Stable Cohomology of the Universal Picard Varieties

and the Extended Mapping Class Group

Johannes Ebert and Oscar Randal-Williams

Received: September 16, 2011 Revised: March 16, 2012

Communicated by Lars Hesselholt

Abstract. We study the moduli spaces which classify smooth surfaces along with a complex line bundle. There are homological stability and Madsen–Weiss type results for these spaces (mostly due to Cohen and Madsen), and we discuss the cohomological calculations which may be deduced from them. We then relate these spaces to (a generalisation of) Kawazumi’s extended mapping class groups, and hence deduce cohomological information about these.

Finally, we relate these results to complex algebraic geometry. We construct a holomorphic stack classifying families of Riemann surfaces equipped with a fibrewise holomorphic line bundle, which is a gerbe over the universal Picard variety, and compute its holomorphic Picard group.

2010 Mathematics Subject Classification: 14H15, 32G15, 14C22, 57R20, 55R40 Keywords and Phrases: Moduli spaces, Picard variety, stable cohomology

1 Introduction

Let Σ_{g,r}^{n} be a connected oriented smooth surface of genus g with r boundary components and P := {p_{1}, …, p_{n}} be n distinct marked points in the interior. Let 𝒟_{g,r}^{n} the group of orientation preserving diffeomorphisms of Σ_{g,r}^{n} which fix the set ∂Σ_{g,r}^{n} ∪ P pointwise and let Γ_{g,r}^{n} := π_{0}𝒟_{g,r}^{n} be the mapping class group. We assume that g ≥ 2, so that by a theorem of Earle and Eells [11] 𝒟_{g,r}^{n} has contractible components and the natural map on classifying spaces B𝒟_{g,r}^{n} → BΓ_{g,r}^{n} is a homotopy equivalence.

Let map_{∂}(Σ_{g,r}^{n}, Bℂ^{×}) denote the space of continuous maps Σ_{g,r}^{n} → Bℂ^{×} which send ∂Σ_{g,r}^{n} ∪P to the basepoint of Bℂ^{×}. We write map_{∂}(Σ_{g,r}^{n}, Bℂ^{×})(k) for the path component consisting of those maps of degree k. The principal object of study in this paper is the moduli space of surfaces of genus g with r boundary components, n marked points and a complex line bundle, originally introduced by Cohen and Madsen [7]:

S_{g,r}^{n} := map_{∂}(Σ_{g,r}^{n}, Bℂ^{×}) ×_{𝒟_{g,r}^{n}} E𝒟_{g,r}^{n}.

Being a Borel construction, this space fits into a defining fibration sequence

map_{∂}(Σ_{g,r}^{n}, Bℂ^{×}) ⟶ S_{g,r}^{n} ⟶ B𝒟_{g,r}^{n}.  (1.1)

The space S_{g,r}^{n} carries a tautological surface bundle

E_{g,r}^{n} := (Σ_{g,r}^{n} × map_{∂}(Σ_{g,r}^{n}, Bℂ^{×})) ×_{𝒟_{g,r}^{n}} E𝒟_{g,r}^{n},

and we denote the bundle projection by π : E_{g,r}^{n} → S_{g,r}^{n}, the vertical tangent bundle by T_{v}, and the n canonical sections by s_{1}, ..., s_{n}. The evaluation map Σ_{g,r}^{n} × map_{∂}(Σ_{g,r}^{n}, Bℂ^{×}) → Bℂ^{×} is 𝒟_{g,r}^{n}-invariant, so gives a map L : E_{g,r}^{n} → Bℂ^{×}. We use the letter L also for the complex line bundle this map induces, and note that it is trivialised over the boundary and marked points in each fibre. The space S_{g,r}^{n} splits into path components S_{g,r}^{n}(k) corresponding to the maps of degree k, and the total space E_{g,r}^{n} has an analogous decomposition. The term “moduli space” is justified by the following observation.

Proposition 1.1. Let X be any paracompact Hausdorff space. Homotopy classes of maps X → S_{g,r}^{n}(k) are in bijection with isomorphism classes of tuples (E, π, s, L, η), where π : E → X is an oriented smooth surface bundle of genus g with r boundary components and the boundary bundle is trivialised; s = (s_{1}, …, s_{n}) are pairwise disjoint cross-sections in the interior; L → E is a line bundle; η is a trivialisation of L over ∂E ∪ s_{1}(X) ∪ … ∪ s_{n}(X), such that with respect to this trivialisation, L has fibrewise degree k.

Definition 1.2. For each k we choose a degree k map L_{0} ∈ map_{∂}(Σ_{g,r}^{n}, Bℂ^{×})(k). The extended mapping class group Γ̃_{g,r}^{n}(k) is the fundamental group of the space S_{g,r}^{n}(k), based at L_{0},

Γ̃_{g,r}^{n}(k) := π_{1}(S_{g,r}^{n}(k), L_{0}).

When k = 0 we make the canonical choice of L_{0}, which is the constant map to the basepoint. In this case the extended mapping class group also has an algebraic description. If we let H_{g,r}^{n} denote the Γ_{g,r}^{n}-module H^{1}(Σ_{g,r}^{n}, ∂Σ_{g,r}^{n} ∪ P; ℤ), there is an isomorphism

Γ̃_{g,r}^{n}(0) ≅ H_{g,r}^{n} ⋊ Γ_{g,r}^{n}.

If (r, n, k) = (1, 0, 0) this definition agrees with that of Kawazumi [20], however we warn the reader that our definition is different from Kawazumi’s for other values of (r, n). For both S_{g,r}^{n}(k) and Γ̃_{g,r}^{n}(k) we adopt the convention of omitting r or n from the notation when they are zero.

1.1 Topological results

The first part of this paper (Sections 2 and 3) is devoted to the study of the cohomology of the group Γ̃_{g,r}(k). We will see in Proposition 2.1 that as long as r + n > 0 the space S_{g,r}^{n}(k) is aspherical, so there is a homotopy equivalence

BΓ̃_{g,r}^{n}(k) ≃ S_{g,r}^{n}(k),

and the cohomology of the space S_{g,r}^{n}(k) translates to the group cohomology of Γ̃_{g,r}^{n}(k). The relation between S_{g}(k) and BΓ̃_{g}(k) turns out to be much more subtle. Due to the moduli-theoretic interpretation of S_{g,r}^{n}(k), the methods of the homotopy theory of moduli spaces apply and yield a great amount of information about these spaces, and hence about BΓ̃_{g,r}^{n}(k) for r + n > 0. Comparison with S_{g}(k) will also be essential for the study of BΓ̃_{g}(k).

Theorem A.

(i) (Independence of k) There are homotopy equivalences

S_{g,r}^{n}(k) ≃ S_{g,r}^{n}(k + 1) for r > 0
S_{g}^{1}(k) ≃ S_{g}^{1}(k + 1)
S_{g}(k) ≃ S_{g}(k + 2g − 2).

(ii) (Harer type stability) Consider the “stabilisation maps”

α(g) : S_{g,r+1}^{n}(k) ⟶ S_{g+1,r}^{n}(k)
β(g) : S_{g,r}^{n}(k) ⟶ S_{g,r+1}^{n}(k)
γ(g) : S_{g,r}^{n}(k) ⟶ S_{g,r−1}^{n}(k)

given by gluing in a pair of pants along the legs, along the waist, or gluing in a disc (and extending the map to Bℂ^{×} trivially over the glued-in part in each case). Then the induced maps in homology H_{∗}(α(g)), H_{∗}(β(g)), H_{∗}(γ(g)) are isomorphisms if 3∗ ≤ 2g − 3, 3∗ ≤ 2g − 1, 3∗ ≤ 2g − 1, respectively.

(iii) (Splitting) The map S_{g,r}^{n}(k) → S_{g,r}(k) × (Bℂ^{×})^{n} given by the tangent spaces at the marked points is a homology isomorphism in degrees 3∗ ≤ 2g − 1.

(iv) (Madsen–Weiss type theorem) Let MTSO(2) be the Madsen–Tillmann– Weiss spectrum. There is a map α_{g,r} : S_{g,r}(k) → Ω_{0}^{∞}(MTSO(2)∧Bℂ_{+}^{×}), which is a homology equivalence in degrees 3∗ ≤ 2g − 3.

We shall say “stable range” for the range of degrees in which a given homology group is independent of g. Most of this theorem has been proved elsewhere. A complete proof of homological stability as formulated in (ii) was given by the second named author [34], but the result has a longer prehistory. For r > 0, it was first proved by Cohen–Madsen [7], but with the weaker stable range 2∗ ≲ g. The range of stability was improved by Boldsen [4], again as long as r > 0. The proof of stability for closing the last boundary in [7] is incorrect and has been partially repaired by those authors in [8], as long as rational coefficients are employed. In the references we have cited only the case n = 0 is covered, but in Section 2.3 we show how to derive the case n > 0. The homological splitting of (iii) is a straightforward adaption of an argument by Bödigheimer and Tillmann [5], and we briefly sketch it in Section 2.3. Part (iv) is a generalisation of the Madsen–Weiss theorem [25] due to Cohen–Madsen [7] (though one needs the result of [34] to deal with the case r = 0), and we recall this result in Section 2.4. We prove (i) in Section 2.2.

Remark 1.3. Parts (ii) and (iii) show that H_{∗}(S_{g}^{n}(k)) ≅ H_{∗}(S_{g}^{n}(k + 1)) in the stable range. However, in general S_{g}(k) ≄ S_{g}(k +1). This can be deduced from Theorem E, which computes the group cohomology of the fundamental groups of these spaces and we see they differ in general.

The space Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}) can be understood by the usual methods of algebraic topology. We defer its definition to Section 2.4, where we will also construct certain cohomology classes

κ_{i,j} ∈ H^{2i+2j}(Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}); ℤ).

Under the map α_{g,r} : S_{g,r}(k) → Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}) the class κ_{i,j} pulls back to

π_{!}(e(T_{v})^{i+1}c_{1}(L)^{j}) ∈ H^{2i+2j}(S_{g,r}^{n}(k); ℤ),

where π : E_{g,r}(k) → S_{g,r}(k) is the tautological surface bundle, T_{v} is the vertical tangent bundle and L is the complex line bundle we described earlier. For simplicity we denote this class by κ_{i,j} too. The class κ_{i,0} may be defined on B𝒟_{g,r}, and here it coincides with the Mumford–Morita–Miller class usually denoted κ_{i}.

Theorem B. There is an isomorphism

H^{∗}(Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}); ℚ) ≅ ℚ[κ_{i,j} | i + j > 0, j ≥ 0, i ≥ −1].

Therefore when r > 0, for degrees 3∗ ≤ 2g − 3 there are isomorphisms

H^{∗}(BΓ̃_{g,r}(k); ℚ) ≅ H^{∗}(S_{g,r}(k); ℚ) ≅ ℚ[κ_{i,j} | i + j > 0, j ≥ 0, i ≥ −1]. (1.2)

We are also able to make low-dimensional integral calculations. The Hodge class λ ∈ H^{2}(S_{g}; ℤ) may be defined as c_{1}(π_{!}^{K}(T_{v}^{∗})), the first Chern class of the push-forward in complex K-theory of the vertical cotangent bundle of the tautological surface bundle, and it is known to satisfy the relation 12λ = κ_{1}. Just as for the κ_{i,j}, the class λ is induced from a class on Ω^{∞}(MTSO(2)∧Bℂ_{+}^{×}) which we also denote λ.

Theorem C. The groups

H^{1}(Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}); ℤ) and H^{3}(Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}); ℤ)

are trivial. The group H^{2}(Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}); ℤ) is free abelian of rank three, with free basis the Hodge class λ, κ_{0,1} and ζ := \frac{1}{2}(κ_{0,1} − κ_{−1,2}).

