j. differential geometry 45 (1997) 331-385

THE GEOMETRY OF THE SPACE OF HOLOMORPHIC MAPS FROM A RIEMANN

SURFACE TO A COMPLEX PROJECTIVE SPACE

SADOK KALLEL & R. JAMES MILGRAM

0. Introduction

In recent years there have been a number of papers on the homology and geometry of spaces of holomorphic maps of the Riemann sphere into complex varieties, [21], [5], [17], [18], [3], [9]. However, the very classic question of the structure of the spaces of holomorphic maps from complex curves M_{g} of genus g ≥ 1 to complex varieties has proved to be very difficult. There is Segal’s stability theorem, [21], which shows that the natural inclusion of the space of based holomorphic maps of degree k into the space of all based maps Hol_{k}^{∗}(M_{g}, V) ↪ Map_{k}^{∗}(M_{g}, V) is a homotopy equivalence through a range of dimensions which increases with k when V = P^{n}, the complex projective space. Also, there is the extension of this result by J. Hurtubise to further V; [11]. But that is about all.

In this paper we begin the detailed study of the topology of the Hol_{k}^{∗}(M_{g}, P^{n}). We are able to completely determine these spaces and their homology when M_{g} is an elliptic curve, and we give an essentially complete determination in the case where M_{g} is hyperelliptic. In particular we determine the rational homology of these spaces when k ≥ 2g−1 in the elliptic and hyperelliptic cases.

Let M_{g} be a genus g complex curve. The key analytic result on the structure of Hol_{k}^{∗}(M_{g}, P^{1}) is Abel’s Theorem which identifies the disjoint pairs of k-tuples of unordered points ⟨r_{1}, …, r_{k}⟩ and ⟨p_{1}, …, p_{k}⟩ on M_{g} that are the roots and poles of a meromorphic function on M_{g} in terms of the Abel-Jacobi embedding of M_{g} into its Jacobi variety, μ: M_{g}⟶J(M_{g}). This extends to give necessary and sufficient conditions for an (n + 1) tuple {V_{i} | V_{i} = ⟨x_{i,1}, …, x_{i,k}⟩, i = 0, …, n} of points in M_{g} with ∩_{0}^{n}{V_{i}} = ∅ to be the root data for a holomorphic map of M_{g} into P^{n}, n ≥ 2, and thus defines an embedding

Received June 22, 1996. The first author holds a Postdoctoral fellowship with CRM, Montréal, and the second author was partially supported by a grant from the NSF and a grant from the CNRS.

Hol_{k}^{∗}(M_{g}, P^{n}) ⊂ (SP^{k}(M_{g}))^{n+1},

where SP^{k}(X) is the k-fold symmetric product of X.

We begin by studying a compactification of the space Hol_{k}^{∗}(M_{g}, P^{n}), which we denote E_{k}^{n}(M_{g}) obtained by taking the closure of the embedding

Hol_{k}^{∗}(M_{g}, P^{n}) ⊂ (SP^{k}(M_{g}))^{n+1}

above. We show in (2.2) that E_{k}^{n}(M_{g}) is given as the total space of a fibering,

(P^{k−g})^{n+1}⟶E_{k}^{n}(M_{g})⟶J(M_{g})

for k ≥ 2g−1, with H^{∗}(E_{k}^{n}(M_{g}); A) = H^{∗}(P^{k−g}, A)^{n+1}⊗H^{∗}(J(M_{g}); A) for A any commutative ring of coefficients. Moreover, for k ≤ 2g − 2, E_{k}^{n}(M_{g}) is stratified by strata which are fiberings

(P^{s})^{n+1}⟶S_{k}^{s}⟶A_{k}^{s}

where the A_{k}^{s} are subspaces of J(M_{g}) determined by the curve. Specifically, W_{k}^{j} ⊂ J(M_{g}) is the subspace of points in the image of the k^{th} Abel-Jacobi map

μ_{k}: SP^{k}(M_{g})⟶J(M_{g})

with (μ_{k})^{−1}(x) = P^{s} where s ≥ max(k−g, 0)+j, and A_{k}^{j} = W_{k}^{j} −W_{k}^{j+1}. Both spaces W_{k}^{j} and A_{k}^{j} are extensively studied in [1], [10] (cf. §5).

This compactification has the property that Hol_{k}^{∗}(M_{g}, P^{n}) is open in E_{k}^{n}(M_{g}), and we write V_{k}^{n}(M_{g}) for the (closed) complement E_{k}^{n}(M_{g}) − Hol_{k}^{∗}(M_{g}, P^{n}). The space E_{k}^{n}(M_{g}) being a closed, compact manifold, Alexander-Poincaré duality now gives

H̃^{2k(n+1)−2ng−∗}(Hol_{k}^{∗}(M_{g}, P^{n}); F) ≅ H_{∗}(E_{k}^{n}(M_{g}), V_{k}^{n}(M_{g}); F)

for k ≥ 2g−1. This then indicates that one can understand Hol_{k}^{∗}(M_{g}, P^{n}) by first studying V_{k}^{n}(M_{g}).

The space V_{k}^{n}(M_{g}) ⊂ E_{k}^{n}(M_{g}) is the union of two pieces. The first, for k > 2g − 1, is a subfibration Z_{k}^{n}(M_{g}) of the form

{ ⋃_{i=0}^{n+1}(P^{k−g})^{i} × P^{k−g−1} × (P^{k−g})^{n−i}} ⟶Z_{k}^{n}(M_{g})⟶J(M_{g})

with analogous definitions of Z_{k}^{n}(M_{g}) for k ≤ 2g − 1 (see Definition 2.6). Moreover, it is easy to determine the relative cohomology groups, H^{∗}(E_{k}^{n}(M_{g}), Z_{k}^{n}(M_{g}); F) for k > 2g − 1, and not too difficult for k ≤ 2g − 2, provided we know enough about the W_{k}^{j}.

In the case of hyperelliptic and elliptic curves the Riemann-Roch Theorem determines the W_{k}^{j} explicitly. Moreover, a complete description of the W_{k}^{j} for all curves of genus ≤ 6 is given in [1, pp.206 - 211].

In §5 we study the way in which the W_{k}^{i} determine

H^{∗}(E_{k}^{n}(M_{g}), Z_{k}^{n}(M_{g}); A)

in detail and obtain a spectral sequence converging to these groups with explicit E^{1}-term.

Proposition 5.9. Suppose that k ≤ g. Then there is a spectral sequence converging to H^{∗}(E_{k}^{n}(M_{g}), Z_{k}^{n}(M_{g}); A) with E^{1}-term

E^{1} =H_{∗}(SP^{k}(M_{g}), SP^{k−1}(M_{g}); A)

⊕ ∐_{i} H_{∗}(W_{k}^{i}, W_{k−1}^{i}; A) ⊗ H̃^{∗}(S^{2i(n+1)}; A),

and in case k ≥ g then

E^{1} =Σ^{2(k−g)(n+1)}H_{∗}(J(M_{g}), W_{k−1}^{k−g}; A)

⊕ ∐_{i>k−g} H_{∗}(W_{k}^{i}, W_{k−1}^{i}; A) ⊗ H̃_{∗}(S^{2i(n+1)}; A).

The second part of V_{k}^{n}(M_{g}) consists of the image of an action map

ν: M_{g} × E_{k−1}^{n}(M_{g})⟶E_{k}^{n}(M_{g})

introduced in (2.4) that puts in redundant roots, and is the main source of difficulty in recovering the homology of Hol_{k}^{∗}(M_{g}, P^{n}) from the homology groups H_{∗}(E_{k}^{n}(M_{g}); A) or H_{∗}(E_{k}^{n}(M_{g}), Z_{k}^{n}(M_{g}); A). In order to handle this part we construct a spectral sequence which converges to the homology of the pair (E_{k}^{n}(M_{g}), V_{k}^{n}(M_{g})) starting with the relative groups

H_{∗}(E_{k}^{n}(M_{g}), Z_{k}^{n}(M_{g}); F).

Theorem 4.3. There is a spectral sequence converging to

H_{∗}(E_{k}^{n}(M_{g}), V_{k}^{n}(M_{g}); F)

for any field F with E^{1}-term

E^{1} = ∐_{{i+j=k \atop i≥1}} H_{∗}(E_{i}^{n}(M_{g}), Z_{i}^{n}(M_{g}); F)

⊗ H_{∗}(SP^{j}(ΣM_{g}), SP^{j−1}(ΣM_{g}); F)
⊕ H_{∗}((SP^{k}(ΣM_{g}), SP^{k−1}(ΣM_{g}); F).

The rest of §4 discusses the structure of d_{1} and various properties such as the multiplicative pairing of the spectral sequences discussed immediately after the proof of (4.3). Of course the structure of the groups H_{∗}(SP^{j}(ΣM_{g}), SP^{j−1}(ΣM_{g}); F) is well known from e.g. [7], [6], and [19]. It is also reviewed in §6.

Next we use these results to clarify the structure of the natural inclusions

Hol_{k}^{∗}(M_{g}, P^{n}) ↪ Map_{k}^{∗}(M_{g}, P^{n}).

The spectral sequence of (4.3) has a natural break at i = 2g − 1 in the sense that i ≥ 2g − 1 implies that the relative homology groups

H_{∗}(E_{i}^{n}(M_{g}), Z_{i}^{n}(M_{g}); F) = \begin{cases}0∗ < 2(i − g)(n + 1), \\ H_{∗−2(i−g)(n+1)}(J(M_{g}); F) & otherwise, \end{cases}

and, when i < 2g −1, the groups depend on the structure of μ_{i}, the W_{i}^{r}, and have to be determined case by case. We call the case i ≥ 2g − 1 the stable range for the spectral sequence and completely determine the differentials in this range. For ∗ > (2n − 1)(k − 2g + 1) only the stable homology contributes to H_{∗}(E_{k}^{n}(M_{g}), V_{k}^{n}(M_{g}); F). Thus, by Poincaré duality, the unstable range only contributes homology above this range. On the other hand it is easily seen that the duals of the stable range classes inject into H_{∗}(Map_{k}^{∗}(M_{g}, P^{n}); F), and many of these classes live considerably above the stable range above. Thus, as was the case with Hol_{k}^{∗}(P^{1}, P^{n}) ([5]) one has considerably more information about the map than was given by Segal’s stability theorem.

In the final sections we study the cases of elliptic and hyperelliptic curves. Here, as indicated, the Riemann-Roch Theorem gives complete control of the W_{k}^{r} and consequently the E^{1}-term of the spectral sequence is within range of calculation. In particular, for elliptic curves the situation is completely understood.

Typical of the results in the elliptic case is

Lemma 9.4. Let I = (b_{1}, b_{2}) be the augmentation ideal in the polynomial ring Q[b_{1}, b_{2}] where dim(b_{i}) = 2, i = 1, 2. Then

H_{∗}(Hol_{k}^{∗}(M_{1}, P^{1}); Q) ≃ {Q[b_{1}, b_{2}]/I^{k+1}} (1, e_{1}, e_{2}, h_{1}, h_{2}, v)
⊕ Q(w_{1}, w_{2}, …, w_{2k−1}),

where dim(e_{i}) = 1, dim(h_{i}) = 2, dim(w_{i}) = 2k − 3 and dim(v) = 3.

Here M_{1} is an arbitrary elliptic curve. Additionally, it turns out that the map

H^{∗}(Map_{k}^{∗}(M_{1}, P^{1}); Q)⟶H^{∗}(Hol_{k}^{∗}(M_{1}, P^{1}); Q)

is surjective for all k ≥ 1 in this case.

For hyperelliptic curves there are some technical questions which seem difficult to handle for finite field coefficients, but with some effort a complete determination of the E^{1}-term in (4.3) with Q-coefficients is given in §10 –15, together with sufficient differentials to completely determine E^{∞} and conclude that H_{∗}(Hol_{k}^{∗}(M_{g}, P^{n}); Q) injects into H_{∗}(Map_{k}^{∗}(M_{g}, P^{n}); Q) for k ≥ 2g − 1 in the hyperelliptic case as well.

As the arguments are pretty involved we summarize the salient points here.

To begin, the Riemann-Roch Theorem gives a complete determination of the W_{i}^{r} as quotients of SP^{i}(M_{g}) via an action discussed in the proof of Lemma 10.2,

SP^{r}(P^{1}) × SP^{l}(M_{g})⟶SP^{l+2r}(M_{g})

induced from the Abel-Jacobi map μ_{2} : SP^{2}(M_{g})⟶J(M_{g}) which fails to be an embedding at only one point where it has inverse image a copy of P^{1}, the P^{1} in the action map above.

These observations give the following result for hyperelliptic curves:

Lemma 10.2. Suppose that M_{g} is hyperelliptic and τ ∈ J(M_{g}) is the hyperelliptic point. Let k ≥ 1 and t ≤ [\frac{k}{2}]. Then we have the following:

(a) for k ≤ g, the space W_{k}^{t} is μ_{k−2t}(SP^{k−2t}(M_{g})) + tτ,

(b) for 2g − 1 > k > g, t > k − g, we also have

W_{k}^{t} = μ_{k−2t}(SP^{k−2t}(M_{g})) + tτ.

Of course, this gives the W_{k}^{t} as quotients, so, in order to obtain information about the W_{k}^{t} we introduce some spectral sequences which take care of the details of the quotienting process in §11 through §13. Using them we are able to determine the rational homology of the W_{k}^{t} as follows.

Lemma 13.7.

(a) The inclusion W_{j} ⊂ J(M_{g}) induces an injection in rational homology H_{∗}(W_{j}; Q) ↪ H_{∗}(J(M_{g}); Q) = Γ(e_{1}, …, e_{2g}) with image the subvector space spanned by the subspaces

{Γ_{s}(e_{1}, …, e_{2g})[M_{g}]^{t} | s + t ≤ j}

where [M_{g}] = ∑_{1}^{g} e_{2i−1}e_{2i} is the image of the fundamental class of M_{g} under the Abel-Jacobi map μ_{∗}.

(b) H_{∗}(W_{j−1}; Q) injects into H_{∗}(W_{j}; Q) under the inclusion so the relative groups are given as

H_{∗}(W_{j}, W_{j−1}; Q) ≅ H_{∗}(W_{j}; Q)/H_{∗}(W_{j−1}; Q).

Next we turn to the homology of the spaces Hol_{k}^{∗}(M_{g}, P^{n}) in the hyperelliptic cases. This involves using the spectral sequences and calculations above. Of course, the E^{1}-term in the spectral sequence (4.3) is very complex, but one is able to recognize in it the direct sum of a family of chain complexes, each of which calculates a part of a certain T or or Ext group of the exterior algebra Γ(e_{1}, …, e_{2g}) modulo the two sided ideal generated by [M_{g}]. This explains §14 which is devoted to the calculation of the relevant T or-groups.

Finally, in §15 we are able to put these results together to obtain our main calculational result, Theorem 15.1. The natural map

H_{∗}(Hol_{k}^{∗}(M_{g}, P^{n}); Q)⟶H_{∗}(Map_{k}^{∗}(M_{g}, P^{n}); Q)

is injective for k ≥ 2g and n > 2 if M_{g} is hyperelliptic.

Acknowledgements

We would like to thank the “Centre de Recherches Mathématiques” at the University of Montréal for their support and hospitality during the period when much of this work was done. The special case of (based) meromorphic functions on the torus; i.e., g = 1, n = 1 was worked out as part of the first author’s Ph.D thesis written under the direction of the second author. We also thank Professor J. Hurtubise for numerous comments and his generous support.

1. Preliminaries on the Abel-Jacobi map

We review some classical definitions and theorems about the algebraic geometry of curves. We begin by defining the Abel-Jacobi map together with the Jacobi variety J(M_{g}) associated to any positive genus Riemann surface. Good references are [1] and [10].

Any Riemann surface M_{g} of genus g ≥ 1 has g independent holomorphic sections of the cotangent bundle τ^{∗}(M_{g}), (holomorphic 1-forms), w_{1}, w_{2}, …, w_{g}, the abelian differentials on M_{g}.

