Tautological characteristic classes II: the Witt class

Let $K$ be an arbitrary infinite field. The cohomology group $H^2(SL(2,K), H_2\,SL(2,K))$ contains the class of the universal central extension. When studying representations of fundamental groups of surfaces in $SL(2,K)$ it is useful to have classes stable under deformations (Fenchel--Nielsen twists) of representations. We identify the maximal quotient of the universal class which is stable under twists as the Witt class of Nekovar. The Milnor--Wood inequality asserts that an $SL(2,{\bf R})$-bundle over a surface of genus $g$ admits a flat structure if and only if its Euler number is $\leq (g-1)$. We establish an analog of this inequality, and a saturation result for the Witt class. The result is sharp for the field of rationals, but not sharp in general.

Theorem B (Theorem 11.6) Let K be an infinite field.(a) The Witt class of any flat SL(2, K)-bundle over an oriented closed surface of genus g has norm ≤ 4(g − 1) + 2. (b) The set of Witt classes of flat SL(2, K)-bundles over an oriented closed surface of genus g contains the set of elements of I 2 (K) of norm ≤ 4(g − 1).
The form of Milnor's inequality we established for general fields is not sharp.An example of nonsharpness is constructed over the field of Laurent series with rational coefficients.But for K = Q we have the sharp result: Theorem C (Theorem 12.2) The set of Witt classes of all representations of π 1 (Σ g ) in SL(2, Q) is equal to the set of elements of I 2 Q with norm ≤ 4(g − 1).
To prove Theorem B we use arithmetic properties of Markov surfaces, established in [GMS], to construct the required representations.The proof of Theorem C uses the classical Milnor-Wood inequality and Meyer's even more classical theorem from the theory of quadratic forms over Q.
The Witt class can be constructed for P SL(2, K), but, in general, it is not equicommutative for that group: that case requires further study.
The paper is divided into four parts, each with its own introduction.
We would like to thank the anonymous referee for numerous helpful suggestions.

I. Moore and Witt classes.
In this part we present the main two protagonists: the Witt and Moore classes.Both are constructed as tautological cohomology classes in the sense of [DJ].
1. Tautological construction of the Witt class.
The Witt class was first defined by Nekovář (cf.[Ne]).It is a cohomology class w ∈ H 2 (SL(2, K), W (K)), where K is an infinite field and W (K) is the Witt group of symmetric bilinear forms over K.A tautological construction of this class is given in [DJ,Section 7].We briefly recall this construction now.Later, in Section 9, we explain how Nekovář modified the class w to w I ∈ H 2 (SL(2, K), I 2 (K)).In this section G = SL(2, K).
The group G acts on P 1 (K).The infinite simplex with vertex set P 1 (K) is a contractible G-simplicial complex that we denote by X.It carries a G-invariant tautological cocycle T , defined as follows.First, we define a 2-cochain on X with values in C 2 X (the chain group of X with integer coefficients) by assigning to a 2-simplex this same 2-simplex treated as an element of C 2 X.This cochain is not closed-we force it to become closed by applying to its coefficient group the quotient map C 2 X → C 2 X/B 2 X.The result is closed (is a cocycle), but it is not G-invariant.We force it to become G-invariant by passing to G-coinvariants, i.e. by applying to it another quotient map C 2 X/B 2 X → (C 2 X/B 2 X) G .This finally gives T .Of course, one fears that the quotient group (C 2 X/B 2 X) G is trivial; however, quite miraculously, it turns out to be isomorphic to W (K), the additive group of the Witt ring.(A basic discussion of the Witt ring can be found in [EKM, Chapter I].) The cocycle T can be pulled back to G via (any) orbit map.In more detail, for any x ∈ P 1 (K) we consider the W (K)-valued 2-cocycle on G defined by (1.1) (g 0 , g 1 , g 2 ) → T (g 0 x, g 1 x, g 2 x).
(If (g 0 x, g 1 x, g 2 x) is a degenerate simplex in X, the right hand side is interpreted as zero.)The cohomology class w ∈ H 2 (SL(2, K), W (K)) of this cocycle does not depend on the choice of x-this is the Witt class.
To be more explicit we recall the standard Witt group notation: for a ∈ K * (:= K \ {0}) we denote by [a] the element of W (K) represented by the 1-dimensional form ax 2 .The symbol [0] is interpreted as 0. The Witt class is represented by the following (homogeneous) cocycle: here v is any non-zero vector in K 2 , and |g i v, g j v| stands for the determinant of the pair of vectors (g i v, g j v).
The cocycle depends on v, but its cohomology class does not.In the non-homogeneous setting we obtain the following cocycle representing the Witt class: ( (In this formula a 21 denotes the 21-entry of the matrix a.)This explicit formula will be very useful later.
For details on all the claims made above we again refer the reader to [DJ], especially to Section 7 therein.
2. Tautological construction of the Moore class.
We will construct a tautological class starting from the action of a group G on the standard model of EG (cf.[Hatcher,Example 1.B.7]).This model is a ∆-complex with n-simplices given by (n + 1)-tuples [g 0 , g 1 , . . ., g n ] of elements of G.The G-action is g[g 0 , g 1 , . . ., g n ] = [gg 0 , gg 1 , . . ., gg n ].The quotient, BG, has n-simplices given by the orbits of G on the set of n-simplices of EG.The standard notation is: [g 1 |g 2 | . . .|g n ] for the orbit of [1, g 1 , g 1 g 2 , . . ., g 1 g 2 . . .g n ].In particular, in BG we have: one vertex []; a loop [g] for each g ∈ G; a triangle [g|h] for every pair (g, h) ∈ G × G, with boundary glued to the edges [g], [h], [gh].
