Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=-1

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=-1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.


Introduction
In this paper, we find simple generating sets for three closely related groups: 1. the hyperelliptic Torelli group SI g , that is, the subgroup of the mapping class group consisting of elements that commute with some fixed hyperelliptic involution and that act trivially on the homology of the surface; 2. the fundamental group of H g , the branch locus of the period mapping from Torelli space to the Siegel upper half-plane; and 3. the kernel of β n , the Burau representation of the braid group evaluated at t = −1 (the representation β n is sometimes known as the integral Burau representation).
Hyperelliptic Torelli group. Let Σ g be a closed oriented surface of genus g and let Mod g be its mapping class group, that is, the group of isotopy classes of orientation-preserving homeomorphisms of Σ g . Let ι : Σ g → Σ g be a hyperelliptic involution; see Figure 1. By definition a hyperelliptic involution is an order two homeomorphism of Σ g that acts by −I on H 1 (Σ g ; Z), and it is a fact there is a unique hyperelliptic involution up to conjugacy by homeomorphisms of Σ g ; we fix one once and for all. The hyperelliptic mapping class group SMod g is the subgroup of Mod g consisting of mapping classes that can be represented by homeomorphisms that commute with ι. The Torelli group I g is the kernel of the action of Mod g on H 1 (Σ g ; Z), and the hyperelliptic Torelli group SI g is SMod g ∩I g .
A simple closed curve x in Σ g is symmetric if ι(x) = x, in which case the Dehn twist T x is in SMod g . If x is a separating curve, then T x ∈ I g ; see Figure 1. The mapping class T x is a Dehn twist about a symmetric separating curve. A bounding pair map, such as T y T −1 z , is the difference of two Dehn twists about disjoint, nonseparating, homologous simple closed curves. The mapping classes T u T u ′ and T v T v ′ are in SMod g and their actions on Theorem A. For g ≥ 0, the group SI g is generated by Dehn twists about symmetric separating curves.
The first two authors proved that Theorem A in fact implies the stronger result that SI g is generated by Dehn twists about symmetric separating curves that cut off subsurfaces of genus 1 and 2 [11,Proposition 1.5].
Theorem A was conjectured by Hain [20,Conjecture 1] and is also listed as a folk conjecture by Morifuji [33,Section 4]. Hain has informed us that he has proven the case g = 3 of Theorem A. His proof uses special properties of the Schottky locus in genus 3.
When we first encountered Hain's conjecture, it appeared to us to be overly optimistic. There is a well-known generating set for I g , namely, the set of bounding pair maps and Dehn twists about separating curves; see Figure 1. There is no reason to expect that an infinite-index subgroup of I g should be generated by the elements on this list lying in the subgroup. Additionally, there are several other natural elements of SI g , and it was not at first clear how to write those in terms of Hain's proposed generators. Consider for instance the mapping class [T u T u ′ , T v T v ′ ] ∈ SI g indicated in Figure 1. Eventually, it turned out this element is a product of six Dehn twists about symmetric separating curves [12], but the curves are rather complicated looking.
Branch locus of the period map. Hain [20] observed that Theorem A has an interpretation in terms of the period map. Let T g be Teichmüller space and h g the Siegel upper half-plane. The period map T g → h g takes a Riemann surface to its Jacobian. It factors through the Torelli space T g /I g , which is an Eilenberg-MacLane space for I g . The induced map T g /I g → h g is a 2-fold branched cover onto its image. The branch locus is the subspace H g ⊂ T g /I g consisting of points that project to the hyperelliptic locus H g in the moduli space of curves. The space H g is not connected, but its components are all homeomorphic and have fundamental group SI g . Thus, Theorem A gives generators for π 1 ( H g ).
Let H c g be the space obtained by adjoining hyperelliptic curves of compact type to H g . Theorem A has the following corollary. See Hain's paper [20] for the details on how to derive Theorem B from Theorem A. The main idea is that when we add to H g a hyperelliptic curve of compact type obtained by degenerating a symmetric separating simple closed curve in a hyperelliptic curve, the effect on π 1 ( H g ) is to kill the generator of π 1 ( H g ) given by the corresponding Dehn twist.
Kernel of the Burau representation. The (unreduced) Burau representation [7] is an important representation of the braid group B n to GL n (Z[t, t −1 ]). Let β n : B n → GL n (Z) be the representation obtained by substituting t = −1 into the Burau representation. Denote the kernel of β n by BI n (the notation stands for "braid Torelli group").
We identify B n with the mapping class group of a disk D n with n marked points, that is, the group of isotopy classes of homeomorphisms of D n preserving the set of marked points and fixing ∂ D n pointwise. For most purposes, we will regard the marked points as punctures. For instance, curves (and homotopies of curves) are not allowed to pass through the marked points. When we say that a simple closed curve is essential in D n , we mean that it is not homotopic to a marked point, an unmarked point, or the boundary.
Theorem C. For n ≥ 1, the group BI n is generated by squares of Dehn twists about curves in D n surrounding odd numbers of marked points.
Just like for Theorem A, our proof gives more, namely that BI n is generated by squares of Dehn twists about curves surrounding exactly 3 or 5 marked points. As pointed out to us by Neil Fullarton, both types of twists are needed. Indeed, the abelianization homomorphism B n → Z maps the square of a Dehn twist about a curve surrounding 2k + 1 marked points to 8k 2 + 4k. Thus the group generated by squares of Dehn twists about curves surrounding 3 marked points maps to 12Z and the group generated by squares of Dehn twists about 5 marked points maps to 40Z. Since gcd (12,40) = 4 the image of the group generated by squares of both types of Dehn twists-hence the image of BI n -contains 4Z.
Hyperelliptic Torelli vs Burau. We now explain the relationship between Theorems A and C. This requires defining the hyperelliptic Torelli group for a surface with boundary.
Let Σ 1 g be the surface obtained from Σ g by deleting the interior of an embedded ιinvariant disk. There is an induced hyperelliptic involution of Σ 1 g which we also call ι. Let Mod 1 g be the group of isotopy classes of homeomorphisms of Σ 1 g that fix ∂Σ 1 g pointwise and let SMod 1 g be the subgroup of Mod 1 g consisting of mapping classes that can be represented by homeomorphisms that commute with ι. Observe that unlike for Mod g , the map ι does not correspond to an element of Mod 1 g . Finally, let I 1 g be the kernel of the action of Mod 1 g on H 1 (Σ 1 g ; Z) and let SI 1 g = SMod 1 g ∩I 1 g . The involution ι fixes 2g + 1 points of Σ 1 g . Regarding the images of these points in Σ 1 g /ι as marked points, we have Σ 1 g /ι ∼ = D 2g+1 . There is a homomorphism L : B 2g+1 → SMod 1 g which lifts a mapping class through the branched cover Σ 1 g → Σ 1 g /ι. Birman-Hilden [8] proved that L is an isomorphism. Under this isomorphism, a Dehn twist about a curve c surrounding an odd number of marked points maps to a half-twist about the (connected) preimage of c in Σ 1 g ; in particular, the Dehn twist about ∂ D 2g+1 acts by −I on H 1 (Σ 1 g ). Similarly, a Dehn twist about a curve c surrounding an even number of marked points maps to the product of the Dehn twists about the two components of the preimage of c.
The representation β 2g+1 decomposes into two irreducible representations. One is the 1-dimensional trivial representation, and the other is conjugate to the composition where the map Mod 1 g → Sp 2g (Z) is the standard representation arising from the action of Mod 1 g on H 1 (Σ 1 g ; Z). The map L therefore restricts to an isomorphism BI 2g+1 ∼ = SI 1 g . Under this isomorphism, squares of Dehn twists about curves surrounding odd numbers of marked points map to Dehn twists about symmetric separating curves. The case n = 2g + 1 of Theorem C is therefore equivalent to the statement that SI 1 g is generated by Dehn twists about symmetric separating curves. The first two authors proved [11,Theorem 4.2] that the kernel of the natural map SI 1 g → SI g is generated by the Dehn twist about ∂Σ 1 g , so this is equivalent to Theorem A.
We can also relate Theorem C for even numbers of punctures to the mapping class group by extending Theorem A to the case of a surface with two boundary components. Briefly, let Σ 2 g be the compact surface of genus g with two boundary components obtained by removing the interiors of two disks in Σ g that are interchanged by ι. Again, there is an induced hyperelliptic involution of Σ 2 g which we will also call ι. The homeomorphism ι interchanges the two boundary components of Σ 2 g . We can define SMod 2 g as before. The Torelli group I 2 g is the kernel of the action of SMod 2 g on H 1 (Σ 2 g , P ; Z), where P is a pair of points, one on each boundary component of Σ 2 g . The hyperelliptic Torelli group SI 2 g = SMod 2 g ∩I 2 g is then isomorphic to BI 2g+2 . The n = 2g + 2 case of Theorem C translates to the fact that SI 2 g is generated by Dehn twists about symmetric separating curves. See [11] for more details.
Prior results. Theorem A was previously known for g ≤ 2. It is a classical fact that I g = 1 for g ≤ 1, so SI g is trivial in these cases. When g = 2, all essential curves in Σ g are homotopic to symmetric curves. This implies that SMod 2 = Mod 2 and SI 2 = I 2 (see, e.g., [17,Section 9.4.2]). The group I 2 is generated by Dehn twists about separating curves; in fact, Mess [32] proved that I 2 is a free group on an infinite set of Dehn twists about separating curves (McCullough-Miller [30] previously showed I 2 was infinitely generated). This implies that SI 2 is generated by Dehn twists about symmetric separating curves.
Theorem A was known for n ≤ 6. The group BI n is trivial for n ≤ 3. Smythe showed [40] that BI 4 ∼ = F ∞ . He also identified the generating set from Theorem C. The group BI 4 is isomorphic to the stabilizer in SI 2 of a nonseparating simple closed curve, and so Smythe's theorem can be considered as a precursor to Mess's theorem. Next, for g ≥ 1 the first two authors proved [11,Theorem 4.2] that BI 2g+1 is isomorphic to SI g × Z, and so by Mess's theorem BI 5 is isomorphic to F ∞ × Z and further it satisfies Theorem C. The first two authors also proved [11, Theorem 1.2] that BI 2g+2 is isomorphic to (BI 2g+1 /Z) ⋉ F ∞ and that each element of the F ∞ subgroup is a product of squares of Dehn twists about curves surrounding odd numbers of marked points and so again by Mess's theorem we obtain that BI 6 is isomorphic to F ∞ ⋉ F ∞ and that it also satisfies Theorem C.
Aside from our Theorem A, little is known about SI g when g ≥ 3. Letting H = H 1 (Σ g ; Z), Johnson [23,24] constructed a Mod g -equivariant homomorphism τ : I g → (∧ 3 H)/H and proved that ker(τ ) is precisely the subgroup K g of I g generated by Dehn twists about (not-necessarily-symmetric) separating curves. Since ι acts by −I on (∧ 3 H)/H, it follows that SI g < K g . Despite the fact that SI g has infinite index in K g , Childers [15] showed that these groups have the same image in the abelianization of I g .
Birman [6] and Powell [35] showed that I g is generated by bounding pair maps and Dehn twists about separating curves; other proofs were given by Putman [36] and by Hatcher-Margalit [21]. One can find bounding pair maps T y T −1 z such that ι exchanges y and z (see Figure 1); however, these do not lie in SI g since ιT y T −1 z ι −1 = T z T −1 y . In fact, since no power of a bounding pair map is in ker(τ ), there are no nontrivial powers of bounding pair maps in SI g .
With Childers, the first two authors proved that SI g has cohomological dimension g − 1 and that H g−1 (SI g ; Z) has infinite rank [13]. This implies SI 3 is not finitely presentable. It is not known, however, whether SI g , or even H 1 (SI g ; Z), is finitely generated for g ≥ 3.
Approach of the paper. The simplest proofs that the mapping class group is generated by Dehn twists or that the Torelli group is generated by separating twists and bounding pair maps rely on the connectivity of certain complexes of curves. One natural complex in our setting has vertices in bijection with the SI g -orbit of the isotopy class of a symmetric nonseparating curve and edges for curves with the minimal possible intersection number. However, we do not know if this complex is connected, so our proof requires a new approach.
First, to prove Theorems A and C, it suffices to prove Theorem C for n = 2g + 1. Indeed, we already said that the n = 2g + 1 case of Theorem C is equivalent to the genus g case of Theorem A and the first two authors proved [11,Theorems 1.4 and 4.2] that the n = 2g + 1 case of Theorem C implies the n = 2g + 2 case of Theorem C.
This isomorphism can be viewed as a finite presentation for Sp 2g (Z) [2] since PB 2g+1 is finitely presented and Θ 2g+1 has a finite normal generating set. There are several known presentations for Sp 2g (Z). Also, there are standard tools for obtaining finite presentations of finite-index subgroups of finitely presented groups (e.g. Reidemeister-Schreier). However, they all explode in complexity as the index of the subgroup grows. And even if we had some finite presentation for Sp 2g (Z) [2], there is no reason to hope that such a presentation would be equivalent to the one given by the (purported) isomorphism PB 2g+1 /Θ 2g+1 ∼ = Sp 2g (Z) [2].
What we do instead is to apply a theorem of the third author in order to obtain an infinite presentation of Sp 2g (Z) [2] with a certain amount of symmetry, and then introduce a new method for changing this presentation into the finite presentation PB 2g+1 /Θ 2g+1 . The tools we construct should be useful in other contexts. In fact, they have already been used by the last two authors to give finite presentations for certain congruence subgroups of SL n (Z) which are reminiscent of the standard presentation for SL n (Z); see [29].