Fix a basepoint p_{0} ∈ M_{g}. Then we can associate to each point p ∈ M_{g} and each path γ between p_{0} and p the vector of integrals

μ_{γ}(p) = (∫_{p_{0}}^{p} w_{1}, …, ∫_{p_{0}}^{p} w_{g}) ∈ C^{g}.

Since any two paths γ and γ′ between p_{0} and p together determine a closed loop based at p_{0}, (which we will denote as L), we have that μ_{γ}(p) is well defined up to vectors of the form

(∫_{L} w_{1}, …, ∫_{L} w_{g}) .

If we choose a set of loops L_{1}, …, L_{2g} which, in homology, form a basis for H_{1}(M_{g}; Z), they give rise to the following g × 2g period matrix

Ω = (\begin{matrix}∫_{L_{1}} w_{1} & ⋯ & ∫_{L_{2g}} w_{1} \\ ⋮ & ... & ⋮ \\ ∫_{L_{1}} w_{g} & ⋯ & ∫_{L_{2g}} w_{g} \end{matrix}) .

Thus, since the w_{i} are closed, the values μ_{γ}(p) depend only on p and not γ in the quotient torus

J(M_{g}) = C^{g}/Ω ≅ (S^{1})^{2g}.

Consequently, they give rise to a well defined map μ: M_{g}⟶J(M_{g}) which is called the Abel-Jacobi map for M_{g}.

At this point we need to introduce symmetric products.

Definition. The n-fold symmetric product SP^{n}(X) of the space X is defined to be the set of unordered n-tuples of points of X; i.e.,

SP^{n}(X) = X^{n}/S_{n},

where S_{n} is the symmetric group on n letters. A point in SP^{n}(X) will be written in the form ∑ m_{i}x_{i} with m_{i} > 0 and ∑ m_{i} = n, or in the form

⟨x_{1}, x_{2}, …, x_{n}⟩.

Remarks. Let X be any CW complex with base point ∗. Then there are inclusions SP^{n}(X) ↪ SP^{n+1}(X) which identify ∑_{i} n_{i}P_{i} with ∑_{i} n_{i}P_{i} + ∗, and we get the increasing sequence of spaces

∗ = SP^{0}(X) ⊂ SP^{1}(X) ⊂ ⋯ ⊂ SP^{n−1}(X) ⊂ SP^{n}(X) ⊂ · · · .

The union of this sequence is the infinite symmetric product SP^{∞}(X), based at ∗. Moreover, if X is path connected, then the homotopy type of SP^{∞}(X) is independent of the choice of ∗.

We can extend the Abel-Jacobi map to the symmetric products of M_{g} by the rule ⟨m_{1}, …, m_{k}⟩ ↦ μ(m_{1}) + ⋯ + μ(m_{k}), obtaining the family of maps

(1.1)  μ_{k}: SP^{k}(M_{g})⟶J(M_{g}).

The map μ is called the Abel-Jacobi map and it, together with the extensions μ_{k}, is a critical piece of the structure data for M_{g}.

It is a remarkable result due to Andreotti that for any M_{g} the symmetric products SP^{k}(M_{g}) are all complex manifolds, indeed, complex algebraic varieties of complex dimension k. To see this note that the symmetric product SP^{k}(C) is diffeomorphic to C^{k} via the map that takes the unordered collection ⟨z_{1}, …, z_{k}⟩ of points in C to the coefficients of the monic polynomial of degree k with the z_{i}, 1 ≤ i ≤ k as roots. Since locally SP^{k}(M_{g}) is modeled on SP^{k}(C), the result follows.

Remark 1.2. In the special case that M_{g} = P^{1} a slight extension of the above argument identifies SP^{k}(P^{1}) with P^{k} which is now regarded as the space of all homogeneous polynomials of degree k in 2 variables.

Remark 1.3. In (1.14) we point out an extension of this result due to Mattuck which for k ≥ 2g − 1 identifies SP^{k}(M_{g}) with the total space of a fibration over (S^{1})^{2g} with fiber P^{k−g}.

Remark 1.4. Another way of thinking about J(M_{g}) is as the Picard group of M_{g}, the space of isomorphism classes of holomorphic line bundles on M_{g}. From this point of view the addition in J(M_{g}) corresponds to the tensor product of line bundles. Under this correspondence, the map μ takes m ∈ M_{g} to the line bundle obtained from the trivial bundle over M_{g} by gluing in a copy of the negative Hopf bundle over m (see [10]).

We can associate to every holomorphic function f ∈ Hol(M_{g}, P^{1}) a divisor (f) defined by (f) = ∑ n_{i}Z_{i} − ∑ m_{j}P_{j} where {Z_{i}}, {P_{i}} are respectively the zeros and poles of f with multiplicities n_{i} and m_{i} respectively. By standard residue calculations, it is easy to see that ∑ n_{i} = ∑ m_{j}. The space of pairs of disjoint divisors on a Riemann surface (ζ, η), subject to the condition deg(ζ) = deg(η), constitutes then the first step in the description of the space of holomorphic maps from the surface to the Riemann sphere P^{1}.

For maps of P^{1} to itself this description is enough for there do exist meromorphic maps having prescribed roots and poles (of equidegree); i.e., any divisor D = ζ − η, degD = 0 is the divisor of a meromorphic function on P^{1}. For the general case g ≥ 1 it turns out that a pair (ζ, η) as above need not necessarily give rise to a meromorphic function and one needs a further condition.

Theorem 1.5 (Abel). Given positive divisors D and D′ on M_{g}, degD = degD′, then there exists an f ∈ Hol(M_{g}, P^{1}) so that (f) = D − D′ if and only if μ(D) = μ(D′).

The f associated to the difference D − D′ is unique provided we specify in advance the image of p_{0} (based maps) and neither D nor D′ contains the basepoint, p_{0}. Additionally, given D −D′, there are unique disjoint positive divisors D_{1}, D′_{1} so that D − D′ = D_{1} − D′_{1} and any f with (f) = D − D′ will have roots precisely the terms in D_{1} and poles in D′_{1}. We make the following definition.

Definition 1.6. The divisor space Div_{k}(X) for a given space X is the set of pairs of disjoint positive divisors on X, i.e.,

(1.7) Div_{k}(X) = {(D, D′) ∈ SP^{k}(X) × SP^{k}(X) | D ∩ D′ = ∅} ,

and more generally

Div_{k}^{n}(X) ={(D_{1}, …, D_{n+1} | D_{i} ∈ SP^{k}(X), 1 ≤ i ≤ n + 1,

D_{1} ∩ D_{2} ∩ ⋯ ∩ D_{n+1} = ∅}. (1.8)

Corollary 1.9. The space of based holomorphic maps of degree k, Hol_{k}^{∗}(M_{g}, P^{1}), is the inverse image of 0 under the subtraction map

s: Div_{k}(M_{g} − p_{0})⟶J(M_{g}),

where the subtraction map s is given by s((D, D′)) = μ(D) − μ(D′).

More generally, and perhaps more usefully, we have

Corollary 1.10. The space of based holomorphic maps of degree k, Hol_{k}^{∗}(M_{g}, P^{n}) is the subspace of Div_{k}^{n}(M_{g} − p_{0}) consisting of (n + 1)- tuples of degree k positive divisors, subject to the following constraint:

μ_{k}(D_{1}) = μ_{k}(D_{2}) = ⋯ = μ_{k}(D_{n+1}).

It was using this formulation of the space Hol_{k}^{∗}(M_{g}, P^{n}) that Segal [21] proceeded to prove his stability result. We will shortly give and use a yet more explicit version of this corollary. But before we do so, here are some of the standard results on the Abel-Jacobi map that we will be using:

(1.11) The orginal map μ: M_{g}⟶J(M_{g}) is an embedding and the (complex) dimension of the image of μ_{d}: SP^{d}(M_{g})⟶J(M_{g}) is d for d ≤ g. In particular μ is onto for d ≥ g (Jacobi inversion theorem).

(1.12) The preimage of any point μ(p) ∈ J(M_{g}), μ_{d}^{−1}(μ(p)) ∈ SP^{d}(M_{g}) is always a complex projective plane P^{m} for some m ≥ 0.

(1.13) For d ≤ 2g−2 the dimension m is less than or equal to \frac{d}{2} (Clifford).

(1.14) For d ≥ 2g − 1 the map μ_{d} makes SP^{d}(M_{g}) into an analytic fiber bundle over J(M_{g}) with fiber P^{d−g} (Mattuck [16]).

In §9 we will give more details on the structure of these maps for g ≤ 5, but now we turn to the construction of an explicit model for the space Hol_{k}^{∗}(M_{g}, P^{n}) from our considerations thus far.

2. A compactified version of Hol_{k}^{∗}(M_{g}, P^{n})

It should be apparent from §1 that we are interested in the inverse image of 0 under s: Div_{k}(M_{g} − x_{0}) → J(M_{g}) for holomorphic maps to P^{1}, and generally in the inverse image of the iterated diagonal

Δ^{n+1}(J(M_{g}) ⊂ \underbrace{J(M_{g}) × ⋯ × J(M_{g})}_{n+1−times}

under μ_{k} × ⋯ × μ_{k} for maps into P^{n} .

Let the space E_{i_{0},i_{1},... ,i_{n}} be defined as the fiber product of μ×⋯×μ and Δ in the diagram below

(2.1)  \begin{matrix}E_{i_{0},i_{1},... ,i_{n}} & ⟶ & SP^{i_{0}}(M_{g}) × ⋯ × SP^{i_{n}}(M_{g}) \\ ↓ & & ↓μ×···×μ \\ J & ⟶^{Δ} & J(M_{g}) × ⋯ × J(M_{g}). \end{matrix}

More explicitly

E_{i_{0},i_{1},... ,i_{n}}

= {(D_{0}, …, D_{n}) ∈ SP^{i_{0}}(M_{g}) × ⋯ × SP^{i_{n}}(M_{g}) | μ(D_{i}) = μ(D_{j})}.

Clearly Hol_{k}^{∗}(M_{g}, P^{n}) ⊂ E_{k,k,...,k} as the subset consisting of all the {(D_{0}, …, D_{n})} where no D_{i} contains ∗ and ⋂_{0}^{n} D_{i} = ∅. Of course, for k sufficiently small Hol_{k}^{∗}(M_{g}, P^{n}) is empty and this is consistent with the fact that Riemann surfaces of positive genus do not admit holomorphic functions with a single pole. However, once Hol_{k}^{∗}(M_{g}, P^{n}) is non-empty, and k is sufficiently large, Hol_{k}^{∗}(M_{g}, P^{n}) will be open in E_{k,...,k} with E_{k,...,k} as its closure. Then E_{k,k,...,k} is the compactification of Hol_{k}^{∗}(M_{g}, P^{n}) mentioned in the title of this section.

Lemma 2.2. For i_{j} ≥ 2g − 1, 0 ≤ j ≤ n, we have a fibering

∏_{j=0}^{n} P^{i_{j}−g} ⟶ E_{i_{0},i_{1},... ,i_{n}} ⟶ J(M_{g}),

and the fiber is totally non-homologous to zero so

H_{∗}(E_{i_{0},i_{1},... ,i_{n}}) ≅ H_{∗}(P^{i_{0}−g}) ⊗ ⋯ ⊗ H_{∗}(P^{i_{n}−g}) ⊗ H_{∗}(J(M_{g})).

Proof. In this range of dimensions i_{j} ≥ 2g−1, one can see by virtue of Mattuck’s Theorem that the space E_{i_{0},i_{1},... ,i_{n}} becomes the total space of a fibration

P^{i_{0}−g} × ⋯ × P^{i_{n}−g} ⟶ E_{i_{0},i_{1},... ,i_{n}} ⟶ J(M_{g})

obtained from the pull-back of a product of Mattuck’s fibrations

P^{i_{0}−g} × ⋯ × P^{i_{n}−g} ⟶ SP^{i_{0}}(M_{g}) × … × SP^{i_{n}}(M_{g}) ⟶ J(M_{g})^{n+1}

by the diagonal inclusion Δ. Since Δ is injective and since the Serre spectral sequence for Mattuck’s fibration collapses at E_{2}, the lemma follows from standard spectral sequence comparison arguments. q.e.d.

Further Properties of the E_{i_{0},...,i_{n}}

First of all, we notice that for any j ≥ 0, we have natural inclusions

E_{i_{0},i_{1},... ,i_{n}} ↪ E_{i_{0},... ,i_{j}+1,... ,i_{n}}

given by adding the basepoint in the (j + 1)^{st} position.

Secondly we observe that the map

μ × ⋯ × μ: SP^{i_{0}}(M_{g}) × ⋯ × SP^{i_{n}}(M_{g})⟶J(M_{g})^{n+1}

is multiplicative by construction, and so it induces a multiplicative pairing on the E_{i_{0},...,i_{n}} which is commutative and associative:

(2.3)  ν^{I,J}: E_{i_{0},...,i_{n}} × E_{j_{0},...,j_{n}}⟶E_{i_{0}+j_{0},...,i_{n}+j_{n}}.

Also, E_{1,1,...,1} = M_{g} and the pairing above thus yields an action

(2.4)  ν: (∐_{k=1}^{∞} SP^{k}(M_{g})) × E_{i_{0},...,i_{n}}⟶ ∐_{k=1}^{∞} E_{i_{0}+k,...,i_{n}+k}

explicitly defined via the diagonal multiplication

SP^{k}(M_{g}) × (SP^{i_{0}}(M_{g}) × ⋯ × SP^{i_{n}}(M_{g}))
⟶SP^{i_{0}+k}(M_{g}) × ⋯ × SP^{i_{n}+k}(M_{g}),

which acts on points as follows

\begin{matrix}( ∑ m_{s}, (⟨m_{11}, …, m_{1,i_{0}}⟩, …, ⟨m_{n1}, …, m_{ni_{n}}⟩)) \\ ↦ (⟨∑ m_{s}, m_{11}, …, m_{1,i_{0}}⟩, …, ⟨∑ m_{s}, m_{n1}, …, m_{ni_{n}}⟩). \end{matrix}

We can then give an explicit reformulation of the description of Hol_{i}^{∗}(M_{g}, P^{n}) in these terms.

Lemma 2.5. Let Ẽ_{i_{0},...,i_{n}} ⊂ E_{i_{0},...,i_{n}} be the subspace in which no m_{rs} is equal to the base point ∗. Then,

Hol_{i}^{∗}(M_{g}, P^{n}) ≅ Ẽ_{i,i,...,i} − Image(ν) ∩ Ẽ_{i,i,...,i}.

The following constructions are now needed for the remainder of our discussion.

Definition 2.6. For each n, the space LE_{i} is the quotient

LE_{i} = E_{i,i,...,i}/ {⋃ E_{i,i,...,i−1,i,...,i}} .

The space QE_{i} is the quotient

LE_{i}/ {Image(ν)} .

Remark 2.7. When i ≥ 2g − 1, the space E_{i,... ,i} is a manifold of dimension 2(n + 1)i − 2ng = 2(i − g)(n + 1) + 2g. However when i < 2g − 1 there is no garantee that E_{i,... ,i} is actually a manifold.

Using Alexander-Poincaré duality, we can now deduce

Lemma 2.8. Assume i ≥ 2g − 1. Then for untwisted coefficients A we have

H_{j}(Hol_{i}^{∗}(M_{g}, P^{n}); A) ≅ H^{2(i−g)(n+1)+2g−j}(QE_{i}; A).

Note at this point that the multiplication μ of (2.3) induces an associative, commutative multiplication on the QE_{i} as well:

μ: QE_{i} × QE_{j}⟶QE_{i+j}.

From Lemma (2.8), it is clear that it is the space QE_{i} that we wish to study in the remainder of this paper. Unfortunately, it is generally very hard to obtain the cohomology of such a space without a careful analysis of the piece we collapse out. So, in order to do this we follow the procedure of [5] and replace the cone on the union above by a more complex but much more structured space.