As long as g ≥ 6, this gives a description of the integral cohomology of S_{g,r}(k) up to degree 3. We now turn to the case r = n = 0. The main obstacle in studying the cohomology of BΓ̃_{g}(k) is that the stabilisation map BΓ̃_{g,1}(k) → BΓ̃_{g}(k) is not a homology isomorphism in a range of degrees. However the failure of homological stability is entirely down to the failure of stability on the third cohomology. Before we state the key result, let us introduce some language.

Definition 1.4. A fibre sequence consist of a connected pointed space (B, b), maps F →^{f} E →^{p} B and a nullhomotopy H : p ∘ f ≃ c_{b} from the composition to the constant map to b. This determines a map F → hofib_{b}(p) which we require to be a weak equivalence. We will often drop the nullhomotopy from the notation.

Define

Ξ : Bℂ^{×} ⟶^{−⊗L_{0}} map(Σ_{g}, Bℂ^{×})(k) ⟶ S_{g}(k),

which is a map that classifies the data (Bℂ^{×} × Σ_{g}, pr_{1}, pr_{1}^{∗}γ ⊗ pr_{2}^{∗}L_{0}), where γ denotes the universal line bundle over Bℂ^{×}.

Theorem D.

(i) For g ≥ 3, there exist fibre sequences

Bℂ^{×} ⟶^{Ξ} S_{g}(k) ⟶^{Π} BΓ̃_{g}(k)

S_{g}(k) ⟶^{Π} BΓ̃_{g}(k) ⟶^{θ} K(ℤ, 3)

(i.e. there exist nullhomotopies of Π∘Ξ and θ∘Π yielding fibre sequences).

(ii) There is a map from BΓ̃_{g,1}(k) to the homotopy fibre of the map θ which induces an isomorphism in integral homology in degrees 3∗ ≤ 2g − 1.

It turns out that the cohomology of Γ̃_{g}(k) behaves in a fairly systematic way in a certain range of degrees, but the systematic way in which it behaves in turn depends on k. The following are our main results on the cohomology of Γ̃_{g}(k).

Theorem E. Suppose g ≥ 6. The group H^{1}(BΓ̃_{g}(k); ℤ) is trivial, and the group H^{2}(BΓ̃_{g}(k); ℤ) is free abelian of rank two and injects into H^{2}(S_{g}(k); ℤ). A free basis for it may be taken to be the Hodge class λ and an element η that maps to

\frac{1}{gcd(2g − 2, g + k − 1)} (kκ_{0,1} + (g − 1)κ_{−1,2})

in H^{2}(S_{g}(k); ℤ). The group H^{3}(BΓ̃_{g}(k); ℤ) is ℤ/ gcd(2g−2, 1−g−k) generated by the class θ ∈ H^{3}(BΓ̃_{g}(k); ℤ) = [BΓ̃_{g}(k), K(ℤ, 3)] of Theorem D.

We are also able to give a complete description of the rational cohomology algebra in the stable range, where the dependence on g and k is less vividly seen.

Theorem F. The rational cohomology H^{∗}(BΓ̃_{g}(k); ℚ) injects into H^{∗}(S_{g}(k); ℚ) in all degrees. There exist classes ν_{i,j} ∈ H^{2i+2j}(BΓ̃_{g}(k); ℚ) defined for i ≥ −1 and j ≥ 0, such that the algebra homomorphism

ℚ[ν_{i,j} | i ≥ −1, j ≥ 0, i + j > 0; (i, j) ≠ (0, 1)] ⟶ H^{∗}(BΓ̃_{g}(k); ℚ)

is an isomorphism in degrees 3∗ ≤ 2g − 3. Moreover, the image of ν_{i,j} in H^{∗}(S_{g}(k); ℚ) is equal to κ_{i,j} modulo the ideal (κ_{0,1}), and ν_{i,0} = κ_{i}.

Remark 1.5. Using different methods, Kawazumi [20, 21, 22] obtained the isomorphism (1.2) when r = 1 and k = 0. Looijenga [24] was able to determine the structure of H^{∗}(BΓ̃_{g}(k); ℚ) as a module over H^{∗}(BΓ_{g}; ℚ), but not the algebra structure. Next, we study the group extension

0 ⟶ H_{g,r}^{n} ⟶ Γ̃_{g,r}^{n}(k) ⟶ Γ_{g,r}^{n} ⟶ 1  (1.3)

obtained by taking the fundamental groups of the spaces in the defining fibration (1.1). The Γ_{g,r}^{n}-modules H_{g,r}^{n} = H^{1}(Σ_{g,r}^{n}, ∂Σ_{g,r}^{n} ∪ P; ℤ) form a coefficient system in the sense of [7] or [4]. We denote this coefficient system simply by H and its rationalisation by H_{ℚ}, thereby explaining notation such as H^{∗}(Γ_{g,r}^{n}; H).

Theorem G. The rational Leray–Hochschild–Serre spectral sequence of the extension (1.3) degenerates at E_{2} in the stable range. If r+n ≤ 1 it degenerates in all degrees. The associated graded algebra in the stable range is

⨁_{p,q} H^{p}(Γ_{g,r}^{n}; ∧^{q}H_{ℚ}) ≅ ℚ[e_{1}, , ..., e_{n}, κ_{1}, …] ⊗ ℚ[x_{i,j} | i + j > 0, j > 0, i ≥ −1],

if r + n > 0, and

⨁_{p,q} H^{p}(Γ_{g}; ∧^{q}H_{ℚ}) ≅ ℚ[κ_{1}, …]⊗ℚ[x_{i,j} | i+j > 0, j > 0, i ≥ −1, (i, j) ≠ (0, 1)],

if r + n = 0. In both cases, e_{i} has bidegree (p, q) = (2, 0), κ_{i} has bidegree (p, q) = (2i, 0) and x_{i,j} has bidegree (p, q) = (2i + j, j).

The proof of this theorem is via the work of Kawazumi [20, 21, 22], who has studied in depth the rational cohomology of Γ̃_{g,1}. He has defined certain classes m_{i,j} ∈ H^{2i+j−2}(Γ_{g,1}, ∧^{j}H) which are permanent cycles in the spectral sequence

E_{2}^{p,q} = H^{p}(Γ_{g,1}; ∧^{q}H_{ℚ}) ⟹ H^{p+q}(Γ̃_{g,1}; ℚ)

and detect classes m̃_{i,j} ∈ H^{2i+2j−2}(Γ̃_{g,1}; ℤ) he has also defined. He then shows that the m_{i,j} generate E_{2}^{∗,∗} in the stable range, so the spectral sequence collapses. If r = 1 and n = 0, Theorem G follows immediately from the more technical statement that Kawazumi’s m̃_{i,j} agree with our κ_{i−1,j}. This is somewhat surprising, as the m̃_{i,j} are defined using an explicit group cocycle which depends crucially on the surface having boundary, whereas κ_{i−1,j} may be defined for closed surfaces. The other cases with r + n > 0 are derived using homological stability.

1.2 Relation to complex geometry

The homotopy types B𝒟_{g}, S_{g}(k) and BΓ̃_{g}(k) bear a close relationship to the geometry of Riemann surfaces, and the preceding purely topological results can be used to obtain more geometrical results. In order to formulate these results, we have to employ the language of stacks. We adopt the convention of writing stacks as Abc and their associated homotopy types as Abc. Let M_{g} be the moduli stack of genus g Riemann surfaces, defined on the site Top of topological spaces (with the ordinary topology of open covers). Elements of M_{g}(X) or, equivalently, maps X → M_{g}, correspond to families of genus g Riemann surfaces π : E → X. It is well-known, by Teichmüller theory, that M_{g} ≅ T_{g}//Γ_{g}, the orbifold quotient of Teichmüller space by Γ_{g}. Moreover, the homotopy type of M_{g} is just B𝒟_{g}. Denote by Hol_{g} the stack over Top which classifies families of Riemann surfaces of genus g, E → X, equipped with a fibrewise holomorphic line bundle L → E. It splits into components Hol_{g}^{k}, where k is the fibrewise degree of L. Recall that for an individual Riemann surface S, the Picard variety Pic(S) is the complex Lie group of isomorphism classes of holomorphic line bundles on S. It splits into components Pic^{k}(S) parametrising isomorphism classes of degree k line bundles. There is a (noncanonical) isomorphism Pic(S) ≅ ℤ × Pic^{0}(S) and Pic^{0}(S) is a complex g-dimensional torus. If π : E → X is a family of genus g Riemann surfaces, there is a family Pic(E/X) → X of complex manifolds whose fibre over x ∈ X is just the Picard variety Pic(E_{x}) of E_{x} := π^{−1}(x). Denote by Pic_{g} the stack over Top which classifies families of Riemann surfaces of genus g, E → X, equipped with a section s : X → Pic(E/X) of the associated bundle of Picard varieties. Again, Pic_{g} splits into components Pic_{g}^{k}, indexed by the fibrewise degree of the line bundle on the total space. Both stacks have a natural map to M_{g} that just remembers the underlying Riemann surface. Furthermore, there is a map Φ_{g}^{k} : Hol_{g}^{k} → Pic_{g}^{k} that takes a complex line bundle to its isomorphism class.

In Theorem 4.5 we show that Φ_{g}^{k} is a gerbe with band ℂ^{×}. As a consequence, after taking homotopy types we get a fibre sequence

Bℂ^{×} ⟶ Hol_{g}^{k} ⟶^{ϕ_{g}^{k}} Pic_{g}^{k}.

In Theorem 4.6 we show that there are equivalences Hol_{g}^{k} ≃ S_{g}(k) and Pic_{g}^{k} ≃ BΓ̃_{g}(k) under which this fibre sequences corresponds to that of Theorem D. Thus the cohomological results of the last section also give information about the cohomology of Hol_{g}^{k} and Pic_{g}^{k}, and of the effect on cohomology of the map Φ_{g}^{k}. Another property of the stacks Hol_{g}^{k} and Pic_{g}^{k} is that they are local quotient stacks (this is the class of stacks which are closest to spaces in the sense that the homotopy type of the stacks reflects geometric properties such as the classification of line bundles and gerbes). Therefore, we are able to prove an analogue of a result of Mestrano–Ramanan on the existence of Poincaré line bundles. It is a well-known—but fairly deep—result that the moduli stack M_{g} is even a holomorphic stack. This will easily imply that Hol_{g}^{k} and Pic_{g}^{k} are holomorphic (local quotient) stacks, so that concepts from holomorphic geometry translate. In particular, the notion of a holomorphic line bundle on these stacks is meaningful. We show that for both these stacks, the holomorphic Picard group is isomorphic to the topological one. The resulting computation of the Picard group of Pic_{g}^{k} is not new: it is due to Kouvidakis [23] and we recover his result, although we give a different description of the generators. We also discuss the relationship between our results and related results in algebraic geometry, in particular those of Melo and Viviani [28].

Acknowledgements

Both authors are grateful to the MFO Oberwolfach and the Mathematics department at Stanford University for their support with research visits while this work was being carried out. The second author is also grateful to M. Melo and F. Viviani for explaining their work. O. Randal-Williams was supported by an EPSRC Studentship, the EPSRC PhD Plus scheme, ERC Advanced Grant No. 228082, and the Danish National Research Foundation through the Centre for Symmetry and Deformation.

2 The spaces of surfaces with line bundles

2.1 Homotopy type

Recall from the introduction the definition of S_{g,r}^{n}, the tautological surface bundle π : E_{g,r}^{n} → S_{g,r}^{n}, its sections s_{1}, ..., s_{n}, the vertical tangent bundle T_{v} and the complex line bundle L → E_{g,r}^{n} trivialised along the boundary and at the marked points.

Proposition 2.1. Let g ≥ 2.

(i) If r + n > 0, then S_{g,r}^{n}(k) is aspherical, and so BΓ̃_{g,r}^{n}(k) ≃ S_{g,r}^{n}(k).

(ii) If r = n = 0, then π_{i}(S_{g}(k)) = 0 for i ≠ 1, 2 and π_{2}(S_{g}(k)) = ℤ.

Proof. We look at the long exact sequence of the defining fibration sequence

map_{∂}(Σ_{g,r}^{n}, Bℂ^{×}) ⟶ S_{g,r}^{n} ⟶ B𝒟_{g,r}^{n}.

