Splitting of the homology of the punctured mapping class group

Let $\Gamma_{g,1}^m$ be the mapping class group of the orientable surface $\Sigma_{g,1}^m$ of genus $g$ with one parametrised boundary curve and $m$ permutable punctures; when $m=0$ we omit it from the notation. Let $\beta_{m}(\Sigma_{g,1})$ be the braid group on $m$ strands of the surface $\Sigma_{g,1}$. We prove that $H_*(\Gamma_{g,1}^m;\mathbb{Z}_2)\cong H_*(\Gamma_{g,1};H_*(\beta_{m}(\Sigma_{g,1});\mathbb{Z}_2))$. The main ingredient is the computation of $H_*(\beta_{m}(\Sigma_{g,1});\mathbb{Z}_2)$ as a symplectic representation of $\Gamma_{g,1}$.


Introduction
Let Σ g,1 be a smooth orientable surface of genus g with one boundary curve ∂Σ g,1 , and let Σ m g,1 be Σ g,1 with a choice of m distinct points in the interior, called punctures. Let Γ g,1 be the mapping class group of Σ g,1 , i.e. the group of isotopy classes of diffeomorphisms of Σ g,1 : diffeomorphisms are required to fix ∂Σ g,1 pointwise. Similarly let Γ m g,1 be the mapping class group of Σ m g,1 , i.e. the group of isotopy classes of diffeomorphisms of Σ m g,1 that fix ∂Σ m g,1 pointwise and permute the m punctures. Forgetting the punctures gives a surjective map Γ m g,1 → Γ g,1 with kernel β m (Σ g,1 ), the m-th braid group of the surface Σ g,1 . We obtain the Birman exact sequence (see [1]) (1) 1 → β m (Σ g,1 ) → Γ m g,1 → Γ g,1 → 1. The associated Leray-Serre spectral sequence E(m) in Z 2 -homology has a second page E(m) 2 k,q = H k (Γ g,1 ; H q (β m (Σ g,1 ); Z 2 )), and converges to H k+q (Γ m g,1 ; Z 2 ). The main result of this article is that this spectral sequence collapses in its second page.
Thus the computation of H * Γ m g,1 ; Z 2 reduces to the computation of the homology of Γ g,1 with twisted coefficients in the representation H * (β m (Σ g,1 ); Z 2 ). We will see that this Γ g,1 -representation splits as a direct sum of symmetric powers of H 1 (Σ g,1 ) with the symplectic action: this is done in Theorem 3.2, which together with Theorem 1.1 is the main result of the article. The strategy of the proof does not generalize to fields of characteristic different from 2 or to the pure mapping class group, in which we consider only diffeomorphisms of Σ m g,1 that fix all punctures. In section 6 we describe in detail a counterexample with coefficients in Q, which can be generalized both to coefficients in a field F p of odd characteristic and to the pure mapping class group. I would like to thank my PhD advisor Carl-Friedrich Bödigheimer for his precious suggestions and his continuous encouragement during the preparation of this work.

Preliminaries
In the whole article Z 2 -coefficients for homology and cohomology will be understood, unless explicitly stated otherwise. In this section we recollect some classical definitions and results about braid groups and mapping class groups.
Note that we require the points of the configuration to lie in the interior of Σ g,1 ; the space F m (Σ g,1 ) is a smooth, orientable 2m-dimensional manifold. The symmetric group S m acts freely on F m (Σ g,1 ) by permuting the labels 1, . . . , n of a configuration; the orbit space F m (Σ g,1 )/S m is called the m-th unordered configuration space of Σ g,1 and is denoted by C m (Σ g,1 ); this space is also a 2m-dimensional orientable manifold (see figure 1).
Applying the construction of the m-th unordered configuration space fiberwise we obtain a bundle C m (F g,1 ) → B Diff(Σ g,1 ; ∂Σ g,1 ) with fiber C m (Σ g,1 ). The space C m (F g,1 ) is a classifying space for the group Γ m g,1 and the Birman exact sequence (1) is obtained by taking fundamental groups of the aspherical spaces (3) C m (Σ g,1 ) → C m (F g,1 ) → B Diff(Σ g,1 ; ∂Σ g,1 ).
In the whole article the genus g ≥ 0 of the surfaces that we consider is supposed to be fixed, and we will abbreviate S = Σ g,1 . We denote by D the open discΣ 0,1 . It will be useful, for many constructions, to choose an embedding D →S near ∂S and to replace Diff(S; ∂S) with its subgroup Diff(S; ∂S ∪ D) of diffeomorphisms of S that fix pointwise both ∂S and D. Saying that D is embedded near ∂S means that there is a compact subsurface S ⊂ S such that S = S D is the union along a segment of S and the closure of D in S. From now on we suppose such an embedding to be fixed and we consider D as a subspace ofS (see Figure 2). In Section 3, Definition 3.3, we will introduce a convenient model T (S) for the spaceS, and in Section 4, Definition 4.1 we will specify an embedding D →S using the model T (S). The construction in Definition 2.1 can be specialised to the surfaces D and S , yielding spaces C m (D) and C m (S ) respectively. We take configurations of points in the interior of the surfaces D and S respectively (since D is already an open surface this remark applies in particular to S ). The inclusion Diff(S; ∂S∪D) ⊂ Diff(S; ∂S) is a homotopy equivalence of topological groups, hence also the induced map is a homotopy equivalence. We will replace the previous construction with the following, homotopy equivalent ones. Note that Diff(S; ∂S ∪ D) acts on S by restriction of diffeomorphisms, and acts trivially on D; therefore F S,D contains subspaces The fiber bundle (3) corresponding to the Birman exact sequence (1) can now be replaced with the following one which takes the union of configurations (see figure 3): µ {P 1 , . . . , P p } ; P 1 , . . . , P m−p = P 1 , . . . , P p , P 1 , . . . , P m−p ∈ C m (S).