3. A model for QE_{i}

Consider the twisted product space

(3.1)  DE(M_{g}) = (⋃ E_{i_{0},i_{1},... ,i_{n}}) ×_{t} SP^{∞}(cM_{g}),

where cT denotes the reduced cone on T, and the twisting t is given by the action above. Precisely, points of DE are of the form

{(D_{0}, …, D_{n}), (t_{1}, z_{1}) … (t_{l}, z_{l})}, D_{i} ∈ SP^{k_{i}}(M_{g}) and μ(D_{i}) = μ(D_{j})}

with the identification that when t_{i} = 0 the point above is identified with

\{(D_{0} + z_{i}, …, D_{n} + z_{i}), (t_{1}, z_{1}) … { \atop (t_{i}, z_{i})} … (t_{l}, z_{l})\},

where the entry (t_{i}, z_{i}) is deleted from the last set of coordinates. Clearly μ(D_{r} + z_{i}) = μ(D_{s} + z_{i}) and the construction makes sense.

The space DE is filtered by the subspaces

DE_{k_{0},k_{1},... ,k_{n}}(M_{g}) = ⋃ ; ⋮^{i_{0}+l≤k_{0}}_{in+l≤kn} E_{i_{0},i_{1},... ,i_{n}} ×_{t} SP^{l}(cM_{g}).

Observe that there are projection maps

p_{k_{0},k_{1},... ,k_{n}}: DE_{k_{0},k_{1},... ,k_{n}} ⟶ E_{k_{0},k_{1},... ,k_{n}}/ {Image(ν)}

where

(v_{1}, …, v_{s}, (t_{1}, w_{1}), …, (t_{r}, w_{r})) ↦ {(v_{1}, …, v_{s})}

and the inverse images of points consist of contractible sets. This implies that the maps p_{k_{0},k_{1},... ,k_{n}} are acyclic and induce isomorphisms

H_{∗}(DE_{k_{0},k_{1},... ,k_{n}}; F) ⟶ H_{∗}(E_{k_{0},k_{1},... ,k_{n}}/ {Image(ν)}; F).

We combine this with the isomorphism in Lemma 2.8 to get

Corollary 3.2. Let k ≥ 2g − 1, then

H̃^{2k(n+1)−2ng−∗}(Hol_{k}^{∗}(M_{g}, P^{n}); F)

≅ H_{∗}(DE\underbrace{k, …, k}_{n+1}/ ⋃_{i} DE_{k,... , \underbrace{k − 1}_{ith−entry},... ,k}; F)

This result is very useful because it is possible, using the filtration of the space DE_{k,...,k} described above, to construct spectral sequences with known E_{1}-terms which converge to the cohomology of the relative spaces above for all k ≥ 1.

4. The spectral sequence

The diagonal action. The diagonal multiplication introduced in (2.3)

SP^{r}(M_{g}) × E_{i_{0},i_{2},... ,i_{n}} ⟶^{ν}E_{i_{0}+r,... ,i_{n}+r}

induces an action in homology, ν_{∗},

\begin{matrix}ν_{∗}: (∐_{r=0}^{∞} H_{∗}(SP^{r}(M_{g}); F)) ⊗ (∐_{j=1}^{∞} H_{∗}(LE_{i}; F)) \\ ⟶ ∐_{j=1}^{∞} H_{∗}(LE_{i+r}; F) \end{matrix} (4.1)

for any field coefficients F. Of course, this quotient action fits together with the original action on the E_{i,...,i} via the following commutative diagram (4.2)

H_{∗}(SP^{r}(M_{g}); F) ⊗ H_{∗}(E_{i,...,i}; F) ⟶^{id⊗p_{∗}} H_{∗}(SP^{r}(M_{g}); F) ⊗ H_{∗}(LE_{i}; F)
↓ν_{∗}  ↓ν_{∗}
H_{∗}(E_{i+r,...,i+r}; F)  ⟶^{p_{∗}}  H_{∗}(LE_{i+r}; F).

Theorem 4.3. There is a spectral sequence converging to H_{∗}(QE_{k}; F) with E^{1} term

\begin{aligned}E^{1} & = ∐_{{i+j=k \atop i≥1}} H̃_{∗}(LE_{i}; F) ⊗ H_{∗}(SP^{j}(ΣM_{g}), SP^{j−1}(ΣM_{g}); F) \\ & ⊕ H_{∗}(SP^{k}(ΣM_{g}), SP^{k−1}(ΣM_{g}); F) \end{aligned}

and d_{1}(Θ ⊗ {|a_{1}| ⋯ |a_{r}|}) = (−1)^{||Θ||||a_{1}||}ν_{∗}(a_{1} ⊗ Θ) ⊗ {|a_{2}| ⋯ |a_{r}|}.

Proof. From the model for QE constructed in §3 we have that

QE_{k} ≅ DE_{k,... ,k}/ ⋃ DE_{k,... ,k−1,... ,k}

where

DE_{k,... ,k}(M_{g}) = ⋃_{i_{r}+j≤k} E_{i_{0},i_{1},... ,i_{n}} ×_{t} SP^{j}(cM_{g}).

Also, LE_{i} = E_{i,...,i}/ {⋃ E_{i,i,...,i−1,i,...,i}} so we can write

QE_{k} = ⋃_{i+j=k} LE_{i} ×_{t} (SP^{j}(cM_{g})/SP^{j−1}(cM_{g})) ,

where the twisting in the description above is given as before by

(h, {(0, w_{1}), (t_{2}, w_{2}), …, (t_{r}, w_{r})}) ∼ (ν(w_{1}, h), {(t_{2}, w_{2}), …, (t_{r}, w_{r})}).

To obtain the desired spectral sequence, introduce the filtration by

F_{r}(QE_{k}) = ⋃_{{i≤r \atop i+j=k}} LE_{i} ×_{t} (SP^{j}(cM_{g})/SP^{j−1}(cM_{g})) ,

and the remainder of the proof of the theorem is direct. q.e.d.

A multiplicative structure for the spectral sequence. The induced multiplication on the QE_{i}’s,

ν̄: QE_{i} × QE_{j}⟶QE_{i+j}

passes to the spectral sequences above and defines a pairing of E^{1}-terms:

(H_{∗}(LE_{i}; F) ⊗ H_{∗}(SP^{j}(ΣM_{g}), SP^{j−1}(ΣM_{g}); F))

⊗ (H_{∗}(LE_{v}; F) ⊗ H_{∗}(SP^{w}(ΣM_{g}), SP^{w−1}(ΣM_{g}); F))

⟶H_{∗}(LE_{i+v}; F) ⊗ H_{∗}(SP^{w+j}(ΣM_{g}), SP^{w+j−1}(ΣM_{g}); F) for which the d_{i}’s act as derivations. For this reason it is often convenient to consider all the spectral sequences above at once. In particular we can describe the direct sum of all the E_{1}-terms as a trigraded ring where an element x ∈ E_{1} has tridegree (i, j, ∗) if and only if

x ∈ H_{s}(LE_{i}) ⊗ H_{∗−s}(SP^{j}(ΣM), SP^{j−1}(ΣM); F).

Remark 4.4. There is a related spectral sequence for H_{∗}(QE_{k}; F) obtained by filtering

QE_{k} = ⋃_{i+j=k} LE_{i} ×_{t} (SP^{j}(cM_{g})/SP^{j−1}(cM_{g}))

in a somewhat different way. Instead of filtering by i in the expression above, filter by the number of distinct t’s in the point

(h, {(t_{1}, w_{1}), (t_{2}, w_{2}), …, (t_{r}, w_{r})}).

In the space SP^{∞}(ΣM_{g}) this filtration results in the Eilenberg-Moore spectral sequence with E_{2}-term Ext_{H_{∗}(SP^{∞}(M_{g});F)}^{∗,∗}(F, F). To describe the resulting spectral sequence most efficiently it is best to include all the QE_{k}’s at once, and what results in our case is a trigraded E_{2}-term,

Ext_{H_{∗,∗}(SP^{∞}(M_{g});F)}^{r,s,∗}(F, ⨁_{i=1}^{∞} H_{∗}(LE_{i}; F)),

where the summand such that r + s = k corresponds to the E_{2}-term of the spectral sequence converging to H^{∗}(QE_{k}; F).

Some remarks on differentials. There are similar spectral sequences for the Div-spaces (cf. 1.6) starting (as in 3.1) with the model

(4.5)  (∐_{j=0}^{∞} \underbrace{SP^{j}(M_{g}) × ⋯ × SP^{j}(M_{g})}_{n+1−times}) ×_{T} SP^{∞}(cM_{g}),

where T is now the diagonal twisting which identifies the point (0, z) in the cone cM with the diagonal element Δ^{n+1}(z) in (M_{g})^{n+1} and then extends this multiplicatively. The associated spectral sequences have E_{1}-term

\begin{matrix}∐_{i+j=k} & [H^{∗}(SP^{i}(M_{g}), SP^{i−1}(M_{g}); F)] & n+1 \\ ⊗ H^{∗}(SP^{j}(ΣM_{g}), SP^{j−1}(ΣM_{g}); F) \end{matrix} (4.6)

and converge to H^{∗}(Div_{k,...,k}(M_{g}); F). (Here the superscript n+1 means the (n + 1)-fold tensor product.) Moreover, as is clear, the inclusions

E_{i,...,i} ⊂ SP^{i}(M_{g}) × ⋯ × SP^{i}(M_{g})

induce maps of spectral sequences here (in cohomology) to the spectral sequences above for the Hol_{k}^{∗} spaces. (Or in homology from the spectral sequences for the Hol_{k}^{∗} spaces to these for the Div-spaces.)

But for the spectral sequences for the Div-spaces, and using H. Cartan’s little constructions [4] to embed the homology into the chain complex one is able to construct an explicit (small) filtered chain complex with associated spectral sequence equal to that in Theorem 4.3, [12]. In particular one knows that E_{∞} = E_{1} for n ≥ 2 for the Div-spaces spectral sequence 4.6.

In the next section we will identify a region of the spectral sequence for the Hol_{k}^{∗} spaces where the induced map of homology spectral sequences is an injection. Hence, in this range for n > 1 the spectral sequence for Hol_{k}^{∗}(M_{g}, P^{n}) collapses at E_{1}. When n = 1 there are differentials, however our knowledge of the differentials in the Div-spectral sequence here implies considerable information about the differentials for the Hol_{k}^{∗} spaces here as well.

The d^{1}-differential for the highest filtration terms of the spectral sequence. One region where the two sequences don’t compare very well is the tail end of Theorem 4.3, the terms

H̃_{∗}(M_{g}; F) ⊗ H_{∗}(SP^{k−1}(ΣM_{g}), SP^{k−2}(ΣM_{g}); F)

and

H_{∗}(SP^{k}(ΣM_{g}), SP^{k−1}(ΣM_{g}); F).

More generally it can happen that μ_{k}: SP^{k}(M_{g})⟶J(M_{g}) is an embedding for 1 ≤ k ≤ m(M_{g}) in which case we have

Lemma 4.7. If the Abel-Jacobi map μ_{k}: SP^{k}(M_{g})⟶J(M_{g}) is an embedding for 1 ≤ k ≤ m(M_{g}), then in this range we have E_{k,...,k} ≅ SP^{k}(M_{g}), E_{k,...,k−1,k,...,k} ≅ SP^{k−1}(M_{g}) included in E_{k,...,k} via the usual base-point embedding

SP^{k−1}(X) ⊂ SP^{k}(X), ⟨x_{1}, …, x_{k−1}⟩ ↦ ⟨x_{1}, …, x_{k−1}, ∗⟩

and the action ν corresponds to the usual multiplication

SP^{r}(M_{g}) × SP^{k−r}(M_{g})⟶SP^{k}(M_{g}).

Consequently, in this range we have LE_{k,...,k} ≅ SP^{k}(M_{g})/SP^{k−1}(M_{g}), 1 ≤ k ≤ m(M_{g}), and the spectral sequence in this region is the corresponding spectral sequence for the quasi-fibration

SP^{∞}(M_{g})⟶SP^{∞}(cM_{g})⟶SP^{∞}(ΣM_{g}).

(The only statement above which might need clarification is the last. But recall that the spectral sequence with field coefficients has E_{2}-term

⨁_{k,t} H^{∗}(SP^{k}(M_{g}), SP^{k−1}(M_{g}); F) ⊗ H^{∗}(SP^{t}(ΣM_{g}), SP^{t−1}(ΣM_{g}); F)

and all the differentials preserve the sum k + t. Moreover, in this region the chain embedding techniques of [12] are valid, so the two spectral sequences have the same (internal) differentials.)

For example, if M_{g} is not hyperelliptic then μ_{2} : SP^{2}(M_{g})⟶J(M_{g}) is an embedding. Also the map μ_{3} : SP^{3}(M_{5})⟶J(M_{5}) is an embedding for most curves of genus 5. (See the discussion in [1, chapter V].)

5. The Jacobi varieties W_{i}^{j} and a spectral sequence for the LE_{i} spaces

Our next step is to analyze the spectral sequence of Theorem 4.3. To do this we must understand the groups H_{∗}(LE_{i}). The LE_{i} are quotients of the spaces E_{i} ≡ E_{i,... ,i} defined in §2 and these latter spaces turn out to be built out of fibrations with projective spaces as fibers. Here there is a stratification of the image of SP^{n}(M_{g}) in J(M_{g}), and over each stratum we get such a fibration, though the dimensions of the fibers vary as we move from stratum to stratum.

Definition 5.1. The image of μ_{d} in J(M_{g}) is written W_{d}. Also the set of points y ∈ W_{d} so that μ_{d}^{−1}(y) = P^{m} with m ≥ r is denoted W_{d}^{r}. Thus we have a decreasing filtration

W_{d} ⊇ W_{d}^{1} ⊇ W_{d}^{2} ⊇ ⋯ ⊇ W_{d}^{r} ⊇ ⋯ ⊇ W_{d}^{[d/2]+1} = ∅.

It is well known that W_{d} = J(M_{g}) whenever d ≥ g, and that the dimension m of a generic fiber P^{m} over W_{d} is 0 when d ≤ g and d − g when d ≥ g.

Examples.

(5.2) The map μ_{1} is always an embedding and so W_{1} ≅ M_{g}.

(5.3) In the genus 1 case the original map μ = μ_{1} is the identity, and J(M_{1}) = M_{1}, while the μ_{d} are fiberings for d ≥ 2. Assume now that g ≥ 2.

(5.4) The map μ_{2} : SP^{2}(M_{g})⟶J(M_{g}) is an embedding unless M_{g} is hyperelliptic (cf §10). In the hyperelliptic case W_{2}^{1} is a single point p and μ_{2}^{−1}(p) = P^{1}. It follows that W_{2} can be identified with SP^{2}(M_{g}) with a single P^{1} blown down. Indeed, we can check that the normal bundle to P^{1} ⊂ SP^{2}(M_{g}) is ξ^{1−g}, the line bundle with self-interesection number 1 − g, and from this it follows that

(1) SP^{2}(M_{2}) is the blowup of J(M_{2}) at a single point (here any genus 2 surface is automatically hyperelliptic).

(2) For g ≥ 3 we have that W_{2} = SP^{2}(M_{g})/(P^{1} ∼ ∗) is a manifold with a single isolated point singularity which looks like the cone on a Lens space L_{g−1}^{3}.

We can introduce the complementary subspace A_{k}^{i} = W_{k}^{i}−W_{k}^{i+1}. By definition, we have that ∀x ∈ A_{k}^{i}, μ_{k}^{−1}(x) = P^{i}. The result of Mattuck quoted in (1.14) takes actually the more general form.

Theorem 5.5 (Mattuck). μ_{k}^{−1}(A_{k}^{i}) ⊂ SP^{k}(M_{g}) is the total space of a locally trivial analytic fibration P^{i} → μ_{k}^{−1}(A_{k}^{i}) → A_{k}^{i}.

Remark 5.6. The following formula relating W_{k}^{i} and W_{k−1}^{i−1} is a special case of [10, (20), p. 53]:

(∗∗)  W_{k}^{i} = ⋂_{m∈μ(M_{g})}(W_{k−1}^{i−1} + m),

that is, W_{k}^{i} is obtained as the intersection of all translates of elements of W_{k−1}^{i−1} by elements of μ(M_{g}) ⊂ J(M_{g}). In particular, this shows that

(∗ ∗ ∗)  W_{k}^{i} ⊂ W_{k−1}^{i−1} + ∗ ⊂ W_{k}^{i−1}.

