IMPROVED STABILITY FOR THE HOMOLOGY OF THE MAPPING CLASS GROUPS OF SURFACES

JOHN L. HARER

0. Introduction

Let F_{g,r} be an oriented surface of genus g with r boundary components. The mapping class group of F_{g,r} is the group Γ_{g,r} of isotopy classes of orientation preserving diffeomorphisms of F_{g,r} which are the identity on its boundary. The rational homology of Γ_{g,r} has been much studied for its own sake and for the fact that

H_{∗}(Γ_{g,r}; Q) ≅ H_{∗}(M_{g,r}; Q)

where M_{g,r} is the moduli space of curves of genus g with r non-zero tangent vectors based at distinct points of the curve. A basic result in the study of this homology is the stability theorem of [Harer3] which states that H_{k}(Γ_{g,r}) is independent of g and r for g ≥ 3k. This range was improved by Ivanov [Ivanov] who showed that stability holds as long as g ≥ 2k. In this paper (from here on assuming rational coefficients) we will show that this range can be improved to g ≥ 3k/2 as long as r ≥ 1. This is best possible because it is known that the stable homology of the mapping class groups includes a polynomial algebra on generators κ_{n}, n = 1, 2, … ([Miller], [Morita]) and it is not hard to see that the class κ_{1}^{n} in H_{2n}(Γ_{g,r}) does not live until genus g ≥ 3n.

The way in which stability is described is in terms of the process of adding a pair of pants (a surface of genus 0 with 3 boundary components) along one or two of its boundary components. If the boundary components of F_{g,r} are numbered 1 to r, this process creates natural inclusions of F_{g,r} in F_{g,r+1} and F_{g+1,r−1} and thereby induces maps of mapping class groups:

ϕ_{i} : Γ_{g,r} → Γ_{g,r+1}, r ≥ 1,
ψ_{i,j} : Γ_{g,r} → Γ_{g+1,r−1}, r ≥ 2.

where the indices indicate where the pants are sewn on. Let τ_{i} denote the Dehn twist on the i^{th} boundary component. If κ_{1} is the generator of H_{2}(Γ_{3,1}) mentioned above, then we define

λ_{n,r} = τ_{r+1} ⊗ κ_{1}^{n}

which lives in H_{2n+1}(Γ_{3n+1,r+1}) modulo the image of H_{2n+1}(Γ_{3n+1,r}) for every r ≥ 1. Also note again that κ_{1}^{n} lives in H_{2n}(Γ_{g,r}) for every g ≥ 3n, r ≥ 0.

1991 Mathematics Subject Classification. Primary 14H10, 20F28; Secondary 14C22, 57M50.

Key words and phrases. mapping class group, moduli space of curves, spin structure.

Supported in part by grants from the National Science Foundation, Sloan Foundation and CNR

The main result of this paper is that the homology group H_{k}(Γ_{g,r}) is independent of g and r as long as g ≥ 3k/2. The precise statement is:

Theorem 0.1. (1) (ϕ_{i})_{∗} : H_{2n+1}(Γ_{3n+1,r}) → H_{2n+1}(Γ_{3n+1,r+1}) is injective

and has cokernel of rank 1 generated by λ_{n,r}.

(2) (ψ_{i,j})_{∗} : H_{2n}(Γ_{3n−1,r+1}) → H_{2n}(Γ_{3n,r}) has cokernel of rank 1 generated by κ_{1}^{n}.

(3) (ϕ_{i})_{∗} : H_{k}(Γ_{g,r}) → H_{k}(Γ_{g,r+1}) is an isomorphism for g ≥ 3k/2 + a_{k}, where

a_{k} = \{ \begin{matrix}−1 & k even \\ 1/2 & k odd \end{matrix}

(4) (ψ_{i,j})_{∗} : H_{k}(Γ_{g−1,r+1}) → H_{k}(Γ_{g,r}) is surjective for g ≥ 3k/2 + b_{k}, where

b_{k} = \{ \begin{matrix}1 & k even \\ −1/2 & k odd \end{matrix}

(5) (ψ_{i,j})_{∗} : H_{k}(Γ_{g−1,r+1}) → H_{k}(Γ_{g,r}) is an isomorphism for g ≥ 3k/2 + c_{k}, where

c_{k} = \{ \begin{matrix}1 & k even \\ 3/2 & k odd \end{matrix}

As a corollary, this combines with the results of [Harer2] to give

Corollary 0.2.

