mor04a]): Generators: [a], a ∈ F^{×}, in dimension 1 and a further generator η in dimension −1.

Relations:

(a) [ab] = [a] + [b] + η · [a] · [b] for all a, b ∈ F^{×}

(b) [a] · [1 − a] = 0 for all a ∈ F^{×} \ {1}

(c) η · [a] = [a] · η for all a ∈ F^{×}

(d) η · h = 0, where h = η · [−1] + 2 ∈ K_{0}^{MW}(F).

Clearly there is a unique surjective homomorphism of graded rings K_{•}^{MW}(F) → K_{•}^{M}(F) sending [a] to {a} and inducing an isomorphism

\frac{K_{•}^{MW}(F)}{⟨η⟩} ≅ K_{•}^{M}(F).

Furthermore, there is a natural surjective homomorphism of graded rings K_{•}^{MW}(F) → I^{•}(F) sending [a] to ⟨⟨a⟩⟩ and η to η. Morel shows that there is an induced isomorphism of graded rings

\frac{K_{•}^{MW}(F)}{⟨h⟩} ≅ I^{•}(F).

The main structure theorem on Milnor-Witt K-theory is the following theorem of Morel:

Theorem 2.3 (Morel, [Mor04b]). The commutative square of graded rings

\xymatrix{K_{•}^{MW}(F) \ar[r] \ar[d] & K_{•}^{M}(F) \ar[d] \\ I^{•}(F) \ar[r] & k_{•}^{M}(F)}

is cartesian.

Thus for each n ∈ ℤ we have an isomorphism

K_{n}^{MW}(F) ≅ K_{n}^{M}(F) ×_{i^{n}(F)} I^{n}(F).

It follows, for example, that for all n, the kernel of K_{n}^{MW}(F) → K_{n}^{M}(F) is isomorphic to I^{n+1}(F) and, for n ≥ 0, the inclusion I^{n+1}(F) → K_{n}^{MW}(F) is given by ⟨⟨a_{1}, …, a_{n+1}⟩⟩ ↦ η[a_{1}] ⋯ [a_{n}]. When n = 0 we have an isomorphism of rings

GW(F) ≅ W (F) ×_{ℤ/2} ℤ ≅ K_{0}^{MW}(F).

Under this isomorphism ⟨⟨a⟩⟩ corresponds to η[a] and ⟨a⟩ corresponds to η[a] + 1. Thus each K_{n}^{MW}(F) has the structure of a GW(F)-module (and hence also of a ℤ [F^{×}]- module), with the action given by ⟨⟨a⟩⟩ · ([a_{1}] ⋯ [a_{n}]) = η[a][a_{1}] ⋯ [a_{n}]. Thus, for all n ≥ 0,

I_{F^{×}} · K_{n}^{MW}(F) = I(F) · K_{n}^{MW}(F) = η · K_{n+1}^{MW}(F) = I^{n+1}(F)

and hence

H_{0}(F^{×}, K_{n}^{MW}(F)) = \frac{K_{n}^{MW}(F)}{I_{F^{×}} · K_{n}^{MW}(F)} ≅ K_{n}^{M}(F).

2.3. Homology of Groups. Given a group G and a ℤ [G]-module M, H_{n}(G, M) will denote the nth homology group of G with coefficients in the module M. B_{•}(G) will denote the right bar resolution of G: B_{n}(G) is the free right ℤ [G]-module with basis the elements [g_{1}| ⋯ |g_{n}], g_{i} ∈ G. (B_{0}(G) is isomorphic to ℤ [G] with generator the symbol [ ].) The boundary d = d_{n} : B_{n}(G) → B_{n−1}(G), n ≥ 1, is given by

d([g_{1}| ⋯ |g_{n}]) = ∑_{i=o}^{n−1}(−1)^{i}[g_{1}| ⋯ |ĝ_{i}| ⋯ |g_{n}] + (−1)^{n}[g_{1}| ⋯ |g_{n−1}]⟨g_{n}⟩.

The augmentation B_{0}(G) → ℤ makes B_{•}(G) into a free resolution of the trivial ℤ [G]- module ℤ, and thus H_{n}(G, M) = H_{n}(B_{•}(G) ⊗_{ℤ[G]} M). If C_{•} = (C_{q}, d) is a non-negative complex of ℤ [G]-modules, then E_{•,•} := B_{•}(G) ⊗_{ℤ[G]} C_{•} is a double complex of abelian groups. Each of the two filtrations on E_{•,•} gives a spectral sequence converging to the homology of the total complex of E_{•,•}, which is by definition, H_{•}(G, C). (see, for example, Brown, [Bro82], Chapter VII). The first spectral sequence has the form

E_{p,q}^{2} = H_{p}(G, H_{q}(C)) ⟹ H_{p+q}(G, C).

In the special case that there is a weak equivalence C_{•} → ℤ (the complex consisting of the trivial module ℤ concentrated in dimension 0), it follows that H_{•}(G, C) = H_{•}(G, ℤ). The second spectral sequence has the form

E_{p,q}^{1} = H_{p}(G, C_{q}) ⟹ H_{p+q}(G, C).

Thus, if C_{•} is weakly equivalent to ℤ, this gives a spectral sequence converging to H_{•}(G, ℤ). In the next theorem, we summarize some of the main results of [Sus84]:

Theorem 2.4. Let T_{n} be the group of diagonal matrices in GL_{n}(F).

(a) The natural maps H_{p}(GL_{n}(F), ℤ) → H_{p}(GL_{n+1}(F), ℤ) are isomorphisms if n ≥ p.

(b) For any fixed p and any n ≥ p there is a natural map ϵ = ϵ_{p,n} : H_{p}(GL_{n}(F), ℤ) → K_{p}^{M}(F) whose kernel is the image of H_{p}(GL_{p−1}(F), ℤ).

(c) For n ≥ p, the composite

\xymatrix{⋀^{p} (T_{n}) \ar[r] & H_{p}(T_{n}, ℤ) \ar[r] & H_{p}(GL_{n}(F), ℤ) \ar[r]^{ϵ} & K_{p}^{M}(F)}

is given by

(a_{1,1}, …, a_{1,n}) ∧ ⋯ ∧ (a_{p,1}, …, a_{p,n}) ↦ ∑_{J=(j_{1},...,j_{p})} {a_{1,j_{1}}, …, a_{p,j_{p}}}

where J ranges over all ordered p-tuples of distinct points of {1, …, n}.

(d) For any n ≥ 3 the composite

\xymatrix{H_{2}(SL_{n}(F), ℤ) \ar[r] & H_{2}(GL_{n}(F), ℤ) \ar[r]^{ϵ} & K_{2}^{M}(F)}

is an isomorphism.

Proof.

(a) [Sus84], Theorem 3.4 (c)

(b) [Sus84], Theorem 3.4 (d)

(c) Let ρ_{k} = ρ_{k,n} : F^{×} → T_{n} denote the inclusion in the kth factor. Now, writing the operation in ⋀^{p} T_{n} as addition, we have

(a_{1,1}, …, a_{1,n}) ∧ ⋯ ∧ (a_{p,1}, …, a_{p,n}) = ∑_{1≤k_{1},...,k_{p}≤n} ρ_{k_{1}}(a_{1,k_{1}}) ∧ ⋯ ∧ ρ_{k_{p}}(a_{p,k_{p}}).