The tautological n-cochain for the G-action on EG assigns to an n-simplex of EG this same simplex treated as an element of C n EG.We turn this cochain into a cocycle by dividing the coefficient group by B n EG-this produces a tautological cocycle in Z n (EG, C n EG/B n EG).We make this cocycle G-invariant by passing to the G-coinvariants U n of the coefficients: The resulting G-invariant cocycle descends to an element T ∈ Z n (BG, U n ).We also get a cohomology However, the values of the cocycle T are usually not contained in this subgroup.Now we specialize to the case of a perfect group G and to n = 2. Recall that G is perfect if it has trivial abelianisation, H 1 G = 0. (We denote by H n G the homology group with integer coefficients, In this special case, the value T ([g|h]) is the class of [g|h] in U 2 , and which is never zero, so that T ([g|h]) ∈ H 2 G .We will show, however, that T is cohomologous to an by the coefficients inclusion).However, there is at most one cohomology class in H 2 (G, H 2 G) with this property: indeed, from the Bockstein sequence we see that the map ι is injective (due to G being perfect, H 1 (G, * ) = 0).It follows that the cohomology class of T − δn in H 2 (G, H 2 G) does not depend on the choice of n.
For a perfect group G, the class u The image of the Moore class under this isomorphism is id H2G .Indeed, for every homology class This property can serve as another (more standard) definition of the Moore class-a point of view that will reappear in Section 6.

II. Central extensions and characteristic classes.
Every cohomology class τ ∈ H 2 (G, U ) corresponds to a central extension G of G with kernel U .This extension can be used to study τ considered as a characteristic class (i.e.evaluated on G-bundles).The first (that we know of) instance of this sort of study is Milnor's paper [Mil].In Section 3 we recall what it means to evaluate τ on a bundle P , and how Milnor expressed the result τ (P ) in term of the lifts to G of the monodromies of P .In Section 4 we use Milnor's expression to prove several formulae computing τ (P ) for a bundle P over a surface in terms of restrictions of P to subsurfaces.This will be used crucially in Section 11.In Section 5 we discuss twists-natural operations that change bundles.We compute how τ of a bundle changes under twists, and derive an algebraic condition on the corresponding central extension G → G that is equivalent to twist-invariance of τ .We call this algebraic condition equicommutativity.In Section 6 we discuss the Moore class again, this time as the universal class; this allows us to prove that for perfect G there exists a universal twist-invariant class.

Characteristic class in terms of monodromies.
In this section G is an arbitrary group, and U is an abelian group.We consider a class τ ∈ H 2 (G, U ).We now recall how this class can be regarded as a characteristic class.Any G-bundle P over a space B has a classifying map, i.e. a map B → BG (unique up to homotopy) such that the pull-back via this map of the universal G-bundle EG → BG is isomorphic to P .The pull-back of τ via the classifying map yields an element τ (P ) ∈ H 2 (B, U )-the characteristic class (corresponding to τ ) of the bundle P .We will be interested in the more specific situation when the base B is a closed surface Σ = Σ g of genus g ≥ 1.The class τ (P ) can then be evaluated on the fundamental class [Σ] to yield an element of U .Evaluation on [Σ] defines an isomorphism H 2 (Σ, U ) → U (e.g. by the universal coefficient theorem), so that there is no loss of information in passing from τ (P ) to τ (P ), [Σ] .
Let P be a G-bundle over the surface Σ, and let τ ∈ H 2 (G, U ). Choose loops a 1 , b 1 , . . ., a g , b g based at the same point that cut Σ into a 4g-gon and generate π 1 (Σ) with the standard presentation ( where, for g ∈ G, we denote by g a lift of g to the central extension G τ of G determined by τ . The central extension G τ mentioned in the lemma can be described as follows.Let τ be represented by a homogeneous cocycle z: G × G × G → U .The associated non-homogeneous cocycle is given by c(g, h) = z(1, g, gh).Then, on the set G × U , we define the multiplication by The cocycle condition is equivalent to associativity.We will typically use the standard (set-theoretic) lift of G to G τ : g = (g, 1).The abelian group U (={1} × U ) is contained in the centre of G τ .To prove this one checks that c(g, 1) = c(1, g) (by setting g = h in the cocycle condition c(g, 1)c(g We will abbreviate (1, u) to u, and use multiplicative notation in U .(Eventually, for U = W (K), we will switch to the additive convention.) Conversely, for any central extension 1 → U → G → G → 1 we may choose a set theoretic lift G ∋ g → g ∈ G and define c: G × G → U by g • h = gh • c(g, h).A change of the lift changes c within its cohomology class.We will always assume that the lift of the neutral element of G is the neutral element of G (more obscurely: 1 = 1).This assumption implies that c(1, g) = c(g, 1) = 1 for all g ∈ G.
A more thorough discussion of central extensions can be found in [Brown, Chapter IV].
The surface Σ can be expressed as a (convex) polygon Q, with 4g sides suitably glued in pairs.We label the vertices of Q by 0, 1, . . ., 4g − 1 (counterclockwise) and the edges (starting by (0, 1) and continuing counterclockwise) by Figure 1: The polygon Q with labels for g = 2.The ∆-complex structure on Σ is determined by the arrows.The fundamental cycle of Σ is the sum of the triangles with the indicated signs.The map We also put a ∆-complex structure (cf.[Hatcher,Section 2.1]) on Σ.We divide Q into triangles, drawing line segments (0, i) for i = 2, 3 . . ., 4g − 2, as shown in Figure 1.Then we order the vertices along each edge, and in each triangle of the triangulation, in a compatible way.The orders are indicated by arrows in Figure 1.The arrows are compatible with the boundary gluings, hence we get a ∆-complex structure on Σ. (Notice that the arrows on the boundary edges cannot all be directed counterclockwise because of the requirement of compatibility with the gluings.) Now we describe a classifying map f : Σ → BG of the bundle P .For convenience, let c ǫi i be the label of the edge (i, i + 1), and let C ǫi i (equal to some A ±1 j or B ±1 j ) be the monodromy along that edge.The map f sends the edge (i, i + 1) to where For ǫ i = +1 we define the triangle ∆ i = (0, i, i + 1) and map it to [g i |C i ]; for ǫ i = −1 we define the triangle ∆ i = (0, i + 1, i) and map it to [g i+1 |C i ].Then the fundamental class of Σ is represented by the cycle 4g−2 i=1 ǫ i ∆ i , mapped by f to the cycle on which the cocycle c evaluates to On the other hand, we see that We apply this inductively; since The lemma is proved.⋄ 4. Gluing formulae.