H_{4}(Γ_{g,r}; Q) = Q^{2}, g ≥ 6, r ≥ 1.

We have not attempted to extend the theorem to the case where r = 0 because our motivation is somewhat different. It is known ([Harer3]) that H_{k}(Γ_{g,r}) is independent of both g and r for g sufficiently large, including the case r = 0. As such our theorem says that the stable homology of the moduli space M_{g,r}, r ≥ 0, may be computed for g large by computing H_{k}(Γ_{g,1}) with g = 3k/2.

The proof of the theorem follows the format of that in [Harer3] except that it is necessary to work with simpler Γ-complexes (BX_{i}, see section 1), to look more carefully at the connectivity of BX_{i}/Γ and to analyze the construction of the classes λ_{n,r} and κ_{1}^{n} inside the associated spectral sequences.

1. Description of the complexes BX_{1} and BX_{2} and the Action of Γ_{g,r}.

Let F = F_{g,r}, r ≥ 1. Label the components of ∂F by ∂_{1}F, ∂_{2}F, ... and let p_{1} and p_{2} be distinct points of ∂_{r}F. The isotopy class < α_{1}, ..., α_{k+1} > of a family of imbedded arcs {α_{i}} which connect p_{1} to p_{2} is called an arc system in F based in {p_{1}, p_{2}} if no α_{i} is null-homotopic and for all distinct i and j, α_{i} and α_{j} meet only in {p_{1}, p_{2}} and are not isotopic rel {p_{1}, p_{2}}. Form a simplicial complex BX_{1} (called AX(p_{1}, p_{2}) in our earlier papers, e.g. [Harer1]) by taking a k-simplex for each < α_{1}, ..., α_{k+1} > such that the union of the α_{i} does not separate F and identifying < α_{1}, ..., α_{k+1} > as a face of < α′_{1}, ..., α′_{l+1} > when there are representatives of the isotopy classes such that {α_{i}} ⊂ {α′_{j}}. The mapping class group Γ_{g,r} acts on BX_{1} in the obvious way. (Remark: If α =< α_{1}, ..., α_{k+1} > we will use the letter α interchangeably to denote the simplex, the arc system and the set {α_{1}, ..., α_{k+1}}.)

We will need a second complex BX_{2} defined as follows. Suppose r ≥ 2; let p_{1} be a point of ∂_{r−1}F and let p_{2} be a point of ∂_{r}F. An arc system based in {p_{1}, p_{2}} is defined just as before (the difference is that the arcs connect distinct components of ∂F). The simplicial complex BX_{2} is defined exactly like BX_{1} using arc systems which do not disconnect F. Again Γ_{g,r} acts on BX_{2} in the obvious way.

An n-dimensional simplicial complex is called spherical if it is homotopy equivalent to a wedge of spheres of dimension n. The following lemma was proven in [Harer4].

Lemma 1.1. (1) BX_{1} is spherical of dimension 2g − 1.

(2) BX_{2} is spherical of dimension 2g.

Considering BX_{1} and BX_{2} simultaneously, every arc system α=< α_{1}, ..., α_{k+1} > determines an element P (α) in the symmetric group Σ_{k+1} as follows. Order the arcs counterclockwise (facing ∂F from F) as they emerge from p_{1}, the order that they encounter p_{2} (measured clockwise) gives P (α). The following lemma is elementary and was proven in [Harer3]:

Lemma 1.2. Two k-cells α, α′ are equivalent under the action of Γ_{g,r} if and only if P (α) = P (α′).