It follows from Corollary 2.7.2 of [Sus84] that the image of ρ_{1}(a_{1}) ∧ ⋯ ∧ ρ_{p}(a_{p}) is {a_{1}, …, a_{p}}. Now if J = (j_{1}, …, j_{p}) is a p-tuple of distinct points of {1, …, n} there exists a permutation sending j_{i} to i. The corresponding permutation on the factors of T_{n} can be realised as conjugation by a permutation matrix in GL_{n}(F). Since the induced automorphism on the integral homology of GL_{n}(F) is trivial, it follows that the image of ρ_{j_{1}}(a_{1}) ∧ ⋯ ∧ ρ_{j_{p}}(a_{p}) is again {a_{1}, …, a_{p}}.

On the other hand, if k_{1}, …, k_{p} are a set of natural numbers smaller than n some two of which are equal, then the image of ρ_{k_{1}}(a_{1}) ∧ ⋯ ∧ ρ_{k_{p}}(a_{p}) lies in the image of H_{p}(GL_{p−1}(F), ℤ) and so is 0.

(d) The Hochschild-Serre spectral sequence of the extension 1 → SL_{n}(F) → GL_{n}(F) → F^{×} → 1 (see the next section) gives, for n ≥ 2, a split exact sequence

0 → H_{0}(F^{×}, H_{2}(SL_{n}(F), ℤ)) → H_{2}(GL_{n}(F), ℤ) → H_{2}(F^{×}, ℤ) → 0.

Since the map H_{2}(GL_{1}(F), ℤ) → H_{2}(F^{×}, ℤ) induced by the determinant is trivially an isomorphism, it follows from the results just cited that ϵ induces an isomorphism H_{0}(F^{×}, H_{2}(SL_{n}(F), ℤ)) ≅ K_{2}^{M}(F) for all n ≥ 2.

It can be shown that, for n ≥ 3, F^{×} acts trivially on H_{2}(SL_{n}(F), ℤ) (see for example Hutchinson, [Hut90], p199) and thus H_{0}(F^{×}, H_{2}(SL_{n}(F), ℤ)) = H_{2}(SL_{n}(F), ℤ) in this case.

□

Remark 2.5. On the other hand, the action of F^{×} on H_{2}(SL_{2}(F), ℤ) is not generally trivial. In fact a result of Suslin ([Sus87], Appendix ) essentially states that there is a natural isomorphism H_{2}(SL_{2}(F), ℤ) ≅ K_{2}^{MW}(F) for which the diagram

\xymatrix{H_{2}(SL_{2}(F), ℤ) \ar[r]^{≅} \ar[d] & K_{2}^{MW}(F) \ar[d] \\ H_{2}(SL_{3}(F), ℤ) \ar[r]^{≅} & K_{2}^{M}(F)}

commutes (for more details see also Mazzoleni, [Maz05] or Hutchinson and Tao, [HT08]). It follows that the kernel of H_{2}(SL_{2}(F), ℤ) → H_{2}(SL_{3}(F), ℤ) is isomorphic to I^{3}(F) and this group measures the nontriviality of the action of F^{×} on the first group.

Remark 2.6. For a different (but related) interpretation of Milnor K-theory in terms of the homology of orthogonal groups see Cathelineau, [Cat04].

We also require the following lemma, essentially due to Suslin, in section 4 below: If G is a subgroup of GL_{r}(F), let G^{0} denote G ∩ SL_{r}(F).

Lemma 2.7. Let G_{1} ⊂ GL_{n}(F) and G_{2} ⊂ GL_{m}(F) be subgroups both of which contain the group, F^{×}, of scalar matrices. Let M ⊂ M(n, m, F) be a vector subspace with the property that G_{1}M = M = MG_{2}. Then the embedding of groups

(\begin{matrix}G_{1} & 0 \\ 0 & G_{2} \end{matrix})^{0} → (\begin{matrix}G_{1} & M \\ 0 & G_{2} \end{matrix})^{0}

Third Homology of SL 7

induces an isomorphism on integral homology. Proof. This is Lemma 9 of [Hut90], which is a modified version of Lemma 1.8 of [Sus87].

□

3. The Hochschild-Serre spectral sequences

For all n ≥ 2, the short exact sequences

\xymatrix{1 \ar[r] & SL_{n}(F) \ar[r] & GL_{n}(F) \ar[r]^{det} & F^{×} \ar[r] & 1}

give Hochschild-Serre spectral sequences of the form

E_{p,q}^{2}(n) = H_{p}(F^{×}, H_{q}(SL_{n}(F), ℤ)) ⇒ H_{p+q}(GL_{n}(F), ℤ)

with differentials

d^{r} : E_{p,q}^{r} → E_{p−r,q+r−1}^{r}.

The edge maps

D_{p} : H_{p}(GL_{n}(F), ℤ) → E_{p,0}^{∞} ⊂ E_{p,0}^{2} = H_{p}(F^{×}, ℤ)

are the maps induced by det : GL_{n}(F) → F^{×}, and thus are surjective since det has a splitting. Thus

E_{p,0}^{2} = E_{p,0}^{3} = ⋯ = E_{p,0}^{∞} for all p

and the differentials d^{r} leaving E_{p,0}^{r} are zero for all r ≥ 2. Of course, the terms E_{p,1}^{r} are zero for all r ≥ 2, p ≥ 0 since H_{1}(SL_{n}(F), ℤ) = 0. We will let

\overline{H}_{q}(SL_{n}(F), ℤ) := E_{0,q}^{∞} = Im(H_{q}(SL_{n}(F), ℤ) → H_{q}(GL_{n}(F), ℤ)).

Thus the E^{∞}-term of the nth spectral sequence has the form

\overline{H}_{4}(SL_{n}(F), ℤ)  ⋯  ⋯  ⋯  ⋯ \overline{H}_{3}(SL_{n}(F), ℤ)  E_{1,3}^{∞}(n)  ⋯  ⋯  ⋯ \overline{H}_{2}(SL_{n}(F), ℤ)  H_{1}(F^{×}, H_{2}(SL_{n}(F), ℤ))  E_{2,2}^{∞}(n)  ⋯  ⋯ 0  0  0  0  0 ℤ  F^{×}  H_{2}(F^{×}, ℤ)  H_{3}(F^{×}, ℤ)  H_{4}(F^{×}, ℤ)

The spectral sequence thus determines a filtration on H_{p}(GL_{n}(F), ℤ), with

F_{p}H_{p}(GL_{n}(F), ℤ) = H_{p}(GL_{n}(F), ℤ)
F_{p−1}H_{p}(GL_{n}(F), ℤ) = F_{p−2}H_{p}(GL_{n}(F), ℤ) = Ker(D_{p})
F_{0}H_{p}(GL_{n}(F), ℤ) = \overline{H}_{p}(SL_{n}(F), ℤ) = E_{0,p}^{∞}.

By Theorem 2.4 (1), the natural maps H_{p}(GL_{n}(F), ℤ) → H_{p}(GL_{n+1}(F), ℤ) are isomorphisms for n ≥ p. Since these maps are compatible with the corresponding maps of Hochschild-Serre spectral sequences, it follows that the induced maps E_{p,q}^{∞}(n) → E_{p,q}^{∞}(n+1) are isomorphisms if p + q ≤ n. In particular, the maps \overline{H}_{3}(SL_{n}(F), ℤ) → \overline{H}_{3}(SL_{n+1}(F), ℤ) are isomorphisms for all n ≥ 3. Furthermore, it follows from the remarks above that the maps E_{p,q}^{r}(n) → E_{p,q}^{r}(n + 1) are isomorphisms for all r ≥ 2 and n ≥ 1 if q ∈ {0, 1}. Observe that the only possible nonzero differential arriving at the (0, 3) position is

d^{2} = d_{2,2}^{2}(n) : E_{2,2}^{2} = H_{2}(F^{×}, H_{2}(SL_{n}(F), ℤ)) → E_{0,3}^{2} = H_{0}(F^{×}, H_{3}(SL_{n}(F), ℤ))

and thus for all n

\overline{H}_{3}(SL_{n}(F), ℤ) = \frac{H_{0}(F^{×}, H_{3}(SL_{n}(F), ℤ))}{Im(d^{2})}.