Let 1 → U → G → G → 1 be an arbitrary central group extension.We choose and fix a set-theoretic lift G → G, to be denoted g → g.We call it the standard lift; we assume that it satisfies 1 = 1.(We will sometimes use g to denote other, non-standard lifts of g.)We denote by c the corresponding cocycle (so that g • h = gh • c(g, h)) and by τ c its cohomology class (τ c ∈ H 2 (G, U )).
Suppose that ξ is a flat G-bundle over an oriented compact surface S of genus g with one boundary component.A boundary framing of ξ will mean the following collection of data: a point s ∈ ∂S; a trivialization of the fibre ξ s ; an element W ∈ G that lifts the monodromy W ∈ G of ξ along ∂S (oriented compatibly with the orientation of S and based at s).We will often abusively say "boundary framing W ", because W is the part of the framing data that presumes the rest of it and appears explicitly in many formulas.Usually we will use W = W -the standard lift of W -and then call our framing a standard framing.
Definition 4.1.The relative class c(ξ, W ) of the bundle ξ with boundary framing W is defined as the element of U given by If the framing is standard (i.e.W = W ), then we put c(ξ) = c(ξ, W ). (The bar over c is a reminder that the class is relative, and some framing is presumed.) Remark 4.2.The name "relative class" is intentionally provocative.We expect that there exists a relative characteristic class τ c related to τ c such that τ c (ξ) evaluated on the relative fundamental class of the base of ξ equals c(ξ).
We will use different lifts to our advantage.On the other hand, changing W to a different lift W u of W results in a change: It is not a priori clear that c(ξ) does not depend on the choice of the collection of loops (x i , y i ); we will check this shortly.
If a closed surface is cut along a separating simple loop into two pieces, a G-bundle over that surface decomposes into two bundles over the pieces.The next lemma describes the relation between the (relative) classes of the three bundles.
Lemma 4.4.Let ξ, ξ ′ be flat G-bundles over oriented surfaces S, S ′ with isomorphic boundary framings W .The isomorphism of boundary framings allows one to glue the bundles ξ, ξ ′ ; the result is a bundle ξ ∪ ξ ′ over Σ = S ∪ ∂ S ′ .We orient Σ compatibly with S and opposite to S ′ .Then Proof.Let (x i , y i ) be a standard collection of loops for S, (x ′ i , y ′ i ) one for S ′ .Then these collections together, in the order (x 1 , y 1 , . . ., x g , y g , y The relative class c(ξ, W ) does not depend on the choice of a standard loop collection (x i , y i ).Proof.Choose any (ξ ′ , S ′ ) with the same boundary framing as (ξ, S) (it can be just another copy of (ξ, S)).
Varying one of these collections while keeping the other fixed we see the claimed independence.
⋄ Now we set the notation for Lemma 4.6.Let ξ be a bundle over S with a standard boundary framing, and let A ∈ G. Then we can change (twist) the framing by A. This means that we change the trivialization ξ s → G by A; then all monodromies M change to A M := AM A −1 .In particular, W changes to A W , and the standard framing of the A-twisted bundle is A W .We denote by A ξ the twisted bundle with this framing.Another natural choice of framing is A W . (Formally, we should use depend on the choice of the lift A of A and will usually be abbreviated to A W .) In general A W = A W , so that we have two natural A-twisted bundles with boundary framing: (ξ, A W ) and A ξ = (ξ, A W ).
We have Proof.Let (x i , y i ) be a standard collection of loops in S, and let (X i , Y i ) be the monodromies of ξ along these loops.The monodromies for the A-twisted ξ are ( A X i , A Y i ).Since the commutator of lifts does not depend on the choice of the lifts, we have [ (Conjugation does not change the central element c(ξ).)We compare this with c( A ξ) using the standard lift A of A: Now the second formula follows from (4.2) and (4.7): ⋄ Now we set up notation for Lemma 4.7 and give another gluing construction, called "boundary connected sum".Let ξ, ξ ′ be bundles with standard boundary framings W , W ′ over S, S ′ .We glue s to s ′ as well as the fibres ξ s , ξ ′ s ′ (via the framing trivializations).Then we glue a triangle ∆ to ∂S ∨ ∂S ′ , one side along ∂S, one along ∂S ′ (all vertices to s = s ′ ), so that the path (∂S)(∂S ′ ) is homotopic (through ∆) to the third side.This third side forms the boundary of the obtained surface Σ of genus g + g ′ .The bundle naturally extends to a bundle ξ ∨ ξ ′ over Σ (all monodromies are already visible in ξ and ξ ′ ).The trivializations at s = s ′ agree and trivialize the new fibre at this point.The boundary monodromy of ξ ∨ ξ ′ is W W ′ ; we use the standard boundary framing, with lift W W ′ .Lemma 4.7.
In the above situation, Proof.Let (x i , y i ) and (x ′ j , y ′ j ) be standard loop collections in S, S ′ .Together they form a standard loop collection in Σ, so that ⋄ One last piece of general calculation: Lemma 4.8.Let ξ be a bundle over a surface with one boundary component and genus 1 with standard boundary framing, standard loop generators (x, y) and monodromies (X, Y ) (with [X, Y ] = W ). Then Proof. (4.12) ⋄ 5. Twists and equicommutativity.
In this section we discuss the twist deformations of flat bundles over surfaces.These twists are associated to the names of Fenchel and Nielsen in the Teichmüller case (cf.[Wol]), and to Goldman in the Lie group case (cf.[Gold86]).
Let P be a (flat) G-bundle over an oriented surface Σ. Choose an oriented simple loop ℓ in Σ, and a base-point b ∈ ℓ.Trivialize P b (by a right-G-equivariant isomorphism P b → G).Then, there is a well-defined element L ∈ G representing the monodromy of P along ℓ.Choose any V ∈ Z G (L) (the centralizer of L in G), and trivialize the bundle P along ℓ (with ambiguity L at b).Then cut Σ and P along ℓ, and glue it back by (left) multiplication by V .(To be precise, we define the right-hand side and the left-hand side of a tubular neighbourhood of ℓ in Σ using orientations.Then, after cutting the bundle, each element p of a trivialized fibre P x at a point x ∈ ℓ is split into a left-right pair p L , p R .We glue p R to V p L .Since the trivialization along ℓ is L-ambiguous at b, the gluing over b is well-defined only for V ∈ Z G (L).) The result is a new (flat) G-bundle P ℓ,V over Σ-the twist of P by V along ℓ.