If r + n > 0, then map_{∂}(Σ_{g,r}^{n}, Bℂ^{×})(k) is an Eilenberg–MacLane space of type BH_{g,r}^{n}. This is immediate and proves the first part. Because Bℂ^{×} is an abelian topological group so is map(Σ_{g}, Bℂ^{×}), and hence map(Σ_{g}, Bℂ^{×})(k) is a product of Eilenberg–MacLane spaces, namely BH_{g} × Bℂ^{×} (however, this product decomposition is not natural, which is responsible for many of the subtleties of that case). The second part follows.

2.2 Independence of the degree

The homotopy equivalences of part (i) of Theorem A are given by tensoring the line bundle with a fixed line bundle, provided by Proposition 2.2 below.

Proposition 2.2. Let E_{g,r}^{n} → B𝒟_{g,r}^{n} be the universal surface bundle, with sections s_{1}, … s_{n}. Then there is a line bundle L → E_{g,r}^{n} with fibrewise degree k and such that both L|_{∂E_{g,r}^{n}} and s_{i}^{∗}L are trivial in each of the following cases:

(i) r > 0, n arbitrary and k = 1.

(ii) r = 0, n = k = 1.

(iii) r = 0, n = 0, k = 2 − 2g.

This proposition can be restated by saying that the map S_{g,r}^{n}(k) → B𝒟_{g,r}^{n} has a section in the listed cases.

Proof. The third case is clear, since we can take the vertical tangent bundle T_{v} (the condition n = 0 is necessary as we might be unable to trivialise T_{v} over the cross-sections in the case n > 0). In the first case, the boundary bundle is trivialised and so the bundle may be assumed to contain a trivialised collar. Inside the collar, there exists an embedded BΓ_{g,r}^{n} × 𝔻^{2} → E_{g,r}^{n} over BΓ_{g,r}^{n} which does not meet the boundary or the marked points. The standard collapse construction defines a map h : E_{g,r}^{n} → BΓ_{g,r}^{n} × 𝔻^{2}/∂𝔻^{2} of fibrewise degree 1. Let H → 𝔻^{2}/∂𝔻^{2} be the Hopf bundle and pick a trivialisation over the point ∂𝔻^{2} and one over the origin of 𝔻^{2}. The bundle L := h^{∗}H, together with the induced trivialisations, does the job in the first case. For second case, let U → BΓ_{g}^{1} be the closed unit disc bundle in the line bundle s^{∗}T, where T is the vertical tangent bundle, and ∂U be the unit sphere bundle. There exists a tubular neighborhood, i.e. a fibrewise embedding U → E_{g}^{1} that maps the zero section to s. The collapse construction, performed fibrewise, yields a fibrewise map E_{g}^{1} → U/∂U of fibrewise degree 1. Consider the rotation action of U(1) on 𝕊^{2} (considered as the Riemann sphere), fixing the points 0, ∞, and the 𝕊^{2}-bundle p : F := EU(1) ×_{U(1)} 𝕊^{2} → BU(1). There is a bundle map U/∂U → F, covering the map ϵ from (2.1). The composition with the collapse is a fibre-preserving map h : E_{g}^{1} → F. The Hopf bundle on 𝕊^{2} admits a U(1)-action that turns it into an equivariant line bundle, therefore inducing a line bundle H′ → EU(1) ×_{U(1)} 𝕊^{2} of fibrewise degree 1. Let t : Bℂ^{×} → EU(1) ×_{U(1)} 𝕊^{2} be the section at 0. Then the bundle H := p^{∗}t^{∗}(H′)^{∗} ⊗ H′ has fibrewise degree 1 and is trivialised over the zero section. To finish the proof, let L := h_{1}^{∗}H.

2.3 Homological stability and its consequences

As we stated in the introduction, proofs of Theorem A (ii) for n = 0 have appeared in several places [7, 8, 4, 34] with various stability ranges. An explicit reference for the version we are using is [34, §1.4]. Now we derive the case n > 0 from the case n = 0. There is a map

ϵ_{i} : S_{g,r}^{n} ⟶ Bℂ^{×},  (2.1)

defined as the classifying map of the line bundle s_{i}^{∗}T_{v} → S_{g,r}^{n} (in other words, ϵ_{i} takes the tangent space at the i-th marked point). The homotopy fibre of ϵ_{i} classifies families of surfaces as classified by S_{g,r}^{n} but with a framing of T_{v} at the i-th marked point: this is homotopy equivalent to S_{g,r+1}^{n−1}. Taking all ϵ_{i} together, we obtain a fibre sequence

S_{g,r+n} ⟶ S_{g,r}^{n} ⟶^{ϵ} (Bℂ^{×})^{n};  (2.2)

analogous to that used by Bödigheimer and Tillmann in [5, eq. (3.2)]. Consider the commutative diagram

[figure: S_{g,r+n}; S_{g,r}^{n}; ϵ; (Bℂ^{×})^{n}; S_{h,s+n}; S_{h,s}^{n}; ϵ; (Bℂ^{×})^{n},]

whose rows are the fibre sequences (2.2) and whose left and middle vertical maps are suitable stabilisation maps. Comparing the Leray–Serre spectral sequences of the two fibre sequences shows homological stability in the case n > 0. The sequence (2.2) can also be used to prove Theorem A (iii), using an argument from [5]. Consider the commutative diagram

[figure: S_{g,r+n}; S_{g,r}^{n}; (Bℂ^{×})^{n}; S_{g,r}; S_{g,r} × (Bℂ^{×})^{n}; (Bℂ^{×})^{n}.]

The bottom sequence is the product sequence, the left vertical map is a composition of stabilisation maps γ(g), and the middle vertical map is the product of ϵ with the forgetful map. Comparing the Leray–Serre spectral sequences of the two fibre sequences establishes that the middle map is a homology isomorphism in degrees 3∗ ≤ 2g − 1.

2.4 Madsen–Weiss type theorem

Consider the vector bundle

γ_{2,n}^{⊥} ⟶ Gr_{2}^{+}(ℝ^{n})

given by the orthogonal complement of the tautological bundle, and write M_{n} for its Thom space. The stabilisation maps Gr_{2}^{+}(ℝ^{n}) → Gr_{2}^{+}(ℝ^{n+1}) pull back γ_{2,n+1}^{⊥} to ϵ^{1} ⊕ γ_{2,n}^{⊥} and so give maps of Thom spaces ΣM_{n} → M_{n+1}. Thus the collection {M_{n}}_{n≥0} forms a spectrum in the sense of stable homotopy theory, which is denoted MTSO(2). Similarly, we define a spectrum MTSO(2)∧Bℂ_{+}^{×} having as its n-th space the smash product M_{n} ∧ Bℂ_{+}^{×}. Homotopy groups of spectra are very hard to compute in general, but in low degrees, the homotopy groups of MTSO(2) ∧ Bℂ_{+}^{×} may be calculated using the Atiyah–Hirzebruch spectral sequence and the known homotopy groups of MTSO(2) up to degree 3, which have, for example, been computed by the first named author in [12]. The result is displayed in the table below.

i  −2 −1 0 1 2  3
π_{i}(MTSO(2))  ℤ 0 ℤ 0 ℤ ℤ/24
π_{i}(MTSO(2) ∧ Bℂ_{+}^{×}) ℤ 0 ℤ^{2} 0 ℤ^{3}  –

Table 1: Some stable homotopy groups.

The infinite loop space Ω^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}) associated to this spectrum is defined to be the colimit

Ω^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}) := colim_{n→∞}Ω^{n}(M_{n} ∧ Bℂ_{+}^{×}),

and the homotopy groups of this space coincide with those of the spectrum in non-negative degrees. From the table we see that Ω^{∞}(MTSO(2)∧Bℂ_{+}^{×}) has π_{0} in bijection with ℤ^{2}. We denote by Ω_{0}^{∞}(MTSO(2)∧Bℂ_{+}^{×}) the path-component of the basepoint. Cohen and Madsen [7, 8] have defined a comparison map

α_{g,k} : S_{g,r}(k) ⟶ Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×})

using Pontrjagin–Thom theory, and shown that it is an integral homology isomorphism in the stable range. Given homology stability, this may also be deduced from the general machines [15, 16] for proving Madsen–Weiss type theorems.

Let us now describe how the classes κ_{i,j} are defined on the space Ω^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}). Under the composition

H̃^{∗+2}(BSO(2)×Bℂ^{×}) ≅ H^{∗}(MTSO(2)∧Bℂ_{+}^{×}) ⟶^{σ_{∗}} H^{∗}(Ω_{0}^{∞}MTSO(2)∧Bℂ_{+}^{×})

of the Thom isomorphism in spectrum cohomology with the cohomology suspension^{1}, the class e^{i+1} ⊗ c_{1}^{j} maps to a class we define to be

κ_{i,j} ∈ H^{2i+2j}(Ω_{0}^{∞}MTSO(2) ∧ Bℂ_{+}^{×}; ℤ).

The description of push-forwards in terms of Pontrjagin–Thom theory shows that these classes pull back under α_{g,k} to the classes

π_{!}(e(T_{v})^{i+1}c_{1}(L)^{j}) ∈ H^{2i+2j}(S_{g,r}(k); ℤ),

as in the introduction.

2.5 Cohomology of the infinite loop space

The rational cohomology of an infinite loop space has a very restricted structure, and is easily deduced from the rational cohomology of the associated spectrum. In turn, the cohomology of a Thom spectrum over a classifying space can be expressed by means of characteristic classes.

Proof of Theorem B. It is well known that the rational cohomology of the zero component infinite loop space Ω_{0}^{∞}X is given by the free graded-commutative algebra on the vector space τ_{∗>0}H^{∗}(X; ℚ) of positive degree elements in spectrum cohomology. (For a proof, note that any rational spectrum splits into a sum of suspensions of Hℚ, and the claim is true for Σ^{n}Hℚ by direct calculation.) The rational spectrum cohomology of MTSO(2) ∧ Bℂ^{×} in positive degrees is the vector space

ℚ⟨u_{−2} · e^{i+1} ∧ c_{1}^{j} | i + j > 0, j ≥ 0, i ≥ −1⟩

where u_{−2} ∈ H^{−2}(MTSO(2); ℚ) denotes the Thom class. The element u_{−2} · e^{i+1} ∧c_{1}^{j} under the cohomology suspension gives the element κ_{i,j} on the infinite loop space.

To prove Theorem C, we first identify H^{2}(Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}); ℤ) as an abstract group.

Lemma 2.3. There are isomorphisms

H^{i}(Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}); ℤ) = \begin{cases}0 & i = 1, 3; \\ ℤ^{3} & i = 2. \end{cases}

1For any spectrum X = {X_{n}}, the evaluation maps Σ^{n}Ω^{n}X_{n} → X_{n} induce maps on cohomology H^{∗+n}(X_{n}) → H^{∗}(Ω^{n}X_{n}), which after taking limits over n gives σ^{∗} : H^{∗}(X) → H^{∗}(Ω^{∞}X).

Documenta Mathematica 17 (2012) 417–450

Proof. By Theorem B we know the statement is true modulo torsion. Thus we must show there is no torsion in cohomology in degrees ∗ ≤ 3, or equivalently that there is no torsion in homology in degrees ∗ ≤ 2. By the isomorphisms listed in Table 1, Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}) is simplyconnected, so in particular H_{1} is zero and hence torsion free. Furthermore

π_{2}(MTSO(2) ∧ Bℂ_{+}^{×}) ≅ H_{2}(Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}); ℤ)

by Hurewicz’ theorem, and hence we see that the second homology is free abelian of rank three.