Lemma 3.1. For all n ≥ 3, E_{0,3}^{∞}(n) = \overline{H}_{3}(SL_{n}(F), ℤ) = H_{0}(F^{×}, H_{3}(SL_{n}(F), ℤ)).

Proof. By the preceding remarks, this amounts to showing that d_{2,2}^{2}(n) = 0 for all n ≥ 3. The natural map H_{p}(SL(F), ℤ) → H_{p}(GL(F), ℤ) is split injective. This follows from letting n go to infinity in the commutative triangle

\xymatrix{H_{p}(SL_{n}(F), ℤ) \ar[rr] \ar[dr] &  & H_{p}(GL_{n}(F), ℤ) \ar[dl]^{α_{p}} \\  & H_{p}(SL_{n+1}(F), ℤ)}

where α_{p} is the map induced on homology by the embedding

α : GL_{n}(F) → SL_{n+1}(F), A ↦ (\begin{matrix}A & 0 \\ 0 & det A^{−1} \end{matrix}) .

It follows that the differential

d^{2} : E_{2,2}^{2} = H_{2}(F^{×}, H_{2}(SL(F), ℤ)) → E_{0,3}^{2} = H_{0}(F^{×}, H_{3}(SL(F), ℤ)).

in the spectral sequence of the extension 1 → SL(F) → GL(F) → F^{×} → 1 is zero, since E_{0,3}^{∞} = E_{0,3}^{2} = H_{3}(SL(F), ℤ). Furthermore, it follows that E_{2,2}^{∞} = Ker(d^{2}) = E_{2,2}^{2} since no other nonzero differentials arrive at or leave this position. By the remarks above, the natural maps E_{p,q}^{∞}(n) → E_{p,q}^{∞} are isomorphisms if either n ≥ p + q or q ≤ 1. In particular we have, for n ≥ 4,

H_{2}(F^{×}, K_{2}^{M}(F)) = E_{2,2}^{2}(n) ⊃ E_{2,2}^{∞}(n) ≅ E_{2,2}^{∞} = H_{2}(F^{×}, K_{2}^{M}(F))

(since H_{2}(SL_{n}(F), ℤ) ≅ H_{2}(SL(F), ℤ) ≅ K_{2}^{M}(F) by 2.4 (4)) so that E_{2,2}^{2}(n) = E_{2,2}^{∞}(n) and hence d_{2,2}^{2}(n) = 0 in the case n ≥ 4. For n = 3 the cokernel of the map H_{4}(GL_{3}(F), ℤ) → H_{4}(GL_{4}(F), ℤ) is K_{4}^{M}(F) by 2.4 (2). By the remarks above, the cokernel of the map F_{2}H_{4}(GL_{3}(F), ℤ) → F_{2}H_{4}(GL_{4}(F), ℤ) is also K_{4}^{M}(F). However, since the composite H_{2}(SL_{2}(F), ℤ) → H_{2}(GL_{2}(F), ℤ) → K_{2}^{M}(F) is surjective, it follows, taking products, that the map

H_{2}(SL_{2}(F), ℤ) × H_{2}(SL_{2}(F), ℤ) → H_{4}(SL_{4}(F), ℤ) → H_{4}(GL_{4}(F), ℤ) → K_{4}^{M}(F)

is composite, and hence that the cokernel of the map F_{0}H_{4}(GL_{3}(F), ℤ) → F_{0}H_{4}(GL_{4}(F), ℤ) maps onto K_{4}^{M}(F). Thus the natural map E_{2,2}^{∞}(3) → E_{2,2}^{∞}(4) = H_{2}(F^{×}, K_{2}^{M}(F)) is surjective. As in the case n ≥ 4, this forces d_{2,2}^{2}(3) = 0.

□

For each n the spectral sequence gives a natural short exact sequence

0 → \overline{H}_{3}(SL_{n}(F), ℤ) → F_{1}H_{3}(GL_{n}(F), ℤ) → H_{1}(F^{×}, H_{2}(SL_{n}(F), ℤ)) → 0.

Now the map of extensions

\xymatrix{1 \ar[r] & SL_{2}(F) \ar[d] \ar[r] & GL_{2}(F) \ar[d] \ar[r]^{det} & F^{×} \ar[r] \ar[d]^{=} & 1 \\ 1 \ar[r] & SL_{3}(F) \ar[r] & GL_{3}(F) \ar[r]^{det} & F^{×} \ar[r] & 1}

induces a map of corresponding Hochschild-Serre spectral sequences and hence a commutative diagram with exact rows and column:

\xymatrix{0 \ar[r] & \overline{H}_{3}(SL_{2}(F), ℤ) \ar[r] \ar[d] & F_{1}H_{3}(GL_{2}(F), ℤ) \ar[r] \ar[d] & H_{1}(F^{×}, H_{2}(SL_{2}(F), ℤ)) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \overline{H}_{3}(SL_{3}(F), ℤ) \ar[r] & F_{1}H_{3}(GL_{3}(F), ℤ) \ar[r] \ar[d] & H_{1}(F^{×}, H_{2}(SL_{3}(F), ℤ)) \ar[r] & 0 \\  &  & K_{3}^{M}(F) \ar[d] \\  &  & 0}

(1)

Note that H_{1}(F^{×}, H_{2}(SL_{3}(F), ℤ)) = H_{1}(F^{×}, K_{2}^{M}(F)) = F^{×} ⊗ K_{2}^{M}(F), and hence there is a natural multiplication map μ : H_{1}(F^{×}, H_{2}(SL_{3}(F), ℤ)) → K_{3}^{M}(F).

Theorem 3.2. The natural inclusion SL_{2}(F) → SL_{3}(F) induces a short exact sequence

\xymatrix{0 \ar[r] & H_{1}(F^{×}, H_{2}(SL_{2}(F), ℤ)) \ar[r] & H_{1}(F^{×}, H_{2}(SL_{3}(F), ℤ)) \ar[r]^{μ} & k_{3}^{M}(F) \ar[r] & 0}

Proof. Recall that there is a natural isomorphism of ℤ[F^{×}]-modules H_{2}(SL_{2}(F), ℤ) ≅ K_{2}^{MW}(F) such that the map H_{2}(SL_{2}(F), ℤ) → H_{2}(SL_{3}(F), ℤ) corresponds to the natural surjection K_{2}^{MW}(F) → K_{2}^{M}(F). Let I = I(F) be the augmentation ideal of the Grothendieck-Witt ring, GW(F), of F. For any n ≥ 0, the exact sequence of ℤ[F^{×}]-modules

0 → I^{n+1} → K_{n}^{MW}(F) → K_{n}^{M}(F) → 0

gives an exact homology sequence

H_{1}(F^{×}, K_{n}^{MW}(F)) → H_{1}(F^{×}, K_{n}^{M}(F)) → H_{0}(F^{×}, I^{n+1}) → H_{0}(F^{×}, K_{n}^{MW}(F)) → K_{n}^{M}(F) → 0.

Now,

H_{0}(F^{×}, K_{n}^{MW}(F)) = \frac{K_{n}^{MW}(F)}{I_{F^{×}} · K_{n}^{MW}(F)} = \frac{K_{n}^{MW}(F)}{I^{n+1}} = K_{n}^{M}(F)

10 KEVIN HUTCHINSON, LIQUN TAO

while

H_{0}(F^{×}, I^{n+1}) = \frac{I^{n+1}}{I_{F^{×}} · I^{n+1}} = \frac{I^{n+1}}{I^{n+2}} ≅ k_{n+1}^{M}(F).