It is possible to phrase the above definition in a slightly more invariant way.Suppose we refrain from choosing a trivialization of P b .Then we still have the monodromy along ℓ.It is an element L in Aut(P b ), the automorphism group of the right G-space P b .(This Aut(P b ) is non-canonically isomorphic to G; possible isomorphisms arise from trivializations of P b .)Then for any V ∈ Z Aut(P b ) (L) the bundle P ℓ,V is well-defined.
Now suppose that we have a central group extension 1 → U → G → G → 1, as in the previous section (with a lift g → g, cocycle c, cohomology class τ ∈ H 2 (G, U )).
Theorem 5.1.Let P be a G-bundle over a closed oriented surface Σ.Let ℓ be an oriented simple loop in Σ, based at b. Let L ∈ G be the monodromy of P along ℓ (with respect to some trivialization of P b ), let V ∈ Z G (L), and let P ℓ,V be the twist of P .Let τ ∈ H 2 (G, U ) be a cohomology class represented by a cocycle c.Then (5.1) Proof.We use the commutator product expression from Lemma 3.1.The basic calculation (valid for any two commuting elements Case 1.The loop ℓ does not separate Σ.Then the standard presentation loops a 1 , . . ., b g in Σ can be chosen so that b 1 = ℓ.If A 1 , . . ., B g are the elements of G representing the monodromies of the bundle P along a 1 , . . ., b g (with B 1 = L), then the monodromies of P ℓ,V along these loops are represented by the same elements except for one change: A 1 gets replaced by A 1 V .In the commutator product expression the first term The claim follows.
Case 2. The loop ℓ separates Σ.Then we cut Σ and P along ℓ into two components, say P 0 over Σ 0 and P 1 over Σ 1 .The assumptions of the theorem induce (isomorphic) boundary framings for P 0 and P 1 (except for lifts of the boundary monodromy L-we take standard lifts).We have P = P 0 ∪ P 1 , P ℓ,V = V P 0 ∪ P 1 .Lemmas 4.4, 4.6 give (5.4) The last equality uses the fact that Proposition 5.3.Let c be a cocycle with cohomology class τ c ∈ H 2 (G, U ), let U → G → G be the corresponding central extension, and let g → g be the lift corresponding to c.The following conditions are equivalent: a) for every commuting pair g, h ∈ G, the lifts g, h ∈ G commute; b) the cocycle c is equicommutative; c) the cohomology class τ c is equicommutative; d) for every commuting pair g, h ∈ G, every lift of g commutes with every lift of h e) every cocycle representing τ c is equicommutative; f) for every commutative subgroup H < G the pre-image of H in G is commutative.
Proof.It is straightforward to see that: -the weak conditions a), b) are equivalent; -the strong conditions d), e), f) are equivalent; -the strong conditions imply the weak conditions; -b) implies c).
To finish the proof we show that c) implies e).It is enough to check that all 2-coboundaries are equicommutative.Let g, h ∈ G be commuting elements, and let n ∈ C 1 (G, U ). Then for every G-bundle P over a closed oriented surface Σ, and for every twist P ℓ,V of that bundle, we have τ (P ℓ,V ) = τ (P ).
In Definition 5.4 one could, equivalently, use the condition Corollary 5.5.A cohomology class is twist-invariant if and only if it is equicommutative.Proof.It follows from Theorem 5.1 that equicommutativity implies twist-invariance.For the converse, let c be a cocycle representing a twist-invariant class τ ∈ H 2 (G, U ).For any pair of commuting elements g, h ∈ G there exists a G-bundle ξ (g,h) over Σ 1 = T 2 with monodromies (along standard generating loops a 1 , b 2 ) equal to g, h.This bundle is a twist of ξ (1,h) , and Theorem 5.1 gives: Now the assumption of twist-invariance implies that c(g, h) = c(h, g).⋄ Corollary 5.6.
The Witt class is twist-invariant.
Proof.We check that the cocycle (1.4) is equicommutative.Let a, b ∈ SL(2, K), and suppose that ab = ba.Then
Unlike the Witt class, the Moore class, in general, is not twist-invariant.This is more fully explained in Section 8.

Universal classes.
In this section we specialize our discussion of characteristic classes to perfect groups.Recall that a group G is perfect if it is equal to its commutator subgroup [G, G].In homological terms this means that is an isomorphism for every abelian group A. In particular, for A = H 2 G we have a well-defined class u G ∈ H 2 (G, H 2 G) that corresponds to id H2G under this isomorphism.(Thus, by Remark 2.2, u G coincides with the class constructed in Section 2.) The class u G is universal in the following sense: for any abelian group A and any class v ∈ H 2 (G, A) there exists a unique homomorphism f : [Brown,Exercise IV.3.7]).Also the central extension 1 (This extension was one of the early reasons for considering the second homology of a group; whence the name "Schur multiplier" for H 2 G.) It is known that all universal central extensions of a perfect group G are canonically isomorphic.Quite often the construction of such an extension and the study of its kernel is the way to calculate H 2 G and to describe u G .We call the class u G the Moore class, because it was investigated by Moore for G = SL(2, K).Definition 6.1.Let G be a perfect group, and let u be a cocycle representing the universal class u G ∈ H 2 (G, H 2 G).Let G [,] be the subgroup of H 2 G generated by the set (6.1) {u(g, h)u(h, g) −1 | g, h ∈ G, gh = hg}.
We put Eq(G) := H 2 G/G [,] .Let w G ∈ H 2 (G, Eq(G)) be the image of the universal class u G by the quotient map q: H 2 G → Eq(G).
The set (6.1) does not depend on the choice of the cocycle u.Indeed, it can be described as the set of commutators, in G, of lifts of pairs of commuting elements g, h ∈ G, as the following calculation shows: The last equality follows from the fact that for commuting g, h the commutator [g, h] is central in G.This commutator does not depend on the choice of lifts of g, h, because all possible lifts are of the form gu, hv with u, v central in G.