Outline of paper
Recall from the introduction that to prove Theorems A, B, and C it is enough to prove Theorem C for n = 2g + 1. Since BI 2g+1 is known to satisfy Theorem C for 0 ≤ g ≤ 2, we can apply induction with g = 2 as the base case. Instead of proving Theorem C directly, we will work with a mild rephrasing, namely, Proposition 2.1 below. After stating this proposition, we give an outline of the proof and a plan for the remainder of the paper.
The Main Proposition. Recall that Θ 2g+1 is the group generated by squares of Dehn twists about curves surrounding odd numbers of marked points. Denote the quotient PB 2g+1 /Θ 2g+1 by Q g . Since Dehn twists about symmetric separating curves in Σ 1 g correspond to squares of Dehn twists about curves surrounding odd numbers of marked points in D 2g+1 , we have Θ 2g+1 BI 2g+1 and so there is a further quotient map Q g → Sp 2g (Z) [2]. The n = 2g + 1 case of Theorem C is then equivalent to the following. Proposition 2.1. For g ≥ 2, the quotient map π : Q g → Sp 2g (Z) [2] is an isomorphism.
The starting point is the following theorem of the first two authors [11,Theorems 1.3 and 4.2], which makes it easy to recognize when certain elements of BI 2g+1 lie in Θ 2g+1 (or, when the corresponding elements of ker π are trivial). We say an element of BI 2g+1 is reducible if it fixes the homotopy class of an essential simple closed curve in D 2g+1 .
This theorem is derived from a version of the Birman exact sequence for SI g . It is used at various points in the proof, specifically Sections 4.4, 4.5, and 5.
To prove Proposition 2.1, it suffices to construct an inverse map φ : Sp 2g (Z) [2] → Q g . Besides Theorem 2.2, there are two main ingredients to the construction. We describe them and at the same time give an outline for the rest of the paper.
1. The first ingredient, Proposition 3.2, is an infinite presentation for Sp 2g (Z) [2]. This presentation has two key properties: first, the set of generators is the union of the stabilizers of nontrivial elements of Z 2g , and second, the action of Sp 2g (Z) on Sp 2g (Z) [2] by conjugation permutes the generators and relations in a natural way. This presentation is obtained by considering the action of Sp 2g (Z) [2] on a certain simplicial complex IB g (Z) which itself admits an Sp 2g (Z) action.
The method for constructing such infinite presentations from group actions is due to the third author, who used it to construct an infinite presentation of the Torelli group [37]. The theory requires the complexes being acted upon to have certain connectivity properties, and our contribution is to verify these properties.
2. Our second ingredient is an action of Sp 2g (Z) on Q g that is compatible (via π) with the action of Sp 2g (Z) on Sp 2g (Z) [2] in a sense made precise by Proposition 4.1. We construct the action by declaring where each generator of Sp 2g (Z) sends each generator of Q g and then checking that all relations in both groups are satisfied. This step uses a mixture of surface topology and combinatorial group theory.
We deal with the above two ingredients in Sections 3 and 4, respectively. In Section 5 we will use Theorem 2.2 to define a homomorphism (Sp 2g (Z) [2]) v → Q g , where (Sp 2g (Z) [2]) v is the stabilizer of some fixed v ∈ Z 2g . Then, we will use the fact that Sp 2g (Z) [2] and Q g have compatible Sp 2g (Z) actions to propagate this to a globally-defined map Sp 2g (Z) [2] → Q g , thus proving Proposition 2.1.
3 An infinite presentation for Sp 2g (Z) [2] In this section, we discuss the first ingredient from Section 2, a special kind of presentation for the group Sp 2g (Z) [2]. The presentation will be derived from a general theorem of the third author about obtaining a presentation of a group from its action on a simplicial complex.

Setup
Our first goal is to give a precise statement of the desired presentation of Sp 2g (Z), namely Proposition 3.2 below.
A presentation theorem. Let G be a group acting without rotations on a simplicial complex X; this means that if an element of G preserves some simplex of X then it fixes that simplex pointwise. For a simplex σ, write G σ for the stabilizer of σ. Also write X (0) for the vertex set of X. There is a homomorphism induced by the various inclusions G v → G. If a ∈ G stabilizes v ∈ X (0) , then we denote by a v the image of a under the inclusion There are some obvious elements in ker(ψ). First, if e is an edge with vertices v and v ′ and a ∈ G e then a v a −1 v ′ ∈ ker(ψ). We call these the edge relators.
for a ∈ G v and b ∈ G w . We call these the conjugation relators. The following theorem of the third author [38] states that under certain circumstances these two types of relators suffice to normally generate ker(ψ).
Theorem 3.1. Let G be a group acting without rotations on a simplicial complex X. Assume that X is 1-connected and X/G is 2-connected. Then where R is the normal closure of the edge and conjugation relators.
Symplectic bases. Let R be either Z or a field and letî be the standard symplectic form on R 2g . A symplectic basis for R 2g is a pair of g-tuples ( a 1 , . . . , a g ; b 1 , . . . , b g ) of elements of R 2g that together form a free basis and satisfŷ where δ ij is the Kronecker delta. A partial symplectic basis is a tuple ( a 1 , . . . , a k ; b 1 , . . . , b ℓ ) of elements of R 2g so that there exist a k+1 , . . . , a g , b ℓ+1 , . . . , b g ∈ R 2g with ( a 1 , . . . , a g ; b 1 , . . . , b g ) a symplectic basis. We allow k = 0 or ℓ = 0 in this notation.
The augmented complex of lax isotropic bases. As we will explain shortly, the action of Sp 2g (Z) [2] on IB g (Z) satisfies the hypotheses of Theorem 3.1 for g ≥ 4. For the special case of g = 3, we will need a different complex obtained by attaching some cells to IB g (Z) (the vertex set of our new complex is the same as for IB g (Z)). We make the following definitions. A simplex {( a 1 ) ± , . . . , ( a k ) ± } of IB g (R) will be called a standard simplex. If ( a 1 , . . . , a k ; b 1 ) is a partial symplectic basis, then the set {( a 1 ) ± , . . . , ( a k ) ± , ( b 1 ) ± } is a simplex of intersection type. If ( a 1 , . . . , a k ; ) is a partial symplectic basis, the sets will be called simplices of additive type. Since ( a 1 , − a 2 , a 3 , . . . , a k ; ) is also a partial symplectic basis, the sets are also simplices of additive type. Similarly, the sets {(± a 1 ± a 2 ) ± , ( a 1 ) ± , ( a 2 ) ± , . . . , ( a k ) ± } and {(± a 1 ± a 2 ± a 3 ) ± , ( a 1 ) ± , ( a 2 ) ± , . . . , ( a k ) ± } are also simplices of additive type for any choice of signs. The augmented complex of lax isotropic bases, denoted IB g (R), is the simplicial complex whose simplices are the standard simplices and the simplices of additive and intersection type.
The presentation of Sp 2g (Z) [2]. The main result of this section is the following.
where R is the normal closure of the edge and conjugation relators.  1. The group Sp 2g (Z) [2] acts without rotations on IB g (Z) and we have an isomorphism of CW complexes: IB g (Z)/ Sp 2g (Z)[2] ∼ = IB g (Z/2).