From now on we do not differentiate between W_{k−1}^{i−1} ⊂ W_{k−1} and W_{k−1}^{i−1} + ∗ ⊂ W_{k}; denoting both by W_{k−1}^{i−1}.

Furthermore, Gunning, [10, p. 54], gives the following dimension estimates for the associated containments.

Lemma 5.7. If the subvariety W_{k−1}^{i−1} is non-empty then:

• dim W_{k}^{i} <dim W_{k−1}^{i−1} whenever 2 ≤ i ≤ k ≤ g.

• dim W_{k−2}^{i−1} < dim W_{k−1}^{i−1} whenever 2 ≤ i ≤ k ≤ g.

Remark 5.8. In chapter V of [1] techniques for determining the W_{k}^{i}’s are extensively discussed, and results for g ≤ 6 are completely given, (p. 206-211).

We now turn to the pull-back spaces E_{k,... ,k} and observe that the fibration of Lemma (5.5) induces in turn a fibration

(P^{i} × ⋯ × P^{i}) ⟶ A_{k}^{i} ⟶^{π}A_{k}^{i}, A_{k}^{i} ⊂ E_{k,... ,k}.

When collapsing A_{k−1}^{i}, we get a quotient

X = A_{k}^{i}/A_{k−1}^{i} ↪ E_{k,... ,k}/E_{k−1,... ,k−1}

and a quotient map X ⟶ LE_{k,... ,k}. The space X projects down via μ to

A_{k}^{i}/A_{k−1}^{i} = W_{k}^{i}/W_{k−1}^{i};

this last equality being a consequence of the inclusion W_{k}^{i+1} ⊂ W_{k−1}^{i} described above.

We then pass to the quotient LE_{k} and observe that since W_{k}^{i} ⊂ W_{k−1}^{i−1}, i ≥ 1, we must further collape, along fibers this time, subsets of the form

⋃_{j≤n+1}(P^{i})^{j−1} × P^{i−1} × (P^{i})^{n+1−j} ↪ (P^{i})^{n+1}.

The fiber (P^{i})^{n+1} has a top 2i(n + 1) cell of the form

e^{2i(n+1)} = e_{1}^{2i} × ⋯ × e_{n+1}^{2i},

where e_{j}^{2i} is the top 2i dimensional cell in the j-th copy P^{i} with boundary mapping to P^{i−1} ⊂ P^{i}. We then see that

(P^{i})^{n+1}/ ∪_{j} (P^{i})^{j−1} × P^{i−1} × (P^{i})^{n+1−j} ≃ e^{2i(n+1)}/∂e^{2i(n+1)} ≃ S^{2i(n+1)}.

The space so obtained is denoted by T_{k}^{i}. In the case when i = 0 we have that W_{k}^{1} ⊂ W_{k−1} ⊂ W_{k}, and hence W_{k}/W_{k−1} ≅ SP^{k}(M_{g})/SP^{k−1}(M_{g}).

Now we filter the Jacobian according to the increasing sequence of W_{i}’s

J(M_{g}) = W_{g} ⊃ W_{g−1} ⊃ ⋯ ⊃ W_{1} = M_{g}.

This induces a filtration on LE_{k} yielding a spectral sequence which by the preceeding discussion has E^{1} term as follows.

Proposition 5.9. Suppose that k ≤ g. Then there is a spectral sequence converging to H̃_{∗}(LE_{k}) with E^{1} term

E^{1} = H_{∗}(SP^{k}(M_{g}), SP^{k−1}(M_{g})) ⊕ ∐_{i} H_{∗}(W_{k}^{i}, W_{k−1}^{i}) ⊗ H̃_{∗}(S^{2i(n+1)})

and in case k ≥ g, then

E^{1} =Σ^{2(k−g)(n+1)}H_{∗}(J(M_{g}), W_{k−1}^{k−g})

⊕ ∐_{i>k−g} H_{∗}(W_{k}^{i}, W_{k−1}^{i}) ⊗ H̃_{∗}(S^{2i(n+1)}).

Remark 5.10. Note that

H_{∗}(LE_{k}) = H_{∗}(J(M_{g})) ⊗ H̃_{∗}(S^{2(k−g)(n+1)}),

whenever k > 2g − 1 for then W_{k−1}^{k−g} = ∅ and W_{k}^{i} = W_{k−1}^{i} = J(M_{g}) otherwise.

Remark 5.11. The spectral sequence of (5.9) turns out to collapse at E^{1} for all cases that we treat in this paper.

6. The structure of symmetric products

In this section, we describe the homology of the symmetric products SP^{n}(M_{g}) for all n and g. Also, since it is required in the spectral sequences of Theorem 4.3 and Proposition 5.9, we give the structure of H_{∗}(SP^{n}(ΣM_{g})) again for all n, g. In the case of SP^{n}(M_{g}) we follow the description given by I.G. Macdonald, [15], while the description for the suspension is taken from [19] and unpublished work of N. Steenrod.

There is an evident pairing SP^{n}(X) × SP^{m}(X)⟶^{+}SP^{n+m}(X) given by addition of points, and this turns SP^{∞}(X) into an associative, abelian monoid with ∗ as a two sided identity. The Dold and Thom theorem states that

(6.1)  SP^{∞}(X) ≃ ∏_{1}^{∞} K(H_{i}(X; Z), i)

is a product of Eilenberg-MacLane spaces if X is path connected [7]. Applying this to X = M_{g} yields

(6.2)  SP^{∞}(M_{g}) ≃ K(Z^{2g}, 1) × K(Z, 2) ≃ (S^{1})^{2g} × P^{∞},

where K(Z, 1) ≃ S^{1} while K(Z, 2) ≃ P^{∞}, the infinite complex projective space. We then find that

(6.3)  H_{∗}(SP^{∞}(M_{g}); Z) ≅ Λ(e_{1}, …, e_{2g}) ⊗ Γ(a),

where Λ(,) denotes the exterior algebra on the stated generators while Γ(a) denotes the divided power algebra on a: the ring with Z-generators a_{i}, i = 0, 1, … and multiplication a_{i}a_{j} = \binom{i+j}{j}a_{i+j}. This accords well with the multiplication of the SP^{n}(M_{g}) described above. Often however it is convenient to work with the cohomology rings so we need the following description.

I.G. Macdonald’s description of H^{∗}(SP^{k}(M_{g}); Z).

Consider the map [M_{g}]^{∗}: M_{g}⟶P^{∞} = K(Z, 2), taking the fundamental class to the dual of the orientation class, and the map

∨e_{i}: M_{g}⟶K(Z^{2g}, 1) ≃ (S^{1})^{2g}.

Both P^{∞} and (S^{1})^{2g} are associative abelian H-spaces with the structure on P^{∞} coming from the identification SP^{∞}(P^{1}) = P^{∞} described earlier.

This allows us to extend [M_{g}]^{∗} and ∨e_{i} to a multiplicative map

θ(k): (M_{g})^{k}⟶P^{∞} × (S^{1})^{2g}

which, by definition factors through the symmetric product SP^{k}(M_{g}). In the limit, as k ↦ ∞ this gives the Dold-Thom equivalence

SP^{∞}(M_{g})⟶K ((Z)^{2g}, 1) × K(Z, 2).

With respect to the maps θ(k) we can describe the cohomology ring H^{∗}(SP^{k}(M_{g}); Z) as follows:

Theorem 6.4 (I.G. Macdonald). The cohomology ring of SP^{k}(M_{g}) over the integers Z is generated by the elements

f_{1} = e_{1}^{∗}, …, f_{i} = e_{2i−1}^{∗}, … f_{g} = e_{2g−1}^{∗}, f′_{1} = e_{2}^{∗}, … f′_{g} = e_{2g}^{∗}, and b

subject to the following relations:

(i) The f_{i}’s and the f′_{i}’s anti-commute with each other and commute with b;

(ii) If i_{1}, …, i_{a}, j_{1}, …, j_{b}, k_{1}, …, k_{c} are distinct integers from 1 to g inclusive, then

f_{i_{1}} ⋯ f_{i_{a}}f′_{j_{1}} ⋯ f′_{j_{b}}(f_{k_{1}}f′_{k_{1}} − b) ⋯ (f_{k_{c}}f′_{k_{c}} − b)b^{q} = 0

provided that

a + b + 2c + q = n + 1.

If k < 2g all the relations above are consequences of those for which q = 0, and if n > 2g − 2 all the relations are consequences of the single relation

b^{k−2g+1} ∏_{i=1}^{g}(f_{i}f′_{i} − b) = 0.

(Actually Macdonald only stated this result in [15] for fields of characteristic zero as coefficients, however, since the relevant invariant maps that he used to prove the result are actually surjective over Z, it is quite direct to extend the result to integers as well.)

The homology of SP^{n}(X) for more general X

The homology of the spaces SP^{n}(X) splits according to a result of Steenrod, [6], (see also [19]) which holds for X any CW -complex:

(6.5)  H_{∗}(SP^{∞}(X); Z) = ⨁_{j} H_{∗}(SP^{j}(X), SP^{j−1}(X); Z).

Consequently, the ring H_{∗}(SP^{∞}(M_{g}); Z) is bigraded, by defining x ∈ H_{∗}(SP^{∞}(X); Z) to have bidegree (i, j) iff

x ∈ H_{j}(SP^{i}(M_{g}), SP^{i−1}(M_{g}); Z) ⊂ H_{∗}(SP^{∞}(M_{g}); Z).

The bigrading is multiplicative in the sense that the product map

SP^{m}(X) × SP^{n}(X)⟶SP^{n+m}(X)

in homology induces a bigraded ring map

H_{i,j}(SP^{n}(X); A) ⊗ H_{r,s}(SP^{m}(X); A)⟶H_{i+r,j+s}(SP^{n+m}(X); A),

where the first degree is the dimension and the second the bidegree.

The bigrading is preserved by the diagonal map, so the cohomology ring is bigraded as well, and the two structures together form a bigraded Hopf algebra. In the case of the SP^{k}(M_{g}) we have that the coproduct on each of the 1-dimensional generators is primitive, while the 2-dimensional generator [M_{g}] in H_{1,2}(SP^{∞}(M_{g}); Z) corresponding to the orientation class of M_{g} has coproduct

(6.6) Δ([M_{g}]) = [M_{g}] ⊗ 1 + ∑_{1}^{g}(e_{2i−1} ⊗ e_{2i} − e_{2i} ⊗ e_{2i−1}) + 1 ⊗ [M_{g}].

The homology of SP^{n}(Σ(M_{g})).

We will need the cohomology and homology of the spaces SP^{n}(Σ(M_{g})) where ΣX denotes the suspension of X. From the Dold-Thom Theorem we have

(6.7)  SP^{∞}(ΣM_{g}) ≃ (∏_{1}^{2g} P^{∞}) × SP^{∞}(S^{3}),

where we can assume that the generator for the homology of each P^{∞} has bidegree (1, 2). Thus, this amounts to describing the homology and cohomology of K(Z, 3) and its associated bigrading.

With coefficients F_{p} for any odd prime p we have, [19],

\begin{matrix}H_{∗}(K(Z, 3); F_{p}) \\ ≅ Λ(|a|, |γ_{p}|, ⋯ |γ_{p^{i}}| ⋯) ⊗ Γ(|a^{p−1}|a|) \\ ⊗ Γ(|γ_{p}^{p−1}|γ_{p}|) ⊗ ⋯ ⊗ Γ(|γ_{p^{i}}^{p−1}|γ_{p^{i}}|) ⊗ · · · . \end{matrix} (6.8)

The generator γ_{p^{i}} corresponds to [SP^{p^{i}}(M_{g})] and |γ_{p^{i}}| has filtration degree p^{i} and dimension 2p^{i} + 1 while |γ_{p^{i−1}}^{p−1}|γ_{p^{i−1}}| has bidegree (p^{i}, 2p^{i} + 2).

In the case of the prime 2 we have

(6.9)  H_{∗}(K(Z, 3); F_{2}) ≅ Γ[|a|, |a_{2}|, …, |a_{2^{i}}|, ...].

Remark 6.10. With F_{p} coefficients, the divided power algebra Γ[a] splits as an algebra into a tensor product of truncated polynomial algebras [4]

Γ[a] ≅ F_{p}[a]/a^{p} ⊗ F_{p}[a_{p}]/(a_{p})^{p} ⊗ · · · .

When F = Q, divided power algebras are isomorphic to polynomial algebras and the answer simplifies greatly; namely

(6.11)  H_{∗}(SP^{∞}(ΣM_{g}); Q) = Q[|e_{1}|, …, |e_{2g}|] ⊗ Λ(a).

The answer is simpler to describe if we use cohomology. Here, H^{∗}(SP^{∞}(S^{3}); F_{p}) is always a tensor product of an exterior algebra on odd dimensional generators and a polynomial algebra on the even dimensional generators. In other words in the formula (6.8) above one replaces all the divided power algebras by polynomial algebras with generators of the same bidegrees to get the cohomology ring description.

We take advantage of the bigrading to write H_{∗}(SP^{∞}(ΣM_{g}); F_{p}) in the form

∐_{1}^{∞} H_{∗}(SP^{n}(ΣM_{g}), SP^{n−1}(ΣM_{g}); F_{p}).

These summands can be written out more completely in terms of the bigrading of SP^{∞}(S^{3}) as follows. First, we write H_{∗}(SP^{∞}(S^{3}); F_{p}) = H_{∗}(SP^{∞}(ΣS^{2}); F_{p}) in this way. The decomposition here has an alternate description, [5]:

(6.12)  H_{∗}(SP^{∞}(S^{3}); F_{p}) = ⨁_{1}^{∞} Σ^{4i}D_{i}^{∗}(p),

where the D_{i} are the Snaith splitting components of the loop space Ω^{2}S^{3} for the prime p, and D_{i}^{∗} means dual, where we index the dual by dim(D_{i,j}) = −j.

Then, going back to (6.7) we see that we can write the term

\begin{matrix}H_{∗}(SP^{n}(ΣM_{g}), SP^{n−1}(ΣM_{g}); F_{p}) \\ = ⨁_{j=0}^{n} Σ^{4j}D_{j}^{∗}(p) ⊗ F_{p}[b_{1}, …, b_{2g}]_{n−j} , \end{matrix} (6.13)

and F_{p}[b_{1}, …, b_{2g}]_{n−j} is the free Z-module on the degree n−j monomials in the variables b_{1}, …, b_{2g}. There are \binom{n−j+2g−1}{2g−1} such monomials.

Example. What follows is a list of generators for

H_{∗}(SP^{j}(ΣM), SP^{j−1}(ΣM); Z_{p})

in the E^{1} term mod(p) of the spectral sequence of Theorem 4.3 together with their tridegrees

(6.14)  \begin{matrix}Generator & trigrading \\ |M| & (0, 1, 3) \\ |e_{i}| & (0, 1, 2) \\ |γ_{p^{i}}| & (0, p^{i}, 2p^{i} + 1) \\ |γ_{p^{i−1}}^{p−1}|γ_{p^{i−1}}| & (0, p^{i}, 2p^{i} + 2). \end{matrix}

Explicit d^{1}-differentials in the spectral sequence.

In Theorem 4.3 the differential d^{1} is implicitly determined. Now that we have the explicit form of the homology groups H_{∗}(SP^{n}(M_{g})) and H_{∗}(SP^{n}(ΣM_{g})) we can make d^{1} explicit. For example we have

d^{1}(|f_{1}|^{s_{1}} ⋯ |f_{2g}|^{k−s_{1}−···−s_{2g−1}})

= ∑_{1}^{2g} f_{i} ⊗ |f_{1}|^{s_{1}} ⋯ |f_{i}|^{s_{i}−1} ⋯ |f_{2g}|^{k−s_{1}−···−s_{2g−1}} (6.15)

and

\begin{matrix}d^{1}(|M_{g}||f_{1}|^{s_{1}} ⋯ |f_{2g}|^{k−s_{1}−···−1} \\ = [M_{g}] ⊗ |f_{1}|^{s_{1}} ⋯ |f_{2g}|^{k−s_{1}−···−1} \\ − (d^{1}(|f_{1}|^{s_{1}} ⋯ |f_{2g}|^{k−s_{1}−···−1}) (1 ⊗ |M_{g}|) . \end{matrix} (6.16)

It is easily checked that this part of the d^{1}-differential is injective to the term H̃^{∗}(M_{g}; F)⊗F[|f_{1}|, …, |f_{2g}|]_{k−1} with cokernel spanned by the monomials of the following form:

(6.17)  {f_{t}|f_{i}|^{j_{i}}|f_{i+1}|^{j_{i+1}} ⋯ |f_{2g}|^{k−1−j_{1}−···−j_{2g−1}}},

where j_{i} > 0, and t > i.