Remark 6.3.Another, more topological description of the set (6.1): it consists of "genus 1 classes", i.e. the classes in H 2 G that are images of the fundamental class of the 2-dimensional torus T 2 under some map T 2 → BG.Indeed, such a map associates to the generators of π 1 (T 2 ) an (arbitrary) commuting pair g, h ∈ G; the image of the fundamental class of T 2 is then u(g, h)u(h, g) −1 by the computation in the proof of Lemma 3.1.Theorem 6.4.Let G be a perfect group.The class w G ∈ H 2 (G, Eq(G)) is a universal equicommutative class in the following sense: for every equicommutative cohomology class v ∈ H 2 (G, A) there exists a unique homomorphism g: Eq(G) → A such that g * w G = v.
Proof.Let f : H 2 G → A be the homomorphism that maps u G to v. Choose a cocycle u representing u G .Then f * u is a cocycle representing v, hence it is equicommutative.It follows that, for every commuting pair g, h ∈ G, we have therefore f factors through the quotient map q: H 2 G → Eq(G), i.e. f = g • q for some g: Eq(G) → A. We get (6.4) The uniqueness statement is proved by contradiction.Suppose two different homomorphisms g, g ′ : ) is, up to a unique isomorphism, the unique universal equicommutative class; this is a standard consequence of universality.
We finish this section by indicating a more general point of view on universality.It will not be used later in this paper.Proposition 6.6.Let G be a perfect group, u G ∈ H 2 (G, H 2 G) its Moore class, and let ϕ: H 2 G → Q be a group epimorphism (coefficient reduction map).We set u G,Q = ϕ * u G .Suppose that a cohomology class v ∈ H 2 (G, A) satisfies the following condition: v(x) = 0 for all x ∈ ker ϕ.Then there exists a unique group homomorphism Proof.Let Ψ: H 2 G → A be the unique map giving Ψ * u G = v.For each x ∈ ker ϕ we have: It follows that there exists a ψ: Now we show that the homomorphism ψ is unique.Suppose that ψ and ψ ′ satisfy the conditions of the theorem.Then ψ • ϕ, ψ ′ • ϕ: H 2 G → A are coefficient maps that map u G to v; thus, these maps are equal, by the universality property of u G .Since ϕ is epimorphic, we deduce ψ = ψ ′ .⋄ III.SL(2, K) In this part we specialize our considerations to the discrete group SL(2, K), where K is an infinite field.This group is perfect, so that the results of Section 6 apply.Our main result is that the (reduced) Witt class is the universal equicommutative class for this group (Theorem 10.1).The proof relies on several known results which we review carefully.The Schur multiplier of SL(2, K), denoted π 1 (SL(2, K)) henceforth to honour Calvin Moore, is classically described by generators and relations; we recall this description in Section 7. The generators are "symbols" {a, b}, a, b ∈ K * .(We use K * to denote K \ {0}.)In the quotient Eq(SL(2, K)) of π 1 (SL(2, K)) the symbols become symmetric: {a, b} = {b, a}.We are thus led to consider the group π 1 (SL(2, K))/sym, defined by adjoining to the classical presentation of π 1 (SL(2, K)) all the symbol symmetry relations {a, b} = {b, a}, a, b ∈ K * .This group is a natural mid-step in the quotient sequence In Section 8 we show that in fact π 1 (SL(2, K))/sym ≃ I 2 (K) (here I 2 (K) is the square of the fundamental ideal I(K) of the Witt ring W (K), cf.[EKM, Chapter I]).A cocycle b representing the universal class u SL(2,K) was given explicitly (though slightly erroneously) by Moore; we recall the correct description in Section 9.There we also present the results of Nekovář, and Kramer and Tent, proving that the image of the Moore cocycle b in I 2 (K) is cohomologous to the (reduced) Witt cocycle.In Section 10 we use this compatibility to show that I 2 (K) ≃ Eq(SL(2, K)) and that the reduced Witt class is (equivalent to) the universal equicommutative class.

Schur multiplier of SL(2, K).
In this section we recall the standard description of the universal central extension and of the Schur multiplier of SL(2, K). (As always, we assume that K is an infinite field.)The classical references are [Moore,Sections 8,9], [Mats], [St,§7].

Symmetrized Schur multiplier and quadratic forms.
The fundamental ideal I(K) of W (K) is generated by non-degenerate symmetric bilinear forms on even-dimensional spaces.Another suitable collection of generators consists of the forms a = 1, −a , a ∈ K * .The ideal I 2 (K) is the square of I(K); it is generated by Pfister forms a, b (for a, b ∈ K * ), where (the last equality valid in W (K)).More on these generating sets (in particular, the relations) can be found in [EKM,I.4].
Putting these two facts together we get the following.
It is known that, in general, the map Φ is not an isomorphism.This means that for some field K and some a, b ∈ K * we have {a, b} = {b, a} in π 1 (SL(2, K)).It follows that for that field K the Moore class is not equicommutative.In topological terms, we may consider the SL( 2 Each step uses symbol symmetry, applies one of the defining relations (7.5), or multiplies a symbol entry by a square.The latter operation is equivalent to multiplication by a symbol of the form {z 2 , y}, as asserted in [Moore,Appendix,(7)]; by (8.1), the symbol {z 2 , y} is trivial in π 1 (SL(2, K))/sym.9. Comparison of the Moore and Witt cocycles.
In this section we recall the explicit form of a cocycle b representing the universal class u SL(2,K) ∈ H 2 (SL(2, K), π 1 (SL(2, K))) as given in [Moore,, with later corrections (cf.[Kr-T, 9.1]).We also describe the image of b under the map Φ: π 1 (SL(2, K)) → I 2 (K).Kramer and Tent show that this image, an I 2 (K)-valued cocycle on SL(2, K), is cohomologous to the Witt cocycle.In the next section we will use this fact to show that the I 2 (K)-valued Witt class is the universal equicommutative class for SL(2, K).
Kramer and Tent do their calculation in the generic case, and argue that this is enough to claim cocycle equality.We present the details in all cases as this allows us to give an explicit formula for a universal equicommutative cocycle.
Every element of SL(2, K) is uniquely represented in one of the forms: This leads to the following definition of a lift SL(2, K) → St(2, K): The corresponding cocycle b was calculated by Moore, with later correction by Schwarze (cf.[Kr-T, 9.1]).We present the formulae for the cocycle b, and for its image under Φ in W (K).