Thus for all n ≥ 0, there is an exact sequence H_{1}(F^{×}, K_{n}^{MW}(F)) → H_{1}(F^{×}, K_{n}^{M}(F)) → k_{n+1}^{M}(F) → 0.

Let A(n) be the statement that the leftmost arrow is an injection. Thus we require, in particular, to prove the statement A(2). For n ≥ 0, let B(n) be the statement that the inclusion I^{n+1} → I^{n} induces the zero map H_{1}(F^{×}, I^{n+1}) → H_{1}(F^{×}, I^{n}). We first show that B(n + 1) ⟹ A(n) for all n ≥ 1: Suppose that B(n + 1) holds. Then the short exact sequence

0 → I^{n+2} → I^{n+1} → k_{n+1}^{M}(F) → 0

gives the exact homology sequence

0 → H_{1}(F^{×}, I^{n+1}) → H_{1}(F^{×}, k_{n+1}^{M}(F)) → k_{n+2}^{M}(F) → 0

(using the fact that H_{0}(F^{×}, I^{n}) = I^{n}/I^{n+1} = k_{n}^{M}(F).) Thus we have

H_{1}(F^{×}, I^{n+1}) ≅ Ker(H_{1}(F^{×}, k_{n+1}^{M}(F)) → k_{n+2}^{M}(F))
= Ker(μ_{n+2} : F^{×} ⊗ k_{n+1}^{M}(F) → k_{n+2}^{M}(F))

where μ = μ_{n+2} denotes the natural multiplication. On the other hand, from the long exact homology sequence associated to 0 → I^{n+1} → K_{n}^{MW}(F) → K_{n}^{M}(F) → 0, A(n) is equivalent to the surjectivity of the connecting homomorphism δ_{n} : H_{2}(F^{×}, K_{n}^{M}(F)) → H_{1}(F^{×}, I^{n+1}). Let ρ_{n} denote the composite H_{2}(F^{×}, K_{n}^{M}(F)) → H_{1}(F^{×}, I^{n+1}) → H_{1}(F^{×}, k_{n+1}^{M}(F)) = F^{×} ⊗ k_{n+1}^{M}(F). Then A(n) holds if and only if the image of ρ_{n} is the kernel of μ_{n+1}. Since K_{n}^{M}(F) is a trivial ℤ[F^{×}]-module, there is a natural homomorphism

⋀^{2} (F^{×}) ⊗ K_{n}^{M}(F) = H_{2}(F^{×}, ℤ) ⊗ K_{n}^{M}(F) → H_{2}(F^{×}, K_{n}^{M}(F))

described at the level of the bar resolution by (a ∧ b) ⊗ z ↦ ([a|b] − [b|a]) ⊗ z.

We calculate the image of (a ∧ b) ⊗ {x_{1}, …, x_{n}} under δ_{n}: As noted this is represented by the element

([a|b] − [b|a]) ⊗ {x_{1}, …, x_{n}} ∈ B_{2}(F^{×}) ⊗_{ℤ[F^{×}]} K_{n}^{M}(F)

which lifts to

([a|b] − [b|a]) ⊗ [x_{1}] ⋯ [x_{n}] ∈ B_{2}(F^{×}) ⊗_{ℤ[F^{×}]} K_{n}^{MW}(F).

This maps, by the boundary homomorphism, to the following element in B_{1}(F^{×})⊗K_{n}^{M}(F):

([b] − [ab] + [a]⟨b⟩ − [a] + [ba] − [b]⟨a⟩) ⊗ [x_{1}] ⋯ [x_{n}]

=  ([a]⟨⟨b⟩⟩ − [b]⟨⟨a⟩⟩) ⊗ [x_{1}] ⋯ [x_{n}]
=  [a] ⊗ ⟨⟨b⟩⟩[x_{1}] ⋯ [x_{n}] − [b] ⊗ ⟨⟨a⟩⟩[x_{1}] ⋯ [x_{n}].

Recall that the inclusion I^{n+1} → K_{n}^{MW}(F) is given by

⟨⟨a_{0}, …, a_{n}⟩⟩ ↦ ⟨⟨a_{0}⟩⟩[a_{1}] ⋯ [a_{n}] = η[a_{0}][a_{1}] ⋯ [a_{n}].

Thus

δ_{n} ((a ∧ b) ⊗ {x_{1}, …, x_{n}}) = [a] ⊗ ⟨⟨b, x_{1}, …, x_{n}⟩⟩ − [b] ⊗ ⟨⟨a, x_{1}, …, x_{n}⟩⟩ ∈ H_{1}(F^{×}, I^{n+1}).

hence

ρ_{n} ((a ∧ b) ⊗ {x_{1}, …, x_{n}}) = a ⊗ {b, x_{1}, …, x_{n}} − b ⊗ {a, x_{1}, …, x_{n}}
= a ⊗ {b, x_{1}, …, x_{n}} + b ⊗ {a, x_{1}, …, x_{n}} ∈ F^{×} ⊗ k_{n+1}^{M}(F).

Since K_{n+2}^{M}(F) is naturally isomorphic to the (n + 2)-nd tensor power of F^{×} modulo the two families of relations

a_{1} ⊗ ⋯ a_{n+1} ⊗ (1 − a_{n+1}) = 0, a_{i} ∈ F^{×}, a_{n+1} ≠ 1
a_{1} ⊗ ⋯ a_{i} ⊗ a_{i+1} ⊗ ⋯ ⊗ a_{n+2} + a_{1} ⊗ ⋯ a_{i+1} ⊗ a_{i} ⊗ ⋯ ⊗ a_{n+2}, a_{i} ∈ F^{×}

it easily follows that Ker(F^{×} ⊗ K_{n+1}^{M}(F) → K_{n+2}^{M}(F)) is generated by the elements

a ⊗ {b, x_{1}, …, x_{n}} + b ⊗ {a, x_{1}, …, x_{n}}.

This shows that the image of ρ_{n} is equal to the kernel of μ_{n+2}, and thus that A(n) holds. Thus, to complete the proof of the theorem, we show that B(n) holds for all n ≥ 1 (and, in particular, that B(3), and hence A(2) holds). For any ℤ[G]-module

H_{1}(G, M) = Ker(I_{G} ⊗_{ℤ[G]} M → I_{G} · M).

Thus

H_{1}(F^{×}, I^{n+1}) = Ker(I_{F^{×}} ⊗_{ℤ[F^{×}]} I^{n+1} → I^{n+2}).

From Lemma 2.1 it easily follows that there is a well-defined homomorphism (for n ≥ 1)

ϕ_{n} : I^{n+2} → I_{F^{×}} ⊗_{ℤ[F^{×}]} I^{n},
⟨⟨a_{0}, …, a_{n+1}⟩⟩ ↦ ⟨⟨a_{0}⟩⟩ ⊗ ⟨⟨a_{1}, …, a_{n+1}⟩⟩.