The group
Sp 2g (Z) [2] acts without rotations on IB g (Z) and we have an isomorphism of CW complexes: Remark. In the above proposition, recall that if a group G acts without rotations on a simplicial complex X, then X/G is a CW complex in a natural way, but is not necessarily a simplicial complex; for instance, consider the usual action of Z on R, where R is triangulated with vertex set Z. The above proposition says that this kind of pathology does not happen for the actions of Sp 2g (Z) [2] on IB g (Z) and on IB g (Z).
Proposition 3.4. If R is either Z or a field, then the complex IB g (R) is homotopy equivalent to a wedge of (g − 1)-spheres. In particular, the complexes IB g (Z) and IB g (Z/2) are both 2-connected for g ≥ 4.

The quotient of the complex of lax isotropic bases
In this section, we prove Proposition 3.3, which describes the restriction to Sp 2g (Z) [2] of the actions of Sp 2g (Z) on IB g (Z) and on IB g (Z). Let r : Z 2g → (Z/2) 2g be the standard projection. Also observe that in (Z/2) 2g there is no difference between a vector and a lax vector. To simplify our notation, we will write the vertices of IB g (Z/2) and IB g (Z/2) simply as vectors. Observe that for a lax vector The proof of Proposition 3.3 has three ingredients. The first is Corollary 3.7 below, which says that the actions in question are without rotations. We require a lemma.
Proof. Since r preserves the algebraic intersection pairing modulo 2, it follows that r takes each symplectic basis for Z 2g to a symplectic basis for (Z/2) 2g . If σ is a standard simplex or a simplex of intersection type, it follows that {r(( v 0 ) ± ), . . . , r(( v k ) ± )} forms a k-simplex of IB g (Z/2) of the same type as σ. If σ is of additive type, then up to reindexing and changing the signs of the v i we can assume that It follows immediately from Lemma 3.6 that r induces simplicial maps ζ : IB g (Z) → IB g (Z/2) and ζ : IB g (Z) → IB g (Z/2) and that both maps take k-simplices onto k-simplices.
Corollary 3.7. The actions of Sp 2g (Z) [2] on IB g (Z) and IB g (Z) are without rotations.
Proof. Since ζ and ζ are Sp 2g (Z)[2]-invariant, Lemma 3.6 implies that the vertices of a simplex of IB g (Z) lie in different Sp 2g (Z) [2]-orbits, and similarly for IB g (Z).
Our second ingredient is Corollary 3.9 below, which says that the images of ζ and ζ contain every simplex of IB g (Z/2) and IB g (Z/2), respectively. This requires the following lemma, which follows easily from a classical theorem of Newman-Smart [34,Theorem 1] that says that the map Sp 2g (Z) → Sp 2g (Z/2) is surjective.
Proof. The corollary follows from Lemma 3.8 if σ is a standard simplex or a simplex of intersection type (in particular the statement for IB g (Z/2) follows from this). If σ is of additive type, then write Our third ingredient is Corollary 3.11 below, which shows that two cells of IB g (Z) that map to the same simplex of IB g (Z/2) differ by an element of Sp 2g (Z) [2], and similarly for IB g (Z). This requires the following lemma.
Proof. Lemma 3.6 implies σ 1 and σ 2 are simplices of the same type and dimension. We deal with the three types in turn. Observe that Case 1 below is sufficient to deal with IB g (Z).
Case 1. The σ i are standard simplices.
} are equal since these are the unique pairs of elements with nontrivial pairing under the symplectic form. If necessary, we replace ( a 1 , b 1 ) with ( b 1 , − a 1 ) in order to ensure that r( a 1 ) = r( a ′ 1 ) and r( b 1 ) = r( b ′ 1 ); this does not change the fact that ( a 1 , . . . , a k ; b 1 ) is a partial symplectic basis. We can further reorder the vertices of σ 1 so that r( [2] can now be found exactly as in Case 1. We can write . . , a ′ ℓ } are the unique minimal linearly dependent sets, respectively. They both must map to the unique minimal linearly dependent set among nonempty subsets of however, after doing this we can only assume that v ′ = h i=1 e i a ′ i for some choices of e i = ±1 (here we are using the fact that there is a linear dependence among and so M ( σ 1 ) = σ 2 , as desired.
Proof of Proposition 3.3. We will deal with IB g (Z); the other case is similar. Corollary 3.7 says that Sp 2g (Z) [2] acts without rotations on IB g (Z). We must identify the quotient. Lemma 3.6 gives an Sp 2g (Z) [2]-invariant projection map ζ : IB g (Z) → IB g (Z/2). This induces a map η : [2] its natural CW complex structure (see the remark after the statement of Proposition 3.3), Lemma 3.6 implies that IB g (Z)/ Sp 2g (Z) [2] is a regular CW complex (i.e. attaching maps are injective) and that for each cell σ of IB g (Z)/ Sp 2g (Z) [2] the map η restricts to a homeomorphism from σ onto a simplex of IB g (Z/2). Corollary 3.9 implies that the image of η contains every simplex of IB g (Z/2), and Corollary 3.11 implies that distinct cells of IB g (Z)/ Sp 2g (Z) [2] are mapped to distinct simplices of IB g (Z/2). We conclude that η is an isomorphism of CW complexes, as desired.

Connectivity of the complex of lax isotropic bases
In this section, we prove Proposition 3.4, which states that for R either a field or Z, the complex IB g (R) is homotopy equivalent to a wedge of (g − 1)-spheres. The proof is similar to a proof of a related result due to Charney; see [14,Theorem 2.9]. Before we begin to prove Proposition 3.4, we recall some basic generalities about posets.
Posets. Let P be a poset. Consider p ∈ P . The height of p, denoted ht(p), is the largest number k such that there exists a strictly increasing chain We will denote by P >p the subposet of P consisting of elements strictly greater than p. Also, if f : Q → P is a poset map, then Finally, the geometric realization of P , denoted |P |, is the simplicial complex whose vertices are elements of P and whose k-simplices are sets {p 0 , . . . , p k } of elements of P satisfying A key example is as follows. Let X be a simplicial complex. Then the set P(X) of simplices of X forms a poset under inclusion and |P(X)| is the barycentric subdivision of X.
We shall require the following version of Quillen's Theorem A [39, Theorem 9.1]. In what follows, when we say that a poset has some topological property, we mean that its geometric realization has that property.
Theorem 3.12. Let f : Q → P be a poset map. For some m, assume that P is homotopy equivalent to a wedge of m-spheres. Also, for all p ∈ P assume that P >p is homotopy equivalent to a wedge of (m − ht(p) − 1)-spheres and that f /p is homotopy equivalent to a wedge of ht(p)-spheres. Then Q is homotopy equivalent to a wedge of m-spheres.
In our application of Theorem 3.12, we will take Q to be IB g (R). The roles of P and f /p will be played by the Tits building T g (R) and the complex of lax partial bases B n (R), both to be defined momentarily. Theorems 3.13 and 3.14 below give that IB g (R) and T g (R) (and the natural map between them) satisfy the hypotheses of Theorem 3.12 with m = g − 1, and so we will conclude that IB g (R) is a wedge of (g − 1)-spheres, as desired.
Buildings. Let F be a field andî the standard symplectic form on F 2g . An isotropic subspace of F 2g is a subspace on whichî vanishes. The Tits building T g (F) is the poset of nontrivial isotropic subspaces of F 2g . The key theorem about the topology of T g (F) is the Solomon-Tits theorem [1, Theorem 4.73].
Theorem 3.13 (Solomon-Tits). If F is a field, then T g (F) is homotopy equivalent to a wedge of (g − 1)-spheres. Also, for V ∈ T g (F) the poset (T g (F)) >V is homotopy equivalent to a wedge of (g − 2 − ht(V ))-spheres.
Complexes of lax partial bases. For R equal to either Z or a field, let B n (R) be the simplicial complex whose k-simplices are sets is a set of elements of R n that forms a basis for a free summand of R n . We then have the following theorem.
Theorem 3.14. If R is either Z or a field, then B n (R) is homotopy equivalent to a wedge of (n − 1)-spheres.
Proof. Let B ′ n (R) be the simplicial complex whose k-simplices are sets { v 0 , . . . , v k } of elements of R n that form a basis for a free summand of R n . In his unpublished thesis [26], Maazen proved that under our assumption that R is either Z or a field, B ′ n (R) is (n − 2)connected, and thus is homotopy equivalent to a wedge of (n − 1)-spheres. For R = Z, there is a published account of Maazen's theorem in [16, Proof of Theorem B, Step 2]. This proof can be easily adapted to work for any Euclidean domain R by replacing all invocations of the absolute value function | · | on Z with the Euclidean function on R; in particular, the proof works for a field. There is a natural map ρ : It follows that ρ induces a surjection on all homotopy groups, so B n (R) is also (n − 2)-connected and thus homotopy equivalent to a wedge of (n − 1)-spheres.
Connectivity of IB g (R). We are almost ready to prove Proposition 3.4, which asserts that IB g (R) is homotopy equivalent to a wedge of (g − 1)-spheres for R equal to Z or a field. We first need the following classical lemma.
This allows us to split the short exact sequence of Z-modules , so Theorem 3.14 says that span/V is homotopy equivalent to a wedge of ht(V )spheres. Theorem 3.13 says that the remaining assumptions of Theorem 3.12 are satisfied for span with m = g − 1. The conclusion of this theorem is that P(IB g (R)), hence IB g (R), is a wedge of (g − 1)-spheres, as desired.