Next we determine a ℤ-basis of H^{2}(Ω_{0}^{∞}(MTSO(2)∧Bℂ_{+}^{×})) ≅ ℤ^{3}, which begins with naming elements. We describe the elements on the moduli space S_{g}(k) as Chern classes of certain index bundles of Cauchy–Riemann operators. A standard exercise in index theory will then show that the classes are indeed induced from Ω^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}) (this exercise has been solved in the section on “universal operators” in [13]). We do this in order to deal with more concrete objects and also to facilitate computations. Recall that H^{2}(BΓ_{g}; ℤ) ≅ ℤ has a generator λ, the Hodge class, that satisfies 12λ = κ_{1}. To define the next element, we need a divisibility result, based on the Grothendieck–Riemann–Roch theorem. Recall the universal surface bundle π : E_{g}(k) → S_{g}(k) with vertical tangent bundle T_{v} and universal line bundle L → E. Consider the Dolbeault operator \overline{∂}_{T^{⊗r}⊗L^{⊗s}} on the tensor product bundle of r copies of the vertical tangent bundle and s copies of L. The Chern character of its index bundle is, by Grothendieck–Riemann–Roch:

ch(Ind(\overline{∂}_{T^{⊗r}L^{⊗s}})) = π_{!}(Td(x)e^{rc_{1}(T_{v})}e^{sc_{1}(L)})

and the degree 2 part is

(6r^{2} + 6r + 1)λ + \frac{1}{2}s^{2}(κ_{0,1} + κ_{−1,2}) + (rs + \frac{1}{2}(s − s^{2}))κ_{0,1}.  (2.3)

The first and third summand are integral, hence so is the middle summand. In other words, we have proved:

Proposition 2.4. The class κ_{0,1} − κ_{−1,2} ∈ H^{2}(S_{g,r}(k); ℤ) is divisible by 2.

Thus we may define

ζ := \frac{1}{2}(κ_{0,1} − κ_{−1,2}) ∈ H^{2}(S_{g,r}(k); ℤ),

as this group is torsion-free and hence κ_{0,1} − κ_{−1,2} is uniquely divisible.

Proof of Theorem C. It is enough to give a map H^{2}(Ω_{0}^{∞}(MTSO(2) ∧ Bℂ_{+}^{×}); ℤ) → ℤ^{3} that maps the tuple B := (λ, κ_{0,1}, ζ) to a basis. To achieve this, we construct three examples of surface bundles equipped with complex line bundles. The genus and the degree of the line bundle are irrelevant for this purpose. Here are the examples:

Example 2.5. For g in the stable range, we consider the universal surface bundle on BΓ_{g}, together with the trivial line bundle on it. We know that H^{2}(BΓ_{g}; ℤ) ≅ ℤ⟨λ⟩, and it is clear that B evaluates to (1, 0, 0).

Example 2.6. Consider the trivial surface bundle π : Bℂ^{×} ×Σ_{g} → Bℂ^{×}, with complex line bundle given by π^{∗}γ. Because this surface bundle is trivial, we have κ_{1} = 0, hence λ = 0. Moreover, κ_{0,1} is χ(Σ_{g}) times a generator. Finally, κ_{−1,2} is zero. If we put g = 2, we obtain that B is mapped to (0, 2, 1).

Example 2.7. Let H_{1} be the Hirzebruch surface (we use the notation of [9]). It is an 𝕊^{2}-bundle over 𝕊^{2}, which is not spin and has κ_{1} = 0 (since the signature of the total space is 0). A basis for H_{2}(H_{1}; ℤ) is given by the fundamental class u of the fibre and the image v of the section 𝕊^{2} → H_{1} at ∞. Let (x, y) be the Poincaré dual basis to (u, v). Using the intersection matrix given in [9], it is not hard to see that the Euler class of the vertical tangent bundle is e = 2x + y. Let L → H_{1} be the line bundle with Chern class y. Again using the intersection matrix, we compute κ_{0,1} = −1 and κ_{−1,2} = −1. Thus B is mapped to (0, −1, 0).

3 Cohomology of the extended mapping class groups

3.1 Proof of Theorems D, E and F

The results of the previous section give the stable cohomology of all the extended mapping class groups Γ̃_{g,r}^{n}, except for the case (r, n) = (0, 0), which is the most interesting of all. Theorem D is the key result about the extended mapping class groups in the case (r, n) = (0, 0).

Proof of Theorem D. All four spaces in the sequences are connected. By definition the map

Π : S_{g}(k) ⟶ BΓ̃_{g}(k)

induces an isomorphism on fundamental groups. Therefore, by Proposition 2.1, the homotopy fibre of Π is a K(ℤ, 2). The mapping space map(Bℂ^{×}, BΓ̃_{g}(k)) is connected (because the source is simply-connected and the target is aspherical). So the map Π ∘ Ξ : Bℂ^{×} → BΓ̃_{g}(k) is nullhomotopic. The choice of a nullhomotopy defines a map F : Bℂ^{×} → hofib_{∗}(Π). To prove that Π and Ξ form a fibre sequence, it suffices to prove that F induces an isomorphism on π_{2}. But hofib_{∗}(Π) → S_{g}(k) induces an isomorphism on π_{2}, so it is enough to prove that Ξ induces an isomorphism on π_{2}. However, Ξ is defined as Bℂ^{×} →^{−⊗L_{0}} map(Σ_{g}, Bℂ^{×})(k) → S_{g}(k) and both maps are π_{2}-isomorphisms. Thus Ξ and Π form a fibre sequence. From now on we replace Π by a fibration and we will show that Γ̃_{g}(k) acts trivially on the cohomology of the fibre, using that g ≥ 3. By Theorem A (iv) and Theorem C, we have the calculation H_{1}(S_{g}(k); ℤ) = 0 for g ≥ 3. The Leray–Serre spectral sequence for the map Π implies that H^{1}(BΓ̃_{g}(k); ℤ/2) = 0, and so Γ̃_{g}(k) can only act trivially on H^{2}(Bℂ^{×}; ℤ) = ℤ. Consequently, the generator of H^{2}(Bℂ^{×}; ℤ) gives an element in the E_{2}-page of the Leray–Serre spectral sequence of the map Π, which transgresses to a class θ ∈ H^{3}(BΓ̃_{g}(k); ℤ). This corresponds to a homotopy class of map θ : BΓ̃_{g}(k) → K(ℤ, 3). Standard obstruction theory provides a homotopy cartesian square

\xymatrix{S_{g}(k) \ar[d] \ar[r] & P K(ℤ, 3) ≃ ∗ \ar[d] \\ BΓ̃_{g}(k) \ar[r]^{θ} & K(ℤ, 3)}

(the left-hand vertical map is the path-fibration of K(ℤ, 3)), which identifies S_{g}(k) with the homotopy fibre of θ. To establish the second part of Theorem D, recall that the stabilization map S_{g,1}(k) → S_{g}(k) is a homology isomorphism in degrees 3∗ ≤ 2g −1 by Theorem A (ii). Together with Proposition 2.1, this completes the proof.

We now prepare to prove Theorem E. Recall the fibre sequence

Bℂ^{×} ⟶^{Ξ} S_{g}(k) ⟶^{Π} BΓ̃_{g}(k)  (3.1)

from Theorem D. Because Π (when replaced by a fibration) is principal, it is simple and so the E_{2}-term of the Leray–Serre spectral sequence takes the form

E_{2}^{p,q} = H^{p}(BΓ̃_{g}; H^{q}(Bℂ^{×})) ≅ \begin{cases}H^{p}(BΓ̃_{g}; ℤ) & q = 0 (mod 2) \\ 0 & q = 1 (mod 2). \end{cases}

The first task will be to compute the edge homomorphism of the spectral sequence, i.e. the map Ξ^{∗} : H^{∗}(S_{g}(k)) → H^{∗}(Bℂ^{×}). We have described this map geometrically: for L_{0} → Σ_{g} the fixed degree k line bundle, this map classifies the trivial surface bundle Bℂ^{×} × Σ_{g} → Bℂ^{×}, together with the line bundle L = pr_{1}^{∗}γ ⊗ pr_{2}^{∗}L_{0}. For this surface bundle and line bundle we immediately compute

e(T_{v}) = (2 − 2g) · 1 ⊗ u
c_{1}(L) = c_{1}(γ) ⊗ 1 + 1 ⊗ k · u,

where u ∈ H^{2}(Σ_{g}; ℤ) is the Poincaré dual to a point, from which it is easy to compute the classes κ_{i,j}.

Lemma 3.1. The classes κ_{i,j} with i > 0 map to zero, κ_{0,j} maps to (2−2g)·c_{1}(γ)^{j} and κ_{−1,j} maps to j · k · c_{1}(γ)^{j−1}. In particular, λ goes to zero, ζ goes to (1−g−k)·c_{1}(γ) and κ_{0,1} to (2−2g)·c_{1}(γ).

Figure 1: E_{2} page of the Serre spectral sequence for (3.1).

Proof of Theorem E. We study the Leray–Serre spectral sequence in cohomology for the fibre sequence (3.1), which has E_{2} page as shown in Figure 1 and which converges to zero in total degrees 1 and 3 by Lemma 2.3. From this we deduce that H^{1}(BΓ̃_{g}(k)) = 0 and we get an exact sequence

0 ⟶ H^{2}(BΓ̃_{g}(k)) ⟶^{Π^{∗}} H^{2}(S_{g}(k)) ⟶^{Ξ^{∗}} H^{2}(Bℂ^{×}) ⟶^{d_{3}} H^{3}(BΓ̃_{g}(k)) ⟶ 0.

By Lemma 3.1, the index of the image of the edge homomorphism in H^{2}(Bℂ^{×}) is gcd(2 − 2g, 1 − g − k). The kernel of the edge homomorphism is free of rank 2, and by elementary algebra it is generated by λ and η.

We now prepare to prove Theorem F, where we describe H^{∗}(BΓ̃_{g}(k); ℚ) in the stable range. We must begin with the definition of the classes ν_{i,j}. Let BΓ_{g}^{1} → BΓ_{g} be the universal surface bundle. There is a homotopy commutative diagram

[figure: Bℂ^{×}; ι; E_{g}(k); π; f; BΓ̃_{g}^{1}(k); p; BΓ_{g}^{1}; Bℂ^{×}; ϵ; S_{g}(k); Π; BΓ̃_{g}(k); BΓ_{g}.]

The two squares on the right are cartesian and π and p are surface bundles. The maps ι and ϵ are the inclusions of the respective homotopy fibres. The first two maps of the bottom sequence form a fibre sequence by Theorem D. Recall that E_{g} = map(Σ_{g}^{1}, Bℂ^{×}) ×_{𝒟_{g}^{1}} E𝒟_{g}^{1}. Evaluation at the basepoint of Σ_{g}^{1} is a 𝒟_{g}^{1}-equivariant map map(Σ_{g}^{1}, Bℂ^{×}) → Bℂ^{×} and so defines a map E_{g} → Bℂ^{×}, which is a left homotopy inverse to ι. As ι has a left homotopy inverse, there is a short exact sequence

0 ⟶ H^{2}(Γ̃_{g}^{1}(k); ℚ) ⟶^{f^{∗}} H^{2}(E_{g}(k); ℚ) ⟶^{ι^{∗}} H^{2}(Bℂ^{×}; ℚ) ⟶ 0.

The rational cohomology class c_{1}(L) − \frac{1}{2−2g} π^{∗}κ_{0,1} maps to 0 ∈ H^{2}(Bℂ^{×}; ℚ) by Lemma 3.1, and thus there exists a unique v ∈ H^{2}(S_{g}^{1}(k); ℚ) with f^{∗}v = c_{1}(L) − \frac{1}{2−2g} π^{∗}κ_{0,1}. Let e(T_{v}) ∈ H^{2}(S_{g}^{1}(k); ℤ) be the Euler class of the vertical tangent bundle of p, and define

ν_{i,j} := p_{!}(e(T_{v})^{i+1}v^{j}) ∈ H^{2i+2j}(BΓ̃_{g}(k); ℚ).