Remark 6.18. This is the only time the d^{1} differential on the elements |f_{1}|^{i_{1}} ⋯ |f_{2g}|^{i_{2g}} is non-trivial, since in the remaining parts of the spectral sequence their images lie in the part which has been collapsed out.

Example 6.19. If g = 1 then the differentials in this region have the form d^{1}(|f_{1}|^{s}|f_{2}|^{k−s}) = f_{1} ⊗ |f_{1}|^{s−1}|f_{2}|^{k−s} + f_{2} ⊗ |f_{1}|^{s}|f_{2}|^{k−s−1} provided s and k − s are both greater than zero and

d^{1}(|T ||f_{1}|^{s}|f_{2}|^{k−s−1}) = f_{1}f_{2} ⊗ |f_{1}|^{s}|f_{2}|^{k−s−1}
− f_{1} ⊗ |T ||f_{1}|^{s−1}|f_{2}|^{k−s−1}
− f_{2} ⊗ |T ||f_{1}|^{s}|f_{2}|^{k−s−2}.

The role of this last differential can be regarded as identifying terms of the form f_{1}f_{2} ⊗ |f_{1}|^{s}|f_{2}|^{k−s−1} with terms involving |T |^{∗}. Also, the first of these differentials is injective but not surjective. The quotient has as a basis the set of images of the elements f_{2} ⊗ |f_{1}|^{s}|f_{2}|^{k−s−1} with s > 0. Consequently it has dimension k − 2.

7. The stable range

In the case where i ≥ 2g, we’ve seen already that the Abel-Jacobi map becomes an analytic fibration, and hence the projection

E_{i,... ,i}⟶J(M_{g})

fibers as

(7.1)  P^{i−g} × … × P^{i−g} ⟶ E_{i,... ,i} ⟶ J(M_{g}).

Because of this and other stabilization properties (§8), we refer to the range i ≥ 2g as stable. In this case, we see that the relative groups

H̃(LE_{i}, A) ≅ H_{∗}(E_{i,i,... ,i}, ⋃ E_{i,...,i−1,...,i}; A)

are given by

\begin{matrix}[H_{∗}(P^{i−g}, P^{i−g−1}; A)]^{n+1} ⊗ H_{∗}(J(M_{g}); A) \\ ≅ Σ^{2(i−g)(n+1)}H_{∗}(J(M_{g}); A). \end{matrix} (7.2)

Dually, in cohomology the relative group is given as

\begin{matrix}b_{0}^{i−g}b_{1}^{i−g} ⋯ b_{n}^{i−g} ⊗ H^{∗}(J(M_{g}); A) \\ = (b_{0} ⋯ b_{n})^{i−g} ⊗ H^{∗}(J(M_{g}); A), \end{matrix} (7.3)

a form which is often of more use in calculations.

Remark 7.4. Of course, in both these forms, the inclusions P^{i−g} ⊂ SP^{i}(M_{g}) induce injections in homology onto Z-direct summands, but even more, from Macdonald’s Theorem 6.4 it follows that the inclusion of pairs

(P^{i−g}, P^{i−g−1}) ⊂ (SP^{i}(M_{g}), SP^{i−1}(M_{g}))

induces an inclusion in homology sending H_{2(i−g)}(P^{i−g}, P^{i−g−1}; Z) isomorphically to the group H_{2(i−g)}(SP^{i}(M_{g}), SP^{i−1}(M_{g}); Z) = Z. (We will expore this fact further below.)

Thus in this stable range the E_{1}-term for Hol_{k}^{∗}(M_{g}, P^{n}) injects into the corresponding E_{1}-term for the Div-space as was asserted at the end of §4. In particular this gives as a corollary to the results of [12],

Lemma 7.5. The spectral sequence of Theorem 4.3 collapses at E_{1} in the stable range for n ≥ 2.

Now we turn to some fairly direct calculations which lead to the determination of the d^{1}-differential in the Hol_{k}^{∗} spectral sequence for n = 1 in the stable range.

More exactly, in the stable range the relative group injects,

H^{∗}(E_{i,i}, E_{i,i−1} ∪ E_{i−1,i}; A) ⊂ H^{∗}(E_{i,i}, A)

as the principal ideal

(7.6)  ((b_{0}b_{1})^{i−2g}W_{0}W_{1}) ⊂ H^{∗}(E_{i,i}; A),

where W_{j} is the polynomial

\begin{aligned}W_{j} & = ∏_{t=1}^{g}(f_{2t−1}f_{2t} − b_{j}) \\ & = (−1)^{g} [b_{j}^{g} − (∑_{1}^{g} f_{2t−1}f_{2t}) b_{j}^{g−1} + ⋯ + (−1)^{g} (∏_{1}^{2g} f_{t})] , \end{aligned} (7.7)

which generates the kernel of the restriction map to H^{∗}(E_{i−1,i}; A) and H^{∗}(E_{i,i−1}; A) respectively according to Macdonald’s Theorem 6.4.

Lemma 7.8. In the stable range the differential d_{1} in the spectral sequence of Theorem 4.3 vanishes for n > 1. For n = 1, it has the form

d_{1}(W_{0}W_{1}b_{0}^{i−2g}b_{1}^{i−2g}) = 2 (∑_{1}^{g} f_{2i−1}f_{2i}) (W_{0}W_{1}b_{0}^{i−2g−1}b_{1}^{i−2g−1})|M_{g}^{∗}|,

and it is multiplicative in the remaining generators. That is to say,

ΘW_{0}W_{1}b_{0}^{i−2g}b_{1}^{i−2g} ⟶^{d_{1}} ± Θd_{1}(W_{0}W_{1}b_{0}^{i−2g}b_{1}^{i−2g}).

Proof. We check the action map M × E_{i−1,...,i−1}⟶E_{i,...,i} in cohomology. The term f_{2r−1}f_{2r} − b_{i} maps to

f_{2r−1} ⊗ f_{2r} − f_{2r} ⊗ f_{2r−1} + 1 ⊗ (f_{2r−1}f_{2r} − b_{i}),

since f_{2r−1}f_{2r} − b = 0 in H^{2}(M_{g}; Z). Hence

W_{i} ↦ ∑_{r=1}^{g}(f_{2r−1} ⊗ f_{2r} − f_{2r} ⊗ f_{2r−1})1 ⊗ W_{i}(r) + 1 ⊗ W_{i},

where, as is evident W_{i}(r) = ∏_{j≠r}(f_{2j−1}f_{2j} − b_{i}). Thus, since f_{i}f_{j} = 0 in H^{∗}(M_{g}; Z) for ⟨i, j⟩ ≠ ⟨2r − 1, 2r⟩ for some r, it follows that

W_{0}W_{1} ↦ ∑_{r=1}^{g} 2b ⊗ (f_{2r−1}f_{2r}W_{0}(r)W_{1}(r))

+ (f_{2r−1} ⊗ f_{2r} − f_{2r} ⊗ f_{2r−1})1 ⊗ (W_{0}(r)W_{1} + W_{0}W_{1}(r))

+ 1 ⊗ W_{0}W_{1}.

Next note that W_{i}b_{i}^{i−2g} = 0 in H^{∗}(E_{i−1,...,i−1}; Z), so the image of (b_{0}b_{1})^{i−2g}W_{0}W_{1} is 2b ⊗ ∑ f_{2r−1}f_{2r}b_{0}^{i−2g−1}b_{1}^{i−2g−1}W_{0}W_{1} since

b_{i}W_{i}(r)f_{2r−1}f_{2r} = W_{i}f_{2r−1}f_{2r}.

This proves the desired result. q.e.d.

Remark 7.9. Actually, what the argument above really determines is the coaction map in cohomology. Given the action map

SP^{r}(M_{g}) × LE_{i}⟶LE_{i+r}

described earlier, and suppose i ≥ 2g; then

U_{i+k} ↦ b^{k} ⊗ \frac{2^{k}}{k!} (∑_{1}^{g} f_{2r−1}f_{2r})^{k} U_{i} ,

where U_{j} is the generating class above for H^{2(j−g)}(LE_{j}; Z) when n = 1.

8. Stabilization

For d ≥ g, any n ≥ 1 and any neighborhood U_{ϵ}(∗), there are n + 1 divisors D_{i} ⊂ SP^{d}(U_{ϵ}(∗)), 0 ≤ i ≤ n with μ(D_{0}) = μ(D_{1}) = ⋯ = μ(D_{n}) and ⋂ D_{i} = ∅. By deforming M_{g} −∗ to M_{g} −U_{ϵ} and then adding in the corresponding divisors, one obtains a stablization map τ which one can iterate

(8.1) Hol_{k}^{∗}(M_{g}, P^{n})⟶^{τ}Hol_{k+d}^{∗}(M_{g}, P^{n})⟶ ⋯ ⟶Hol_{k+rd}^{∗}(M_{g}, P^{n})

obtaining in the limit a space lim_{r↦∞} Hol_{k+rd}^{∗}(M_{g}, P^{n}) which is homotopy equivalent to any component in the mapping space Map_{∗}^{∗}(M_{g}, P^{n}). Note that this last statement is a consequence of Segal’s stabilization theorem, [21].

The inclusion τ descends to a map of spectral sequences and by an argument similar to one in [5], we see the corresponding map on the E_{1}-term is given by

∪ (b_{0}b_{1} ⋯ b_{n}) ⊗ id:

H^{∗}(LE_{i}; Z) ⊗ H^{∗}(SP^{j}(ΣM_{g}), SP^{j−1}(ΣM_{g}); Z) (8.2)

⟶H^{∗}(LE_{i+1}; Z) ⊗ H^{∗}(SP^{j}(ΣM_{g}), SP^{j−1}(ΣM_{g}); Z).

This relation between inclusion at the level of mapping spaces and cupping with b_{0} ⋯ b_{n} in the Poincaré duals is quite important and follows basically by checking normal bundles (cf [12]). In any case the content of the preceeding remarks can be summarized in

Lemma 8.3. In the stable range the cohomology of Hol_{k}^{∗}(M_{g}, P^{n}) is isomorphic to the cohomology of Map_{k}^{∗}(M_{g}, P^{n}) via the natural inclusion.

This is again direct from the stabilization via cupping with b_{0} ⋯ b_{n} and Segal’s stability theorem.

The differentials which we know appear come from comparison with the Div-space spectral sequence. These are the differential d_{1} already described and the further differentials which only hold mod(p) for odd p,

\begin{aligned}d(|γ_{p^{i}}|^{∗}) & = \frac{1}{p^{i}} (∑_{1}^{g} f_{2j−1}f_{2j})^{p^{i}} , \\ d(|γ_{p^{i}}^{p−1}|γ_{p^{i}}|^{∗}) & = [\frac{1}{p^{i}} (∑_{1}^{g} f_{2j−1}f_{2j})^{(p−1)p^{i}} ] |γ_{p^{i}}|^{∗}. \end{aligned} (8.4)

The last differential in (8.4) is called the Kudo differential, [14]. There are no differentials for p = 2 (see next remark). Similarly, for n > 2 all the differentials are zero for all p and hence, even over the integers.

Remark 8.5. There is a sequence of Serre fibrations

Ω^{2}(P^{n})⟶Map^{∗}(M_{g}, P^{n})⟶ (Ω(P^{n}))^{2g} ⟶^{f^{†}}Ω(P^{n})

induced from the cofibration sequence associated to M_{g};

S^{1}^{\underrightarrow{f}} ⋁^{2g} S^{1} ↪ M_{g} ⟶ S^{2}

which gives Map^{∗}(M_{g}, P^{n}) as the total space of a principal Ω^{2}(P^{n}) fibration with classifying map f^{†}. Then the discussion above shows by dualizing that in the Serre spectral sequence of the fibration

(8.6)  Ω^{2}(P^{n})⟶Map^{∗}(M_{g}, P^{n})⟶ (Ω(P^{n}))^{2g}

we have (for n = 1) the differentials d^{1}(|b|^{∗}) = 2 ∑_{1}^{g} f_{2j−1}f_{2j}, the transgressive mod(p) differentials from |b^{p^{i}}|^{∗} to the divided power of (∑_{1}^{g} f_{2j−1}f_{2j}), and the Kudo differentials for p odd. Of course, for p = 2 the differentials in the Serre spectral sequence are totally trangressive from the fiber to the base. Hence, there are no differentials and E_{2} = E_{∞}. Similarly, for n > 2 there are no differentials for any p and E_{2} = E_{∞} in this spectral sequence even with integer coefficients though the only way we know to prove this is via the results of [13].

9. The genus one case

When g = 1, then μ: M_{1}⟶J(M_{1}) is a holomorphic homeomorphism identifying the Jacobi variety with T itself. Also, the stable range starts immediately in this case and hence, for all n ≥ 1 the Abel-Jacobi map μ is a fibration

P^{n−1} ⟶ SP^{n}(T)^{\underrightarrow{μ}}T.

In the spectral sequence for QE_{k}(n) with n ≥ 2 (dual to Hol_{k}^{∗}(T, P^{n})) there are no differentials except for those at the tail end (6.15) and (6.16), but these are the only differentials and E^{1} = E^{∞} away from this region, while E^{2} = E^{∞} for the entire spectral sequence. In the case n = 1 the only differential is again d_{1} which now also has a stable component given in (5.9), and again E^{2} = E^{∞}.

It also follows that duality gives the isomorphism

(9.1)  H̃^{2k(n+1)−2n−∗}(Hol_{k}^{∗}(T, P^{n}); F) ≅ H_{∗}(QE\underbrace{k, …, k}_{n+1}; F)

for all k ≥ 2, with Hol_{1}^{∗}(T, P^{n}) being empty for all n.

We now determine these homology groups explicitly for n = 1. First, in the case where F has characteristic 2 there are no stable differentials and E^{1} = E^{∞} in this range. Second, in case F = Q we have

E^{1}(Hol_{k}^{∗}) = ∐_{i=2}^{k} Σ^{4(i−1)}H_{∗}(T; Q)

⊗ (Q[|e_{1}|, |e_{2}|]_{k−i} ⊕ |[T]| ⊗ Q[|e_{1}|, |e_{2}|]_{k−i−1}) (9.2)

⊕ H̃_{∗}(T; Q) ⊗ (Q[|e_{1}|, |e_{2}|]_{k−1} ⊕ |[T]| ⊗ Q[|e_{1}|, |e_{2}|]_{k−2})
⊕ Q[|e_{1}|, |e_{2}|]_{k} + |T |Q[|e_{1}|, |e_{2}|]_{k−1},

where Q[|e_{1}|, |e_{2}|]_{r} is the (r + 1)-dimensional subspace spanned by the monomials of degree r; that is: |e_{1}|^{r}, |e_{1}|^{r−1}|e_{2}|, …, |e_{2}|^{r}. The only differential is d_{1}, generated by

d_{1}(Σ^{4(k−j−1)}e_{1}e_{2}|T |) = Σ^{4(k−j)}1,

and the unstable differentials of (6.15) and (6.16). Thus one has directly (compare (6.19) for the last term) that

E^{2} = ∐_{i=2}^{k} Σ^{4(i−1)}H̃_{∗}(T; Q) ⊗ Q[|e_{1}|, |e_{2}|]_{k−i}
⊕ Σ^{4(i−1)}|T |(1, e_{1}, e_{2})Q[|e_{1}|, |e_{2}|]_{k−i−1}

⊕ Q{e_{2}|e_{1}|^{k−1}, e_{2}|e_{1}|^{k−2}|e_{2}|, …, e_{2}|e_{1}||e_{2}|^{k−2}} (9.3)

⊕ |T |(1, e_{1}, e_{2})Q[|e_{1}|, |e_{2}|]_{k−2}
= E^{∞}.