We can apply this construction at the same time to each couple of fibers of the bundles C m (F S,D ) and C m−p (F S ) lying over the same point of B Diff(S; ∂S ∪ D).
We obtain a map If we see the domain of µ F as a fibered product of bundles over B Diff(S; ∂S ∪ D) then the map µ F is also a map of bundles over B Diff(S; ∂S ∪ D), and the corresponding map on fibers is precisely µ.
We recall now the structure of H * (C m (D)). The cohomology of C m (D) with coefficients in Z 2 was first computed by Fuchs in [8]: in section 3 we will generalise Fuchs' argument to surfaces with boundary of positive genus. In [5, Chap.III] Cohen considers the space m≥0 C m (D) as an algebra over the operad of little 2-cubes, and describes its Z 2 -homology as follows: Here ε ∈ H 0 (C 1 (D)) is the fundamental class, and for all k, m ≥ 0 we denote by Q : H k (C m (D)) → H 2k+1 (C 2m (D)) the first Dyer-Lashof operation. In particular Q j ε is the generator of The isomorphism in equation (5) is an isomorphism of bigraded rings. The left-hand side is a ring with the Pontryagin product, and the right-hand side is a polynomial ring in infinitely many variables ε, Qε, Q 2 ε, . . . . The bigrading is given by the homological degree * , that we call the degree, and by the index m of the connected component on which the homology class is supported (informally, the number of points involved in the construction of the homology class), that we call the weight.
In this article we will only need the isomorphism in equation (5) to hold as an isomorphism of bigraded Z 2 -vector spaces. In particular for all choices of natural numbers (α j ) j≥0 with all but finitely many α j = 0, we will consider the monomial j Q j ε αj , corresponding to a homology class in some bigrading ( * , m): we will only use the fact that the set of these monomials is a bigraded basis for the left-hand side of equation (5). We end this section recalling the classical notions of symmetric product and onepoint compactification. Definition 2.6. For a space M we denote by M ∞ its one-point compactification; the basepoint is the point at infinity, that we denote by ∞.
We will consider in particular the one-point-compactification C m (S) ∞ of C m (S).

Homology of configuration spaces of surfaces
We want to study H * (β m (Σ g,1 )) = H * (C m (S)), which appears as the homology of the fiber in the Leray-Serre spectral sequence associated to (1), (3) or (4). In particular we are interested in H * (C m (S)) as a Z 2 -representation of the group Γ g,1 .
In [10] Löffler and Milgram implicitly proved that H * (β m (Σ g,1 )) = H * (C m (S)) is a Z 2 -symplectic representation of the mapping class group. By Z 2 -symplectic we mean the following: The natural action of Γ g,1 on H induces a surjective map Γ g,1 → Sp 2g (Z 2 ). A representation of Γ g,1 over Z 2 is called Z 2symplectic if it is a pull-back of a representation of Sp 2g (Z 2 ) along this map.
In [3] Bödigheimer, Cohen and Taylor computed H * (C m (S)) as a graded Z 2 -vector space. Their method provides all Betti numbers, but the action of Γ g,1 cannot be easily deduced: their descripition of H * (C m (S)) depends on a handle decomposition of S, which is not preserved by diffeomorphisms of S, not even up to isotopy. In this section and in the next one we will prove the following theorem; to the best of the author's knowledge it does not appear in the literature.
Here we mean the following: (i) on the left-hand side the bigrading is given by homological degree * and by the direct summand, indexed by m, on which the homology class is supported, i.e. by the number m of points involved in constructing the homology class; we call * the degree and m the weight, and write ( * , m) for the bigrading; (ii) for j ≥ 0, Q j ε is the image in H 2 j −1 (C 2 j (S)) of a generator of the group H 2 j −1 (C 2 j (D)) Z 2 under the natural map induced by the embedding D → S, and Z 2 Q j ε | j ≥ 0 is the polynomial ring on infinitely many variables ε, Qε, Q 2 ε, . . . ; (iii) H = H 1 (Σ g,1 ) is identified with H 1 (C 1 (S)) in a natural way, and lives in degree 1 and weight 1; Sym • (H) denotes the symmetric algebra on H; (iv) degrees and weights are extended on the right-hand side by the usual multiplicativity rule; (v) the action of Γ g,1 on the right is the tensor product of the trivial action on the factor Z 2 [Q j ε | j ≥ 0], and of the action on Sym • (H) which is induced by the Z 2 -symplectic action on H.