Connectivity of IB g (R)
We now prove Proposition 3.5, which asserts that the complexes IB 3 (Z) and IB 3 (Z/2) are 1-connected and 2-connected, respectively. The proof is more complicated than the one for Proposition 3.4, and so we begin with an outline. For R either Z or a field, denote by IB α g (R) the subcomplex of IB g (R) consisting of standard simplices and simplices of additive type. The proof of Proposition 3.5 consists of three steps: 1. We show that the map π k (IB α 3 (R)) → π k ( IB 3 (R)) is surjective for k = 1, 2.
Pushing into IB α 3 (R). As above, the first step of the proof is to show that IB α g (R) carries all of π 1 ( IB g (R)) and π 2 ( IB g (R)). We in fact have a more general statement.
Proof. Let S be a simplicial complex that is a combinatorial triangulation of a k-sphere (recall that a combinatorial triangulation of a manifold is a triangulation where the link of each d-simplex is a triangulation of a (k − d − 1)-sphere) and let f : S → IB g (R) be a simplicial map. It is enough to homotope f such that its image lies in IB α g (R). If the image of f is not contained in IB α g (R), then there exists some simplex σ of S such that f (σ) is The key observation is that Link IBg(R) {( a) ± , ( b) ± } can only contain standard simplices, and moreover all of its vertices are lax vectors ( v) ± such thatî( a, v) =î( b, v) = 0. Indeed, if a simplex {( w 0 ) ± , . . . , ( w k ) ± } spans a simplex with the edge {( a) ± , ( b) ± }, then necessarily {( a) ± , ( b) ± , ( w 0 ) ± , . . . , ( w k ) ± } is a simplex of intersection type, which implies the observation.
Theî-orthogonal complement of span( a, b) is isomorphic to a 2(g − 1)-dimensional free symplectic module over R, so we deduce that we conclude that there exists a combinatorially triangulated (k − d)-ball B with ∂B = Link S (σ) and a simplicial map We can therefore homotope f so as to replace f | Star S (σ) with F | B . This eliminates σ without introducing any new d-dimensional simplices mapping to 1simplices of intersection type. Doing this repeatedly homotopes f so that its image contains no simplices of intersection type.
Let Y g be the join of g copies of S 0 , so Y g ∼ = S g−1 . If x i and y i are the vertices of the ith copy of S 0 in Y g , then the simplices of Y g are the nonempty subsets σ ⊂ {x 1 , y 1 , . . . , x g , y g } such that σ contains at most one of x i and y i for each 1 ≤ i ≤ g. Given a symplectic basis B = ( a 1 , . . . , a g ; b 1 , . . . , b g ) for F 2g , we obtain a poset map α B : P(Y g ) → T g (F) as follows.
We then define The resulting map α B : Y g → T g (F) is a (g − 1)-sphere in T g (F). We have the following theorem [1, Theorem 4.73].
Theorem 3.18. Let Q and P be connected posets and f : Q → P a poset map. Assume that f /p is m-connected for all p ∈ P . Then the induced map f * : π k (Q) → π k (P ) is an isomorphism for 1 ≤ k ≤ m.
We will also need the following easy lemma. Lemma 3.19. Any subset of (Z/2) n \ {0} with cardinality at most 4 has one of the forms: where in each set the v i are linearly independent vectors in (Z/2) n .
Proof. Consider V ∈ T g (Z/2), and let d = dim(V Proof of Proposition 3.5. We already explained how the 1-connectivity of IB 3 (Z) and of IB 3 (Z/2) follow from Lemma 3.16. It remains to show that IB 3 (Z/2) is 2-connected. Lemma 3.16 says that the inclusion map IB α 3 (Z/2) ֒→ IB 3 (Z/2) induces a surjection on π 2 . We will prove that it induces the zero map as well.
By Lemma 3.21, it suffices to show that for each symplectic basis B of (Z/2) 6 , the map is nullhomotopic. In the fourth picture in Figure 2, we indicate an explicit nullhomotopy of α B (Y 3 ) in IB 3 (Z/2). Specifically, we realize α B (Y 3 ) as the boundary of a 3-ball formed by four simplices of intersection type: We conclude that the inclusion IB α 3 (Z/2) ֒→ IB 3 (Z/2) induces the zero map on π 2 , as desired.

The symplectic group action
We now discuss the second ingredient for the proof of Proposition 2.1, namely, the group action of Sp 2g (Z) on Q g . The group Sp 2g (Z) acts on Sp 2g (Z) [2] by conjugation. We wish to lift this to an action of Sp 2g (Z) on Q g in a natural way.

Setup
Our first task is to give a precise description of the action we would like to obtain (Proposition 4.1). Recall that Q g = PB 2g+1 /Θ 2g+1 . Define Q g = B 2g+1 /Θ 2g+1 , and let ρ : PB 2g+1 → Q g and ρ : B 2g+1 → Q g π : Q g → Sp 2g (Z) [2] π : Q g → Sp 2g (Z) be the quotient maps. The first two parts of Proposition 4.1 below posit the existence of an action of Sp 2g (Z) on the Q g that is natural with respect to the actions of Sp 2g (Z) on Sp 2g (Z) [2] and Q g on Q g . We will require our action to have one extra property, which requires some setup. Let c 23 be the curve in D 2g+1 ∼ = Σ 1 g /ι shown in Figure 3 and let (PB 2g+1 ) c 23 be its stabilizer. Next, define Finally, let ( v 23 ) ± be the lax vector of H 1 (Σ 1 g ; Z) represented by one component of the preimage of c 23 in Σ 1 g and let (Sp 2g (Z)) ( v 23 ) ± denote the stabilizer. Proposition 4.1. Let g ≥ 3. Assume that BI 2h+1 = Θ 2h+1 for all h < g. There then exists an action of Sp 2g (Z) on the group Q g with the following three properties: 1. for Z ∈ Sp 2g (Z) and η ∈ Q g we have π(Z · η) = Zπ(η)Z −1 , 2. for ν ∈ Q g and η ∈ Q g we have π(ν) · η = νην −1 , and 3. the action of (Sp 2g (Z)) ( v 23 ) ± on Q g preserves Ω 23 .
The second statement of Proposition 4.1 already completely specifies the desired action on a finite-index subgroup of Sp 2g (Z), the image in Sp 2g (Z) of B 2g+1 . The second statement also immediately implies that BI 2g+1 /Θ 2g+1 is central in Q g (of course, our goal is to show that this quotient is trivial).
We will prove Proposition 4.1 in five steps. First, in Section 4.2 we give explicit finite presentations for Q g and Sp 2g (Z). Then in Section 4.3 we propose an action of Sp 2g (Z) on Q g by declaring where each generator of Sp 2g (Z) sends each generator of Q g . Next, in Section 4.4 we check that the proposed action respects the relations of Q g , and in Section 4.5 we check that the proposed action respects the relations of Sp 2g (Z). Finally, in Section 4.6, we verify that the resulting action of Sp 2g (Z) on Q g satisfies the three properties listed in Proposition 4.1.

Presentations for Q g and Sp 2g (Z)
In this section, we give explicit finite presentations Sp 2g (Z) ∼ = S Sp | R Sp and Q g ∼ = S Q | R Q . The trick here is to find just the right balance: more generators for Sp 2g (Z) will mean that checking the well-definedness of our action with respect to the Sp 2g (Z) relations is easier (relations are smaller), but checking the well-definedness with respect to the Q g relations is harder (more cases to check), and vice versa.
Generators for Q g . Since Q g is a quotient of PB 2g+1 , any set of generators for PB 2g+1 descends to a set of generators for Q g . We identify PB 2g+1 with the pure mapping class group of a disk D 2g+1 with 2g+1 marked points p 1 , . . . , p 2g+1 , that is, the group of homotopy classes of homeomorphisms of D 2g+1 that fix each p i and each point of the boundary; see [17,Section 9.3]. For concreteness, we take D 2g+1 to be a convex Euclidean disk and the p i to lie on the vertices of a regular (2g + 1)-gon, appearing clockwise in cyclic order; see Figure 4. Choose this identification so that if c ij is one of the curves c 12 , c 23 , or c 45 in Figure 3, then c ij is the boundary of a convex region containing p i and p j and no other p k .
More generally, for any subset A of {1, . . . , 2g + 1} we denote by c A the simple closed curve in D 2g+1 that bounds a convex region of D 2g+1 containing precisely {p i | i ∈ A} in its interior; this curve is unique up to homotopy in D 2g+1 . We will write c ij or c i,j for c {i,j} , etc., when convenient. The curves c 1234 and c 45 are shown in Figure 4.
Artin proved that PB 2g+1 is generated by the Dehn twists about the curves in the set The resulting generating set for Q g is where by definition the element of Q g associated to s c is ρ(T c ).
Relations for Q g . Our set of relations R Q for Q g will consist of the four families of relations below. Recall that Q g is defined as Q g = PB 2g+1 /Θ 2g+1 . We first give a finite presentation for PB 2g+1 , and then add relations for normal generators of Θ 2g+1 inside PB 2g+1 . There are many presentations for the pure braid group, most notably the original one due to Artin [4]. We will use here a modified version of Artin's presentation due to the second author and McCammond [28, Theorem 2.3]. There are three types of defining relations for PB 2g+1 , as follows; refer to Figure 4. We will write i 1 < · · · < i n to refer to the cyclic clockwise ordering of labels.