We compute

Π^{∗}ν_{i,j} = π_{!}(e(T_{v})^{i+1}(c_{1}(L) − \frac{1}{χ}π^{∗}κ_{0,1})^{j})

= ∑_{p+q=j} \binom{j}{p}(−\frac{1}{χ})^{q}π_{!}(e(T_{v})^{i+1}c_{1}(L)^{p}π^{∗}(κ_{0,1}^{q}))

= ∑_{p+q=j} \binom{j}{p}(−\frac{1}{χ})^{q}κ_{i,p}κ_{0,1}^{q},

bearing in mind that the degree zero classes are κ_{−1,1} = k, κ_{0,0} = 2 − 2g.

Proof of Theorem F. To see that Π^{∗} is rationally injective, we use the fibre sequence (3.1) and the fact from Theorem E that the element θ ∈ H^{3}(BΓ̃_{g}(k); ℤ) which classifies this fibre sequence is torsion. This implies that the rational Leray–Serre spectral sequence collapses, so that Π^{∗} is rationally injective. We certainly have an algebra homomorphism

ℚ[ν_{i,j} | i ≥ −1, j ≥ 0, i + j > 0; (i, j) ≠ (0, 1)] ⟶ H^{∗}(BΓ̃_{g}(k); ℚ) (3.2)

and by the calculation above Π^{∗}ν_{i,j} ≡ κ_{i,j} mod (κ_{0,1}), so the algebra ℚ[ν_{i,j} | i ≥ −1, j ≥ 0, i + j > 0; (i, j) ≠ (0, 1)] maps injectively into H^{∗}(S_{g}(k); ℚ) in the stable range, and so (3.2) is injective too. Counting dimensions and using the collapse of the Leray–Serre spectral sequence above shows (3.2) is also surjective in the stable range.

Remark 3.2. The computation above shows that ν_{0,1} = 0 and ν_{−1,2} = \frac{1}{g−1} gcd(2g − 2, g − 1 + k)η.

3.2 Structure of the extended mapping class groups

From the results proved so far and some facts on the cohomology of the mapping class groups, we can illuminate the structure of the extended mapping class groups. If we write H_{g,r}^{n} := H^{1}(Σ_{g,r}^{n}, ∂Σ_{g,r}^{n} ∪ P; ℤ), then there is a homotopy equivalence map_{∂}(Σ_{g,r}^{n}, Bℂ^{×})(k) ≃ BH_{g,r}^{n} as long as r + n > 0, and so extensions

0 ⟶ H_{g,r}^{n} ⟶ Γ̃_{g,r}^{n}(k) ⟶ Γ_{g,r}^{n} ⟶ 1.  (3.3)

If (r, n) = (0, 0) there is a square of isomorphisms

\xymatrix{π_{1}(map(Σ_{g}, Bℂ^{×})(k), L_{0}) \ar[d]^{Hurewicz} & π_{1}(map(Σ_{g}, Bℂ^{×})(0), ∗) ≅ H_{g} \ar[d]^{Hurewicz} \ar[l]_{−⊗L_{0}} \\ H_{1}(map(Σ_{g}, Bℂ^{×})(k); ℤ) & H_{1}(map(Σ_{g}, Bℂ^{×})(0); ℤ) \ar[l]_{−⊗L_{0}}}  (3.4)

and so we still have the sequence (3.3).

Lemma 3.3. The action of Γ_{g,r}^{n} on H_{g,r}^{n} in the extension (3.3) is the usual one.

Proof. Let γ ∈ 𝒟_{g,r}^{n} be a diffeomorphism. It acts on map_{∂}(Σ_{g,r}^{n}, Bℂ^{×})(k) by precomposition, which does not typically fix the point L_{0}. Instead, we have the equation γ^{∗} ∘(−⊗L_{0}) = (−⊗γ^{∗}L_{0})∘γ^{∗}. As γ has degree 1, γ^{∗}L_{0} is homotopic to L_{0}, and so the action of γ on H_{1}(map(Σ_{g}, Bℂ^{×})(k); ℤ) corresponds to the usual action on H_{g} under the isomorphism of (3.4).

We ask when the extension (3.3) is split, i.e. admits a homomorphic section. For all values of r, n, k in which Proposition 2.2 applies, there is a splitting, showing triviality in these cases. For (r, n) = (0, 0), the following proposition shows that this result is sharp (in the stable range).

Proposition 3.4. Let g ≥ 6 and n = r = 0. Then the extension (3.3) splits if and only if k ≡ 0 (mod 2g − 2).

Proof. Let σ be a splitting, giving a map σ : BΓ_{g} → BΓ̃_{g}(k). If g ≥ 6, then it is known that H^{3}(BΓ_{g}; ℤ) = 0 and so σ^{∗}θ = 0. Therefore σ lifts to a cross-section τ : BΓ_{g} → S_{g}(k) by elementary obstruction theory. This means that there is a line bundle on the universal surface bundle E_{g} → BΓ_{g} of fibrewise degree k. But Bödigheimer and Tillmann have shown [5] that the fibrewise degree of a line bundle on E_{g} is divisible by the Euler number 2 − 2g.

Remark 3.5. Morita [31] has proved that H^{2}(Γ_{g}; H_{g}) ≅ ℤ/(2 − 2g). One can show that the map ℤ → H^{2}(Γ_{g}; H_{g}) that sends k to the isomorphism class of the extension H_{g} → Γ̃_{g}(k) → Γ_{g} is a surjective homomorphism.

3.3 Proof of Theorem G

The rational Leray–Serre spectral sequence of the fibre sequence

BH_{g,r}^{n} ⟶ BΓ̃_{g,r}^{n}(k) ⟶ BΓ_{g,r}^{n}  (3.5)

has the form

E_{2}^{p,q} = H^{p}(BΓ_{g,r}^{n}; ∧^{q}(H_{g,r}^{n} ⊗ ℚ)) ⟹ H^{p+q}(BΓ̃_{g,r}^{n}; ℚ),  (3.6)

where the action of Γ_{g,r}^{n} is given by the usual action on the cohomology of Σ_{g,r}^{n}. The proof that this collapses begins with the case (r, n) = (0, 0), where we need the following result.

Proposition 3.6. (Deligne [10, Proposition 2.1]) Let f : X → Y be a fibration, F be a local coefficient system of ℚ-vector spaces on X, m ∈ ℕ and u ∈ H^{2}(X; ℚ) be given. Denote X_{y} := f^{−1}(y) and suppose that for each y ∈ Y and each i ≥ 0, the multiplication by (u^{i})|_{X_{y}} induces an isomorphism H^{m−i}(X_{y}; F|_{X_{y}}) → H^{m+i}(X_{y}; F|_{X_{y}}) (the “Lefschetz condition for (X, f, F, u, m)”). Then the Leray–Serre spectral sequence E_{2}^{p,q} ⇒ H^{p+q}(X; F) for f collapses at the E_{2}-stage.

To apply this criterion, let ω ∈ H^{2}(BH_{g}) be the symplectic class. It is easy to see that multiplication by ω^{k} induces an isomorphism H^{g−k}(BH_{g}; ℚ) → H^{g+k}(BH_{g}; ℚ). The next lemma therefore shows that (3.5) satisfies the Lefschetz condition for the constant coefficient system ℚ, m = g and the class η ∈ H^{2}(BΓ̃_{g}(k)) from Theorem E.

Lemma 3.7. The restriction of η ∈ H^{2}(BΓ̃_{g}(k)) to BH_{g} is a non-trivial multiple of ω.

Proof. As the Γ_{g}-invariant part of H^{2}(BH_{g}; ℤ) = ∧^{2}H_{g} is the subspace spanned by the symplectic form ω, the restriction of η is certainly some multiple of ω. By a theorem of Morita [30, Proposition 4.1], H^{1}(Γ_{g}; H_{g}) = 0. Thus from the spectral sequence (3.6) we find an exact sequence

0 ⟶ H^{2}(Γ_{g}; ℚ) ⟶ H^{2}(Γ̃_{g}(k); ℚ) ⟶ H^{2}(H_{g}; ℚ)^{Γ_{g}} ⟶^{d_{3}} · · · .

As η and κ_{1,0} are linearly-independent, the image of η is non-trivial.

Remark 3.8. In fact, one may calculate that the restriction of η is \frac{2−2g}{gcd(2−2g,g+k−1)} · ω.

This shows that the spectral sequence (3.6) collapses if r = n = 0.

Proof of the collapse of (3.6) for r + n = 1 or r > 1 and n = 0. If g : Z → X is a map, then the Lefschetz condition for a fibration X → Y implies the Lefschetz condition for the pullback fibration g^{∗}X → Z. Because the two squares

\xymatrix{BΓ̃_{g,1}(k) \ar[r] \ar[d] & BΓ̃_{g}^{1}(k) \ar[r] \ar[d] & BΓ̃_{g}(k) \ar[d] \\ BΓ_{g,1} \ar[r] & BΓ_{g}^{1} \ar[r] & BΓ_{g}}

are cartesian, this observation shows the collapse of the spectral sequence for r + n = 1. Now we use homological stability to extend this result to the cases n = 0 and r ≥ 1, so we only obtain the collapsing in the stable range. We use the homological stability theorem of Boldsen [4] for the mapping class group with twisted coefficients. The coefficient system ∧^{j}H_{ℚ} has degree j and so the map

H^{2i+j}(Γ_{g,r}; ∧^{j}H_{ℚ}) ⟶ H^{2i+j}(Γ_{g,1}; ∧^{j}H_{ℚ}) is an isomorphism for 2i + j ≤ ⌊\frac{2g}{3} ⌋ − j. Hence the map of spectral sequences corresponding to Γ̃_{g,1}(k) → Γ̃_{g,r}(k) is an isomorphism in the stable range on E_{2}, and hence the spectral sequence for Γ̃_{g,r}(k) also collapses.

Computation of the associated graded algebras for n = 0, r > 0. The case r = 1, k = 0 has been proved by Kawazumi [20, 21, 22]. He defined certain cohomology classes [20]

m̃_{i,j} ∈ H^{2i+2j−2}(Γ̃_{g,1}(0); ℤ),

in the framework of group cohomology, as pushforwards π_{!}(e^{i} · ω^{j}) where the element e ∈ H^{2}(Γ_{g,1}^{1}, Γ_{g,1} × ℤ; ℤ) is the group level analogue of the Euler class of the vertical tangent bundle, and ω̃ ∈ H^{2}(Γ̃_{g,1}^{1}, Γ̃_{g,1} × ℤ; ℤ) is represented by a cocycle he defines manually. He does not phrase it in this way, but it is the Euler class of the relative central extension

[figure: ℤ; H_{1}(Σ_{g,1}^{1}) ⋊ Γ_{g,1}^{1}; H_{1}(Σ_{g,1}) ⋊ Γ_{g,1}^{1}; ℤ; (H_{1}(Σ_{g,1}^{1}) ⋊ Γ_{g,1}) × ℤ; Γ̃_{g,1}(0) × ℤ]  (3.7)

The homomorphism ρ : Γ_{g,1} × ℤ → Γ_{g,1}^{1} implicit in the diagram is obtained by gluing in an annulus with a Dehn twist, see [20, p. 140]. Here H_{1}(Σ_{g,1}^{1}) denotes the homology of the surface with the marked points removed. The top sequence arises from the short exact sequence 0 → ℤ → H_{1}(Σ_{g,1}^{1}) → H_{1}(Σ_{g,1}) → 0 of Γ_{g,1}^{1}-modules. The bottom sequence is induced from the top one via ρ and it is split via the Γ_{g,1}-equivariant inclusion H_{1}(Σ_{g,1}) → H_{1}(Σ_{g,1}^{1}) obtained by boundary connected sum with a punctured disc. Diagram (3.7) has a topological interpretation. Recall the U(1)-principal bundle S_{g,1}^{1} → E_{g,1} (that is, the frame bundle of the universal line bundle) and recall also that over the boundary S_{g,1} × 𝕊^{1} this bundle is trivialised. The diagram

[figure: 𝕊^{1}; S_{g,1}^{1}; E_{g,1}; 𝕊^{1}; ∂E_{g,1} × 𝕊^{1}; ∂E_{g,1}; BΓ̃_{g,1}(0) × 𝕊^{1}]

consisting of aspherical spaces induces the diagram (3.7) on fundamental groups. Therefore ω̃ coincides with the Chern class c_{1}(L). Thus we have proved:

Proposition 3.9. There is an equality m̃_{i,j} = κ_{i−1,j} ∈ H^{∗}(Γ̃_{g,1}(0); ℤ).