Note that for any bi-homogeneus element in the right-hand side, the weight is greater or equal than the degree: indeed factors of the form Q j ε have weight strictly higher than their degree, whereas factors belonging to H or to its symmetric powers have equal weight and degree. Note that in the case g = 0 the group Γ 0,1 is trivial and the previous theorem reduces to equation (5). We point out that in [4] Bödigheimer and Tillmann have essentially proved that for a field F of any characteristic the F-vector space is isomorphic, as a bigraded Γ g,1 -representation over F, to the tensor product of the ring F[ε], with trivial action, and some other bigraded representation: here ε denotes, in analogy with the notation of Theorem 3.2, the standard generator of H 0 (C 1 (D)). In this section we will prove that there is an isomorphism of bigraded Z 2 -vector spaces as in Theorem 3.2; in the next section we will deal with the action of Γ g,1 .
Since we work with coefficients in a field, it is equivalent to compute homology or cohomology, and in this section we will prefer to compute H * (C m (S)) for all bigradings ( * , m). We will mimic the method used by Fuchs [8] to compute the Z 2 -cohomology of C m (D). As already mentioned in section 2, our computation recovers a known result, but it has the advantage of being quite elementary and of providing a part of the geometric insight that we will need in the next section.
In the whole section we assume m ≥ 0 to be fixed. Since the space C m (S) is homeomorphic to the interior of a compact 2m-manifold with boundary, by Poincaré-Lefschetz duality we have where in the right hand side we consider reduced homology of the one-point compactification (see Definition 2.6). We introduce a space T (S) which is homeomorphic toS, the interior of S. The construction corresponds to a handle decomposition of S with one 0-handle and 2g 1-handles.
: all these intervals are canonically diffeomorphic to [0, 1] by projecting on the second coordinate, rescaling linearly by a factor 2g and translating. We define a bijection between the two sets of left and right intervals: for 1 ≤ i ≤ g, the interval I l 2i−1 corresponds to I r 2i , and the interval I r 2i−1 corresponds I l 2i . The space Q(S) is the quotient of the square [0, 1] 2 obtained by identifying each couple of corresponding intervals in the canonical way. For 1 ≤ i ≤ g we call U i the image of I l 2i−1 in the quotient Q(S), and we call V i the image of I l 2g in Q(S). Note that the image in Q(S) of the set {0, 1} × {P 0 , . . . , P 2g } consists of two points P odd and P even : for ε ∈ {0, 1} and 0 ≤ i ≤ 2g the point (ε, P i ) is mapped to P even if ε + i is even, and is mapped to P odd otherwise. The spaces U i ⊂ Q(S) and V i ⊂ Q(S) are intervals with endpoints P even and P odd ; the interiors of these intervals are disjoint. Each U i and V i is the homeomorphic image of some left interval I l j , and inherits from the latter a parametrisation by [0, 1]. We call U i and V i the interiors of the intervals U i and V i respectively. The space Q(S) is homeomorphic to the compact surface S; we call T (S) the interior of Q(S). From now on we will identify S with Q(S) andS with T (S); consequently we will identify C m (S) with the space of configurations of m points in T (S).
Our next aim is to define a structure of CW -complex on the space C m (S) ∞ : the only 0-cell will be the point ∞, whereas the other cells will be introduced in the following definition.  satisfying the following equality In symbols we write h = (l, x, u, v). We omit from the notation x if it vanishes: this happens precisely when l = 0. The dimension of h is defined as m + l. For a tuple h let e h be the subspace of C m (S) of configurations of m points inS such that the following conditions hold (see picture 6): • for all 1 ≤ i ≤ g, exactly u i points lie on U i and exactly v i points lie on V i ; • there are exactly l vertical lines in the open square ]0, 1[ 2 ⊂S of the form {s i } ×]0, 1[ for some 0 < s 1 < · · · < s l < 1, containing at least one point of the configuration. From left to right, these lines contain exactly x 1 , . . . , x l points respectively.
The space e h is homeomorphic to the interior of the following multisimplex: where the simplex ∆ r is the subspace of [0, 1] r of sequences 0 ≤ τ 1 ≤ · · · ≤ τ r ≤ 1 (the numbers τ 1 , . . . , τ r are the local coordinates of the simplex). The homeomorphism is given as follows: • the local coordinates of the ∆ l -factor correspond to the positions s 1 , . . . , s l of the vertical lines in (0, 1) 2 containing points of the configuration; • the local coordinates of the ∆ xi -factor correspond to the positions of the x i points lying on the vertical line {s i } × (0, 1); • the local coordinates of the ∆ ui -factor correspond to the positions of the u i points lying on U i , which is canonically identified with (0, 1); similarly for the ∆ vi -factor, with v i and V i replacing u i and U i . Note that the dimension of ∆ h agrees with the dimension of h from Definition 3.4.
so that the image of ∂∆ h is contained in the union of all subspaces e h for tuples h of lower dimension than h, together with the 0-cell ∞.
The construction of the map Φ h is as follows: (1) we identify the one-point compactification (S) ∞ ofS = T (S) as the quotient of Q(S) collapsing the boundary to a point ∞; (2) we consider the m-fold symmetric product SP m (S) ∞ (see Definition

2.5): it contains as an open
, that we can then further project to SP m (Q(S)), then to SP m (S) ∞ and then to C m (S) ∞ : the composition is the map Φ h .