The essential point here is that relations (a),(c) and (d) are clearly respected, while in I_{F^{×}} ⊗ I^{n} we have

⟨⟨a_{0}⟩⟩ ⊗ ⟨⟨a_{1}, …, a_{n+1}⟩⟩ = ⟨⟨a_{0}⟩⟩ ⊗ ⟨⟨a_{1}⟩⟩ · ⟨⟨a_{2}, …, a_{n+1}⟩⟩
= ⟨⟨a_{0}⟩⟩⟨⟨a_{1}⟩⟩ ⊗ ⟨⟨a_{2}, …, a_{n+1}⟩⟩
= ⟨⟨a_{1}⟩⟩ ⊗ ⟨⟨a_{0}⟩⟩ · ⟨⟨a_{2}, …, a_{n+1}⟩⟩
= ⟨⟨a_{1}⟩⟩ ⊗ ⟨⟨a_{0}, a_{2}, …, a_{n+1}⟩⟩

so that the relation (b) is also satisfied. Since the diagram

\xymatrix{I_{F^{×}} ⊗_{ℤ[F^{×}]} I^{n+1} \ar[rr] \ar@{-->}[dr] &  & I^{n+2} \ar[dl]_{ϕ_{n}} \\  & I_{F^{×}} ⊗_{ℤ[F^{×}]} I^{n}}

commutes, statement B(n) now follows, and the theorem is proven. □

Lemma 3.3. The square

\xymatrix{F_{1}H_{3}(GL_{3}(F), ℤ) \ar[r] \ar[d]^{ϵ} & H_{1}(F^{×}, H_{2}(SL_{3}(F), ℤ)) \ar[d]^{μ} \\ K_{3}^{M}(F) \ar[r] & k_{3}^{M}(F)}

commutes.

Proof. We set ST_{n} := Ker(det : T_{n} → F^{×}) so that T_{n} decomposes as T_{1} × ST_{n}, where T_{1} = {(a, 1, …, 1) | a ∈ F^{×}} ⊂ T_{n}. The inclusion ST_{3} → SL_{3}(F) induces a map

⋀^{2} (ST_{3}) = H_{2}(ST_{3}, ℤ) → H_{2}(SL_{3}(F), ℤ) ≅ K_{2}^{M}(F)

which, by 2.4 (3), is given by

x ∧ y ↦ −{x_{2}, y_{3}} − {x_{3}, y_{2}} − 2 · ({x_{2}, y_{2}} + {x_{3}, y_{3}}) (2)

where x = (x_{1}, x_{2}, x_{3}), y = (y_{1}, y_{2}, y_{3}) with ∏ x_{i} = ∏ y_{i} = 1. The Hochschild-Serre spectral sequence of the extension

1 → ST_{3} → T_{3} → F^{×} → 1

maps to the corresponding spectral sequence for GL_{3}(F). The induced filtration on H_{•}(T_{3}, ℤ) is compatible with the Künneth decomposition of the integral homology of T_{3} = T_{1} × ST_{2} and the related decomposition of ⋀^{•} (T_{3}). In particular we have a map of short exact sequences

\xymatrix{0 \ar[r] & ⋀^{3} (ST_{3}) \ar[r] \ar[d] & F_{1}⋀^{3} (T_{3}) \ar[r] \ar[d] & T_{1} ⊗ ⋀^{2} (ST_{3}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \overline{H}_{3}(SL_{3}(F), ℤ) \ar[r] & F_{1}H_{3}(GL_{3}(F), ℤ) \ar[r] & T_{1} ⊗ H_{2}(SL_{3}(F), ℤ) \ar[r] & 0}

The upper sequence is split by the natural inclusion T_{1} ⊗ ⋀^{2} (ST_{2}) → ⋀^{3} (T_{3}). Using 2.4 (3) again, the composite

T_{1} ⊗ ⋀^{2} (ST_{3}) → F_{1}⋀^{3} (T_{3}) → F_{1}H_{3}(GL_{3}(F), ℤ) → K_{3}^{M}(F)

is given by

(a, 1, 1) ⊗ ((x_{1}, x_{2}, x_{3}) ∧ (y_{1}, y_{2}, y_{3})) ↦ {a, x_{2}, y_{3}} + {a, x_{3}, y_{2}}.

In particular, this composite map is surjective. On the other hand, formula (2) shows that the composite

T_{1} ⊗ ⋀^{2} (ST_{3}) → T_{1} ⊗ H_{2}(SL_{3}(F), ℤ) → K_{3}^{M}(F)

sends (a, 1, 1) ⊗ ((x_{1}, x_{2}, x_{3}) ∧ (y_{1}, y_{2}, y_{3})) to the element

−{a, x_{2}, y_{3}} − {a, x_{3}, y_{2}} − 2 · ({a, x_{2}, y_{2}} + {a, x_{3}, y_{3}}) .

In view of diagram (1), this proves the lemma. □

Corollary 3.4. For any infinite field F, the natural inclusion SL_{2}(F) → SL_{3}(F) induces an exact sequence

\xymatrix{\overline{H}_{3}(SL_{2}(F), ℤ) \ar[r] & \overline{H}_{3}(SL_{3}(F), ℤ) \ar[r]^{ϵ} & 2 · K_{3}^{M}(F) \ar[r] & 0}

Proof. This follows from the snake lemma applied to Diagram (1), Theorem 3.2 and Lemma 3.3.

□

4. Homology Stability for H_{3}(SL_{n}(F), ℤ)

By Lemma 3.1, \overline{H}_{3}(SL_{3}(F), ℤ) = H_{0}(F^{×}, H_{3}(SL_{3}(F), ℤ)). In this section we prove that the natural action of F^{×} on H_{3}(SL_{3}(F), ℤ) is trivial (and hence that \overline{H}_{3}(SL_{3}(F), ℤ) = H_{3}(SL_{3}(F), ℤ)). Let X_{q} = X_{q}(F) be the set of all (q + 1)-tuples (x_{0}, …, x_{q}) of points of ℙ^{2}(F) in general position (in the sense that no three of the x_{i} are collinear in ℙ^{2}(F)). Thus X_{q} is naturally a GL_{3}(F)-set. We let C_{q} = C_{q}(F) := ℤ[X_{q}] be the corresponding permutation module. Let d : C_{q} → C_{q−1} be the standard simplicial boundary map

(x_{0}, …, x_{q}) ↦ ∑_{i=0}^{q}(−1)^{i}(x_{0}, …, x̂_{i}, …, x_{q}).

The natural augmentation ϵ : C_{•} → ℤ gives a resolution of ℤ by GL_{3}(F)-modules; i.e., ϵ is a weak equivalence of complexes of ℤ [GL_{3}(F)]-modules. (This is standard: see, for example, section 1 of [Hut90].) Restricting the action to SL_{3}(F), there is a resulting spectral sequence of the form

E_{p,q}^{1} = H_{p}(SL_{3}(F), C_{q}) ⟹ H_{p+q}(SL_{3}(F), ℤ)

with differentials d^{r} : E_{p,q}^{r} → E_{p+r−1,q−r}^{r}. The differential d^{1} : E_{p,q}^{1} → E_{p,q−1}^{1} is the map induced on homology by the boundary map d. Since the C_{q} are permutation modules, the terms E_{p,q}^{1} and the differentials d^{1} can be explicitly computed: We recall the following basic principles (see, for example, [Hut90]): If G is a group and if X is a G-set, then Shapiro’s Lemma says that

H_{p}(G, ℤ[X]) ≅ ⨁_{y∈X/G} H_{p}(G_{y}, ℤ),

the isomorphism being induced by the maps

H_{p}(G_{y}, ℤ) → H_{p}(G, ℤ[X])

described at the level of chains by

B_{p} ⊗_{ℤ[G_{y}]} ℤ → B_{p} ⊗_{ℤ[G]} ℤ[X], z ⊗ 1 ↦ z ⊗ y.

Let X_{i}, i = 1, 2 be transitive G-sets. Let x_{i} ∈ X_{i} and let H_{i} be the stabiliser of x_{i}, i = 1, 2. Let ϕ : ℤ[X_{1}] → ℤ[X_{2}] be a map of ℤ[G]-modules with

ϕ(x_{1}) = ∑_{g∈G/H_{2}} n_{g}gx_{2}, with n_{g} ∈ ℤ.