In the remaining case where F = F_{p^{n}} has characteristic p an odd prime, as explained in (6.6), (6.7) we can write the E_{1}-term as

∐_{i+j=k} Σ^{4(i−1)}H^{∗}(T; F) ⊗ ∑_{l=0}^{j} Σ^{4l}D_{j}^{∗}(p) ⊗ F[|b_{1}, b_{2}]_{j−l},

where D_{j}^{∗} = 0 if j ≠ pk or pk + 1 and with the usual special considerations at the tail end of the filtration. Incidently, the case j = pk + 1 implies that Σ^{4j}D_{j}^{∗} = |M|Σ^{4(j−1)}D_{j−1}^{∗}. The differential is as before, and we have that in the stable range

E^{2} = ∐_{i=0}^{k} Σ^{4(i+j−1)}H̃_{∗}(T; F) ⊗ D_{j}^{∗} ⊗ F[b_{1}, b_{2}]_{k−i−j}
⊕ Σ^{4(i+j−2)}|T |F[b_{1}, b_{2}]_{k−i−j−1}
= E^{∞}.

In each case the homology injects into H_{∗}(Map_{k}^{∗}(T, P^{1}); F). Thus the geometric interpretation of each of the homology and cohomology classes in H_{∗}(Hol_{k}^{∗}(T, P^{1}); F) can be regarded as the same as that for the corresponding class in the mapping space.

For example with rational coefficients we have the following corollary to 9.3 and 9.1.

Lemma 9.4. Let I = (|e_{1}|, |e_{2}|) be the augmentation ideal in Q[|e_{1}|, |e_{2}|]. Then

H_{∗}(Hol_{k}^{∗}(T, P^{1}); Q) ≅ Q(1, e_{1}, e_{2}, h_{2,1}, h_{2,2}, v_{3}) ⊗ Q[|e_{1}|, |e_{2}|]/I^{k−1}
⊕ Q{e_{2}|e_{1}|^{k−1}, …, e_{2}|e_{1}||e_{2}|^{k−2}}.

In particular the Poincaré series for H_{∗}(Hol_{k}^{∗}(T, P^{1}); Q) is

(1+2x+2x^{2} +x^{3})(1+2x^{2} +3x^{4} +⋯+(k−1)x^{2(k−2)})+(k−2)x^{2(k−1)−1}.

Note how this compares with the corresponding Poincaré series for the mapping space H_{∗}(Map^{∗}(T, P^{1}); Q):

\frac{(1 + 2x + 2x^{2} + x^{3})}{(1 − x^{2})^{2}}.

The case n ≥ 2. Any component in the mapping space is given as the total space of the Serre fibration

Ω^{2}S^{2n+1}⟶E⟶(S^{1} × ΩS^{2n+1})^{2}

with

H_{∗}(Map^{∗}(T, P^{n}); F) = H_{∗}(Ω^{2}S^{2n+1}; F) ⊗ H_{∗}(T; F) ⊗ F[g_{2n}, g′_{2n}].

Consequently, with Q as coefficients the Poincaré series for the mapping space has the form

\frac{(1 + x)^{2}(1 + x^{2n−1})}{(1 − x^{2n})^{2}},

while for Hol_{k}^{∗}(T, P^{n}) we have

(1+x)^{2}(1+x^{2n−1})(1+2x^{2n}+3x^{4n}+⋯+(k−1)x^{(k−2)2n})+(k−2)x^{2(k−1)n−1}.

Remark 9.5. Notice that in this case all the calculations were independent of the particular elliptic curve T that we were studying, even though different T are not holomorphically isomorphic. In fact this is not unexpected since we have

Theorem 9.6. Let T be any non-singular elliptic curve. Then the homeomorphism type of Hol_{k}^{∗}(T, P^{n}) is independent of T.

Proof. Let T and T′ be two given tori with corresponding lattices Λ and Λ′ ⊂ C. Then there is a map Λ ⟶ Λ′ which induces a homeomorphism

T = C/Λ^{\underrightarrow{f}}T′ = C/Λ′,

and such that the following commutes for all n;

SP^{n}(T) ⟶^{SP^{n}(f)} SP^{n}(T′)
↓μ  ↓μ
T  ⟶^{f}  T′.

It follows therefore that E_{i,i,...i} ≅ E′_{i,i,... ,i} where E_{i,i,...i} (resp. E′_{i,i,... ,i}) is the pushout as in §3

E_{i,i,... ,i} ⟶ SP^{i}(T) × ⋯ × SP^{i}(T)
↓  ↓
T  ^{\underrightarrow{Δ}}  T × ⋯ × T

(respectively, T is replaced by T′). It is now clear that

Hol_{n}^{∗}(T, P^{n}) ≅ Ẽ_{i,...i} − Image(ν) ≅ Ẽ′_{i,...i} − Image(ν) ≅ Hol_{n}^{∗}(T, P^{n}),

and the proof is complete. q.e.d.

10. The decomposition for hyperelliptic curves

The surface M_{g} is hyperelliptic if and only if there is a degree-two (branched) holomorphic map f: M_{g}⟶P^{1}. This map f fits in the following diagram

PSfrag replacements

M_{g}

P^{2g−2}
P^{1}
s

where s is the canonical map, [8, p. 247].

From the point of view of the Abel-Jacobi maps, M_{g} is hyperelliptic if and only if we have a cofibering of the form

P^{1} ↪ SP^{2}(M_{g})⟶SP^{2}(M_{g})/P^{1} = W_{2} ⊂ J(M_{g}).

The holomorphic embedding P^{1} ↪ SP^{2}(M_{g}) is now constructed by associating to z ∈ P^{1} the pair f^{−1}(z) ∈ SP^{2}(M_{g}) where the points are counted with multiplicity, i.e., the ramification points are counted twice. The image τ ∈ J(M_{g}) of P^{1} under μ_{2} is called the hyperelliptic point of M_{g}.

Lemma 10.2. Suppose that M_{g} is hyperelliptic and τ ∈ J(M_{g}) is the hyperelliptic point. Let k ≥ 1 and t ≤ [\frac{k}{2}]. Then:

(a) for k ≤ g, the space W_{k}^{t} is μ_{k−2t}(SP^{k−2t}(M_{g})) + tτ,

(b) for 2g − 1 > k > g, t > k − g, we also have

W_{k}^{t} = μ_{k−2t}(SP^{k−2t}(M_{g})) + tτ.

Proof. Note that SP^{r}(P^{1}) = P^{r} so the pairing

SP^{2l}(M_{g}) × SP^{k−2l}(M_{g})⟶SP^{k}(M_{g})

induces an action

ν: P^{l} × SP^{k−2l}(M_{g})⟶^{=}SP^{l}(P^{1}) × SP^{k−2l}(M_{g})⟶SP^{k}(M_{g}),

and the image of

ν(⟨p_{1}, …, p_{l}⟩, ⟨m_{1}, …, m_{k−2l}⟩)

under the Abel-Jacobi map is lτ + μ_{k−2l}(⟨m_{1}, …, m_{k−2l}⟩. From this follows directly the fact that W_{k}^{t} is at least as large as asserted in (a). The inclusion argument for (b) is a similar dimension check when we recall that generically μ_{g+s}^{−1}(j) = P^{s}, so we are checking for points with inverse image P^{t} where t > s and ν(P^{t}×SP^{g+s−2t}(M_{g})) ⊂ μ_{g+s}^{−1}(W_{g+s}^{t}).

The converse is given on page 13 of [1] for k ≤ g where it is pointed out that as a consequence of the geometric version of the Riemann-Roch Theorem every P^{t} inverse image under the Abel-Jacobi map for SP^{k}(M_{g}) has the form μ^{−1}(tτ) + m_{1} + ⋯ + m_{k−2t} where no two of the m_{i}’s are conjugate in M_{g} under the hyperelliptic involution. To obtain the remaining part of the converse one uses the residual series isomorphism

W_{k}^{t} ≅ W_{2g−2−k}^{g−k+t−1}, D ↔ K − D,

which holds for all curves M_{g}, k ≥ g. q.e.d.

For later use we also specify an appropriate base point for the construction. Let ∗ ∈ M_{g} be one of the ramification points of f: M_{g}⟶P^{1}. Then 2μ(∗) = τ, and we choose ∗ as the base point in M_{g}. Also the base point of P^{1} is ⟨∗, ∗⟩, and the inclusion f: P^{1}⟶SP^{2}(M_{g}) is based. With respect to this choice, the composite inclusion

SP^{k}(M_{g}) ↪ SP^{k+1}(M_{g}) ↪ SP^{k+2}(M_{g})

commutes with the P^{1}-action. Moreover, in the actual construction of the Abel-Jacobi map μ given at the beginning of §1 we can take our base point in M_{g} to be ∗, so that μ(∗) = 0 ∈ J(M_{g}), and the hyperelliptic point τ = 0 as well. This has the advantage that the inclusion W_{k}^{t} + τ ⊂ W_{k+2}^{t+1} becomes simply W_{k}^{t} ⊂ W_{k+1}^{t+1}.

11. Some preliminary constructions

Let V be a space with a commutative, associative multiplication. For example

V = SP^{∞}(X, ∗), V = ∐_{1}^{∞} SP^{n}(X), V = ∐_{1}^{∞} SP^{i}(X) × SP^{i}(X),

etc. If f: X⟶V is any continuous map, then f defines an action

μ_{f}: ∐ SP^{k}(X) × V ⟶V

by μ_{f}(⟨x_{1}, …, x_{k}⟩, v) = f(x_{1})f(x_{2}) ⋯ f(x_{k})v.

Suppose now that V, f_{1} : P^{1}⟶V and V′, f_{2} : P^{1}⟶V′ are given with V, V′ as above. Then we can form the bicomplex

(11.1)  B(f_{1}, f_{2}) : = V ×_{f_{1}} (∐_{k} SP^{k}(I × P^{1})) ×_{f_{2}} V′,

defined by the equivalence relation built from the basic relations

\begin{matrix}(v, & ⟨(t_{1}, p_{1}), …, (t_{k}, p_{k})⟩, v′) \\ ∼ \begin{cases}(v_{1} · f_{1}(p_{i}), ⟨(t_{1}, p_{1}), …, { \atop (t_{i}, p_{i})}, …, (t_{k}, p_{k})⟩, v′) & if t_{i} = 0, \\ (v_{1}, ⟨(t_{1}, p_{1}), …, { \atop (t_{i}, p_{i})}, …, (t_{k}, p_{k})⟩, f_{2}(p_{i}) · v′) & if t_{i} = 1. \end{cases} \end{matrix}

Remark 11.2. This is just the 2-sided bar construction which is associated to the homological functor T or_{A}(M_{A},_{A} N). But here, left and right modules are equivalent since A is commutative.

Remark 11.3. This is an unbased construction. If f_{1} and f_{2} are based maps with f_{i}(∗) = id, i = 1, 2, then there is an associated reduced construction, B̃(f_{1}, f_{2}). Also, the fact that we are using P^{1} plays no real role in the definition. It is just there because that is all we use in the applications of this construction to our study of Hol_{k}^{∗}(M_{g}, P^{n}) for M_{g} hyperelliptic.

Example 11.4. Suppose that V = ∐ SP^{k}(X) and

f_{1} : P^{1}⟶SP^{i}(X).

Then write

V_{j} = ∐_{k=0}^{∞} SP^{j+ik}(X), 0 ≤ j < i

so we have separate actions on each V_{j}.

Example 11.5. Suppose f_{2} : P^{1}⟶∗ where ∗ has the obvious commutative and associative action. Then

B(f_{1}, f_{2}) = V ×_{f_{1}} ∐ SP^{n}(I × P^{1}) ×_{f_{2}} ∗ ≃ V ×_{f_{1}} ∐ SP^{n}(cP^{1})

is a space that we have seen many times before.

Example 11.6. Δ_{n+1} : P^{1}⟶(P^{1})^{n+1} induces an action on (∐ SP^{k}(P^{1}))^{n+1} which restricts, in a manner similar to that in example (1) to an action on

E_{n}(i_{1}, …, i_{n+1}) = ∐_{k=0}^{∞} SP^{i_{1}+k}(P^{1}) × ⋯ × SP^{i_{n+1}+k}(P^{1})

for each (n+1)-tuple (i_{1}, …, i_{n+1}) of non-negative integers with at least one of the i_{j} = 0.

12. Models for the spaces E_{i,...,i}, LE_{i,...,i} and W_{k}^{t} when M_{g} is hyperelliptic

We specialize to

\begin{matrix}BH(f, n) \\ = (∐_{m} SP^{m}(M_{g})) ×_{f} (∐_{m} SP^{m}(I × P^{1})) \\ ×_{f_{2}} E_{n}(0, 0, …, 0), \end{matrix} (12.1)

where f: P^{1} ↪ SP^{2}(M_{g}) embeds P^{1} as the inverse image under the Abel-Jacobi map of the hyperelliptic point, 0, for some hyperelliptic structure on M_{g}, and f_{2} is the structure map Δ_{n+1} of (11.6).

This construction specializes, as in (11.5), to give two disjoint subspaces,

\begin{aligned}(12.2)  BH_{0}(f, n) & = V_{0} ×_{f} (∐_{m} SP^{m}(I × P^{1})) ×_{f_{2}} E_{n}(0, …, 0), \\ (12.3)  BH_{1}(f, n) & = V_{1} ×_{f} (∐_{m} SP^{m}(I × P^{1})) ×_{f_{2}} E_{n}(0, …, 0), \end{aligned}

where V_{0} = ∐ SP^{2k}(M_{g}) and V_{1} = ∐ SP^{2k+1}(M_{g}). Each BH_{ϵ}(f, n) breaks up into further components as follows. We define a grading by letting

\begin{matrix}(⟨m_{1}, …, m_{l}⟩, ⟨(t_{1}, p_{1}), …, (t_{r}, p_{r})⟩, (w_{s,1}, …, w_{s,n+1})) \\ ∈ SP^{l}(M_{g}) × SP^{r}(I × P^{1}) × ∏_{1}^{n+1} SP^{s}(P^{1}) \end{matrix} (12.4)

have grading degree l + 2r + 2s. This grading is obviously preserved by the identifications and hence induces the desired decomposition of the quotient BH_{ϵ}(f). We write G_{s}(f, n) for the component of points of grading degree s.

This is the general case, however, there is a special case BH_{ϵ}(f, −1) which is also needed where V_{2} ∼ ∗ consists of a single point. In this case the grading degree is not preserved by the identifications, and the grading only gives a filtration of BH_{ϵ}(f, −1). We write F_{k}(BH_{ϵ}(f, −1)) for the filtering subspace consisting of all points of grading degree ≤ k. Moreover, in this case the identification has the form (v, (t, x), ∗) ∼ (vf(x), ∗) when t = 0 and (v, (t, x), ∗) ∼ (v, ∗) when t = 1.

Lemma 12.5. We have the following homotopy equivalences in the case where M_{g} is hyperelliptic:

(a) G_{k}(BH_{ϵ}(f, n)) ≃ E_{k,...,k} for k ≡ ϵ mod (2) provided k ≤ g.

(b) F_{k}(BH_{ϵ}(f, −1)) ≃ W_{k}, k ≡ ϵ mod (2) and k < g.

(These constructions are modeled on the analogous constructions in [20] and the proofs are the same.)

Remark 12.6. The point of these constructions is to handle the algebraic complexities of systematically collapsing out P^{n}’s. As one sees from the lemma, the effect is to introduce SP^{n}(ΣP^{1})’s which provide a measure of the geometry of the collapsed spaces as the collapsing gets more and more complex.