In the last step we used the open inclusion
We conclude that the collection of the e h 's, together with the 0-cell ∞, gives a cell decomposition of C m (S) ∞ , with characteristic maps of cells Φ h . We compute the reduced cellular chain complex of C m (S) ∞ with coefficients in Z 2 . It is a chain complex in Z 2 -vector spaces with basis given by tuples h, which correspond to the cells e h . • l ≥ 2 and h is obtained from h by decreasing l by 1, setting In this case we say that h is an inner boundary of h, and we have • l ≥ 1 and h is obtained from h by decreasing l by 1, choosing a splitting of In this case we say that h is a left, outer boundary of h, and we have • l ≥ 1 and h is obtained from h by decreasing l by 1, choosing a splitting of In this case we say that h is a right, outer boundary of h, and we have It may indeed happen that h is both a left and a right outer boundary of h, namely when all numbers (x i ) 1≤i≤l are equal; then the two contributions cancel each other, so that [h : h ] = 0 as stated.
This is also referred as the i-th horizontal face. All other faces of codimension 1 of the multisimplex ∆ h are called vertical. We note that Φ h restricts to a cellular map ∂ hor i ∆ h → C m (S) ∞ on every horizontal face, where ∂ hor i ∆ h is given the cell structure coming from the multisimplicial structure: every subface of ∂ hor The same holds for vertical faces, where the restriction of Φ h is the constant map to ∞: in this case the local index over e h is zero. The index [h : h ] of the map Φ h : ∂∆ h → C m (S) ∞ on e h splits as a sum of local indices: We can filter the reduced chain complexC * (C m (S) ∞ ) by giving filtration norm For example the tuple in Figure 6 has norm 6. By Lemma 3.5 the norm is weakly decreasing along differentials. Denote by F p ⊂ C * (C m (S) ∞ ) the subcomplex generated by tuples of norm ≤ p, and let F p /F p−1 be the p-th filtration stratum. Then F p /F p−1 is isomorphic, as a chain complex, to a direct sum of copies ofC * (C p (Σ 0,1 ) ∞ ): there is one copy for each partition The isomorphism does not preserve the degrees but shifts them by p. Indeed in F p /F p−1 all outer differentials vanish (see Lemma 3.5): in particular the numbers u i , v i do not change along the differentials of F p /F p−1 . Therefore F p /F p−1 splits as a direct sum of chain complexes indexed by all partitions (m − p) = g i=1 (u i + v i ) as above. It is then immediate to identify the inner faces with the ones one would have in the case g = 0, i.e. for the surface Σ 0,1 , after shifting degrees by p. We note thatC * (C p (Σ 0,1 ) ∞ ) is exactly the chain complex described by Fuchs in [8]: we recall Fuchs' computation of the cohomology of configuration spaces of the disc, and abbreviate D forSigma 0,1 as in section 2.
Definition 3.6. Consider a partition of p into powers of 2 p = j≥0 α j 2 j , and let α = (α j ) j≥0 be the sequence of multiplicities. We understand that only finitely many α j 's are strictly positive. The associated symmetric chain inC * (C p (D) ∞ ), denoted by κ(α), is the sum of all • every x i is a power of 2; • for all j ≥ 0 there are exactly α j indices i such that x i = 2 j .
In [8] Fuchs shows that a graded basis forH * (C p (D) ∞ ) is given by the collection of all classes [κ(α)] associated to sequences α = (α j ) j≥0 which satisfy the equality p = j≥0 α j 2 j . By Poincaré-Lefschetz duality this corresponds to a basis for H * (C p (D)). The dual basis of H * (C p (D)) happens to be the basis of monomials This basis consists of all monomials of weight p, using the isomorphism (5) in its full meaning (i.e. as an isomorphism of rings, where Q denotes the first Dyer-Lashof operation). We will not need this finer result in this article, so in the following the expression Q α will only denote the (unique) homology class in H p−l (C p (D) such that the following holds: for all α = (α j ) j≥0 with j≥0 α j = p, the algebraic intersection between We now go back to the filtered chain complexC * (C m (S) ∞ ). The E 1 -page of the associated Leray spectral sequence contains on the p-th column the homology of F p /F p−1 ; as we have seen the homology of this filtration stratum is the direct sum of several copies of the homology ofC * (C p (D)), one copy for each partition We define a chain κ(p, α, u, v) inC m+l (C m (S)): it is the sum of all tuples h of the form (l, x, u, v), for varying x, which satisfy the three properties listed in Definition 3.6. We call such a chain a generalised symmetric chain; we will see in the following that it is a cycle, and we will denote by [κ(p, α, u, v)] the associated homology class inH m+l (C m (S) ∞ ). If any of α, u and v vanishes, we omit it from the notation.
A generalised symmetric chain κ(p, α, u, v) is not only a cycle when projected to its filtration quotient F p /F p−1 , as the E 1 -page of the spectral sequence tells us, but also in the chain complexC * (C m (S) ∞ ) itself. To prove this fact, first note that an inner boundary of a tuple h preserves the norm; hence the fact that κ(p, α, u, v) is a cycle in F p /F p−1 guarantees the fact that all inner boundaries of κ(p, α, u, v) cancel each other inC * (C m (S) ∞ ), that is, is equal to the sum of all outer boundaries of the tuples involved. Note now that outer boundaries of a generalised symmetric chain also cancel out: the left outer boundary of a tuple h = (l, x, u, v) in the generalised symmetric chain cancels against the right outer boundary of the tuple h = (l, x , u, v), with x l = x 1 and x i = x i+1 for 1 ≤ i ≤ l − 1. If all x i happen to be equal, then h = h and we are in the situation described before the proof of Lemma 3.5.