Then the induced map ϕ_{•} : H_{•}(H_{1}, ℤ) → H_{•}(H_{2}, ℤ) is given by the formula

ϕ_{•}(z) = ∑_{g∈H_{1}\G/H_{2}} n_{g}cor_{g^{−1}H_{1}g∩H_{2}}^{H_{2}}res_{g^{−1}H_{1}g∩H_{2}}^{g^{−1}H_{1}g} (g^{−1} · z) (3)

There is an obvious extension of this formula to non-transitive G-sets.

4.1. The case q = 0: We have X_{0} = ℙ^{2}(F). SL_{3}(F) acts transitively on X_{0} and the stabiliser of ẽ_{1} is the group

S_{0} := \{( \begin{matrix}det(A)^{−1} & ∗ & ∗ \\ 0 & a & b \\ 0 & c & d \end{matrix} ) | A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(F)\} .

By Lemma 2.7 above, the natural inclusion

GL_{2}(F) → S_{0}, A ↦ ( \begin{matrix}det A^{−1} & 0 \\ 0 & A \end{matrix} )

induces a homology isomorphism H_{p}(GL_{2}(F), ℤ) ≅ H_{p}(S_{0}, ℤ), and thus

E_{p,0}^{1} = H_{p}(SL_{3}(F), C_{0}) ≅ H_{p}(S_{0}, ℤ) ≅ H_{p}(GL_{2}(F), ℤ).

The edge homomorphisms

H_{p}(GL_{2}(F), ℤ) = E_{p,0}^{1} → E_{p,0}^{∞} → H_{p}(SL_{3}(F), ℤ)

are the maps on integral homology induced by the natural inclusion

GL_{2}(F) → SL_{3}(F), A ↦ ( \begin{matrix}det A^{−1} & 0 \\ 0 & A \end{matrix} )

and so E_{p,0}^{∞} is isomorphic to the image of H_{p}(GL_{2}(F), ℤ) in H_{p}(SL_{3}(F), ℤ).

4.2. The case q = 1: X_{1} consists of pairs (x_{0}, x_{1}) of distinct points of ℙ^{2}(F). The action of SL_{3}(F) is again transitive and the stabiliser of (ẽ_{1}, ẽ_{2}) is the group

S_{1} := \{( \begin{matrix}a & 0 & ∗ \\ 0 & b & ∗ \\ 0 & 0 & c \end{matrix} ) | abc = 1 in F^{×} \} .

By Lemma 2.7 again, the inclusion ST_{3} → S_{1} induces an isomorphism on integral homology and so

E_{p,1}^{1} = H_{p}(SL_{3}(F), C_{1}) ≅ H_{p}(S_{1}, ℤ) ≅ H_{p}(ST_{3}, ℤ) ≅ H_{p}(F^{×} × F^{×}, ℤ).

4.3. The case q = 2: X_{2} consists of ‘triangles’ – triples of non-collinear points – in ℙ^{2}(F). Again, the action of SL_{3}(F) is transitive, and the stabiliser of (ẽ_{1}, ẽ_{2}, ẽ_{3}) is the group of diagonal matrices, ST_{3}. Thus

E_{p,2}^{1} = H_{p}(SL_{3}(F), C_{2}) ≅ H_{p}(ST_{3}, ℤ) ≅ H_{p}(F^{×} × F^{×}, ℤ).

Third Homology of SL 15

4.4. The case q = 3: X_{3} consists of 4-tuples, (x_{0}, x_{1}, x_{2}, x_{3}) of points in ℙ^{2}(F) in general position.

Lemma 4.1. Let T̃_{3} be the multiplicative group {(a, b, c) ∈ F^{3} | abc ≠ 0}. Let D_{3} ⊂ T̃_{3} be the subgroup generated by {(a, b, c) | abc = 1} and Δ(F^{×}) := {(a, a, a) | a ∈ F^{×}}. Then the ‘determinant map’ \overline{det} : T̃_{3} → F^{×}/(F^{×})^{3}, (a, b, c) ↦ abc (mod (F^{×})^{3}) induces an isomorphism T̃_{3}/D_{3} ≅ F^{×}/(F^{×})^{3}. Lemma 4.2. Let x = (x_{0}, x_{1}, x_{2}, x_{3}) ∈ X_{3}, and choose v_{i} ∈ F^{3} \ {0} with x_{i} = ṽ_{i}. Let

V := (v_{0}|v_{1}|v_{2}) ∈ GL_{3}(F).

Then V^{−1}v_{3} ∈ T̃_{3}. Let

λ(x) := \overline{det}(V^{−1}v_{3}) · det V ∈ F^{×}/(F^{×})^{3}.

Then λ(x) is well-defined (i.e. independent of the choice of the v_{i}) and x, y ∈ X_{3} are in the same SL_{3}(F)-orbit if and only if λ(x) = λ(y).

Proof. Observe first that (ẽ_{1}, ẽ_{2}, ẽ_{3}, ṽ) ∈ X_{3} if and only if v ∈ T̃_{3}. Since the stabiliser of (ẽ_{1}, ẽ_{2}, ẽ_{3}) is ST_{3} ⊂ SL_{3}(F), it follows from Lemma 4.1 that (ẽ_{1}, ẽ_{2}, ẽ_{3}, ṽ) and (ẽ_{1}, ẽ_{2}, ẽ_{3}, w̃) are in the same SL_{3}(F)-orbit if and only if \overline{det}(v) = \overline{det}(w). Since SL_{3}(F) acts transitively on X_{2}, it follows that there exists A ∈ SL_{3}(F) with

A · (ẽ_{1}, ẽ_{2}, ẽ_{3}, w̃) = x = (ṽ_{0}, ṽ_{1}, ṽ_{2}, ṽ_{3}).

Thus there exists c_{1}, c_{2}, c_{3} ∈ F^{×} with Ae_{i} = c_{i}v_{i−1} for i = 1, 2, 3 It follows that

A = [c_{1}v_{0}|c_{2}v_{1}|c_{3}v_{2}] = V · (\begin{matrix}c_{1} & 0 & 0 \\ 0 & c_{2} & 0 \\ 0 & 0 & c_{3} \end{matrix})

. Furthermore, c_{1}c_{2}c_{3} = 1/ det(V) since det(A) = 1. We have Aw = cv_{3} for some c ∈ F^{×}. Thus

V · (\begin{matrix}c_{1} & 0 & 0 \\ 0 & c_{2} & 0 \\ 0 & 0 & c_{3} \end{matrix}) w = cv_{3} ⟹ (\begin{matrix}c_{1} & 0 & 0 \\ 0 & c_{2} & 0 \\ 0 & 0 & c_{3} \end{matrix}) w = c(V^{−1}v_{3}) ∈ T̃_{3}.

Applying \overline{det} to this now gives

(c_{1}c_{2}c_{3})\overline{det}(w) = c^{3}\overline{det}(V^{−1}v_{3}) ⟹ \overline{det}(w) = \overline{det}(v^{−1}v_{3}) det(V) ∈ F^{×}/(F^{×})^{3}. □

It follows that the SL_{3}(F)-orbits of X_{3} are indexed by F^{×}/(F^{×})^{3}, where ā ∈ F^{×}/(F^{×})^{3} corresponds to the orbit of the element

[ā] := (ẽ_{1}, ẽ_{2}, ẽ_{3}, [1, 1, a]).

The stabiliser of [ā] is Δ(F^{×}) ∩ SL_{3}(F) = μ_{3}(F). Thus

E_{p,3}^{1} = H_{p}(SL_{3}(F), X_{3}) ≅ ⨁_{ā∈F^{×}/(F^{×})^{3}} H_{p}(μ_{3}(F), ℤ) · [ā].