Recall that the species of an arc system α = < α_{1}, ..., α_{k+1} > is the genus of a regular neighborhood of α in F. Notice that it depends only on P (α), not on F or how α lies in F. It therefore makes sense to talk about the species of the permutation P (α). Define C_{i,k}^{s} to be the subgroup of C_{k}(BX_{i}/Γ_{g,r}) spanned by those arc-systems which have species s. It is not hard to see that a permutation P on k + 1 letters occurs if and only if it has species s with

max(k − g + 1, 0) ≤ s ≤ [(k + 1)/2]

in BX_{1} and

max(k − g, 0) ≤ s ≤ [k/2]

in BX_{2}.

The following is therefore immediate:

Lemma 1.3. In F_{g,r}, (1)

C_{k}(BX_{1}/Γ_{g,r}) = ⨁_{s=max(k−g+1,0)}^{[(k+1)/2]} C_{1,k}^{s}

(2)

C_{k}(BX_{2}/Γ_{g,r}) = ⨁_{s=max(k−g,0)}^{[k/2]} C_{2,k}^{s}

In [Harer2] we proved:

Theorem 1.4. (1) H_{k}(BX_{1}/Γ_{g,r}) = 0 for k ≤ g.

(2) H_{k}(BX_{2}/Γ_{g,r}) = 0 for k ≤ g + 1.

Here, however, we will need a related but weaker result. It is weaker in that we will derive it from the facts:

H_{k}(BX_{1}/Γ_{g,r}) = 0, k ≤ g − 1,

and

H_{k}(BX_{2}/Γ_{g,r}) = 0, k ≤ g.

These two statements are much easier, their proofs can be found in the first part of the proof of 4.1 in [Harer2].

Lemma 1.5. (1) For p ≤ g − 1 the sequence

⨁_{s=1}^{[(p+2)/2]} C_{1,p+1}^{s} → ⨁_{s=0}^{[(p+1)/2]} C_{1,p}^{s} → ⨁_{s=0}^{[p/2]} C_{1,p−1}^{s}.

is exact.

(2) For p ≤ g the sequence

⨁_{s=1}^{[(p+1)/2]} C_{2,p+1}^{s} → ⨁_{s=0}^{[p/2]} C_{2,p}^{s} → ⨁_{s=0}^{[(p−1)/2]} C_{2,p−1}^{s}.

is exact.

Proof: Denote the permutation which sends 1 to i_{1}, …, k to i_{k} by [i_{1}, …, i_{k}]. These symbols now describe uniquely the cells of BX_{i}/Γ_{g,r}. Furthermore, the boundary operator is given by deletion and shift:

∂[i_{1}, …, i_{k}] = [τ_{1}(i_{2}), …, τ_{1}(i_{k})] − [τ_{2}(i_{1}), τ_{2}(i_{3}), …, τ_{2}(i_{k})] + ⋯

where

τ_{n}(i) = \{ \begin{matrix}i & if i < n \\ i − 1 & if i > n. \end{matrix}

Filtering by species gives the double complexes:

C_{1,g−1}^{0} ← C_{1,g}^{1} ← C_{1,g+1}^{2} ← ⋯ ← C_{1,2g−2}^{g−1} ← C_{1,2g−1}^{g}
↓ ↓ ↓ ↓
C_{1,g−2}^{0} ← C_{1,g−1}^{1} ← C_{1,g}^{2} ← ⋯ ← C_{1,2g−3}^{g−1}
↓  ↓  ↓
⋮  ⋮  ⋮
↓  ↓  ↓
C_{1,2}^{0} ← C_{1,3}^{1} ← C_{1,4}^{2} ← ⋯
↓ ↓ ↓
C_{1,1}^{0} ← C_{1,2}^{1} ← C_{1,3}^{2}
↓  ↓
C_{1,0}^{0} ← C_{1,1}^{1}

and

C_{2,g}^{0} ← C_{2,g+1}^{1} ← C_{2,g+2}^{2} ← ⋯ ← C_{2,2g−1}^{g−1} ← C_{2,2g}^{g}
↓ ↓ ↓ ↓
C_{2,g−1}^{0} ← C_{2,g}^{1} ← C_{2,g+1}^{2} ← ⋯ ← C_{2,2g−2}^{g−1}
↓  ↓  ↓
⋮  ⋮  ⋮
↓  ↓  ↓
C_{2,2}^{0} ← C_{2,3}^{1} ← C_{2,4}^{2}
↓  ↓
C_{2,1}^{0} ← C_{2,2}^{1}
↓
C_{2,0}^{0}