13. Some spectral sequences for the BH_{ϵ}(f, n)

We can bifilter G_{k}(BH_{ϵ}(f, n)) and F_{k}(BH_{ϵ}(f, −1)) by the number of SP^{t}(I × P^{1}) terms appearing, obtaining spectral sequences converging to the various spaces above as well as certain useful quotients. The E^{1}-terms are as follows:

∐_{k+2l+2t=v} H_{∗}(SP^{k}(M_{g}))

⊗ H_{∗}(SP^{l}(ΣP_{+}^{1}), SP^{l−1}(ΣP_{+}^{1})) ⊗ (H_{∗}(P^{t}))^{n+1} (13.1)

converging to H_{∗}(E_{v,...,v}) with field coefficients and v < g,

(13.2)  ∐_{k+2l≤v} H_{∗}(SP^{k}(M_{g})) ⊗ H_{∗}(SP^{l}(ΣP_{+}^{1}), SP^{l−1}(ΣP_{+}^{1}))

with k ≡ v mod (2) converging to H_{∗}(W_{v}) for v < g, and

(13.3)  ∐_{k+2l=v} H_{∗}(SP^{k}(M_{g})) ⊗ H_{∗}(SP^{l}(ΣP_{+}^{1}), SP^{l−1}(ΣP_{+}^{1}))

converging to H_{∗}(W_{v}/W_{v}^{1}). (The notation ΣP_{+}^{1} means the reduced suspension on the union of P^{1} with a disjoint base point denoted +, it has the homotopy type of the wedge Σ(P^{1}) ∨ S^{1}.)

If we relativize we obtain a spectral sequence with E^{1}-term

∐_{k+2(l+t)=v} H_{∗}(SP^{k}(M_{g}), SP^{k−1}(M_{g}))

⊗ H_{∗}(SP^{l}(ΣP_{+}^{1}), SP^{l−1}(P_{+}^{1})) ⊗ H̃_{∗}(S^{2(n+1)t}) (13.4)

⟹ H̃_{∗}(LE_{v,...,v}).

All these spectral sequences are algebraic in the sense of [12], and can be modeled by simply replacing the terms H_{∗}(SP^{l}(ΣP^{1})) by the cobar construction on

H_{∗} (∐ SP^{l}(P_{+}^{1})) = Z[t] ⊗ Γ(b).

In particular, the relative spectral sequence splits as a direct sum of spectral sequences each of the form

[ ∐_{k+2l=v−2t} H_{∗}(SP^{k}(M_{g}), SP^{k−1}(M_{g})) ⊗ H_{∗}(SP^{l}(ΣP_{+}^{1}), SP^{l−1}(ΣP_{+}^{1}))]

⊗ H̃_{∗}(S^{2t(n+1)})
(13.5)  ⟹ Σ^{2t(n+1)} (H_{∗}(W_{v−2t}/(W_{v−2t}^{1} ∪ W_{v−2t−1})))

for any n > 0.

In the case where the coefficients are the rationals, these spectral sequences become quite simple. First H_{∗}(SP^{∞}(ΣP_{+}^{1}); Q) = Λ(h_{1}, h_{3}), and

H_{∗}(SP^{i}(ΣP^{1}), SP^{i−1}(ΣP^{1}); Q) = 0

for i ≥ 3. Second, when we write

H_{∗}(SP^{k}(M_{g}), SP^{k−1}(M_{g}); Z) = ⨁_{s=0}^{k} Λ_{s}(e_{1}, …, e_{2g}) ⊗ [M_{g}]^{k−s},

for k < g, we see that the action map

∗ [P^{1}]: H_{∗}(SP^{k}(M_{g}), SP^{k−1}(M_{g}); Z)
⟶H_{∗+2}(SP^{k+2}(M_{g}), SP^{k−1}(M_{g}); Z)

is just multiplication by − ∑_{j=1}^{g} e_{2j−1}e_{2j} since

[P^{1}] = [M_{g}] − ∑_{1}^{g} e_{2j−1}e_{2j}

in H_{∗}(SP^{2}(M_{g}); Z). Third, from [2], this map is injective in homology for k ≤ g − 2. Thus we have

Lemma 13.6.

\begin{matrix}H_{∗}(LE_{v,...,v}; Q) \\ = ∑_{t=0}^{[v/2]} \frac{Σ^{2t(n+1)}H_{∗}(SP^{v−2t}(M_{g}), SP^{v−2t−1}(M_{g}); Q)}{im( ∗ ( ∑_{j=1}^{g} e_{2j−1}e_{2j}))} \end{matrix}

for v ≤ g, which has Poincaré series

∑_{t=0}^{[v/2]} x^{2t(n+1)}S_{v−2t},

where S_{r} = ∑_{l≤r} [\binom{2g}{l} − \binom{2g}{l−1}] x^{2r−l}.

Similarly, we can analyze H_{∗}(W_{j}; Q).

Lemma 13.7.

(a) The inclusion W_{j} ⊂ J(M_{g}) induces an injection in rational homology H_{∗}(W_{j}; Q) ↪ H_{∗}(J(M_{g}); Q) = Λ(e_{1}, …, e_{2g}) with image the subvector space spanned by the subspaces

{Λ_{s}(e_{1}, …, e_{2g})[M_{g}]^{t} | s + t ≤ j} ,

where [M_{g}] = ∑_{1}^{g} e_{2i−1}e_{2i} is the image of the fundamental class of M_{g} under the Abel-Jacobi map μ_{∗},

(b) H_{∗}(W_{j−1}; Q) injects into H_{∗}(W_{j}; Q) under the inclusion, so the relative groups are given as

H_{∗}(W_{j}, W_{j−1}; Q) ≅ H_{∗}(W_{j}; Q)/H_{∗}(W_{j−1}; Q).

Proof. We have that the E^{1}-term of the spectral sequence above is given as

\begin{aligned}H_{∗}(SP^{j}(M_{g}); Q) & ⊕ H_{∗}(SP^{j−2}(M_{g}); Q)(1, h_{1}, h_{3}) \\ & ⊕ ∑_{2≤2l≤j} H_{∗}(SP^{j−2l}(M_{g}); Q)(1, h_{1}, h_{3}, h_{1}h_{3}). \end{aligned}

d_{1}(Θh_{1}) = i_{∗}(Θ) − Θ where Θ ∈ H_{∗}(SP^{j−2l}(M_{g})) and

i: SP^{j−2l}(M_{g})⟶SP^{j−2l+2}(M_{g})

is a base point inclusion. Similarly

d_{1}(Θh_{1}h_{3}) = d_{1}(Θh_{1})h_{3} ± (Θh_{1})d_{1}(h_{3})

where, of course, this last term is just multiplication by [M_{g}]−∑ e_{2i−1}e_{2i}. Ignoring the last piece one sees that the d_{1}-differential on h_{1} reduces the calculation of E^{2} to the following complex:

\begin{matrix}H_{∗}(SP^{j−2}(M_{g}); Q)h_{3} & ⟶H_{∗}(SP^{j}(M_{g}); Q), \\ Θh_{3} & ↦ Θ([M_{g}] − ∑_{1}^{g} e_{2i−1}e_{2i}) \end{matrix}

from which the first statement of the lemma follows. The second statement is then immediate. q.e.d.

The remaining groups H_{∗}(LE_{v,...,v}; Q) in the hyperelliptic case.

It is convenient to write v = g + s, 1 ≤ s < g − 1 when v > g. In this case we have that J(M_{g}) is filtered by W_{0} ⊂ W_{1} ⊂ ⋯ W_{g−1} ⊂ J(M_{g}), and over points in W_{g−s−2l} − W_{g−s−2l−2} the inverse image in E_{v,...,v} is (P^{s+l})^{n+1}, while the inverse image over J(M_{g}) − W_{g−s−2} is (P^{s})^{n+1}. Thus, we obtain a spectral sequence with E^{1}-term:

H_{∗}(J(M_{g}), W_{g−s−2}) ⊗ (H_{∗}(P^{s}))^{n+1}

⊕ ∑_{l=1}^{[g−s/2]} H_{∗}(W_{g−s−2l}, W_{g−s−2l−2}) ⊗ (H_{∗}(P^{s+l}))^{n+1} (13.8)

⟹ H_{∗}(E_{g+s,...,g+s})

with field coefficients.

Similarly, the corresponding E^{1}-term for H_{∗}(LE_{g+s,...,g+s}) is given by

H_{∗}(J(M_{g}), W_{g−s−1}) ⊗ (H_{∗}(P^{s}, P^{s−1}))^{n+1}

(13.9) ⊕ ∑_{l=1}^{[g−s/2]} H_{∗}(W_{g−s−2l}, W_{g−s−2l−1}) ⊗ (H_{∗}(P^{s+l}, P^{s+l−1}))^{n+1}
⟹ H_{∗}(LE_{g+s,...,g+s}).

Note here that in the formula above the term involving the relative groups

H_{∗}(W_{g−s−2l−1}, W_{g−s−2l−2})

is zero. That is because, in this region of J(M_{g}) and this range for v the inverse image of a point is P^{s+l} in both E_{g+s,...,g+s} and E_{g+s−1,...,g+s−1}.

Again, because of this gap the spectral sequence collapses and E^{1} = E^{∞}. Of course the calculations become considerably more complex with F_{p} coefficients.

14. The Ext-groups for Λ_{g}/S_{g}

By inspection, in the spectral sequence of (5.9) we find the that d_{1}-differential gives rise to a series of complexes that occur in the calculation of Ext-groups for the ring V_{g} defined below. In this section we calculate these Ext-groups and apply the results in §15 to determine the structure of the rational homology of the space Hol_{k}^{∗}(M_{g}, P^{n}) with M_{g} hyperelliptic and n ≥ 2.

Notation 14.1.

(1) Λ_{g} = Λ(e_{1}, …, e_{2g}) = H_{∗}(J(M_{g}), Q),

(2) f_{g} = ∑_{i=1}^{g} e_{2i−1}e_{2i} ∈ Λ_{g},

(3) S_{g} ⊂ Λ_{g} is the ideal (f_{g}), and

(4) Λ_{g}/S_{g} = V_{g} is the quotient.

V_{g} occurs quite often in both our study of Hol_{d}^{∗}(M_{g}, P^{n}) and Map_{d}^{∗}(M_{g}, P^{n}). In particular we need the groups Ext_{Λ_{g}}^{∗,∗}(V_{g}, Q) which we will describe as modules over Ext_{Λ_{g}}^{∗,∗}(Q, Q) = Q[h_{1}, …, h_{2g}] where h_{i} = |e_{i}|^{∗}, 1 ≤ i ≤ 2g.

In [2] it is shown that

∗f_{g}: (Λ_{g})_{i}⟶(Λ_{g})_{i+2}

is injective for i ≤ g − 1 and surjective for i ≥ g − 1. Consequently, (V_{g})_{i} = 0 for i > g while (V_{g})_{i} = Q^{N(g,i)} for 0 ≤ i < g where

N(g, i) = \begin{cases}\binom{2g}{i} − \binom{2g}{i−2} & for 2 ≤ i ≤ g, \\ 2g & for i = 1, \\ 1 & for i = 0. \end{cases}

As a special case note that

V_{1} = Λ_{1}/{e_{1}e_{2}},

so there is a short exact sequence of Λ_{1}-modules

0⟶Q^{{e_{1}e_{2} \atop ↪}} Λ_{1}⟶V_{1}⟶0,

where Q denotes the “trivial” module given by the augmentation map ϵ: Λ_{g}⟶Q, ϵ(e_{i}) = 0, ϵ(1) = 1. This exact sequence gives rise to the long exact sequence of Ext-groups:

⋯⟶Ext_{Λ_{1}}^{i−1,j−2}(Q, Q)⟶^{δ}Ext_{Λ_{1}}^{i,j}(V_{1}, Q)

⟶Ext_{Λ_{1}}^{i,j}(Λ_{1}, Q)⟶Ext_{Λ_{1}}^{i,j}(Q, Q)⟶^{δ} ⋯,

and since Ext_{Λ_{1}}^{i,j}(Λ_{1}, Q) = \begin{cases}0 & (i, j) ≠ (0, 0), \\ Q & (i, j) = (0, 0), \end{cases} we have

Ext_{Λ_{1}}^{∗,∗}(V_{1}, Q) = Qe_{0,0} ⊕ Ext_{Λ_{1}}^{∗,∗}(Q, Q)P_{1,2}
= Qe_{0,0} ⊕ Q[h_{1}, h_{2}]P_{1,2} ,

where P_{1,2} is a generating element in Ext_{Λ_{1}}^{1,2}(V_{1}, Q).

Before we can describe the structure of the modules Ext_{Λ_{g}}^{∗,∗}(V_{g}, Q) for g > 1, we need some preliminary constructions.

First, we will often be dealing with a situation where we have a ring R_{i+1} = R_{i} ⊗_{Q} R_{1} given as the tensor product of augmented rings, together with a module, M, over R_{i}. M becomes a module over R_{i+1} via the following obvious composition

(R_{i} ⊗ R_{1}) ⊗ M ⟶^{id⊗id}(R_{i} ⊗ R_{1}) ⊗ (M ⊗_{Q} Q)⟶^{1⊗T ⊗1}(R_{i} ⊗ M) ⊗ (R_{1} ⊗ Q)

⟶^{μ_{i}⊗ϵ}M ⊗_{Q} Q = M.

Then, by change of rings,

(14.2)  Ext_{R_{i+1}}(M, Q) = Ext_{R_{i}}(M, Q) ⊗ Ext_{R_{1}}(Q, Q).

Second, we will need to consider modules of the following kind: given a module M over the ring R and an exact sequence of R-modules

0⟶K ↪ (R)^{N}⟶M⟶0,

then K is written ΩM and is unique up to direct sum with a free Rmodule. The following result is well known.

Lemma 14.3. Ext_{R}^{i,j}(ΩM, N) ≅ Ext_{R}^{i+1,j}(M, N) for all i ≥ 1, and

Ext_{R}^{0,∗}(ΩM, N)⟶Ext_{R}^{1,∗}(M, N)

is surjective for all R-modules N.

We say ΩM is minimal if the natural map

Ext_{R}^{0,∗}(ΩM, Q)⟶Ext_{R}^{1,∗}(M, Q)

is an isomorphism as well. Of course, ΩM is minimal if and only if the map R^{N}⟶M induces isomorphisms of Ext^{0,∗}( , Q)-groups.

Consider the following module:

(14.4)  M_{1} = I ⊂ Q[h_{1}, h_{2}];

it is the augmentation ideal, I = (h_{1}, h_{2}). Moreover, M_{1} = ΩQ and is even minimal for R = Q[h_{1}, h_{2}]; thus

Ext_{Q[h_{1},h_{2}]}^{i,j}(M_{1}, Q) = Ext_{Q[h_{1},h_{2}]}^{i+1,j}(Q, Q)

for all j and i ≥ 0. Moreover, Ext_{Q[h_{1},h_{2}]}(Q, Q) = Λ(|h_{1}|^{∗}, |h_{2}|^{∗}); thus

Ext_{Q[h_{1},h_{2}]}^{i,j}(M_{1}, Q) = \begin{cases}Q\{|h_{1}|^{∗}, |h_{2}|^{∗}\} & if (i, j) = (0, 2), \\ Q\{|h_{1}|^{∗}|h_{2}|^{∗}\} & if (i, j) = (1, 4). \end{cases}

It follows that a minimal resolution for M_{1} over Q[h_{1}, h_{2}] has the form

Q[h_{1}, h_{2}]h_{1,2}⟶Q[h_{1}, h_{2}]{[h_{1}], [h_{2}]}⟶M_{1}⟶0,

where [h_{i}] ↦ h_{i} and h_{1,2} ↦ h_{1}[h_{2}] − h_{2}[h_{1}].

Next we consider M_{1} as a module over

Q[h_{1}, h_{2}, h_{3}, h_{4}] = Q[h_{1}, h_{2}] ⊗ Q[h_{3}, h_{4}]

as above, and define M_{2} as the kernel of the surjective homomorphism

π_{2} : Q[h_{1}, h_{2}, h_{3}, h_{4}]{b_{1}, b_{2}}⟶M_{1}⟶0,

π_{2}(b_{i}) = h_{i}. Clearly M_{2} = ΩM_{1} as a module over Q[h_{1}, h_{2}, h_{3}, h_{4}] and is minimal as well. Consequently we have

Ext_{Q[h_{1},...,h_{4}]}^{0,∗}(M_{2}, Q) = Ext_{Q[h_{1},...,h_{4}]}^{1,∗}(M_{1}, Q)
= Ext_{Q[h_{1},h_{2}]}^{2,∗}(Q, Q) ⊕ Ext_{Q[h_{1},h_{2}]}^{1,∗}(Q, Q)
⊗ Ext_{Q[h_{3},h_{4}]}^{1,∗}(Q, Q)
= Q^{5}

with generators |h_{1}|^{∗}|h_{2}|^{∗}, |h_{1}|^{∗}|h_{3}|^{∗}, |h_{1}|^{∗}|h_{4}|^{∗}, |h_{2}|^{∗}|h_{3}|^{∗}, |h_{2}|^{∗}|h_{4}|^{∗}. Likewise,

Ext^{1}(M_{2}, Q) = Ext^{2}(M_{1}, Q)
≅ Ext_{Q[h_{1},h_{2}]}^{2}(Q, Q) ⊗ Ext_{Q[h_{3},h_{4}]}^{1}(Q, Q)
⊕ Ext_{Q[h_{1},h_{2}]}^{1}(Q, Q) ⊗ Ext_{Q[h_{3},h_{4}]}^{1}(Q, Q)
= Q^{4}.