We say that the natural filtration of the cohomology of Γ̃_{g,r}^{n} is the Leray filtration coming from the fibre sequence (3.5). Kawazumi shows that the class m̃_{i+1,j} ∈ H^{∗}(BΓ̃_{g,1}) is detected in the Leray–Serre spectral sequence (3.6) by a certain class

m_{i+1,j} ∈ H^{2i+j}(Γ_{g,1}; ∧^{j}H)

which he constructs. Equivalently, we may say that κ_{i,j} = m̃_{i+1,j} has natural filtration precisely j. In [22, Theorem 3.2] he then shows that rationally the associated graded algebra to the natural filtration is

Gr(H^{∗}(Γ̃_{g,1}(0); ℚ)) = H^{∗}(Γ_{g,1}; ℚ) ⊗ ℚ[m_{i+1,j} | j > 0],

which proves Theorem G in this case. On the other hand, by Theorem A (i), there is an equivalence BΓ̃_{g,1}(k) ≃ BΓ̃_{g,1}(0) so this proves the result for all k. Furthermore, homological stability for twisted coefficients extends this to all r > 0 by the comparison maps BΓ̃_{g,r}(k) → BΓ̃_{g,1}(k).

Proof of collapsing and associated graded algebra for n > 0 and r ≥ 0. As in Section 2.3 we have a fibre sequence

BΓ_{g,r+n} ⟶ BΓ_{g,r}^{n} ⟶^{e_{1}×···×e_{n}} Bℂ^{×} × ⋯ × Bℂ^{×}

and as long as r > 0 the composition BΓ_{g,r+n} → BΓ_{g,r}^{n} → BΓ_{g,r} induces a cohomology isomorphism with coefficients in ∧^{j}H_{ℚ} in a stable range. Thus

H^{∗}(Γ_{g,r}; ∧^{j}H_{ℚ})[e_{1}, ..., e_{n}] ⟶ H^{∗}(Γ_{g,r}^{n}; ∧^{j}H_{ℚ})

is an isomorphism in a stable range. The same argument with the fibre sequence

BΓ̃_{g,r+n}(k) ⟶ BΓ̃_{g,r}^{n}(k) ⟶^{e_{1}×···×e_{n}} Bℂ^{×} × ⋯ × Bℂ^{×}

shows that H^{∗}(Γ̃_{g,r}(k); ℚ)[e_{1}, ..., e_{n}] → H^{∗}(Γ̃_{g,r}^{n}(k); ℚ) is also an isomorphism in the stable range, and by counting dimensions we see that the spectral sequence for Γ̃_{g,r}^{n}(k) must collapse. This leaves the case r = 0, n > 0. We consider the fibre sequence

BΓ_{g,1}^{n−1} ⟶ BΓ_{g}^{n} ⟶^{e_{1}} Bℂ^{×}

which gives a spectral sequence

Ê_{2}^{∗,∗} := H^{∗}(Γ_{g,1}^{n−1}; ∧^{q}H_{ℚ})[e_{1}] ⟹ H^{∗}(Γ_{g}^{n}; ∧^{q}H_{ℚ}).

The output of this spectral sequence is the input of the spectral sequence

E_{2}^{p,q} := H^{p}(Γ_{g}^{n}; ∧^{q}H_{ℚ}) ⟹ H^{p+q}(Γ̃_{g}^{n}(k); ℚ)

which we want to show collapses. In the stable range

H^{∗}(Γ̃_{g}^{n}(k); ℚ) ≅ ℚ[e_{1}, ..., e_{n}, κ_{1}, κ_{2}, ...] ⊗ ℚ[κ_{i,j} | i + j > 0, j > 0, i ≥ −1] and

⨁_{p,q} H^{p}(Γ_{g,1}^{n−1}; ∧^{q}H_{ℚ}) ≅ ℚ[e_{2}, ..., e_{n}, κ_{1}, κ_{2}, ...]⊗ℚ[x_{i,j} | i+j > 0, j > 0, i ≥ −1]

and hence by counting dimensions we see that both spectral sequences must collapse. From the collapse of the first spectral sequence we find that the associated graded is as claimed.

The case (r, n) = (0, 0). The last part of Theorem G to be proved is the claim about the associated graded to the natural filtration of H^{∗}(Γ̃_{g}(k); ℚ). For this we consider the fibre sequences

Σ_{g} ⟶ BΓ_{g}^{1} ⟶^{p} BΓ_{g} Σ_{g} ⟶ BΓ̃_{g}^{1}(k) ⟶^{p̃} BΓ̃_{g}(k).

The first one satisfies the assumptions of Theorem 3.6 for any rational coefficient system that is induced from BΓ_{g}, by taking u to be the Euler class of the vertical tangent bundle. The second sequence is the pullback of the first. Therefore the spectral sequence for BΓ_{g}^{1} → BΓ_{g} with coefficients in ∧^{r}H_{ℚ}:

E_{2}^{p,q} = H^{p}(BΓ_{g}; H^{q}(Σ_{g}) ⊗ ∧^{r}H_{ℚ}) ⟹ H^{p+q}(BΓ_{g}^{1}; ∧^{r}H_{ℚ})

collapses at the E_{2}-stage and

p^{∗} : H^{∗}(Γ_{g}; ∧^{q}H_{ℚ}) ⟶ H^{∗}(Γ_{g}^{1}; ∧^{q}H_{ℚ})

is injective. Hence the natural filtration on H^{∗}(Γ̃_{g}(k); ℚ) and the filtration induced by the injection p̃^{∗} agree, and the claimed description of the associated graded follows from Theorem F.

4 Relation to complex algebraic geometry

4.1 Definitions and results

We first summarise our results, and will give proofs below. We assume that the reader is familiar with the basic vocabulary of the language of topological stacks. If not, they are advised to read [18], [32] and the relevant parts of [14].

Definition 4.1. Let Hol_{g} denote the following stack, defined on the site Top. An object of Hol_{g}(X), X ∈ Top, is a triple (E, π, L), consisting of a family of genus g Riemann surfaces π : E → X and a fibrewise holomorphic line bundle L → E. An isomorphism (E, L) → (E′, L′) is an isomorphism of families f : E → E′ and an isomorphism h : L → f^{∗}L′ of holomorphic line bundles. We denote by Hol_{g}^{k} ⊂ Hol_{g} the open and closed substack consisting of those bundles having fibrewise degree k.

There is an obvious forgetful map Hol_{g} → M_{g}, which just remembers the underlying family of Riemann surfaces. The definition of the Picard stack Pic_{g} is a little more involved. Let π : E → X be a family of genus g Riemann surfaces, that is, an element of M_{g}(X). We define the associated Picard bundle Pic(E/X) → X as follows. Let O denote the sheaf of continuous, fibrewise holomorphic functions on E, and O^{×} the subsheaf of nowhere zero functions. The exponential sequence of sheaves ℤ → O → O^{×} gives an exact sequence of sheaves on X,

0 ⟶ R^{1}π_{∗}ℤ ⟶ R^{1}π_{∗}O ⟶ R^{1}π_{∗}O^{×} ⟶ R^{2}π_{∗}ℤ ⟶ 0.

This is because the sequence 0 → π_{∗}ℤ → π_{∗}O → π_{∗}O^{×} → 0 is isomorphic to 0 → ℤ → ℂ → ℂ^{×} → 0 and hence exact. Moreover, R^{2}π_{∗}O = 0. These are all sheaves of continuous sections of certain bundles of groups on X,

0 ⟶ [H^{1}(F_{x}; ℤ)] ⟶ [H^{1}(F_{x}; O)] ⟶ Pic(E/X) ⟶ ℤ × X ⟶ 0

where [H^{1}(F_{x}; F)] is the bundle of fibrewise first cohomologies with coefficients in the sheaf F, and Pic(E/X) is a bundle of abelian groups isomorphic to ℤ × 𝕋^{2g}. The fibre of Pic(E/X) over x ∈ X is the Picard group Pic(π^{−1}(x)). The group π_{1}(X) acts trivially on the set of path components, and we denote by Pic^{k}(E/X) the degree k component.

Definition 4.2. Let Pic_{g} denote the following stack, defined on the site Top. An object of Pic_{g}(X) is a family of Riemann surfaces π : E → X and a section s : B → Pic(E/X). An isomorphism (E, s) → (E′, s′) is an isomorphism of families f : E → E′ such that f^{∗}(s′) = s. We denote by Pic_{g}^{k} ⊂ Pic_{g} the open and closed substack consisting of those pairs (E, s) where s takes values in Pic^{k}(E) ⊂ Pic(E).

Definition 4.3. Let Φ_{g}^{k} : Hol_{g}^{k} → Pic_{g}^{k} be the morphism that sends a fibrewise holomorphic line bundle to its isomorphism class.

Let X be a space and β : X → Pic_{g}^{k} be a map, which is a family of Riemann surfaces π : E → X and a section s : X → Pic^{k}(E/X). We can ask whether there exists a (fibrewise) holomorphic line bundle L → E such that for each x ∈ X, the isomorphism class of the line bundle L|_{E_{x}} is s(x). Such a line bundle will be called topological Poincaré line bundle for β. It is plain to see that topological Poincaré line bundles for β are the same as lifts X → Hol_{g}^{k} of β along Φ_{g}^{k}. This observation can be used to generalise the notion of a Poincaré line bundle to stacks, as follows: a Poincaré line bundle for a map X → Pic_{g}^{k} is a lift to Hol_{g}^{k}. Remark 4.4. In the classical literature, the notion “Poincaré line bundle” is used slightly differently. Namely, for a single Riemann surface S, one asks for a line bundle L → Pic^{k}(S) × S, such that for each ℓ ∈ Pic^{k}(S), the restriction L|_{{ℓ}×S} is in the isomorphism class ℓ. More generally, for a family E → X, one asks for a holomorphic line bundle L → E ×_{X} Pic^{k}(E/X) with that property.

Theorem 4.5. Let β : X → Pic_{g}^{k} be given. Then the following properties hold:

(i) Each x ∈ X has an open neighborhood U such that the restriction β|_{U} admits a Poincaré line bundle.

(ii) If L_{i}, i = 0, 1 are two Poincaré line bundles, then each x ∈ X has a neighborhood U, such that (L_{0})|_{U} ≅ (L_{1})|_{U}.

(iii) If L is a Poincaré line bundle, then the group of automorphisms of L is isomorphic to map(X, ℂ^{×}).

More technically, the map Φ_{g}^{k} is a gerbe with band ℂ^{×}.

In fact, the three statements in Theorem 4.5 are precisely the gerbe axioms [6, §V.2 ], translated to the present context. Probably Theorem 4.5 is a special case of a statement that is well-known among algebraic geometers who are also fond of stacks, though we were unable to find a complete proof in the literature. The proof below applies only to line bundles on curves of genus g ≥ 2. A map of stacks X → Y of stacks is a universal weak equivalence if for each space Z and each map Z → Y, the induced map X ×_{Y} Z → Z is a weak equivalence of topological spaces. A homotopy type of the stack X is a universal weak equivalence X → X with source a space X. Each topological stack admits a homotopy type, and two homotopy types are unique up to homotopy equivalence. See Noohi [32] for these notions (he writes “classifying space” instead of “homotopy type”). We say shortly “the homotopy type of the stack X is X” if there exists a space X′, a weak equivalence X′ ≃ X and a universal weak equivalence X′ → X.

Theorem 4.6. The homotopy type of the stack Hol_{g}^{k} is S_{g}(k), the homotopy type of Pic_{g}^{k} is BΓ̃_{g}(k), and under these equivalences the gerbe Φ_{g}^{k} corresponds to Π.