The sequences of the lemma are part of the complex obtained by chopping off the rows above C_{1,p+s}^{s}. The results may be identified with the complexes in genus p + 1 and p respectively. Theorem 1.4 (with k ≤ g − 1 + i) now implies the lemma.

The action of the mapping class group Γ_{g,r} on BX_{i} gives rise to a spectral sequence (E_{p,q}^{r}, d_{p,q}^{r}) converging to 0 for p + q ≤ dim(BX_{i}) with

E_{p,q}^{1} = ⊕_{σ_{p}}H_{q}(Γ_{σ_{p}})

where the sum is over representatives of the orbits of the p-cells under the action of Γ_{g,r} and Γ_{σ_{p}} denotes the stabilizer of σ_{p}. The augmentation gives E_{−1,q}^{1} = H_{q}(G), and the d^{1}-differential is induced from the boundary map of BX_{i}/Γ_{g,r}. We will use these spectral sequences to study H_{∗}(Γ_{g,r}).

2. Description of the classes λ_{n,r} and κ_{1}^{n}

In this section we address the creation of two special types of classes in H_{∗}Γ_{g,r}. As always, suppose that F_{g,r} is a surface of genus g with r boundary components. Number the boundary components from 1 to r and let τ_{i} denote the Dehn twist on the i^{th} component. Decompose F_{3n+1,r+1} into n + 1 subsurfaces, n homeomorphic to F_{3,1} and one to F_{1,n+r+1}, where τ_{1}, …, τ_{r+1} lie in F_{1,n+r+1}. If κ_{1} is the generator of H_{2}Γ_{3,1}, then we may take a κ_{1} in each copy of F_{3,1} and define

λ_{n,r} = τ_{r+1} ⊗ κ_{1}^{n}

which lives in H_{2n+1}Γ_{3n+1,r+1} modulo the image of H_{2n+1}Γ_{3n+1,r} for every r ≥ 1. The second set of special classes are the classes κ_{1}^{n} which live in a similar way in H_{2n}Γ_{g,r} for every g ≥ 3n.

It will be necessary to describe how the classes λ_{n,r} and κ_{n} occur in the spectral sequences associated to the actions of the mapping class groups on BX_{1} and BX_{2}. Before doing so we make the following:

Conventions:

(1) When β = < β_{1}, …, β_{n} > is an arc-system in BX_{1} or BX_{2}, the result of splitting F_{g,r} open along β will be denoted F_{β} and the component of ∂F_{β} which lies “to the right” of β_{i} as it emerges from p_{1} will be denoted σ_{i}. Also denote the one lying “to the left” of β_{i} by σ_{i−1}. Note that the σ_{i} are not all distinct. For example, in BX_{1}, if P (α) = [2134], then σ_{0} = σ_{1} = σ_{2}, but σ_{0}, σ_{3} and σ_{4} are all distinct.

(2) If C is a simple closed curve in F and C′ is any simple closed curve isotopic to C, we will denote the Dehn twist on C or C′ as well as the homology class they represent in H_{1}(Γ_{g,r}) by the same letter C. Thus, for example, τ_{i} denotes the i^{th} component of ∂F, the Dehn twist on any simple closed curve isotopic to it, and its class in H_{1}(Γ_{g,r}).

The class λ_{n,r} can be seen in the spectral sequences associated to the action of Γ_{3n+1,r+1} on BX_{2} as follows. The E^{1} term of the spectral sequence is

E_{p,q}^{1} = ⊕_{σ_{p}}H_{q}(Γ_{σ_{p}})

where the sum is over the orbits of the p-cells σ_{p} in BX_{2}. Recall that the 1-cells of BX_{2}/Γ_{3n+1,r+1} are in 1 − 1 correspondence with the permutations [12] and [21].