To obtain M_{3} regard M_{2} as a module over Q[h_{1}, …, h_{6}] = R_{3}, and define M_{3} = ΩM_{2}, the kernel in the exact sequence

0⟶M_{3} ↪ R_{3}^{5}⟶M_{2}⟶0,

where the map to M_{2} takes the basis for R_{3}^{5} to the five generators of M_{2} corresponding to the description above. M_{3} is again minimal.

In general M_{n} is given as a module over R_{n} = Q[h_{1}, …, h_{2n}], indeed is a submodule of a free R_{n}-module, and M_{n+1} is ΩM_{n} where M_{n} is now regarded as a module over R_{n+1} = R_{n} ⊗ Q[h_{2n+1}, h_{2n+2}].

Note that for M_{1} we have Ext^{0}(M_{1}, Q) = Ext^{0,2}(M_{1}, Q), and for M_{2}, Ext^{0}(M_{2}, Q) = Ext^{0,4}(M_{2}, Q). In other words, for these modules Ext^{0} is concentrated in a single bidegree. Also, note that Ext_{R_{n}}(Q, Q) = Λ(|h_{1}|^{∗}, …, |h_{2n}|^{∗}) has each generator |h_{i}|^{∗} occurring in bidegree (1, 2), so all the elements in Ext^{i} occur in a single bidegree (i, 2i) here as well. In fact we have by a direct induction

Lemma 14.5. The action map

Ext_{R_{n}}^{∗,∗}(Q, Q) ⊗ Ext_{R_{n}}^{0,∗}(M_{n}, Q)⟶Ext_{R_{n}}^{i,j}(M_{n}, Q)

is surjective in all degrees, and Ext_{R_{n}}^{0,j}(M_{n}, Q) = 0 unless j = 2n. Consequently,

Ext_{R_{n}}^{i,j}(M_{n}, Q) = 0

unless j = 2n + 2i.

As an immediate corollary we see that the M_{i}, each being minimal, are all unique, since, in as much as all the generators of M_{i} occur in the same degree, the map of the minimal free module onto M_{i} is well defined up to an isomorphism, hence the same is true of the kernel, M_{i+1}.

With this backround discussion complete we are able to state our main result.

Theorem 14.6. Suppose g ≥ 2. Then

Ext_{Λ_{g}}^{i,j}(V_{g}, Q) = \begin{cases}Q & if (i, j) = (0, 0), \\ Q & if (i, j) = (1, 2), \\ (M_{g})_{i−2,j+i−4} & if i ≥ 2. \end{cases}

In particular, as a module over Ext_{Λ_{g}}(Q, Q)

Ext_{Λ_{g}}(V_{g}, Q) = Q ⊕ Q ⊕ M_{g},

where the first generator gives Ext^{0,0}, the second gives Ext^{1,2} and the rest is shifted by (2, −i + 4).

Example 14.7. We have

Ext_{Λ_{2}}(V_{2}, Q) = Qe_{0,0} ⊕ QP_{1,2} ⊕ Q[h_{1}, …, h_{4}]{b_{1}, b_{2}, …, b_{5}}/R,

where b_{i} ∈ Ext_{Λ_{2}}^{2,4}(V_{2}, Q), and R is the set of four relations

R = {h_{5}b_{1}, h_{6}b_{1}, h_{5}b_{3} − h_{6}b_{2}, h_{5}b_{5} − h_{6}b_{4}} .

Proof. A resolution of V_{g} over Λ_{g} starts in the following way

0⟶K_{g} ↪ Λ_{g}P_{1,2} ⟶^{∗f_{g}}Λ_{g}e_{0,0}⟶V_{g}⟶0,

and we have already seen that (K_{g})_{i} = 0 for i ≤ g + 1. Thus it follows that

Ext_{Λ_{g}}^{r,s}(V_{g}, Q) = 0 whenever r ≥ 2 and s ≤ g + 1.

On the other hand, since K_{g} has N(g, g) generators in g + 2, we have that

Ext_{Λ_{g}}^{2,g+2}(V_{g}, Q) = Q^{N(g,g)}.

We now give an inductive analysis of the V_{g}. In particular we assume the theorem is true for V_{g−1} and begin by establishing a relation between V_{g} and V_{g−1}. Actually, we start with g = 1 where the theorem is not quite true as stated, but Ext_{Λ_{1}}(V_{1}, Q) is given as an extension

0⟶M_{1}⟶Ext_{Λ_{1}}(V_{1}, Q)⟶Qe_{0,0} ⊕ QP_{1,2}⟶0,

which turns out to be sufficient to start the induction. There is a surjection

π_{g}: V_{g}⟶V_{g−1}

induced by the projection p_{g}: Λ_{g}⟶Λ_{g−1},

p_{g}(e_{i}) = \begin{cases}e_{i} & for i ≤ i ≤ 2g − 2, \\ 0 & fori = 2g-1or i = 2g, \end{cases}

which takes f_{g} to f_{g−1}. Let N_{g} be the kernel of π_{g}, and K_{g} be the kernel of p_{g}. Then we can write

K_{g} ≅ Λ_{g−1}{e_{2g−1}, e_{2g}, e_{2g−1}e_{2g}}

and

K_{g}/f_{g}K_{g} = V_{g−1} ⊗ Q{e_{2g−1}, e_{2g}, e_{2g−1}e_{2g}}

as a module over Λ_{g}. However, the surjection

p̄_{g}: K_{g}/f_{g}K_{g}⟶N_{g}⟶0

is not an isomorphism. In fact we have

Lemma 14.8. The kernel of p̄_{g} is a direct sum of trivial modules over Λ_{g} concentrated in degree g + 1,

Ker(p̄_{g}) = Ker(p̄_{g})_{g+1} ≅ Q^{N(g−1,g−1)}

as a module over Λ_{g}.

Proof. For 2 ≤ i ≤ g − 1, using the expansion

\binom{r}{i} = \binom{r − 2}{i} + 2\binom{r − 2}{i − 1} + \binom{r − 2}{i − 2}

valid for r ≥ i ≥ 2, we have

\begin{aligned}Dim(N_{g})_{i} & = [\binom{2g}{i} − \binom{2g}{i − 2}] − [\binom{2g − 2}{i} − \binom{2g − 2}{i − 2}] \\ & = \{2\binom{2g − 2}{i − 1} + \binom{2g − 2}{i − 2}\} − \{2\binom{2g − 2}{i − 3} + \binom{2g − 2}{i − 4}\} . \end{aligned}

This shows that p̄_{g} is an isomorphism in this range. Likewise,

Dim(N_{g})_{g} = \binom{2g}{g} − \binom{2g}{g − 2}

= \binom{2g − 2}{g} + 2\binom{2g − 2}{g − 1} − 2\binom{2g − 2}{g − 3} − \binom{2g − 2}{g − 4}

= [\binom{2g − 2}{g − 2} − \binom{2g − 2}{g − 4}] − 2 [\binom{2g − 2}{g − 1} − \binom{2g − 2}{g − 3}] ,

using the equality \binom{2g−2}{g} = \binom{2g−2}{g−2}, which gives the isomorphism in dimension g. However, (N_{g})_{g+1} = 0 while

Dim((K_{g}/f_{g}K_{g})_{g+1}) = Dim((V_{g−1})_{g−1}) = N(g − 1, g − 1),

and the lemma follows since (K_{g}/f_{g}K_{g})_{t} = 0 for t ≥ g + 2. q.e.d.

Consequently we have a short exact sequence of Λ_{g}-modules

0⟶Q^{N(g−1,g−1)}⟶K_{g}/f_{g}K_{g}⟶N_{g}⟶0

and a long exact sequence of Ext-modules

\begin{aligned}⋯ & ⟶^{δ}Ext_{Λ_{g}}(N_{g}, Q)⟶Ext_{Λ_{g}}(K_{g}, Q) \\ & ⟶^{i^{∗}} ∐_{i=1}^{N(g−1,g−1)} Ext_{Λ_{g}}(Q, Q)χ_{i} ⟶^{δ} ⋯ \end{aligned} (*)

where the generator, χ_{i} for each of the Ext_{Λ_{g}}(Q, Q) terms occurs in bidegree (0, g + 1).

By change of rings we have

Ext_{Λ_{g}}(K_{g}, Q) ≅ Ext_{Λ_{g−1}}(V_{g−1}, Q)

⊕ Ext_{Λ(e_{2g−1},e_{2g})}({e_{2g−1}, e_{2g}, e_{2g−1}e_{2g}}, Q)

= Ext_{Λ_{g−1}}(V_{g−1}, Q) ⊗ Q[h_{2g−1}, h_{2g}](b_{1}, b_{2})/R,

where R is the relation h_{2g}b_{1} − h_{2g−1}b_{2} = 0. By our inductive assumption this implies that the generators of Ext_{Λ_{g}}(K_{g}, Q) as a module over Ext_{Λ_{g}}(Q, Q) occur in bidegrees (0, 1), (1, 3), and (2, g+2). But in these bidegrees Ext_{Λ_{g}}(Q, Q)e_{0,g+1} is identically zero. It follows that the map i^{∗} in (∗) is zero so

δ: ∐_{i=1}^{N(g−1,g−1)} Ext_{Λ_{g}}(Q, Q)χ_{i}⟶Ext_{Λ_{g}}(N_{g}, Q)

is an injection, and therefore

Ext_{Λ_{g}}(N_{g}, Q) = ∐_{i=1}^{N(g−1,g−1)} Ext_{Λ_{g}}(Q, Q)e_{1,g+1}^{i} ⊕ Ext_{Λ_{g}}(K_{g}, Q)

as modules over Ext_{Λ_{g}}(Q, Q) up to a possible extension problem. The remainder of the proof is direct. The exact sequence

0⟶N_{g}⟶V_{g} ⟶^{π_{g}}V_{g−1}⟶0

gives us the long exact sequence of Ext-groups

\begin{matrix}⋯ Ext_{Λ_{g}}^{i}(N_{g}, Q) & ⟶^{δ}Ext_{Λ_{g−1}}^{i+1}(V_{g−1}, Q) ⊗ Q[h_{2g−1}, h_{2g}] \\ ⟶Ext_{Λ_{g}}(V_{g}, Q)⟶ · · · . \end{matrix}

The map δ on the piece Ext_{Λ_{g}}(K_{g}, Q) in Ext_{Λ_{g}}(N_{g}, Q) injects to

Ext_{Λ_{g−1}}(V_{g−1}, Q) ⊗ I,

where I ⊂ Q[h_{2g−1}, h_{2g}] is the augmentation ideal (h_{2h−1}, h_{2g}). The resulting quotient is Ext_{Λ_{g−1}}(V_{g−1}, Q) and so the calculation reduces to the determination of the map

δ: ∐_{i=1}^{N(g−1,g−1)} Ext_{Λ_{g}}(Q, Q)χ_{i}⟶M_{g−1}

from the inductive assumption. However, we know Ext_{Λ_{g}}^{∗,j}(V_{g}, Q) = 0 for ∗ ≥ 2 and j ≤ g + 1, and also

Ext_{Λ_{g−1}}^{2,g+1}(V_{g−1}, Q) = Q^{N(g−1,g−1)},

so it follows that δ must give an isomorphism from (1, g+1) to (2, g+1). Since M_{g−1} is generated over Ext_{Λ_{g−1}}(Q, Q) by the elements in this dimension, consequently δ is surjective and the kernel is minimal and hence M_{g}, with the generators all occuring in bidegree (2, g + 2). The induction is complete. q.e.d.

15. The rational homology of Hol_{k}^{∗}(M_{g}, P^{n}) for M_{g}

hyperelliptic

We now use the results of §10-§14 to prove the following theorem.

Theorem 15.1. The map

H_{∗}(Hol_{k}^{∗}(M_{g}, P^{n}); Q)⟶H_{∗}(Map_{k}^{∗}(M_{g}, P^{n}); Q)

is injective for k ≥ 2g and n > 2.

Proof. Consider the subquotient of the E^{1} term in the spectral sequence of (5.9) converging to H_{∗}(QE_{v,...,v}; Q) defined as the direct sum

∑_{v=2s}^{g+s−1} H_{∗}(W_{v−2s}, W_{v−2s−1}) ⊗ H̃_{∗}(S^{2s(n+1)})

⊗ H_{∗}(SP^{k−v}(ΣM_{g}), SP^{k−v−1}(ΣM_{g}); Q)

for a given s = 0, 1, 2, … with the term

H_{∗}(J(M_{g}), W_{g−s+1}) ⊗ H̃_{∗}(S^{2s(n+1)})
⊗ H_{∗}(SP^{k−g−s}(ΣM_{g}), SP^{k−v−s−1}(ΣM_{g}); Q)

added but playing a special role. We have determined that

H_{∗}(W_{j}, W_{j−1}) = ∑_{l=0}^{[j/2]}(V_{g})_{j−2l}[M]^{l},

where [M] ∼ ∑_{1}^{g} e_{2i−1}e_{2i} in H_{∗}(J(M_{g})). Also,

H_{∗}(SP^{k}(ΣM_{g}), SP^{k−1}(ΣM_{g}))
= Q[|e_{1}|, …, |e_{2g}|]_{k} ⊕ Q[|e_{1}|, …, |e_{2g}|]_{k−1}|[M_{g}]|

and the (sub-quotient) d_{1} differentials are given by d_{1}(|[M_{g}]|) = [M_{g}], d_{1}(|e_{i}|) = e_{i}, including the map from the term H_{∗}(W_{g−s+1}, W_{g−s}) ⊗ ⋯ to the extreme term

H_{∗}(J(M_{g}), W_{g−s}) ⊗ · · · .

If we first apply the differential to |[M_{g}]|, we obtain the complex

{ ∑_{v=2s}^{g+s−1} (V_{g})_{v−2s} ⊗ Q[|e_{1}|, …, |e_{2g}|]_{k−v}} ⊗ H̃_{∗}(S^{2s(n+1)}),

with differentials as specified as well as an appropriate quotient of the extreme term with differentials mapping to it from the v = g+s−1-term in the sum above.

Note that the complex in brackets above is a piece of a direct summand,

∑_{w=0}^{g−2} (V_{g})_{w} ⊗ Q[|e_{1}|, …, |e_{2g}|]_{k−2s−w}

of the complex (V_{g}) ⊗ Q[|e_{1}|, …, |e_{2g}|] which has, as its homology the groups

T or_{∗}^{Λ_{g}}(Q, V_{g}).

Here, the extreme term contains the next term in the resolution as a direct summand, but contains H_{∗}(J(M_{g}), W_{g−s−1}) ⊗ ⋯ as well as the complementary summand. On the other hand we have already determined these T or-groups in §14 and have shown, in particular that T or_{l,m}^{Λ_{g}}(Q, V_{g}) = 0 unless m = m(g, l) is

m(g, l) = \begin{cases}l + g & if l ≥ 2, \\ 2 & if l = 1, \\ 0 & if l = 0. \end{cases}

But, in the main part of the complex the m’s which appear are always less than m(g, l) so they contribute nothing by E^{2}, while at E^{2} the term H_{∗}(J(M_{g}), W_{g−s}) ⊗ ⋯, which is a surjective image of the stable term H̃(J(M_{g})) ⊗ ⋯, contributes a quotient, consequently still a surjective image of the stable term, and the result follows for n > 2 by the collapsing of the stable spectral sequence at E^{2}. q.e.d.

384 sadok kallel & r. james milgram

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