In particular, if ζ_{3} ∉ F then E_{p,3}^{1} = 0 for p > 0 and, in any case,

E_{0,3}^{1} ≅ ⨁_{ā∈F^{×}/(F^{×})^{3}} ℤ · [ā] = ℤ[F^{×}/(F^{×})^{3}].

4.5. The case q = 4: Given a, b, c, d ∈ F^{×} satisfying (b − a)(d − c)(ad − bc) ≠ 0, then

∥\begin{matrix}a & b \\ c & d \end{matrix}∥ := (ẽ_{1}, ẽ_{2}, ẽ_{3}, [1, a, c], [1, b, d]) ∈ X_{4}.

Let Y_{4} be the set of all such 5-tuples. Every element of X_{4} belongs to the SL_{3}(F)-orbit of an element of this type. Furthermore, two elements

∥\begin{matrix}a & b \\ c & d \end{matrix}∥ , ∥\begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix}∥ ∈ Y_{4}

belong to the same SL_{3}(F)-orbit if and only if there exist r, s ∈ F^{×} satisfying

rs ∈ (F^{×})^{3},  \frac{a′}{a} = \frac{b′}{b} = r,  \frac{c′}{c} = \frac{d′}{d} = s.

Thus, letting Z_{4} equal Y_{4} modulo this equivalence relation we have

E_{p,4}^{1} = ⨁_{∥\begin{matrix}a & b \\ c & d \end{matrix}∥∈Z_{4}} H_{p}(μ_{3}(F), ℤ) · ∥\begin{matrix}a & b \\ c & d \end{matrix}∥.

Thus the E^{1}-term of the spectral sequence has the form

⨁_{Z_{4}} ℤ · ∥\begin{matrix}a & b \\ c & d \end{matrix}∥  ⋯  ⋯  ⋯ ℤ[F^{×}/(F^{×})^{3}]  ⋯  ⋯  ⋯ ℤ  ST_{3}  H_{2}(ST_{3}, ℤ)  ⋯ ℤ  ST_{3}  H_{2}(ST_{3}, ℤ)  H_{3}(ST_{3}, ℤ) ℤ  F^{×}  H_{2}(GL_{2}(F), ℤ)  H_{3}(GL_{2}(F), ℤ)

4.6. Some E^{∞}-terms. To compute H_{3}(SL_{3}(F), ℤ), we need to calculate the terms E_{p,q}^{∞} with p + q = 3. As already observed E_{3,0}^{∞} is the image of H_{3}(GL_{2}(F), ℤ). We will show that the other E^{∞}-terms on the line p + q = 3 are 0. We have E_{0,3}^{∞} = E_{0,3}^{2} = Coker(d^{1}) : E_{0,4}^{1} → E_{0,3}^{1}.

Lemma 4.3. The map

d^{1} : ⨁ ℤ · ∥\begin{matrix}a & b \\ c & d \end{matrix}∥ = E_{0,4}^{1} → E_{0,3}^{1} = ⨁_{a∈F^{×}/(F^{×})^{3}} ℤ · [a]

is surjective.

Proof.

d^{1} (∥\begin{matrix}a & b \\ c & d \end{matrix}∥) = d^{1}(ẽ_{1}, ẽ_{2}, ẽ_{3}, [1, a, c], [1, b, d])
= λ(ẽ_{2}, ẽ_{3}, [1, a, c], [1, b, d]) − λ(ẽ_{1}, ẽ_{3}, [1, a, c], [1, b, d])
+λ(ẽ_{1}, ẽ_{2}, [1, a, c], [1, b, d]) − λ(ẽ_{1}, ẽ_{2}, ẽ_{3}, [1, b, d]) + λ(ẽ_{1}, ẽ_{2}, ẽ_{3}, [1, a, c])
= [(b − a)(d − c)] − [ab(a − b)(ad − bc)] + [(c − d)(ad − bc)cd] − [bd] + [ac].

It follows, in particular, that if b^{2} ≠ a^{2}, then

d^{1} (∥\begin{matrix}a & b \\ b & a \end{matrix}∥) = [(b − a)^{2}] = [(b − a)^{−1}]

so that d^{1} is surjective as claimed (and E_{0,3}^{2} = 0). □

Lemma 4.4. The spectral sequence admits an F^{×}-action compatible with the action of F^{×} on H_{n}(SL_{3}(F), ℤ). The induced F^{×}-action on the terms E_{p,1}^{r} and E_{p,2}^{r} is trivial for all p and all r ≥ 1.

Proof. Let δ : F^{×} → GL_{3}(F) be the map a ↦ diag(1, 1, a) (a splitting of the determinant map). Recall that the spectral sequence E_{•,•}^{r}(3) is the second spectral sequence associated to the double complex

E_{p,q}^{0}(3) := B_{p}(SL_{3}(F)) ⊗_{ℤ[SL_{3}(F)]} C_{q}.

Let F^{×} act on E_{p,q}^{0} 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 makes E_{•,•}^{0} into a double complex of ℤ[F^{×}]-modules and the induced actions on E_{p,q}^{1} = H_{p}(SL_{3}(F), C_{q}) are the natural actions derived from the GL_{3}(F)-action on C_{q} and the short exact sequence 1 → SL_{3}(F) → GL_{3}(F) → F^{×} → 1. At the level of chains, the isomorphisms H_{p}(ST_{3}, ℤ) ≅ H_{p}(S_{q}, ℤ) ≅ H_{p}(SL_{3}(F), C_{q}) for q = 1, 2 are given by

B_{p}(ST_{3}) ⊗_{ℤ[ST_{3}]} ℤ → B_{p}(S_{q}) ⊗_{ℤ[S_{q}]} ℤ → B_{p}(SL_{3}(F)) ⊗_{ℤ[SL_{3}(F)]} C_{q}
[g_{1}| ⋯ |g_{p}] ⊗ 1  ↦  [g_{1}| ⋯ |g_{p}] ⊗ e(q)

where e(1) := ẽ_{1} and e(2) := (ẽ_{1}, ẽ_{2}). If the domain here is endowed with the trivial ℤ[F^{×}]-module structure, then this is a module homomorphism, since δ(F^{×}) centralizes ST_{3} and stabilizes the e(q), q = 1, 2. □

Corollary 4.5. There is an exact sequence of ℤ[F^{×}]-modules

H_{3}(GL_{2}(F), ℤ) → H_{3}(SL_{3}(F), ℤ) → Q → 0

where I_{F^{×}}^{2}Q = 0.

Proof. Since E_{0,3}^{∞} = 0 and E_{3,0}^{∞} is the image of H_{3}(GL_{2}(F), ℤ), the spectral sequence yields an exact sequence of of the type described where Q fits into a short exact sequence (of ℤ[F^{×}]-modules)

0 → E_{2,1}^{∞} → Q → E_{1,2}^{∞} → 0.

By Lemma 4.4, I_{F^{×}}E_{2,1}^{∞} = I_{F^{×}}E_{1,2}^{∞} = 0 and thus I_{F^{×}}^{2}Q = 0. □

Lemma 4.6. For all infinite fields F, H_{0}(F^{×}, H_{3}(SL_{3}(F), ℤ)) = H_{3}(SL_{3}(F), ℤ).

Proof. This amounts to showing that the natural action of F^{×} on H_{3}(SL_{3}(F), ℤ) is trivial. Since the action of F^{×} on H_{3}(GL_{2}(F), ℤ) is trivial, it is enough to prove that Q = 0. Now

\overline{H}_{3}(SL_{3}(F), ℤ) ≅ H_{0}(F^{×}, H_{3}(SL_{3}(F), ℤ)) = \frac{H_{3}(SL_{3}(F), ℤ)}{I_{F^{×}} · H_{3}(SL_{3}(F), ℤ)}.