The spectral sequence considered above collapses on its first page and we have the following lemma: Lemma 3.8. The homology H * (C m (S) ∞ ) has a graded basis given by the classes [κ(p, α, u, v)] associated to generalised symmetric chains of weight m. Definition 3.9. We can see the U i 's and V i 's as properly embedded 1-manifolds inS; by Poincaré-Lefschetz duality they represent classes inH 1 (S) ∞ H 1 (S), and in particular they form a basis of the latter cohomology group. We call u i , v i ∈ H 1 S the dual basis.
We establish a bijection between monomials in the tensor product of Theorem 3.2 and the basis of H * (C m (S)) H * (C m (S) ∞ ) in Lemma 3.8: the class is associated with the monomial This shows an isomorphism of bigraded Z 2 -vector spaces from which we conclude that there exists an isomorphism as in Theorem 3.2 at least as bigraded Z 2 -vector spaces: the two bigraded vector spaces have the same dimension in all bigradings.

Action of Γ g,1
We now turn back to homology of C m (S). In the first subsection we describe geometrically some homology classes, in order to give some intuition for the following subsections. In the second subsection we prove Theorem 3.2 in bigradings ( * , m) with * = m. In the third subsection we extend the proof to all other bigradings.

4.1.
Geometric examples of homology classes. In this subsection we consider the case g = 2, hence S denotes the surface Σ 2,1 . We construct some homology classes in H 2 (C 2 (S)): our aim is to get a first understanding of why this homology group is isomorphic to Sym 2 (H) = H ⊗2 /S 2 . In the following c and d will always denote two simple closed curves on S, with corresponding homology classes [c], [d] ∈ H.
Example 1. Suppose that c and d are as in Figure 7: c and d are disjoint and non-separating. We can consider inside C 2 (S) the torus c × d of configurations in which one of the two points runs along c, while the other runs along d. We associate to the fundamental class τ 1 ∈ H 2 (C 2 (S)) of this torus the tensor product     We obtain an embedding Σ 1,1 → C 2 (S); the boundary ∂Σ 1,1 , seen as a curve in C 2 (S), is homotopic to a curve γ in which one point spins 360 • around the other. We note that the curve γ ⊂ C 2 (S) is homotopic to a double covering of a curve γ ⊂ C 2 (S), in which the two points exchange their positions after spinning 180 • around each other. All curves ∂Σ 1,1 , γ, γ and the homotopies relating them are supported on the closure of C 2 (U) in C 2 (S). We can therefore find a map from a Möbius band M to the closure of C 2 (U) in C 2 (S), such that the images of the curves ∂M and ∂Σ 1,1 in C 2 (S) coincide. The union Σ 1,1 ∪ ∂ M along the boundary is then a closed non-orientable surface of genus 3, i.e. the connected sum of a torus and a projective plane. The surface Σ 1,1 ∪ ∂ M is equipped with a map to C 2 (S), hence its fundamental class with coefficients in Z 2 yields a homology class τ 3 ∈ H 2 (C 2 (S)); thus we have managed to adapt the construction from Example 1 to the case of two intersecting curves. We represent τ 3 as [c] · [d] ∈ Sym 2 (H).   Example 5. Suppose that c and d are as in Figure 11: c bounds a subsurfaceŠ, and d is cut by c into two arcs, one of which, denoted by e, lies inŠ.
In this example the complications from Examples 3 and 4 arise at the same time.
To define the class τ 5 ∈ H 2 (C 2 (S)), we start with the torus c × d and we perform two surgeries with two different Möbius bands to solve the two intersections of c and d: the homology class that we obtain is represented by a non-orientable surface of genus 4, i.e. the connected sum of a torus and 2 projective planes. To show that τ 5 vanishes, we start with the 3-manifold with boundaryŠ × d ⊂ SP 2 (S) and we perform a surgery. We identify with e the arč this is a properly embedded arc in the 3-manifold with boundaryŠ × d, and a tubular neighborhood of it, after a suitable homotopy, can be identified with a trivial U-bundle U×e. We remove this solid cylinder from the 3-manifold and glue a trivial M-bundle M×e, by applying an argument similar as the one in Examples 3 and 4. We obtain a 3-manifold with boundary endowed with a map to C 2 (S); the boundary of this 3-manifold is precisely the surface used to represent τ 5 : therefore τ 5 = 0 ∈ H 2 (C 2 (S)), and this is consistent with the representation

Figure 11
If we consider more than 2 points, the examples become more and more complicated, and proving the theorem with this geometric approach seems rather difficult.
We already know that these Z 2 -vector spaces have the same dimension: indeed Sym m (H) is precisely the summand in bigrading (m, m) in equation (6), using that a monomial whose weight is equal and not bigger than its degree cannot contain factors of the form Q i ε.
The construction of ψ m is rather long and technical and involves a few definitions.  Curves are seen as maps S 1 →S, and the intersection of two curves is the intersection of their images. Here and in the following S 1 is the unit circle in C.
Two m-tuples of curves are isotopic if there is an ambient isotopy of S relative to ∂S ∪ D transforming one m-tuple into the other. In particular C m is more than countable.