Triangle relations: s
3. Crossing relations: [s c ij , s c js s crs s −1 c js ] = 1 if i < r < j < s.
We now add relations coming from Θ 2g+1 . This group is normally generated in PB 2g+1 by the squares of Dehn twists about the convex curves in D 2g+1 surrounding odd numbers of marked points; indeed any two Dehn twists about curves surrounding the same marked points are conjugate in PB 2g+1 ; cf. [17, Section 1.3]. We need to add one relation for each of these elements.
The last relation comes from the following relation in the pure braid group: see [17,Section 9.3].
Transvections in Sp 2g (Z). We now turn to the symplectic group. The transvection on v ∈ Z 2g is the element τ v ∈ Sp 2g (Z) given by whereî is the symplectic form. Note that τ v = τ − v . The group Sp 2g (Z) is generated by transvections on primitive elements of Z 2g . Also, if c is a simple closed curve in Σ 1 g , then the image of the Dehn twist T c ∈ Mod 1 g in Sp 2g (Z) is τ [c] for any choice of orientation of c.
Transvections and simple closed curves. Consider a simple closed curve a in D 2g+1 surrounding an even number of marked points. We construct a transvection associated to a as follows. The preimage of a in Σ 1 g is a pair of disjoint nonseparating simple closed curvesã 1 andã 2 that are homologous (with respect to some choice of orientation). The transvection associated to a is then τ [ã 1 ] = τ [ã 2 ] . We pause now to record the following lemma. Proof. We must determine the image of T a under B 2g+1 L → Mod 1 g → Sp 2g (Z), where L is the lifting map from Section 1. The preimage in Σ 1 g of a is a pair of disjoint simple closed homologous curvesã 1 andã 2 , and L(T a ) = Tã 1 Tã 2 . By the definition of τ v , both Tã 1 and Tã 2 map to τ v ∈ Sp 2g (Z), and the lemma follows.
Generators for Sp 2g (Z). Denote by a 0 the convex simple closed curve c 1234 . Also, for 1 ≤ i ≤ 2g, set a i = c i,i+1 . Humphries (see [17,Section 4]) proved that one can choose connected componentsã i of a i in Σ 1 g such that Mod 1 g is generated by the Dehn twists about the curvesã 0 , . . . ,ã 2g (in fact, any set of choices will do). Since Mod 1 g surjects onto Sp 2g (Z), it follows that the transvections associated to a 0 , a 1 , . . . , a 2g generate Sp 2g (Z).
In order to simplify our presentation for Sp 2g (Z) we need to add some auxiliary generators to Sp 2g (Z). Consider the following curves: The resulting generating set for Sp 2g (Z) is where the element t a ∈ Sp 2g (Z) associated to the generator t a is the transvection associated to a as above. It is remarkable that all of the curves in the generating set are convex.
Relations for Sp 2g (Z). Our set of relations R Sp for Sp 2g (Z) will consist of the six families of relations below. Since Sp 2g (Z) ∼ = Mod 1 g /I 1 g , we obtain a presentation for Sp 2g (Z) by starting with a presentation for Mod 1 g and adding one relation for each normal generator of I 1 g in Mod 1 g . Wajnryb gave a finite presentation for Mod 1 g with Humphries' generating set {Tã 0 , . . . , Tã 2g }; see Wajnryb's original paper [42] and the erratum by Birman and Wajnryb [9]. The image of Tã i in Sp 2g (Z) is t a i and so we obtain the first part of our presentation for Sp 2g (Z) by replacing each Tã i in Wajnryb's presentation with t a i . We have the following list of relations, derived from the Wajnryb's standard presentation [17,Theorem 5.3]. Here i(·, ·) denotes the geometric intersection number of two curves.

Disjointness relations: t
In the lantern relation, we have replaced some complicated expressions from Wajnryb's relations with some of our auxiliary generators. Thus, similar to the reference [17, Theorem 5.3], we need to add relations that express each of these generators in terms of the t a i .

Auxiliary relations:
The auxiliary generators were introduced exactly so that we could break up the lantern relation into these shorter auxiliary relations. This feature will be used in Section 4.5. By work of Johnson [22], the group I 1 g is normally generated by a single bounding pair map of genus 1 when g ≥ 3. Thus, to obtain our presentation for Sp 2g (Z), we simply need one more relation.
6. Bounding pair relation: t a 0 = t b 0 , where t b 0 is as in the 3-chain relation above.
Generators and intersection numbers. In choosing auxiliary generators for Sp 2g (Z), we were careful not to introduce too many new generators; by inspection, we see that all generators satisfy the following useful property, used several times below.

Construction of the action
Sp and s ∈ S Q . The goal of this section is to construct an element t ⊚ s ∈ Q g that satisfies the naturality property where w denotes the image of an element of the free group on S Q or S Sp in the corresponding group; this is Proposition 4.5 below. We will show in Sections 4.4 and 4.5 that there is an action of Sp 2g (Z) on Q g defined by t · s = t ⊚ s and in Section 4.6 we will show that this action satisfies Proposition 4.1.
Analogy with transvections. For a transvection τ w ∈ Sp 2g (Z) and a square of a transvection τ 2 v ∈ Sp 2g (Z) [2], we have Since transvections generate Sp 2g (Z) and squares of transvections generate Sp 2g (Z) [2], the action of Sp 2g (Z) on Sp 2g (Z) [2] is completely described by this formula. If we write w + ( v) for τ w ( v), then this formula becomes τ w τ 2 v τ −1 w = τ 2 w + ( v) . In other words, the action of Sp 2g (Z) on Sp 2g (Z) [2] is given by an "action" of Z 2g on itself. Our strategy is to give an analogous action of the set of curves in D 2g+1 on itself, and use this to define each t a ⊚ s c .
An action of curves on curves. If d and e are two collections of pairwise disjoint simple closed curves in D 2g+1 in minimal position, we define Surger(d, e) to be the collection of simple closed curves obtained from d ∪ e by performing the surgery shown in the left-hand side of Figure 5 at each point of d ∩ e and then discarding any inessential components. Note that this definition does not depend on any orientations of the elements of d or e.
Next, for two simple closed curves a and c in D 2g+1 we define |î|(a, c) to be the absolute value of the algebraic intersection number of any two connected components of the preimages of a and c in Σ 1 g . These curves do not have a canonical orientation, so the algebraic intersection is not itself well defined. Also, let na denote n parallel copies of the curve a. Note that Surger(i(a, c)a, c) = T a (c).
We now give our "action" of C(S Sp ) on C(S Q ). For a ∈ C(S Sp ) and c ∈ C(S Q ), we define a + (c) = Surger(|î|(a, c) Tã(c) Figure 6: A "fake half-twist." The preimage in Σ 1 g of a isã ∪ ι(ã), andc is one component of the preimage of c . Thus |î|(a, c) = 1 and a + (c) = Surger(a, c) is as shown.
It is important here that the curves a and c are in minimal position.
The effect of the action on homology. In the case that a surrounds two marked points and intersects c in two points, then a + (c) is precisely the image of c under the positive half-twist H a about a; see Figure 5. By definition, t a is the image in Sp 2g (Z) of Tã, wherẽ a is one component of the preimage of a in Σ 1 g . But since Tã is the image of H a in SMod 1 g , it follows that t a ([c]) is represented by one of the components of the preimage of a + (c) in Σ 1 g . This naturality property is precisely the reason for our definition a + (c). When a surrounds four or more marked points, the situation is more subtle. Consider the curves a = a 0 and c = c 45 ; these curves and their preimagesc andã ∪ ι(ã) in Σ 1 g are shown in Figure 6. The transvection t a is the image in Sp 2g (Z) of Tã. This Dehn twist does not lie in SMod 1 g , and so does not project to a homeomorphism of D 2g+1 . However, the curve Tã(c), which represents t a ([c]), still projects to a simple curve in D 2g+1 , namely a + (c). So we again have the same naturality property as in the previous paragraph, that t a ([c]) is represented by a lift of a + (c), even though a + (c) is not derived from c via a homeomorphism of D 2g+1 . We now show that this naturality property holds in general. Proof. We only treat the case of a + (c) with the other case being completely analogous. We begin with the first statement, that a + (c) is connected and surrounds an even number of marked points. The geometric intersection number i(a, c) is equal to 0, 2, or 4; this is because c ij is the boundary of a regular neighborhood of the straight line segment connecting p i to p j , and a straight line segment can intersect a convex curve in 0, 1, or 2 points (cf. Lemma 4.3). We treat each of the three cases in turn.
If i(a, c) = 0, then |î|(a, c) = 0. Thus, a + (c) is equal to c, which is a simple closed curve. If i(a, c) = 2, then we claim that |î|(a, c) = 1. Indeed, the arc of a crossing through c necessarily separates the two marked points inside c from each other, creating two bigons, each containing one marked point. The preimage of one bigon in Σ 1 g is a square whose four corners are the four intersection points of the preimages of a and c. We know that the hyperelliptic involution ι interchanges the two lifts of each curve and that ι rotates the square by π. Our claim follows. It thus remains to check that Surger(a, c) is a simple closed curve surrounding an even number of marked points, which is immediate from Figure 7.
If i(a, c) = 4, then we claim that |î|(a, c) is equal to either 0 or 2, depending on whether the arcs of c divide the marked points inside a into two sets of even cardinality or odd cardinality, respectively. The curve a divides the convex region bounded by c into three connected components: one square and two bigons, each with one marked point. Consider the union of the square and one bigon. The preimage in Σ 1 g is a rectangle made up of three squares; there is one central square (the preimage of the bigon) and two other squares with edges glued to the left and right edges of the central square. Since each intersection point in D 2g+1 lifts to two intersection points in Σ 1 g , and since we already see 8 intersection points on the boundary of this rectangle, we conclude that this picture contains all of the intersection points of preimages of a and c. Also, by construction the horizontal sides of the rectangle belong to preimages of c. The involution ι acts on this rectangle, rotating it by π though the center. We also know that ι interchanges the two preimages of each of a and c. Therefore, it suffices to count the intersections of the bottom of the rectangle with the vertical sides of the rectangle belonging to a single component of the preimage of a. Again because ι exchanges the two components of the preimage of a, two of the vertical segments belong to one component, and two to the other. Thus, if we choose one component of the preimage of a, it intersects the bottom edge of the rectangle in precisely two points. It immediately follows that |î|(a, c) is equal to either 0 or 2, as claimed. By the claim, it suffices to check that Surger(2a, c) is a simple closed curve surrounding an even number of marked points, which is again immediate from Figure 7.
We now address the second statement of the lemma. The preimage in Σ 1 g of |î|(a, c)a ∪ c is a symmetric configuration (that is, preserved by ι). It contains both preimages of c and |î|(a, c) parallel copies of each preimage of a. We orient these so all preimages of a represent the same element of H 1 (Σ 1 g ; Z). We do the same for c; there are two choices, and we use the one that is consistent with the surgery in Figure 5. When we perform surgery on this configuration, we therefore obtain a symmetric representative of the homology class This symmetric representative is the preimage of a + (c) and so the first statement of the lemma implies that this representative has exactly two connected components that are interchanged by the hyperelliptic involution. It follows that each component, in particular a + (c), represents But (up to sign) this is equal to t a ([c]), and the lemma is proven.
Definition of t ⊚ s. We can now define the elements t ±1 a ⊚ s c ∈ Q g for t a ∈ S Sp and s c ∈ S Q : t a ⊚ s c = ρ T a + (c) and t −1 a ⊚ s c = ρ T a − (c) . These are both well-defined elements of Q g since T a ± (c) only depends on the homotopy class of a ± (c), and we already said that the latter is a well-defined simple closed curve.
Naturality. We now verify the naturality property (1) from the start of this section.
Proposition 4.5. For any t a ∈ S Sp and s c ∈ S Q and ǫ ∈ {−1, 1}, we have Proof. To simplify notation, we will treat the case ǫ = 1; the other case is essentially the same. Let a + (c) be one component in Σ 1 g of the preimage of a + (c). We have that