We summarize: On the other hand, we have the Witt class w ∈ H 2 (SL(2, K), W (K)), given by the cocycle w defined by (1.4): where e = 1 0 .We now express this cocycle in the parametrization of SL(2, K) used by Moore.Lemma 9.1.
Proof.Notice that g 1 (u, t)e = x(u)h(t)e = t 0 , so that |e, g 1 (u, t)e| = 0. Therefore, the value of w(g, h) is zero if any of the arguments g, h is of the form g 1 (u, t).Even so, for later use we need the following easily checked formulae: Let us turn to case 1: (9.9) ⋄ Nekovář [Ne,§2] noticed that w is cohomologous to an I 2 (K)-valued cocycle; it is enough to add the coboundary of the following cochain: In the parametrization used by Moore: We are ready to calculate δn and see that Φ * b = w + δn.Using the formula (δn)(g, h) = n(g) − n(gh) + n(h) we get: Comparing these four formulae, (9.5) and (9.7) we obtain the following proposition.
The phrasing of Proposition 9.2 is slightly awkward, because Φ * b has coefficients in I 2 (K), while w has coefficients in W (K). Fortunately, it is not hard to check that the relevant cohomology groups embed: Proof.Let Q = W (K)/I 2 (K).Consider the short exact sequence of coefficient groups: and the associated long exact sequence Definition 9.4.The reduced (or I 2 (K)-valued ) Witt class w I ∈ H 2 (SL(2, K), I 2 (K)) is defined by w I = ι −1 * (w); it is equal to Φ * (u SL(2,K) ) and represented by the cocycle Φ * b, explicitly given by (9.5).
For practical purposes, one can ignore the difference between the classes w and w I , mainly because of the following corollary of Proposition 9.2.
The class w I is equicommutative.
Proof.A cocycle c representing w I treated as a W (K)-valued cocycle (via the embedding I 2 (K) → W (K)) is cohomologous to the (standard) Witt cocycle w.The latter is equicommutative, hence, by Proposition 5.3, so is c. ⋄ 10.The Witt class is universal equicommutative.
We prove what is in the title of this section.
Theorem 10.1.(Theorem A) Let K be an infinite field.The group Eq(SL(2, K)) is isomorphic to I 2 (K), and the Witt class w I ∈ H 2 (SL(2, K), I 2 (K)) is the universal equicommutative class.
Proof.Consider the following diagram of coefficient groups. (10.2) The diagonal arrows are the unique maps deduced from the universal properties of u SL(2,K) and w SL(2,K) , applied to the Witt class w I .Uniqueness of the universal map implies commutativity of the diagram.Namely, the left triangle commutes because φq 1 = Φ maps u SL(2,K) to w I (Proposition 9.2), hence is equal to the diagonal map d 1 .Similarly, we have w I = (d 2 ) * w SL(2,K) = (d 2 ) * (q 2 q 1 ) * u SL(2,K) , hence d 2 q 2 q 1 = d 1 = φq 1 ; but q 1 is surjective, therefore we deduce d 2 q 2 = φ, i.e. the commutativity of the right triangle.Now d 2 is an isomorphism, because φ is an isomorphism (Proposition 8.1) and q 2 is surjective.The isomorphism d 2 maps the universal equicommutative class w SL(2,K) to w I .⋄ Another corollary of the proof is that q 2 is an isomorphism: the quotient of π 1 (SL(2, K)) by the symbol symmetry relations is already equal to Eq(SL(2, K)).

IV. Witt range.
What are the possible values of the Witt class for SL(2, Q)-bundles over surfaces?We know that these values reside in I 2 Q, which is a direct sum of 4Z (the real, signature part) and an infinite direct sum of Z/2 (p-adic parts, one per odd prime).More details of this description are given in Section 12.While the real part of the Witt class is related to the Euler class, hence non-trivial (cf.[DJ,Section 13]), the p-adic parts are more mysterious-perhaps trivial?Using (1.4) we did some computer calculations in FriCAS that indicated non-triviality of the p-adic parts.In Section 12 we give a complete description of the range of the Witt class over Q, proving that the Milnor-Wood inequality (restricting the real part in terms of the genus of the base surface) is the only restriction-the p-adic parts can be arbitrary.The challenging part of this result is the construction of sufficiently many non-trivial bundles.This is done in Section 11 in much greater generality, for arbitrary infinite fields K.We analyse SL(2, K)-bundles over simple surfaces (pair of pants; genus 1 surface with one boundary component) using some results on the Markov equation (quoted from [GMS]).Then we use the gluing results from Section 4. The formula (1.4) is used throughout to control the Witt class.Our final result, Theorem 11.6, gives a large subset of Witt classes in I 2 (K).This subset is quite close to the one defined by the boundedness restriction for the Witt class; the difference is discussed in Section 13.
An early paper where many SL(2, Q)-bundles were constructed by gluing is [Takeuchi].
In this section K will be an arbitrary infinite field.Let w be the Witt class.We will use the representing cocycle where e = 1 0 , X 21 denotes the 21-entry of the matrix X, and [0] is interpreted as 0. Additive convention will be used for the cocycle.
Remark 11.1.If either X, Y or XY is diagonal (or even upper-triangular), then w(X, Y ) = 0. We will use notions and notation discussed at the beginning of Section 4, in particular the notion of standard framing, and the notion of "relative class" w (cf.Definition 4.1).
Lemma 11.2.Let K be an infinite field.Then, for any α, β ∈ K * there exists a flat SL(2, K)-bundle ξ with a standard framing, over the oriented genus 1 compact surface with one boundary component, such that w(ξ) = [α] + [β] ∈ W (K).Moreover, for every z ∈ −αβK * 2 (with finitely many exceptions) the bundle ξ may be chosen so that its boundary monodromy is diagonal with eigenvalues z, z −1 .