If Y → X is a gerbe with band ℂ^{×}, the map X → Y on homotopy types has homotopy fibre a K(ℤ, 2) at every point, and the associated fibration is principal (c.f. [33]). Isomorphism classes of line bundles on a stack are in bijection with the sheaf cohomology group H^{1}(X; \underline{ℂ^{×}}). Similarly, isomorphism classes of gerbes with band ℂ^{×} are in bijection with the sheaf cohomology group H^{2}(X; \underline{ℂ^{×}}). The connecting homomorphism H^{i}(X; \underline{ℂ^{×}}) → H^{i+1}(X; ℤ) may fail to be an isomorphism on a general stack. However, if X is a local quotient stack (see [18] or [14] for this notion), the connecting homomorphism is an isomorphism. The reason is that the sheaf of continuous functions on a local quotient stack is acyclic. This is well-known, but we do not know a reference and include a proof as Lemma 4.19. The image of the isomorphism class of a gerbe G on X in H^{3}(X; ℤ) is called Dixmier–Douady class. So on a local quotient stack, gerbes are classified by their Dixmier–Douady class. Finally, the Dixmier–Douady class of G agrees with the characteristic class of the induced principal fibration.

Proposition 4.7. Both Hol_{g} and Pic_{g} are local quotient stacks.

Together with the observation that a gerbe is trivial once there exists a section, we obtain a corollary, using Theorem E.

Corollary 4.8. There exists a Poincaré line bundle for the identity map on Pic_{g}^{k} if and only if gcd(2g − 2, 1 − g − k) = 1.

This is a precise topological analogue with the result by Mestrano and Ramanan [27] that takes place in the holomorphic category. The proof of Proposition 4.7 will show the stronger result that both stacks are even holomorphic local quotient stacks. On a holomorphic stack X, we can talk about holomorphic line bundles. Let Pic_{hol}(X) be the group of isomorphism classes of holomorphic line bundles. There is a comparison map Pic_{hol}(X) → Pic_{top}(X), which we will show to be an isomorphism in Theorem 4.17. This theorem, combined with the purely topological Theorem E and Lemma 3.7, gives another proof of the main result of Kouvidakis’ paper [23].

4.2 The gerbe of holomorphic maps

Here we prove Theorem 4.5, which begins with a trivial observation:

Lemma 4.9. Tensor multiplication with the cotangent bundle induces a commutative diagram

\xymatrix{Hol_{g}^{k} \ar[r] \ar[d]^{Φ_{g}^{k}} & Hol_{g}^{k+2g−2} \ar[d]^{Φ_{g}^{k+2g−2}} \\ _{Pick g} \ar[r] & _{Pic{k+2g−2 \atop g}}}

whose horizontal arrows are isomorphisms.

Therefore, it is enough to prove Theorem 4.5 for large values of k. In the sequel, we assume that k > 2g − 2. The third part of Theorem 4.5 is obvious, because the automorphism group of a holomorphic line bundle on a compact Riemann surface is ℂ^{×}. The other two parts are in the lemmata below.

Lemma 4.10. The map Φ_{g}^{k} : Hol_{g}^{k} → Pic_{g}^{k} admits local sections.

Proof. Let X be a space, x ∈ X and β : X → Pic_{g}^{k} be a map, giving a family of Riemann surfaces E → X and a section s : X → Pic^{k}(E/X). We use a classical construction from the geometry of algebraic curves. The fibrewise kfold symmetric product is denoted by Sym^{k}(E/X) → X. It is well-known and not hard to see that this is a fibre bundle with smooth complex manifolds as fibres. Recall the classical divisor–line-bundle correspondence [17, p. 129 ff.]. This construction yields a fibre-preserving map η : Sym^{k}(E/X) → Pic^{k}(E/X). As long as k ≥ g, η is surjective (the Jacobi inversion theorem; because any line bundle of degree k then has a nontrivial holomorphic section). It is a classical result that for k > 2g − 2, η is a submersion with fibres isomorphic to ℂℙ^{k−g}. More precisely, Mattuck [26] has shown that for an individual Riemann surface S, the map Sym^{k}(S) → Pic^{k}(S) is a projective bundle (the structural group is ℙ Gl_{k+1−g}(ℂ)). It follows that η : Sym^{k}(E/X) → Pic^{k}(E/X) is a proper surjective submersion. In particular, η has local sections and by passing to a smaller X we can pick a lift of s; denoted by t : X → Sym^{k}(E/X); this is a section of the bundle Sym^{k}(E/X) → X. Given a divisor D on a Riemann surface S, the classical construction gives an actual line bundle (and not merely an isomorphism class) when one specifies local holomorphic functions on S that define D, i.e. have the same zeroes (with mutliplicity). Observe that such local functions can be picked continuously when the divisor varies continuously. This argument shows that after passage to an even smaller X, we can find a line bundle L → E whose restriction to E_{y} corresponds to the divisor t(y), for all y ∈ X. This L is the desired Poincaré line bundle.

Lemma 4.11. Let β : X → Pic_{g}^{k} and let L_{i} → E, i = 0, 1, be two Poincaré line bundles for β. Then each x ∈ X has a neighborhood U such that (L_{0})|_{U} ≅ (L_{0})|_{U}.

Proof. Consider the line bundle H := Hom(L_{0}, L_{1}) → E. It has fibrewise degree 0, and the hypothesis states that for each b ∈ B, the bundle H|_{E_{b}} is trivial. Now look at the fibrewise Cauchy–Riemann operator for H. As the kernels all have dimension 1, they form a line bundle on B by basic Fredholm theory. By passing to a neighborhood U of a given b ∈ B, we can pick a nowhere zero section of the kernel bundle. This is nothing else than an isomorphism L_{0}|_{U} ≅ L_{1}|_{U}.

Remark 4.12. One can show a stronger statement. Namely, the fibre bundle η : Sym^{k}(E/B) → Pic^{k}(E/B) with fibre ℂℙ^{k−g} has structure group ℙ Gl_{k−g+1}(ℂ) (if k > 2g − 2). There exists a cartesian diagram:

\xymatrix{Hol_{g}^{k} \ar[r] \ar[d] & ∗// Gl_{k−g+1}(ℂ) \ar[d] \\ _{Pick g} \ar[r] & ∗//ℙ Gl_{k−g+1}(ℂ).}

The lower horizontal map classifies the bundle η and the upper horizontal map classifies the vector bundle on Hol_{g}^{k} whose fibre at (C, L) is the space of holomorphic sections of L. We do not make use of this result, but leave a remark. The Dixmier–Douady class of the gerbe Φ_{g}^{k} is an element of H^{3}(Pic_{g}^{k}; ℤ) ≅ H^{3}(Pic_{g}^{k}; ℤ), and under the homotopy equivalence Pic_{g}^{k} ≃ BΓ̃_{g}(k) it corresponds to the generator θ ∈ H^{3}(BΓ̃_{g}(k); ℤ). Since the order of the gerbe ∗// Gl_{k−g+1}(ℂ) → ∗//ℙ Gl_{k−g+1}(ℂ) is k − g + 1, this can be used to give another proof that gcd(k − g + 1, 2g − 2)θ = 0.

Proof of Proposition 4.7. An atlas for Pic_{g}^{k} is given as follows. Let π : E_{g} → T_{g} be the universal family of Riemann surfaces on Teichmüller space. The total space of Pic(E_{g}/T_{g}) is a complex manifold of dimension 4g − 3. There is a map μ : Pic(E_{g}) → Pic_{g} that fits into a cartesian diagram

\xymatrix{Pic(E_{g}) \ar[r] \ar[d] & Pic_{g} \ar[d] \\ T_{g} \ar[r] & M_{g}.}

Therefore μ is an atlas, and so Pic_{g} is a Deligne–Mumford stack. The corresponding result for Hol_{g}^{k} follows from this and [18, Corollary 6.3].

4.3 Proof of Theorem 4.6

We will use the basic notions of homotopy theory for stacks as developed in [14] or [32]. The homotopy equivalence goes via a zig-zag, and we have to introduce an intermediate stack for this purpose.

Definition 4.13. Let Cx_{g} denote the following stack, defined on the site Top. An object of Cx_{g}(Y) consists of an oriented smooth surface bundle π : E → Y with genus g fibres, and a complex line bundle L → E. An isomorphism (E, L) → (E′, L′) is an isomorphism of surface bundles f : E → E′ and an isomorphism h : L → f^{∗}L′ of complex line bundles. We denote by Cx_{g}^{k} ⊂ Cx_{g} the open and closed substack consisting of those bundles having fibrewise degree k.

There is a diagram

\xymatrix{Hol_{g} \ar[r] \ar[d] & Cx_{g} \ar[d] & S_{g} \ar[l] \ar[dl] \\ M_{g} \ar[r] & ∗//𝒟_{g}}

where the left horizontal maps forget the holomorphic structure and the right horizontal one is given by taking the induced bundle. The vertical morphisms forget the line bundle data. It is clear that the diagram commutes (up to 2-isomorphism).

Lemma 4.14. The map S_{g} → Cx_{g} is a universal weak equivalence.

Proof. Since by definition

S_{g} = E𝒟_{g} ×_{𝒟_{g}} map(Σ_{g}, Bℂ^{×}),

we can rewrite the map S_{g} → Cx_{g} as the composition

E𝒟_{g} ×_{𝒟_{g}} map(Σ_{g}, Bℂ^{×}) ⟶ map(Σ_{g}, Bℂ^{×})//𝒟_{g} ⟶ Cx_{g}.  (4.1)

The first map is a universal weak equivalence by general stack-theoretic principles, namely [14, Proposition 2.5]. To analyze the second map, consider a space X and a map X → Cx_{g}, representing L → E →^{π} X, where π : E → X is a surface bundle and L → E a complex line bundle. Look at the auxiliary space S of pairs (x, f), where x ∈ X and f : L|_{π^{−1}(x)} → π^{−1}(x) × ℂ^{∞} is a bundle monomorphism. The topology on S is the unique one such that the projection S → X is a locally trivial fibre bundle and such that the preimage of x has the compact-open topology. The fibre over x is the space of bundle monomorphisms L|_{π^{−1}(x)} → π^{−1}(x) × ℂ^{∞}, which is contractible. It is a routine verification to identify the fibre product X ×_{Cx_{g}} map(Σ_{g}; Bℂ^{×})//𝒟_{g} with S. Thus the second map in (4.1) has local sections and it is a universal weak equivalence. This finishes the proof

Lemma 4.15. The map Hol_{g} → Cx_{g} is a universal weak equivalence.