Well-definedness with respect to Q g relations
For t ∈ S ±1 Sp and s ∈ S Q , we have now defined an element t ⊚ s in Q g . Recall our goal is to show that the formula t a · s c = t a ⊚ s c defines an action of Sp 2g (Z) on Q g . However, at this point if we use this formula we do not even know that (t a ) −1 · (t a · s c ) is equal to s c .
Let F (S Q ) denote the free group on S Q . For each t ∈ S ±1 Sp what we do have now is a homomorphism F (S Q ) → Q g given by s → t ⊚ s for s ∈ S Q (so it makes sense to write t ⊚ w for w ∈ F (S Q )). The next proposition says that each of these homomorphisms respects the relations of Q g , which is to say that each of these homomorphisms induces an endomorphism of Q g . To put it another way, the free monoid F (S ±1 Sp ) acts on the group Q g . In the next section we will show that this monoid action descends to a group action of Sp 2g (Z) on Q g . Proposition 4.6. Let g ≥ 3 and assume BI 2h+1 = Θ 2h+1 for h < g. For all t ∈ S ±1 Sp and r ∈ R Q , we have t ⊚ r = 1.
We introduce the following terminology, which will also be used in Section 4.5. Let d be an essential simple closed curve in D 2g+1 . An element of Q g is said to be reducible along d if it is the image of an element of PB 2g+1 that preserves the isotopy class of d. The next lemma is an immediate consequence of Theorem 2.2.
We need one more basic lemma about reducible elements, Lemma 4.9 below, but first we require a subordinate lemma, which follows immediately from Proposition 4.5.
Lemma 4.8. Let r ∈ F (S Q ) be a relator for Q g and let t ∈ S ±1 Sp . Then π(t ⊚ r) = 1. Lemma 4.9. Assume BI 2h+1 = Θ 2h+1 for h < g. Let t a ∈ S Sp , let ǫ = ±1, and let r ∈ F (S Q ) be a relator for Q g . Suppose there is an essential simple closed curve d in D 2g+1 disjoint from a and from each curve c of C(S Q ) such that s ±1 c appears in r. Then t ǫ a ⊚ r = 1.
Proof. Write r = s ǫ 1 c i 1 j 1 · · · s ǫn c injn with ǫ i = ±1. By hypothesis each c i k j k is disjoint from d. By definition of t ǫ a ⊚ r, we have: As a is disjoint from d and each c i k j k is disjoint from d, it follows from the definition of the action that each t ǫ a ⊚ s c i k j k is reducible along d (that is, if we surger two curves that are disjoint from d, the result is disjoint from d). Since the set of elements of Q g that are reducible along d forms a subgroup of Q g , it follows that t ǫ a ⊚ r is reducible along d. By Lemma 4.8, π(t ǫ a ⊚ r) = 1. Lemma 4.7 thus implies that t ǫ a ⊚ r = 1.
The next two lemmas give generating sets for two kinds of subgroups of PB 2g+1 . The first follows from the fact that any inclusion D n → D 2g+1 respecting marked points induces an inclusion on the level of mapping class groups [17,Theorem 3.18].
Lemma 4.10. Let ∆ be a convex subdisk of D 2g+1 containing n marked points in its interior. Then the subgroup of PB 2g+1 consisting of elements with representatives supported in ∆ is isomorphic to PB n and is generated by the Dehn twists T c ij with p i , p j ∈ ∆.
Lemma 4.11. Let 1 ≤ i, j ≤ n be consecutive integers modulo n. Then the stabilizer in PB n of the curve c ij is generated by the Dehn twists about curves in the set Proof. Let (PB n ) c ij denote the stabilizer in PB n of c ij , and let γ ij denote the straight line segment connecting p i to p j . Any element of the group (PB n ) c ij has a representative that preserves γ ij . Any such homeomorphism descends to a homeomorphism of the disk with n − 1 marked points obtained from D n by collapsing γ ij to a single marked point. This procedure gives rise to a short exact sequence: cf. [17,Proposition 3.20]. The curve c ij lies in C ij , and the Dehn twists about the other curves in C ij map to the Artin generators for PB n−1 . The lemma follows.
Finally, in light of Lemma 4.11, we need to understand t ⊚ w, where t ∈ S ±1 Sp and w is an element of F (S Q ) mapping to ρ(T c ijk ) ∈ Q g . We can obtain an explicit such w using the relation in PB 2g+1 mentioned immediately after the list of relators for Q g were introduced. Indeed, if s c ijk ∈ F (S Q ) is the element s c ij s c jk s c ik , this relation tells us that s c ijk = ρ(T c ijk ). We observe that, as an element of F (S Q ), this s c ijk depends on the order of {i, j, k} (not just their cyclic order), though its image in Q g only depends on the cyclic order. Lemma 4.12. Consider 1 ≤ i, j, k ≤ 2g + 1 with i < j < k (up to cyclic permutation). If a = c A ∈ C(S Sp ) satisfies i(a, c ij ) = 0, then t ±1 a ⊚ s c ijk is reducible along c ij .
Proof. We will deal with t a ⊚ s c ijk ; the proof for t −1 a ⊚ s c ijk is similar. If |î|(a, c jk ) = |î|(a, c ik ) = 0 (which holds in particular when i(a, c jk ) = i(a, c ik ) = 0), then we have a + (c jk ) = c jk and a + (c ik ) = c ik , so Since T c ijk fixes c ij , the lemma follows. We can therefore assume that at least one of |î|(a, c jk ) and |î|(a, c ik ) is nonzero. The proof now divides into three cases depending on A ∩ {i, j, k}. Observe that A is either disjoint from or contains {i, j}. Also, since either |î|(a, c jk ) or |î|(a, c ik ) is nonzero, we cannot have {i, j, k} ⊂ A.
The first case is A ∩ {i, j, k} = {k}; see the top row of Figure 8. In this case, |î|(a, c jk ) = |î|(a, c ik ) = 1, so a + (c jk ) and a + (c ik ) are as in the top row of Figure 8. The key to this step of the proof (as well as the subsequent ones) is the lantern relation in the mapping class group (see [17,Proposition 5.1]). This is a relation between seven Dehn twists that lie in a sphere with four boundary components in any surface; the four-holed sphere in this case is shaded in the top row of Figure 8. The associated lantern relation is We can therefore compute that in order, the equalities use the definition of s c ijk , the definition of t a ⊚ w, the definition of t a ⊚ s c ℓm , the above lantern relation, and the fact that T c i and T c j are trivial. Since the curves d and e are disjoint from c ij , it follows that t a ⊚ s c ijk is reducible along c ij , as desired.
The second case is A ∩ {i, j, k} = {i, j}; refer now to the middle row of Figure 8. In this case, we again have |î|(a, c jk ) = |î|(a, c ik ) = 1, so a + (c jk ) and a + (c ik ) are as shown. Just like in the previous case, we can prove that t a ⊚ s c ijk is reducible along c ij using the indicated lantern relation.
The final case is A ∩ {i, j, k} = ∅; refer to the bottom row of Figure 8. Since at least one of |î|(a, c jk ) and |î|(a, c ik ) is nonzero, we cannot have i(a, c ijk ) = 0. Using Lemma 4.3, we deduce that a and c ijk are as shown. We know that a must surround an even number of marked points, so the parities of the numbers of marked points inside the dotted circles on the bottom row must be the same. If this parity is even, then |î|(a, c ik ) = |î|(a, c jk ) = 0 (cf. the proof of Lemma 4.4). This is excluded by our assumptions (it was dealt with in the first paragraph of this proof), so this parity must be odd. It then follows that |î|(a, c jk ) = |î|(a, c ik ) = 2. Therefore a + (c ij ) and a + (c jk ) and a + (c ik ) are as in the bottom row of Figure 8. Just like in the case A ∩ {i, j, k} = {k}, we can prove that t a ⊚ s c ijk is reducible along c ij using the indicated lantern relation.
Proof of Proposition 4.6. The proof will be broken into two steps. For the first, let R PB ⊂ R Q be the subset consisting of the disjointness, triangle, and crossing relations. As was observed in Section 4.2, we have PB 2g+1 ∼ = S Q | R PB .
Step 1. For t ∈ S ±1 Sp and r ∈ R PB , we have t ⊚ r = 1. Write t = t ǫ a with ǫ = ±1. By Lemma 4.9, it suffices to find an essential simple closed curve d in D 2g+1 disjoint from a and from each curve of C(S Q ) that appears in r.
Denote by ∆ r the convex hull of curves of C(S Q ) that appear in r. Examining the relations in R PB , we see that ∆ r contains at most 4 marked points, and hence there are at least 3 marked points outside of ∆ r .
It follows from Lemma 4.3 that the intersection of a with the closure of the exterior of ∆ r is a union of at most two arcs. These two arcs partition the marked points outside of ∆ r into at most three sets. We deduce that one of the following holds: 1. some pair of marked points can be connected by an arc α disjoint from a ∪ ∆ r , or 2. the convex hull of of a ∪ ∆ r contains at least one marked point in its exterior.
In the first case, we can take d to be the boundary of a regular neighborhood of α. In the second case, we can take d to be the boundary of the convex hull of a ∪ ∆ r .
Step 2. For t ∈ S ±1 Sp and r ∈ R Q an odd twist relator, t ⊚ r = 1. Again, write t = t ǫ a with ǫ = ±1. Consider B ⊂ {1, . . . , 2g + 1} with 3 ≤ |B| ≤ 2g + 1 and |B| odd. There is an odd twist relator r B corresponding to B and its image under the map F (S Q ) → PB 2g+1 is T 2 c B . We need to show t ⊚ r B = 1. It follows from Step 1 that if two elements w and w ′ of F (S Q ) have the same image in PB 2g+1 , then t ⊚ w = t ⊚ w ′ . Thus, we may replace the odd twist relator r B with any element of F (S Q ) whose image in PB 2g+1 is T 2 c B . First we treat the case where there is a marked point p k exterior to both a and c B . Let A k = {1, . . . ,k, . . . , 2g + 1}. By Lemma 4.10, we can write T 2 c B as a product of Dehn twists (and inverse Dehn twists) about the c ij where i, j = k. This implies that there is a product B is a relator for Q g . Since a and each c ij with i, j = k is disjoint from c A k , Lemma 4.9 gives that t ⊚ r ′ B , hence t ⊚ r B , is equal to 1, as desired. Next suppose all marked points lie interior to either a or c B . The proof of this case is similar, but we will have to contend with curves that surround three marked points, not just two, so Lemma 4.9 does not apply directly. To begin, we claim that there exist i, j ∈ B that are consecutive in B (cyclically ordered) such that c ij is disjoint from a.
If there are at least three marked points of D 2g+1 exterior to a, then it follows from Lemma 4.3 that a and B satisfy the claim. The only remaining case for the claim is where g = 3 (so D 2g+1 = D 7 ) and a surrounds 6 marked points. In this case i and j can be taken to be any two marked points that lie inside a and c B and are consecutive in B.
It remains to show that given i, j consecutive in B with c ij disjoint from a, we have t ⊚ r B = 1. By Lemma 4.11, the element T 2 c B is the image in PB 2g+1 of a product r ′ B of s c ∈ F (S Q ) with c ∈ C ij (here we are using the definition of s c ijk given before Lemma 4.12). It follows from Lemma 4.12 that t ⊚ r ′ B is reducible along c ij . It then follows from Lemmas 4.7 and 4.8 that t ⊚ r ′ B , hence t ⊚ r B , is equal to 1.