Observe that the composite ⋀^{3} (ST_{3}) → H_{3}(GL_{2}(F), ℤ) → \overline{H}_{3}(SL_{3}(F), ℤ) → 2K_{3}^{M}(F) is surjective, since for example the element (a, 1, a^{−1}) ∧ (b, b^{−1}, 1) ∧ (c, c^{−1}, 1) maps to {a^{−1}, b^{−1}, c} + {a^{−1}, b, c^{−1}} = −2{a, b, c}. Since the image of β : H_{3}(GL_{2}(F), ℤ) → \overline{H}_{3}(SL_{3}(F), ℤ) also contains the image of H_{3}(SL_{2}(F), ℤ) → \overline{H}_{3}(SL_{3}(F), ℤ), β is surjective by Theorem 3.2. In the diagram

\xymatrix{ & 0 \ar[d] \\  & I_{F^{×}} · H_{3}(SL_{3}(F), ℤ) \ar[d] \ar[dr]^{α} \\ H_{3}(GL_{2}(F), ℤ) \ar[r] \ar[dr]^{β} & H_{3}(SL_{3}(F), ℤ) \ar[r] \ar[d] & Q \ar[r] & 0 \\  & \overline{H}_{3}(SL_{3}(F), ℤ) \ar[d] \\  & 0}

β is surjective and hence so is α. It follows that Q = I_{F^{×}}Q. Thus Q = I_{F^{×}}^{2}Q = 0 using Corollary 4.5. □

Theorem 4.7. If F is an infinite field then

H_{3}(SL_{3}(F), ℤ) ≅ H_{3}(SL_{4}(F), ℤ) ≅ ⋯ ≅ H_{3}(SL(F), ℤ)

and there is a natural exact sequence

H_{3}(SL_{2}(F), ℤ) → H_{3}(SL_{3}(F), ℤ) → 2K_{3}^{M}(F) → 0.

Proof. In view of Lemma 3.1, Corollary 3.4 and Lemma 4.6 it remains to show that the action of F^{×} on H_{3}(SL_{n}(F), ℤ) is trivial for all n ≥ 4. This can be accomplished in the same way as the case n = 3, but the computations are simpler: Consider the spectral sequence associated to the action of SL_{n}(F) on the complex C_{•}(n) where C_{q}(n) has as a basis the set of (q + 1)-tuples of points of ℙ^{n−1}(F) in general position. The E^{∞} terms on the base are the image of the homology of GL_{n−1}(F) as before. It is now easy to show by direct computation, using formula (3) above, that the terms E_{0,3}^{2}, E_{1,2}^{2} and E_{2,1}^{2} are all zero (this is not the case when n = 3). Alternatively one can show as above that F^{×} acts trivially on E_{1,2}^{1} and E_{2,1}^{1} and argue as in the case n = 3. □

5. Concluding Remarks: Indecomposable K_{3}.

We let K_{•}(F) be the Quillen K-theory of the field F. There is a natural homomorphism of graded rings K_{•}^{M}(F) → K_{•}(F), which is an isomorphism in dimensions at most 2. Let K̄_{•}^{M}(F) denote the image of this homomorphism.The kernel of the surjective homomorphism K_{3}^{M}(F) → K̄_{3}^{M}(F) is known to be 2-torsion, and it is an open question whether this homomorphism is an isomorphism. The indecomposable K_{3} of the field F is the group K_{3}(F)^{ind} := K_{3}(F)/K̄_{3}^{M}(F). Suslin has shown that the Hurewicz homomorphism of algebraic K-theory yields a surjective homomorphism K_{3}(F) → H_{3}(SL(F), ℤ) whose kernel is {−1} · K_{2}(F) ⊂ K̄_{3}^{M}(F) ([Sus90], Corollary 5.2 or see Knudson, [Knu01]). It follows that there is an induced surjective map H_{3}(SL_{3}(F), ℤ) = H_{3}(SL(F), ℤ) → K_{3}(F)^{ind}. In view of the results above, we have

Lemma 5.1. For any infinite field F, the natural homomorphism H_{3}(SL_{2}(F), ℤ) → K_{3}(F)^{ind} is surjective.

Proof. We have the following diagram (defining the maps γ and δ) with exact row and column

\xymatrix{ & K_{3}^{M}(F) \ar[d] \ar[dr]^{γ} \\ H_{3}(SL_{2}(F), ℤ) \ar[r] \ar[dr]^{δ} & K_{3}(F)/(\{−1\} · K_{2}(F)) \ar[r] \ar[d] & 2K_{3}^{M}(F) \ar[r] & 0 \\  & K_{3}(F)^{ind} \ar[d] \\  & 0}

Now Suslin has shown that the composite

\xymatrix{K_{3}^{M}(F) \ar[r] & K_{3}(F) \ar[r]^{H} & H_{3}(GL(F), ℤ) \ar[r] & K_{3}^{M}(F)}

(where H is the Hurewicz homomorphism) is just multplication by 2 ([Sus84], Corollary 4.4). Since, by the arguments above, the diagram

\xymatrix{K_{3}(F) \ar[r] \ar[dr]^{H} & H_{3}(SL(F), ℤ) \ar[r] \ar[d] & 2K_{3}^{M}(F) \ar[d] \\  & H_{3}(GL(F), ℤ) \ar[r] & K_{3}^{M}(F)}

commutes, it follows that γ is multiplication by 2 and so is surjective. This implies the surjectivity of δ. □

Regarding the kernel of the map H_{3}(SL_{2}(F), ℤ) → K_{3}(F)^{ind}, Suslin has asked:

Is it the case that H_{0}(F^{×}, H_{3}(SL_{2}(F), ℤ)) coincides with K_{3}(F)^{ind} for all (infinite) fields F? (see for example Sah, [Sah89]).

In [Sah89], Sah proves that the answer is affirmative when F satisfies (F^{×})^{6} = F^{×}, and also for F = ℝ. More generally, he states that the natural map H_{0}(F^{×}, H_{3}(SL_{2}(F), ℤ)) → K_{3}(F)^{ind} is an isomorphism modulo the Serre class of torsion abelian groups annihilated by a power of 6. Since then, P. Elbaz-Vincent, [EV98], proved that for any semi-local ring R with infinite residue fields there is an isomorphism

K_{3}(R)^{ind} ⊗ ℚ ≅ H_{0}(R^{×}, H_{3}(SL_{2}(R), ℚ)).

More recently, B. Mirzaii, [Mir07], has improved on Sah’s result by showing that if F is an infinite field, then

K_{3}(F)^{ind} ⊗ ℤ[1/2] ≅ H_{0}(F^{×}, H_{3}(SL_{2}(F), ℤ[1/2]))

and that if F^{×} is 2-divisible, then

K_{3}(F)^{ind} ≅ H_{3}(SL_{2}(F), ℤ).

In view of the results above, we note that an affirmative answer to Suslin’s question for a given infinite field F is equivalent to the truth of the following three statements:

(i) The image of H_{3}(SL_{2}(F), ℤ) in H_{3}(SL_{3}(F), ℤ) is isomorphic to K_{3}(F)^{ind}.

(ii) The map H_{3}(GL_{2}(F), ℤ) → H_{3}(GL_{3}(F), ℤ) is injective, and (iii) The differential d_{2,2}^{2}(2) is zero.

6. Acknowledgements

The work in this article was partially funded by the Science Foundation Ireland Research Frontiers Programme grant 05/RFP/MAT0022.

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School of Mathematical Sciences, University College Dublin

E-mail address: kevin.hutchinson@ucd.ie,  lqtao@ucd.ie