Proof.To construct the bundle as in the lemma, with boundary monodromy Z, we need to find X, Y ∈ SL(2, K) such that [X, Y ] = Z.We heavily rely on the classical description of the space of solutions of the commutator equation [X, Y ] = Z; we use the version from [GMS], though some results go back as early as to Fricke.To start, if [X, Y ] = Z, then the Fricke identity says that the scalars x 1 = tr X, x 2 = tr Y , x 3 = tr XY and m = tr Z + 2 satisfy the Markov equation ) satisfies tr Y = tr ZY = x 2 ; then there exists a unique X ∈ SL(2, K) that satisfies tr X = x 1 , tr XY = x 3 and [X, Y ] = Z (cf.[GMS,Lemma 3.5]).Explicitly, this X is given by We will work out the case The condition det Y = 1 gives the following form of Y : (11.4) The solution depends on a unique parameter c ∈ K * .(We have c = 0, since otherwise 1 = det Y would imply 1 = zx 2 2 (1 + z) −2 = x 2 2 m −1 , contradicting (c).)Condition (c) ensures that m − x 2 2 = 0; therefore we may plug (11.4) into (11.2) and determine X: Further direct computations show that Let ξ c be the bundle defined by these (X, Y ).Finally, using Lemma 4.8 and Remark 11.1 we get (11.7) If our solution (x 1 , x 2 , x 3 ) satisfies two further conditions (e) x 1 x 2 − (1 + z −1 )x 3 = 0, x 1 (1 + z) − x 2 x 3 = 0; then we modify c to a new parameter (11.8) (C runs through K * as c does), and then we get (11.9) To summarize, to realize [α] + [β] ∈ W (K) (for given α, β ∈ K * ) as w(ξ c ) we may choose z = −αβλ 2 , C = α (for some λ ∈ K * ) and apply the above construction-provided that we find a solution (x 1 , x 2 , x 3 ) of M m that satisfies (c) and (e).In [GMS,proof of Proposition 3.6] the following solution of M t+2 is given: (We have corrected a sign mistake in x 3 ; ζ ∈ K * \ {±1} is arbitrary.)Putting t = z + z −1 we get: Let us now discuss the conditions (c) and (e) for the solution (S ′ ).For a fixed ζ, these are three nontrivial polynomial inequalities on z, hence they hold except for some finite set E of values of z. (Since the general procedure described above requires z = ±1, we include these two values in E as well.)Clearly, for all but finitely many values of λ we have z ∈ E; this proves the lemma.
Suppose that the triple (t 1 , t 2 , t 3 ) does not satisfy Markov's equation M 4 .Then for every c ∈ K * there exist a unique pair of matrices (L, M ) in SL(2, K) that satisfies the following conditions: Moreover, regardless of the choice of A, B in condition (a), we then have (Recall that A Λ is our notation for the conjugate AΛA −1 .) Corollary 11.4.Let ξ, ξ ′ be flat SL(2, K)-bundles with standard boundary framings and diagonal boundary mon- Then, for every z ∈ −αβλ 1 λ 2 K * 2 (with finitely many exceptions) there exist matrices A, B ∈ SL(2, K) such that A ξ∨ B ξ ′ has diagonal boundary monodromy z 0 0 z −1 = ±I and with standard boundary framing satisfies (11.12) Proof (of the Corollary).We use Lemma 11.3 with Λ 1 , Λ 2 equal to the boundary monodromies of ξ, ξ ′ .We choose λ 3 = z = −αβλ 1 λ 2 λ 2 and c = αλ 1 for some λ ∈ K * .(There is a finite set of values that λ 3 has to avoid: ±1 and the values for which (t 1 , t 2 , t 3 ) would satisfy M 4 ; we avoid them for all but finitely many choices of λ ∈ K * .)To prove (11.12) we use Lemmas 4.6 and 4.7 (that describe the change of w under twists and ∨), the remark about vanishing of w for diagonal matrices, and (11.11): (11.13)We claim that c = 0. Indeed, if c were 0 then L would be upper-triangular (with eigenvalues λ ±1 1 ), and M = L −1 Λ 3 would also be upper-triangular (with eigenvalues λ ±1 2 ).Direct calculation shows that then [L, M ] would be upper-triangular and of trace 2 (with both diagonal entries equal to 1).The Fricke identity for the traces (t 1 = tr L, t 2 = tr M, t 3 = tr Λ 3 ) would show that this triple satisfies the Markov equation (M 4 )-contradiction.

Thus, we have
To finish the discussion of the existence and uniqueness question we remark that any matrix with determinant 1 and trace t i is conjugate (in SL(2, K)) to Λ i .We now pass to the calculation of w.The equation (We know that A 21 = 0, for otherwise A and A Λ 1 = L would be upper-triangular, contradicting L 21 = c = 0.) Similarly, (11.17) ⋄ One of the ways to look at the Witt ring W (K) is the following.Any element of W (K) is represented by a unique (up to isomorphism) anisotropic form over K.The dimension of this anisotropic representative defines a norm • on W (K) (cf.[MH,3.1.7,3.1.8]).On the other hand, any element x ∈ W (K) can be represented-in many ways-as a finite sum where n i ∈ Z, a i ∈ K * .The symbolic norm x s is the minimum-over all such representations-of the expression i∈I |n i |.
For each x ∈ W (K) we have x = x s .
Proof.If x = i∈I n i [a i ], then the form x = i∈I |n i | (sgn n i )a i represents x.This form has a largest anisotropic direct summand-the unique (up to isomorphism) representative of x, of dimension x .It follows that (11.19) x ≤ dim( Therefore x ≤ x s .On the other hand, a diagonalization of the anisotropic representative of x expresses x as a sum of x symbols [a i ], so that x s ≤ x .⋄ The diameter of W (K) with respect to the above norm is sometimes called the u-invariant of K (cf.[Lam,XI.6]).However, it is too often infinite (e.g. for K = Q), hence a refinement is widely used: u(K) is the diameter of the set W t (K) of torsion elements in W (K) (cf.[EKM, Chapter VI] or [Lam,Definition XI.6.24]).It is classically known that u(K) = 4 for local (non-archimedean) and for global fields (cf.[Lam,Examples XI.6.2,XI.6.29] or [EKM,Example 36.2]).
(a) This part is straightforward.The closed orientable genus g surface Σ g has a ∆-complex structure with 4g − 2 triangles (cf. the proof of Lemma 3.1).The value of the Witt cocycle w(ξ) (of any flat SL(2, K)bundle ξ over Σ g ) on each of these triangles has norm ≤ 1.This implies claim a).