Well-definedness with respect to Sp 2g (Z) relations
Proposition 4.6 implies that the free monoid F (S ±1 Sp ) acts on Q g ; we write this action as (t, η) → t ⊙ η. By definition, t ⊙ η is equal to t ⊚ w where w ∈ F (S Q ) and η = w is the image of w in Q g .
Let R Sp denote R Sp ∪ {tt −1 | t ∈ S Sp }, thought of as a subset of the free monoid on S ±1 Sp . The next proposition says that the monoid action of F (S ±1 Sp ) on Q g respects the relations in R Sp ; in other words, the monoid action descends to a group action of Sp 2g (Z) on Q g . supported in the interior of d. Continuing inductively, we deduce that r ⊙ s c is reducible along d. Since s c is also reducible along d, we have that s c (r ⊙ s c ) −1 is reducible along d. By Proposition 4.5, π(s c )π(r ⊙ s c ) −1 = 1 in Sp 2g (Z). Then by Lemma 4.7, s c (r ⊙ s c ) −1 is equal to the identity in Q g , as desired.
The third case is when there is a straight line segment connecting consecutive marked points in D 2g+1 and disjoint from c and each curve of C(S Sp ) that appears in r. Let d = c kℓ denote the boundary of a regular neighborhood of this line segment. The argument is similar to the previous case. The only difference is that when we factor the preimage of t n ⊚ s c in PB 2g+1 , we must use Dehn twists about curves that surround two or three marked points and are disjoint from d (that such curves suffice follows from Lemma 4.11). However, we can use the same argument, applying Lemma 4.12 as necessary.
Proof of Proposition 4.13. Examining the relators in R Sp and the elements of C(S Q ) one by one-see Figure 9 for a representative collection-we claim that, with a single exception, each relator r ∈ R Sp satisfies the reducibility criterion with each c ∈ C(S Q ). When g ≥ 4, one can always find a pair of consecutive marked points lying outside each curve in a given relator. The only c then that fails part (2) of the reducibility criterion is one surrounding those two points, but this curve satisfies part (1) of the reducibility criterion. Thus, the claim is a finite check. For each such non-exceptional choice of r ∈ R Sp and c ∈ C(S Q ), Lemma 4.14 applies, and we have that r ⊙ s c = s c .
The exceptional case is where g = 3 and r is the relator r (vii) corresponding to auxiliary relation (vii) and c ∈ C(S Q ) is c 47 . (This is the main place where the auxiliary generators in our presentation for Sp 2g (Z) come into play; if we were to use Wajnryb's presentation without our auxiliary generators, the lantern relation would fail the reducibility criterion with every element of C(S Q ) when g = 3.) It thus remains to show r (vii) ⊙ s c 47 = s c 47 .
Let c 0 denote c 1234567 . Using the relation in PB n from which we derived the odd twist relators for Q g and the fact that T c 0 is central in PB 7 , it follows that PB 7 is generated by the T c with c ∈ (C(S Q ) ∪ {c 0 }) \{c 47 }. Therefore, it suffices to show that r (vii) ⊙ ρ(T c 0 ) = ρ(T c 0 ).
Each of the curves u, u ′ , and v of C(S Sp ) appearing in r (vii) is disjoint from c 17 , as is c 0 . As in the third case in the proof of Lemma 4.14, we can use Lemmas 4.11 and 4.12 with c 17 as the reducing curve to argue that r (vii) ⊙ ρ(T c 0 ) = ρ(T c 0 ), as desired.