An element of C m is called multicurve; by abuse of notation the m-tuple (c 1 , . . . , c m ) will often represent its class in C m , i.e. the corresponding multicurve. See picture 12 We denote by Z 2 C m the free   We now prove that the map j m lifts along (ι m ) * to a Γ g,1 -equivariant mapψ m as in the diagram. Since (ι m ) * is injective by Lemma 4.4, it suffices to prove that j m lands in the image of (ι m ) * , and this last statement does not depend on how Γ g,1 acts on these groups. We will prove by induction on m the following technical lemma: in C D m (S) which is supported on N (the word homology denotes here a (m + 1)-singular chain whose boundary is the difference between the two cycles).
For m = 0 both Z 2 C 0 and H 0 (C 0 (S)) are isomorphic to Z 2 and there is nothing to show. For m = 1 we have a canonical identification H 1 (C 1 (S)) H Sym 1 (H), so we takeψ 1 = p 1 ; obviously for all c 1 representing a class in C 1 , the homology class p 1 (c 1 ) ∈ H is represented by a cycle supported on c 1 , and in this case the cycles ι * ψ (c 1 ) and p 1 (c 1 ) = j m (c 1 ) coincide.
Let now m ≥ 1 and in the following fix a class (c 1 , . . . , c m+1 ) ∈ C m+1 . Definition 4.6. We introduce several variations of the notion of configuration space; see Figure 13. We have the following inclusions:   We note that p restricts to a homeomorphism 1,m → m+1 . Moreover both m+1 ⊂ C  ] · c that spins around c 1 to be white. We then note that the right vertical map can be rewritten, after using excision to tubular neighborhoods of 1,m and m+1 respectively, and the Thom isomorphism, as a map The latter map is multiplication by 2, after identifying m+1 and 1,m along p: indeed the normal bundle of 1,m is a double covering of the normal bundle of m+1 , hence the Thom class of the first disc bundle corresponds to twice the Thom class of the second disc bundle. We are working with coefficients in Z 2 , so multiplication by 2 is the zero map. Therefore the image of the cycle [c 1 ] ⊗ c along the diagonal of the square in diagram To prove Lemma 4.5 we need to find a good cycle and a good homology, namely two that are supported on N : a priori both c and the homology between (ι m+1 ) * (c ) and [c 1 ] · c are only supported onS. This can be done by replacing, in the whole argument of the proof, the surfaceS with the surface N . We can define configuration spaces as in Definition 4.6 also for the open surface N , and we can repeat the argument considering N as the ambient surface: indeed we only needed a surface containing D and all curves c 1 , . . . , c m+1 . It is crucial that the action of Γ g,1 is not involved in the statement of Lemma 4.5, as N ⊂ S is not preserved, even up to isotopy, by diffeomorphisms of S. Lemma 4.5 is proved. We now have to prove the following lemma to conclude the proof of Theorem 3.2 in bigradings (m, m). We can therefore compute the map (· ∩ e h ) •ψ m as the map The latter map coincides with the composition Note that H p−l (C p (D)) ⊗ H m−p (C m−p (S )) is the tensor product of the trivial representation H p−l (C p (D)), and of the representation H m−p (C m−p (S )), which by the results of the previous section is isomorphic to the symplectic representation Sym m−p (H). We will prove the following lemma, from which Theorem 3.2 follows: Proof. Note that the statement of the lemma does not depend on the the action of Γ g,1 : we have a map from the right-hand side to the left-hand side of equation (13), we already know that it is Γ g,1 -equivariant, we only need to show that it is a linear isomorphism. Note also that Lemma 3.8 implies that the two vector spaces have the same dimension. Fix l ≤ p ≤ m and α = (α j ) j≥0 , and let [a] = Q α ε = ∞ j=0 (Q j ε) αj be a generator of H p−l (C p (D)), hence l = j α j and p = j α j 2 j . Fix also u = (u 1 , . . . , u g ) and v = (v 1 , . . . , v g ), and let  We give (C p (D) × C m−p (S )) ∞ the cell complex structure of the smash product C p (D) ∞ ∧ C m−p (S ) ∞ . Here C p (D) ∞ is given the cell structure of C p ((0, 1) 2 ) ∞ coming from the natural identification D =]1/4, 3/4[×]1/2, 1[ ∼ =]0, 1[ 2 , which is obtained by rescaling and translating. Moreover we choose any diffeomorphismS ∼ =S that restricts to the identity on all U i 's and V i 's, and give C m−p (S ) ∞ the cell structure of C m−p (S) ∞ . Recall that C m (S) ∞ can be filtered according to the norm of cells: a cell e h associated with the tuple h = (l, x, u, v) has norm l i=1 x i , and the norm is weakly decreasing along boundaries. In the previous section we just considered the associated filtration of the reduced chain complexC * (C m (S) ∞ ), whereas now we consider the closed subcomplex F p C m (S) ∞ ⊂ C m (S) ∞ , which is the union of all cells of norm ≤ p. The crucial observation is that µ ∞ restricts to a cellular map To see this, fix a tupleh = (l,x,ũ,ṽ) of normp ≤ p and of dimensionl + m, and consider the open cell cell eh ⊂ F p C m (S) ∞ . Ifp < p, then eh ∩ (C p (D) × C m−p (S )) is empty. Ifp = p, then where h = (l,x) and h = (0,ũ,ṽ).