Proof. The map Hol_{g} → Cx_{g} can be factored into three maps

Hol_{g} ⟶^{ϕ_{1}} H̃ol_{g} ⟶^{ϕ_{2}} qHol_{g} ⟶^{ϕ_{3}} Cx_{g},

each of which is a homotopy equivalence. The stack qHol_{g} parametrises families of Riemann surfaces together with complex line bundles, and the map ϕ_{3} is the forgetful map. It is a universal weak equivalence because the diagram

\xymatrix{qHol_{g} \ar[d] \ar[r] & Cx_{g} \ar[d] \\ M_{g} \ar[r] & ∗//𝒟_{g}}

is a fibre square and the bottom map is a universal weak equivalence by Teichmüller thory. Now we define the stack H̃ol_{g}. A map X → H̃ol_{g} is an element L → E → X of qHol_{g}(X), together with a family of fibrewise differential operators D : Γ(E; L) → Γ(E; Λ_{v}^{0,1} ⊗ L) such that

D(fs) = \overline{∂}f ⊗ s + fD(s)  (4.2)

for each s ∈ Γ(E; L) and f ∈ C^{∞}(E), where differentiation is understood to be in the fibrewise sense. The map ϕ_{2} forgets the differential operator. The condition (4.2) is convex, which implies that each smooth line bundle admits such an operator and that the space of these operators is convex. Therefore ϕ_{2} is a universal weak equivalence. The map ϕ_{1} associates to each holomorphic line bundle the Cauchy–Riemann operator on that line bundle. We claim that ϕ_{1} is an isomorphism of stacks. This amounts to showing that a holomorphic structure on a line bundle is determined by its Cauchy–Riemann operator (which is a tautology) and that any family of operators satisfying (4.2) induces the structure of a holomorphic line bundle on L, i.e. there exist locally nonzero solutions of Ds = 0. The argument we give is due to Atiyah and Bott [2, p. 555] (they consider the case of higher-dimensional vector bundles, which is more complicated, but for a fixed Riemann surface, which is easier). Let (L → E →^{p} X, D) be an element of H̃ol_{g}(X). For x ∈ X and y ∈ p^{−1}(x), we can pick a neighborhood U of x and a map α : U × 𝔻 → E over X which is an open embedding, fibrewise holomorphic and satisfies α(x, 0) = y; furthermore we require L to be trivial over U × 𝔻. We wish to find a section s of L over α(U × 𝔻) that is nowhere zero and satisfies Ds = 0 over U × 𝔻_{\frac{1}{2}} . To this end, pick a fibrewise smooth section s_{0} of L over α(U × 𝔻) and look for a function f that satisfies D(e^{f}s_{0}) = 0. So we have to solve the PDE

0 = e^{−f}D(e^{f}s_{0}) = \overline{∂}f ⊗ s_{0} + Ds_{0}.

Write Ds_{0} = −β ⊗ s_{0} for a (0, 1)-form β; this reduces the problem to the equation

\overline{∂}(f) = β.

Since all that matters is a local section on U × 𝔻_{\frac{1}{2}} , we can multiply β with a cut-off function and thus assume that β has compact support. Now we pick a fibrewise holomorphic embedding U × 𝔻 → U × ℂℙ^{1} and a bundle map from the trivial bundle on U × 𝔻 into the tautological line bundle on U×ℂℙ^{1}. By Riemann–Roch, the Cauchy–Riemann operator on the tautological line bundle (it has degree −1) is invertible. Hence its inverse is continuous as well. Therefore we can find a continuous solution of \overline{∂}f = β over U × 𝔻. The arguments given so far amount to the construction of an inverse map to ϕ_{1} and thus the proof is complete.

Now we will show how to compare Pic_{g}^{k} with BΓ̃_{g}(k). The first observation is that the homotopy type Pic_{g}^{k} is aspherical, as the map Pic_{g}^{k} → M_{g} is representable and a torus bundle, and the homotopy type of M_{g} is BΓ_{g}, so aspherical.

Lemma 4.16. After taking homotopy types, the map Φ_{g}^{k} : Hol_{g}^{k} → Pic_{g}^{k} is a first Postnikov approximation.

Proof. As the homotopy type of Pic_{g}^{k} is aspherical, it is enough to show that the map Φ_{g}^{k} induces an isomorphism on fundamental groups of homotopy types at all basepoints. By [33, Example 4.4 and Theorem 5.2], there is an exact sequence

⋯ π_{n}(Bℂ^{×}, ∗) ⟶ π_{n}(Hol_{g}^{k}, x) ⟶ π_{n}(Pic_{g}^{k}, ϕ_{g}^{k}(x)) ⟶ π_{n−1}(Bℂ^{×}, ∗) ⋯

on homotopy groups. As Bℂ^{×} is simply connected, the claim follows.

We have the diagram of solid arrows

\xymatrix{S_{g}(k) \ar[r]^{≃} \ar[d]^{Π} & Cx_{g}^{k} & Hol_{g}^{k} \ar[l]_{≃} \ar[d]^{Φ_{g}^{k}} \\ BΓ̃_{g}(k) \ar@{.>}[rr] &  & _{Pick g}}

By taking associated homotopy types, and using obstruction theory, there exists a dotted map (on homotopy types) which is a weak equivalence. Composing with Pic_{g}^{k} → Pic_{g}^{k}, we obtain the map in the statement of

4.4 Holomorphic versus topological Picard groups

Theorem 4.17. For both stacks Pic_{g}^{k} and Hol_{g}^{k}, the comparison map c_{1} : Pic_{hol}(−) → H^{2}(− ; ℤ) is an isomorphism.

Proof of surjectivity. We know that H^{2}(Hol_{g}^{k}; ℤ) ≅ ℤ^{3} generated by λ, κ_{0,1} and ζ, by Theorem A (ii), Theorem C, and Theorem 4.6. The index bundles of the Cauchy–Riemann operators on powers of the tangent bundle and the line bundle are holomorphic vector bundles, and in equation (2.3) we computed their first Chern classes. To realise λ as a first Chern class, put r = s = 0. To realise λ + ζ, put (r, s) = (0, 1). Finally, s = r = 1 yields 13λ + ζ + κ_{0,1}, which completes the proof for X = Hol_{g}^{k}. The result for Pic_{g}^{k} follows from this and Lemma 4.18 below.

Lemma 4.18. Let X and Y be holomorphic local quotient stacks and X → Y a ℂ^{×}-gerbe. If the comparison map Pic_{hol}(X) → Pic_{top}(X) is surjective, then so is Pic_{hol}(Y) → Pic_{top}(Y).

Proof. Any line bundle (holomorphic or topological) L → X has an action of ℂ^{×} coming from the gerbe structure. As usual, z ∈ ℂ^{×} acts by multiplication with z^{w} for a uniquely determined w ∈ ℤ, called the weight of L. It is not difficult to see that w : Pic_{top}(X) → ℤ is a homomorphism and coincides with the edge homomorphism H^{2}(X; ℤ) → H^{2}(Bℂ^{×}; ℤ) derived from the Leray– Serre spectral sequence. Moreover, a line bundle L → X descends to a line bundle on Y if and only if w(L) = 0. Therefore we get a diagram with exact rows:

\xymatrix{ & Pic_{hol}(Y) \ar[r] \ar[d] & Pic_{hol}(X) \ar[r]^{w} \ar@{->>}[d] & ℤ \\ 0 \ar[r] & Pic_{top}(Y) \ar[r] & Pic_{top}(X) \ar[r]^{w} & ℤ.}

The bottom sequence is exact at the left by a look at the Leray–Serre spectral sequence. A fragmentary version of the 5-lemma holds in this situation and shows that the left vertical map is onto.

Proof of injectivity. This begins with a result by Arbarello and Cornalba [1]. They showed that Pic_{hol}(M_{g}^{n}) → H^{2}(M_{g}^{n}; ℤ) is an isomorphism for all n ≥ 0. The long exact cohomology sequence of the exponential sequence, using that H^{1}(M_{g}^{n}; ℤ) = 0, shows H^{1}(M_{g}^{n}; O) = 0 for all n ≥ 0. We use the Leray spectral sequence for the map f : Pic_{g}^{k} → M_{g}, which takes the form

E_{2}^{p,q} := H^{p}(M_{g}; R^{q}f_{∗}O) ⟹ H^{p+q}(Pic_{g}^{k}; O).

As f is a proper map with connected fibres, f_{∗}O = O, and so E_{2}^{1,0} = 0. Let π : M_{g}^{1} → M_{g} denote the universal curve. There is a map i : M_{g}^{1} → Pic_{g}^{k} over M_{g} given by (Σ, p) ↦ (Σ, K + (k − 2g + 2) · p) where K denotes the canonical class of Σ. It satisfies f ∘ i = π, and

i^{∗} : R^{1}f_{∗}O ⟶ R^{1}π_{∗}O

is known to be an isomorphism. We argue that H^{0}(M_{g}; R^{1}π_{∗}O) = 0. Otherwise, there would be a nontrivial section of the holomorphic vector bundle R^{1}π_{∗}O, producing a nontrivial family of sections of Pic^{0}(M_{g}^{1}/M_{g}), parametrised by ℂ. This would produce a line bundle on M_{g}^{1} which has fibrewise degree 0, but is nontrivial in some fibre, contradicting the result of Arbarello and Cornalba just quoted. Therefore H^{1}(Pic_{g}^{k}; O) = 0. To derive that H^{1}(Hol_{g}^{k}; O) = 0 as well, consider the Leray–Serre spectral sequence of the gerbe Hol_{g}^{k} → Pic_{g}^{k}, which has the form E_{2}^{p,q} = H^{p}(Pic_{g}^{k}; H^{q}(∗//ℂ^{×}; O)). Since ℂ^{×} is reductive, this E_{2}-term is concentrated in the bottom row and because H^{0}(∗//ℂ^{×}; O) = ℂ, the proof is complete.

4.5 Relation to algebraic geometry

In the algebraic setting there exists a smooth algebraic stack Pic_{g}^{k} over the moduli stack of curves M_{g}, which is an algebraic analogue of our Hol_{g}^{k} → M_{g}. There is a natural copy of the multiplicative group 𝔾_{m} inside the endomorphisms of every object of Pic_{g}^{k}, hence it may be rigidified to a stack P_{g}^{k} which is an algebraic analogue of our Pic_{g}^{k}. This new stack is smooth and Deligne–Mumford, and the quotient map is representable. We have not proved, but expect to be true, that the associated analytic stacks to Pic_{g}^{k} and P_{g}^{k} are Hol_{g}^{k} and Pic_{g}^{k} respectively. In any case, there is certainly an analytic map (P_{g}^{k})^{an} → Pic_{g}^{k}, and so we may consider the composition

Pic(P_{g}^{k}) ⟶ Pic_{hol}((P_{g}^{k})^{an}) ⟶ Pic_{hol}(Pic_{g}^{k}),  (4.3)

and similarly for Pic_{g}^{k}. In [28], Melo and Viviani use algebro-geometric methods to study the Picard groups of P_{g}^{k} and Pic_{g}^{k}, and also of their compactifications. Comparing with [28, Theorem 4.2 and Corollary 4.4], we see—by computation of both sides— that the composition (4.3) and its analogue for Pic_{g}^{k} are both isomorphisms.

4.6 A technical result on local quotient stacks

Lemma 4.19. Let X be a local quotient stack. Then the sheaf C_{X} of continuous complex-valued functions is acyclic, and so the homomorphism c_{1} : Pic_{top}(X) → H^{2}(X; ℤ) is an isomorphism.

Proof. This follows from a twofold application of the descent spectral sequence. If X and Y are stacks, Y → X a representable surjective map and F a sheaf on X, then there is a spectral sequence

E_{1}^{p,q} = H^{q}(Y_{p}; F_{p}) ⟹ H^{p+q}(X; F),

where Y_{p} := Y ×_{X} Y ×_{X} … ×_{X} Y (p factors) and F_{p} is the pullback of F to Y_{p}. If X = X//G is a global quotient of a space by a compact Lie group, then the descent spectral sequence is

E_{2}^{p,q} = H_{cts}^{p}(G; H^{q}(X, C_{X})) ⟹ H^{p+q}(X; C_{X});

this is zero if q > 0 since the sheaf C_{X} is fine. Thus the spectral sequence collapses to the continuous group cohomology H_{cts}^{∗}(G; map(X, ℂ)), which vanishes in positive degrees as the group G is compact and the coefficient module is a locally convex topological vector space with a continuous G-action. For details of that argument, consult [3, Proposition 6.3]. If X is merely a local quotient stack, there is an open cover by substacks X_{i} each of which is a global quotient stack. Since fibre products of global quotients are global quotient stacks, the descent spectral sequence for the map ∐_{i} X_{i} → X has E_{1}^{p,q} = 0 for q > 0. The E^{p,0}-line is just the Čech complex for the open cover. Because there exist partitions of unity in this situation, see [14, Appendix A], this complex is acyclic.

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Johannes Ebert

Mathematisches Institut

der Westfälischen

Wilhelms-Universität Münster Einsteinstraße 62 DE-48149 Münster Germany johannes.ebert@uni-muenster.de Oscar Randal-Williams Institut for Matematiske Fag Universitetsparken 5 DK-2100 København Ø Denmark o.randal-williams@math.ku.dk