Completing the proof of Proposition 4.1
By Propositions 4.6 and 4.13, there is an action of Sp 2g (Z) on Q g given by the formula t · s = t ⊚ s.
It remains to check that this action has all three properties stipulated by Proposition 4.1. We already mentioned that property (1), namely, that π(Z · η) = Zπ(η)Z −1 for Z ∈ Sp 2g (Z) and η ∈ Q g , follows directly from Proposition 4.5.
Property (2) asserts that π(ν) · η = νην −1 for ν ∈ Q g and η ∈ Q g . The half-twists about a 1 , . . . , a 2g are the usual generators for the braid group B 2g+1 . The half-twist H a i maps to t a i ∈ Sp 2g (Z), so to prove property (2) it is enough to show that t ǫ a i · s c jk = ρ(H ǫ a i ) s c jk ρ(H −ǫ a i ) for all choices of i, j, k. Since s c jk = ρ(T c jk ), the right-hand side is equal to s H ǫ a i (c jk ) = ρ(T H ǫ a i (c jk ) ), and so it remains to show that (a i ) ǫ (c jk ) = H ǫ a i (c jk ), where (a i ) ǫ is interpreted as either (a i ) + or (a i ) − . For any choices of i, j, and k, we have i(a i , c jk ) is either 0 or 2, and there is only one configuration in each case up to homeomorphisms of D 2g+1 . In the case i(a i , c jk ) = 0, we have (a i ) ± (c jk ) = H ±1 a i (c jk ) = c jk . It remains to check the case (i, j, k) = (1, 2, 3). As mentioned in Section 4.3, we have in this case (a 1 ) ǫ (c 23 ) = H ǫ a 1 (c 23 ). We now turn to property (3) of Proposition 4.1, that the action of (Sp 2g (Z)) ( v 23 ) ± on Q g preserves Ω 23 , the subgroup consisting of all elements that are reducible along c 23 . We will use the fact that (Sp 2g (Z)) ( v 23 ) ± is generated by the set That Ξ generates is an immediate consequence of the semidirect product decomposition for the stabilizer in Sp 2g (Z) of a primitive lax vector that is given in the proof of Lemma 3.10; this fact can also be proven in much the same way as the level 2 version, Lemma 5.1 below.
Let Υ be the image in Q g of the generating set for (PB 2g+1 ) c 23 from Lemma 4.11. It is enough to show that for x ∈ Ξ and y ∈ Υ, the element x · y is reducible along c 23 . First, if x = −I, then it follows immediately from property (2), the fact that T ∂ D 2g+1 is central in PB 2g+1 and the fact that π • ρ(T ∂ D 2g+1 ) = −I that −I · y = T ∂ D 2g+1 yT −1 ∂ D 2g+1 = y. Next, we may assume that x = −I; note then that x ∈ S Sp . For the elements y ∈ Υ that lie in S Q , the reducibility along c 23 of x · y is obvious from the description of our action in Section 4.3. For the others, it is an immediate consequence of Lemma 4.12. This completes the proof.

The proof of the Main Proposition
In this section, we prove Proposition 2.1 by induction on g using Propositions 3.2 and 4.1. The base case is g = 2, which we already said is known to be true. So we can assume that g ≥ 3 and that the quotient map Q h → Sp 2h (Z) [2] is an isomorphism for h < g. Equivalently, we are assuming that BI 2h+1 = Θ 2h+1 for all h < g and we want to prove that the quotient map π : Q g → Sp 2g (Z) [2] is an isomorphism. The map π is a surjection, so it is enough to construct a homomorphism φ : Sp 2g (Z) [2] → Q g such that φ • π = 1.
Let X g denote IB g (Z) when g ≥ 4 and IB g (Z) when g = 3. We will construct the map φ in two steps. First in Lemmas 5.2 and 5.3 we will use Proposition 4.1 to construct a homomorphism φ : * ( v) ± ∈X (0) g Sp 2g (Z) [2] ( v) ± → Q g (recall that Proposition 4.1 requires the assumption that BI 2h+1 = Θ 2h+1 for all h < g).
Then we will show that φ takes the edge and conjugation relators from Proposition 3.2 to the identity (Lemmas 5.4 and 5.5), so it induces a homomorphism φ : Sp 2g (Z) [2] → Q g . Finally, we will check that φ • π is equal to the identity (Lemma 5.6), completing the proof.
Lemma 5.1. The following restrictions of π • ρ are all surjective: Proof. Let A i = {2i, 2i + 1} and B i = {1, . . . , 2i}. We can choose and orient a connected component of the preimage in Σ 1 g of each of c A 1 , c B 1 , . . . , c Ag , c Bg in order to obtain a symplectic basis ( a 1 , . . . , a g ; b 1 , . . . , b g ) for H 1 (Σ 1 g ). Note that c A 1 = c 23 , c B 1 = c 12 , and c A 2 = c 45 , so we can choose the orientations such that v 23 = a 1 , v 12 = b 1 , and v 45 = a 2 . Next, let E i = {2, 3, 2i, 2i + 1} and F i = {1, 4, 5, . . . , 2i}. The oriented lifts of c E i and c F i lie in the homology classes a 1 ± a i and a 1 ± b i . The two signs here depend on the choice of the a i and b i . To simplify the notation, we assume both signs are positive. We now proceed in three steps, corresponding to the three statements of the lemma.
Under this isomorphism, π • ρ(T c 123 ) = (−I, id); indeed, the preimage in Σ 1 g of the subdisk bounded by c 123 is homeomorphic to Σ 1 1 and T c 123 lifts to a hyperelliptic involution of this subsurface. It is therefore enough to show that the composition of the restriction of π • ρ to (PB 2g+1 ) {c 23 ,c 12 } with the projection map (Sp 2g (Z) [2]) {( v 23 ) ± ,( v 12 ) ± } → Sp 2g−2 (Z) [2] is surjective. This map factors as where the map ξ 2g+1 is obtained by collapsing the disk bounded by c 123 to a single marked point; this makes sense because (PB 2g+1 ) {c 23 ,c 12 } ⊆ (PB 2g+1 ) c 123 . The map ξ 2g+1 is surjective because every homeomorphism of D 2g−1 can be homotoped such that it fixes a disk surrounding the first marked point, and we have already stated that β 2g−1 is surjective. This completes the proof of the first statement.
Since Y ∈ Sp 2g (Z) [2], each ℓ i and m i is even. For 2 ≤ i ≤ g, set n i = m i ℓ i − ℓ i − m i . Define Z to equal The key property of Z is that ZY ( b 1 ) = b 1 . Since τ 2 a i , τ 2 b i , τ 2 a 1 + a i , and τ 2 a 1 + b i are the images under π • ρ of the Dehn twists about c A i , c B i , c E i , and c F i , we have an explicit b ∈ (PB 2g+1 ) c 23 with π • ρ(b) = Z. Using this b, we can modify Y so that Y ( b 1 ) = b 1 , so Y ∈ (Sp 2g (Z) [2]) {( a 1 ) ± ,( b 1 ) ± } . We have thus reduced the second statement to the first.
Just like before, each ℓ i and m i is even. Define n i = m i ℓ i − ℓ i − m i and Z to equal τ ℓ 1 a 1 τ −ℓ 2 a 1 τ ℓ 2 a 1 + a 2 τ n 3 a 1 τ ℓ 3 a 1 + a 3 τ −m 3 The key property of Z is that Z(Y ( b 1 )) = b 1 . As before, each square of a transvection appearing in Z is the image of a Dehn twist about a curve disjoint from c 23 and c 45 , so there is a b ∈ (PB 2g+1 ) {c 23 ,c 45 } with π • ρ(b) = Z. Using this b, we can modify Y so that The collapsing map D 2g+1 → D 2g−1 described in the first step induces a collapsing map Σ 1 g → Σ 1 g−1 whereby a torus with one boundary component (the preimage of the disk bounded by c 123 ) is collapsed to a point. There is an induced splitting H 1 (Σ 1 g ) ∼ = H 1 (Σ 1 1 ) ⊕ H 1 (Σ 1 g−1 ) ∼ = Z 2 ⊕ Z 2g−2 . Under this identification, v 45 lies in the Z 2g−2 factor. The map (PB 2g+1 ) {c 23 ,c 12 } → (Z/2) ⊕ Sp 2g−2 (Z) [2]  where ξ 2g+1 is the restriction of the map described in Step 1, where d 23 ⊆ D 2g−1 is the image of c 45 under the collapsing map, and where the rightmost vertical map is projection onto the second factor. The leftmost vertical map is surjective as in Step 1, and the rightmost vertical map is obviously surjective. Recall that Y lies in (Sp 2g (Z) [2]) { a 1 , b 1 , a 2 } ∼ = (Z/2) ⊕ (Sp 2g−2 (Z) [2]) ( v 45 ) ± . Since T c 123 ∈ (PB 2g+1 ) {c 23 ,c 12 ,c 45 } maps to the generator of the first factor, we have reduced the problem to the surjectivity of the bottom horizontal map. This is equivalent to Step 2, so we are done.
Lemma 5.2. The map φ ( v) ± does not depend on the choice of Z.
We will require the following easy consequence of Proposition 4.1(1).
Well-definedness of φ. The individual maps φ ( v) ± together define the map φ as in the start of the section. In order to check that φ descends to a well-defined homomorphism φ : Sp 2g (Z) [2] → Q g , we must check that φ respects the edge and conjugation relations for Sp 2g (Z) [2] as in Proposition 3.2. First we deal with the edge relations.
Lemma 5.4. If ( v) ± , ( w) ± ∈ X (0) g are joined by an edge e and Y ∈ (Sp 2g (Z) [2]) e , then Proof. Recall that all simplices of additive type have dimension at least 2, and so we only need to consider standard edges and edges of intersection type. Let e 1 = {( v 23 ) ± , ( v 45 ) ± } and e 2 = {( v 23 ) ± , ( v 12 ) ± }. The edge e 1 is a standard edge in X g and e 2 is an edge of intersection type. The keys to this lemma are the following two facts, the first of which can be proved in the same way as Cases 1 and 2 of Corollary 3.11 and the second of which is a consequence of the classification of surfaces, cf. [17, Section 1.3]: 1. the group Sp 2g (Z) acts transitively on the set of standard edges and on the set of edges of intersection type in X g , and 2. for k ∈ {0, 2}, the group B 2g+1 acts transitively on the set of ordered pairs of distinct homotopy classes of simple closed curves that intersect k times and surround two marked points each.