Therefore µ ∞ (eh) is {∞} in the first case, and in the second case it is contained in the union {∞} ∪ e h × e h , which is also contained in the (l + m)-skeleton of (C p (D) × C m−p (S )) ∞ . Consider now the generalised symmetric chain κ(p , α , u , v ) representing a class inH m+l (C m (S) ∞ ) = H m−l (C m (S)), with α = (α j ) j≥0 , u = (u 1 , . . . , u g ) and v = (v 1 , . . . , v g ); in particular l = j≥0 α j . Suppose moreover p ≤ p. If p < p, the previous argument shows that µ ∞ * (κ(p , α , u , v )) = 0 in the reduced cellular chain complex, and in particular the corresponding homology class is mapped to zero. Suppose now p = p: then the previous argument shows that the homology class Indeed each tuple h in the cycle κ(p , α , u , v )| is mapped by µ ∞ * to a corresponding pair of tuples h ⊗ h in the cycle κ(α ) ⊗ κ(u , v ), so even at the level of chains we have To finish the proof we consider the collection of all strings of the form (p, α = (α j ) j≥0 , u = (u 1 , . . . , u g ), v = (v 1 , . . . , v g )) satisfying l = j α j ,p = j α j 2 j and (m−p) = i (u i +v i ); we choose a total order on the set of these strings, such that the parameter p is weakly increasing along this order; we associate to each string its corresponding class in H m−l (C m (S)) of the form µ * ([a] ⊗ [b]) and its corresponding class [κ(p, α , u , v )] ∈H m+l (C m (S) ∞ ). Then the matrix of algebraic intersections between these two sets of classes is an upper-triangular matrix with 1's on the diagonal, and in particular it is invertible. This shows that the set of classes of the form µ * ([a]⊗[b]) is a basis for H m−l (C m (S)).
One could expect that the basis given by classes of the form [a]⊗[b] ∈ H m−l (C m (S)) is also dual to the basis of classes [κ(p, α, u, v)] ∈H m+l (C m (S) ∞ ), i.e., the matrix considered in the end of the previous proof is not only upper-triangular but also diagonal. This is however not true, as the following example shows. Let g = 1, m = 2, p = 1, p = 2 and consider the classes [a] = ε ∈ H 0 (C 1 (D)), [b] = u 1 ∈ H = H 1 (C 1 (S )). Moreover let the generalised symmetric chain κ(p , α , u , v ) be defined by α = (α j ) j≥0 with α 1 = 1 and all other α j = 0, u = (u 1 = 0) and v = (v 1 = 0).  The corresponding cell e h intersects once, transversely, the cycle µ * (a ⊗ b), which is represented by the curve of configurations of two points inS, one of which is fixed at a whereas the other runs along b: there is exactly one position on b lying under a. Hence the algebraic intersection between these two classes is 1, and since p > p this is an entry strictly above the diagonal in the matrix considered in the proof of Lemma 4.9. The proof of Theorem 3.2 can be easily generalised to surfaces with more than one boundary curve. Let Σ g,n be a surface of genus g with n ≥ 1 parametrised boundary curves and let Γ g,n be the group of connected components of the topological group Diff(Σ g,n ; ∂Σ g,n ): then there is an isomorphism of bigraded Z 2 -representations where the action of Γ g,n on the right-hand side is induced by the natural action on H 1 (Σ g,n ). For n ≥ 2 the intersection form on the vector space H 1 (Σ g,n ) is degenerate, but it is still invariant under the action of Γ g,n , so there is still a map from Γ g,n to the subgroup of GL 2g+n−1 (Z 2 ) fixing this bilinear form, and in this sense we can say that the representation in (14) is symplectic. The proof of the isomorphism (14) is almost verbatim the same; the main difference is in the construction of the model T (Σ g,n ) forΣ g,n : we divide the vertical segments {0, 1} × [0, 1] ⊂ [0, 1] 2 into 2g + n − 1 equal parts, that we call I l i and I r i according to their order; we identify, for each i > 2g, the interval I l i with the interval I r i ; the other couples of intervals, yielding the genus, are identified just as before. One can further generalise to non-orientable surfaces with non-empty boundary: it suffices, in the above construction, to glue some of the intervals I l i and I r i reversing their orientation. We leave all details of these generalisations to the interested reader.

Proof of Theorem 1.1
We will prove Theorem 1.1 by induction on m. The case m = 0 is trivial. For all m ≥ 0 we let E(m) be the Leray-Serre spectral sequence associated with the bundle (4): its second page has the form E(m) 2 k,q = H k (B Diff(S; ∂S ∪ D); H q (C m (S))) = H k (Γ g,1 ; H q (C m (S))).