(b) Let q ∈ I 2 (K) be an element of norm ≤ 4g − 4. Then we can find α If the norm of q is smaller than 4g − 4, we add some extra trivial terms [1] + [−1] to obtain the above form; since q ∈ I 2 (K) we know that dim q is even, and that the following product formula holds: Now we find, using Lemma 11.2, bundles ξ i (for i < g), (over genus 1 oriented surfaces with one boundary component), with standard boundary framing, with w(ξ Induction then gives: The goal of this section is to describe the range of the Witt class for all representations of π 1 (Σ g ) (the fundamental group of the genus g orientable surface) in SL(2, Q).
(Each I 2 Q p is isomorphic to Z/2; each non-trivial a p is interpreted as 1.)The torsion part of I 2 Q is isomorphically mapped onto the subgroup For part (a) we use Meyer's theorem (cf.[MH,Corollary II.3.2]): a quadratic Q-form of dimension greater than 4 is Q-isotropic if it is R-isotropic.Let x ∈ I 2 Q, σ(x) = 4h > 0. Let q be an anisotropic representative of x.If dim q > 4h, then q is R-isotropic, hence, by Meyer's theorem, also Q-isotropiccontradiction.The rest of the statement is seen as in part (b): the condition i a i ∈ Q * 2 , i.e. d ± = 1, characterizes elements in I 2 Q, while positivity of a i is equivalent to σ = 4h.The claim for signature −4h can be deduced by switching from x to −x. ⋄ Using Lemma 12.1 we can now give a complete description of possible Witt classes of SL(2, Q)-bundles.
Theorem 12.2.(Theorem C) The set of Witt classes of all representations of π 1 (Σ g ) in SL(2, Q) is equal to the set of elements of I 2 Q with norm ≤ 4(g − 1).
Proof.The classical Milnor-Wood inequality states that the Euler class of a flat SL(2, R)-bundle over Σ g has absolute value ≤ g − 1.For an SL(2, Q)-bundle (treated as a flat SL(2, R)-bundle) this Euler class is equal to 1 4 of the signature of the Witt class (cf.[DJ,Theorem 13.4]); therefore, the Witt class of an SL(2, Q)bundle over Σ g has signature of absolute value ≤ 4(g − 1).Then it has also norm ≤ 4(g − 1), by Lemma 12.1-except, possibly, for g = 1.But for g = 1 we know that the Witt class is 0 (by equicommutativity and Lemma 3.1), so that the norm bound holds also in this case.
Conversely, by Theorem 11.6, every element of I 2 Q of norm ≤ 4(g − 1) is realizable as the Witt class of some SL(2, Q)-bundle over Σ g .⋄ Remark 12.3.Assume g ≥ 2. Then in the statement of Theorem 12.2 one can replace "with norm ≤ 4(g − 1)" by "with signature of absolute value ≤ 4(g − 1)" (as is evident from the proof).
13.The easy norm bound is not sharp.
The Witt class w(ξ) of a flat SL(2, K)-bundle ξ over Σ g has norm ≤ 4g − 2. For K = R and the Euler class this bound can be improved to 4g − 4 (the already mentioned Milnor-Wood).It is unclear whether this stronger estimate ( w(ξ) ≤ 4g −4) holds for the general Witt class; we have not found any counterexamples.The question is meaningful for fields with u > 4. We now present an example of an element of I 2 (K) with norm 6 that is not realizable as w(ξ) over Σ 2 .
Then q ∈ I 2 (K), q = 6 and q is not realizable as the Witt class of an SL(2, K)-bundle over Σ 2 .
Now 1, 1 is anisotropic over F 7 , which finishes the proof.⋄ Not all forms of norm 6 are susceptible to this argument, however.For example, consider (13.4) q ′ = 1, 1, 1, 5, x, −5x over the same field Q((x)).This form is in I 2 Q((x)), has norm 6, but 2q ′ is isotropic, hence has norm ≤ 10.We do not know whether q ′ is realizable as the Witt class over Σ 2 .
then (a) is fulfilled).The matrices Y satisfying (d) can be found explicitly.If Y = a * * d , then tr Y = tr ZY = x 2 are equivalent to a linear system of equations on a, d with unique solution

⋄
Proof (of Lemma 11.3).Suppose that (L, M ) is a pair satisfying (a), (b) and (c Theorem 11.6.(Theorem B) Let K be an infinite field.(a) The Witt class of any flat SL(2, K)-bundle over an oriented closed surface of genus g has norm ≤ 4(g − 1) + 2. (b) The set of Witt classes of flat SL(2, K)-bundles over an oriented closed surface of genus g contains the set of elements of I 2 (K) of norm ≤ 4(g − 1).
σ: W (Q) → W (R) ≃ Z the signature map, normalized by σ([1]) = 1.Then the torsion elements of W (Q) are the ones of signature zero.Lemma 12.1.(a) Elements of I 2 Q of signature ±4h (where h ≥ 1) have norm 4h.The set of such elements can be described as(12.4){ 4h i=1 [a i ] ∈ W (Q) | ±a i > 0, i a i ∈ Q * 2 }.(b) Non-trivial elements of I 2 Q of signature 0 have norm 4.The set of such elements can be described as(12.5){[a] + [b] + [c] + [d] ∈ W (Q) | abcd ∈ Q * 2 ,and exactly two of a, b, c, d are positive}.Proof.Let us start with the proof of part (b).Let x ∈ I 2 Q, x = 0, σ(x) = 0. Since u(Q) = 4 we know that x ≤ 4. As the dimension function takes even values on IQ (by definition), hence also on I 2 Q, an anisotropic representative of x is 4-or 2-dimensional.If x = a, b , however, we get 1 = d ± ( a, b ) = −ab; then ab = −1 in Q * /Q * 2 , and x = a, −a = 0 in W (Q). Therefore, an anisotropic representative of x is 4-dimensional.Thus, we have x = [a] + [b] + [c] + [d] for some a, b, c, d ∈ Q * that satisfy abcd ∈ Q * 2 (this is equivalent to x ∈ I 2 Q), and exactly two of a, b, c, d are positive (equivalent to σ(x) = 0).