From Theorem 3.2 we know that this spectral sequence is concentrated on the rows q = 0, . . . , m. We want to prove the vanishing of all differentials appearing in the pages E(m) r with r ≥ 2; the r-th differential takes the form ∂ r : E(m) r k,q → E(m) r k−r,q+r−1 . In particular any differential ∂ r exiting from the row q = m is trivial, because it lands in a higher, hence trivial row. Fix now q = m − l < m, in particular l ≥ 1; by Theorem 3.2, and in particular by Lemma 4.9, we have a splitting of H k (Γ g,1 ; H m−l (C m (S))) as We fix now l ≤ p ≤ m and show the vanishing of all differentials ∂ r exiting from the summand with label p in the previous equation. Consider the map µ F from Definition 2.3 as a map of bundles over the space B Diff(S; ∂S ∪ D): ; therefore the spectral sequence associated with C p (D) × C m−p (F S ) is isomorphic, from the second page on, to the tensor product of H * (C p (D)) and the spectral sequence associated with the bundle C m−p (F S ); the latter spectral sequence is isomorphic, in our notation, to the spectral sequence E(m − p). In particular µ F induces a map of spectral sequences µ F * : H * (C p (D)) ⊗ E(m − p) → E(m); that in the second page, on the (m − l)-th row and k-th column, restricts to the inclusion of one of the direct summands in equation (15): H k (Γ g,1 ; µ * (H p−l (C p (D)) ⊗ H m−p (C m−p (S )))) ⊂ H k (Γ g,1 ; H m−l (C m (S))) .
In particular if we prove the vanishing of all differentials ∂ r in the first spectral sequence, then also all differentials ∂ r exiting from this direct summand in the second spectral sequence E(m) must vanish. The differentials in the spectral sequence H * (C p (D)) ⊗ E(m − p) are obtained by tensoring the identity of H * (C p (D)) with the differentials of the spectral sequence E(m − p); as p ≥ l ≥ 1 we know by inductive hypothesis that the latter vanish. Theorem 1.1 is proved. One can generalise Theorem 1.1 to orientable or non-orientable surfaces with nonempty boundary, following the generalisation of Theorem 3.2 discussed at the end of section 4.

A rational counterexample
In this section we prove that a statement as in Theorem 3.2 cannot hold if we consider homology with coefficients in Q. We do not know if the analogue of Theorem 1.1 holds in homology with coefficients in Q or in fields of odd characteristic. We point out that H * (C m (Σ g,1 ); Q) has been computed as a bigraded Q-vector space by Bödigheimer, Cohen and Milgram in [2], and more recently by Knudson in [9]. A description of these homology groups as a Γ g,1 -representation seems to be still missing in the literature. We will prove the following theorem: Theorem 6.1. Let g ≥ 2 and m ≥ 2; then H 2 (C 2 (Σ g,1 ); Q) is not a symplectic representation of Γ g,1 .
Proof. Let again S = Σ g,1 . We will use the following strategy: • we define a homology class [a] ∈ H 2 (C 2 (S); Q) represented by a cycle a; • we prove that [a] = 0 by computing the algebraic intersection of [a] with a homology class inH 2 (C m (S) ∞ ; Q): note that the manifold C m (S) is orientable, hence Poincaré-Lefschetz duality holds also with rational coefficients; • we define another homology class [b] ∈ H 2 (C m (S); Q) and show that [b] is mapped to [b] + 2[a] by some element in the Torelli group I g,1 .
Recall that the Torelli group I g,1 is the kernel of the surjective homomorphism Γ g,1 → Sp 2g (Z) induced by the action of Γ g,1 on H 1 (Σ g,1 ; Z); if an element of the Torelli group acts non-trivially on some class in H 2 (C 2 (S); Q), the latter cannot be a symplectic representation of Γ g,1 .
We consider again T (S) as model forS. Let c be an simple closed curve representing the homology class u 1 , and assume that c intersects U 1 once transversely and is disjoint from all other U i 's and from all V i 's. Let c be a parallel copy of c.
We consider the torus a = c × c of configurations in C 2 (S) having one point lying on c and one lying on c ; we let [a] ∈ H 1 (C 2 (S); Q) be the fundamental class of a associated with one orientation of a. The cell e h intersects once, transversely the torus a, therefore the algebraic intersection [a] ∩ [κ(0, u, v)] Q is ±1, where the signs depends on how we have chosen orientations on a, e h and C 2 (S) itself. In particular [a] = 0. The action of Γ g,1 on isotopy classes of non-separating simple closed curves is transitive, so we can repeat the construction of the torus a with any other copy of parallel, non-separating simple closed curves c and c , and the resulting class [a] will always be non-trivial in H 2 (C 2 (S); Q). See Figure 16 to visualize the following discussion. Let d and d be disjoint, nonseparating simple closed curves such that the following holds: if we cut S along d and d we separate S into two pieces, one of which is a subsurfaceS Σ 1,2 of genus 1, with boundary curves d and d . Here we use that the genus of S is at least 2. Suppose now that c is a non-separating simple closed curve inS, and let c , c be two parallel copies of c in Σ 1,2 , one on each side of a small tubular neighborhood of c. Then d, d , c , c are the boundary of a subsurfaceŠ Σ 0,4 ⊂S. Orient all curves d, d , c, c , c in such a way that the following equalities hold: The previous proof works verbatim if we replace Q by any field of odd characteristic. Moreover all the arguments in the previous proof can be adapted to show that H * (F 2 (S); F) is not a symplectic representation of Γ g,1 (see Definition 2.1), where F is any field, including Z 2 . Therefore, at least for orientable surfaces Σ g,1 of genus g ≥ 2, being a symplectic representation of the mapping class group seems to be a peculiarity of the homology of unordered configuration space and of the characteristic 2.