Consider a choice of arc system α = < α_{1}, α_{2} > with P (α) = [12]. As above F_{α} is the result of splitting F_{3n+1,r+1} open along α, it has genus 3n with r+1 boundary components. These components are τ_{3}, …, τ_{r+1} from ∂F_{g,r} and two new ones σ_{0} and σ_{1} indexed as indicated above. Identify F_{α} with the complement of a regular neighborhood of α in F_{3n+1,r+1}. The key property of this α is that there is an element f of Γ_{3n+1,r+1} which is the identity on F_{α} and takes α_{2} to α_{1}. In fact,

f = τ_{1} ∘ x^{−1}

where x is a certain non-separating simple closed curve.

Let C_{∗} be the chain complex of the “Simplex on Γ_{3n+1,r+1}”, that is the simplicial complex whose k-simplices are the symbols of the form [f_{0}, …, f_{k}] with {f_{i}} elements of Γ_{3n+1,r+1}. Choose a realization K_{n} of κ_{1}^{n} in C_{2n} consisting of simplices whose entries lie in the mapping class group of F_{α}. Since f is the identity on F_{α}, an unravelling of the spectral sequence shows that

(< α_{1}, α_{2} > − < α_{1} > ⊗ [1, f]) ⊗ K_{n}

represents f ⊗ κ_{1}^{n}. Finally, x ⊗ κ_{1}^{n} is easily seen to lie in the image of Γ_{3n+1,r}, so f ⊗ κ_{1}^{n} is equivalent to λ_{n,r}.

Next we describe κ_{1}^{n}. Since there is a natural inclusion of Γ_{3n,1} in Γ_{g,r} for all g ≥ 3n and r ≥ 1, it suffices to do the description for g = 3n and r = 1. Consider the spectral sequence associated to the action of Γ_{3n,1} on BX_{1}. Its E^{1} term is

E_{p,q}^{1} = ⊕_{σ_{p}}H_{q}(Γ_{σ_{p}})

where the sum is over the orbits of the p-cells σ_{p} in BX_{1}. Once again consider an arc-system α = < α_{1}, α_{2} > with P (α) = [12], F_{α} has genus 3n − 2 and r + 2 boundary components. Let F_{0} be a subsurface of F_{3n,1} which is genus 3n − 2 with 1 boundary component, disjoint from α and is such that α does not separate the complementary surface F_{3n,1} − F_{0} (which is genus 2 with 2 boundary components).

The class λ_{n−1,2} lives in H_{2n−1} of the stabilizer of α, and we may represent it as

σ_{0} ⊗ K_{n−1}

where K_{n−1} represents κ_{1}^{n−1} in C_{∗} with elements which are the identity outside F_{0}. The class κ_{1}^{n} is now represented by

(< α_{1}, α_{2} > − < α_{1} > ⊗ ω_{1}− < α_{2} > ⊗ ω_{2}) ⊗ K_{n−1}

where ω_{i} is a 2−chain in the stabilizer of α_{i} which has boundary σ_{0}.

We will make use of these descriptions in the next section.

3. Proof of the Main Theorem

In this section we give the proof of the main theorem of the paper. The proof of stability for H_{k}(Γ_{g,r}) is by induction on k beginning with H_{1} and H_{2}. The theorem follows easily for these cases by our earlier work [Harer1] since H_{1}(Γ_{g,r}) = 0 for g ≥ 2, H_{2}(Γ_{2,r}) = 0 and H_{2}(Γ_{g,r}) is rank 1 generated by κ_{1} for g ≥ 3.

Step 1: First we show that

(ϕ_{i})_{∗} : H_{2n+1}(Γ_{3n+1,r}) → H_{2n+1}(Γ_{3n+1,r+1})

has cokernel of rank 1 with generator λ_{n,r} = τ_{r+1} ⊗ κ_{1}^{n}. Look at E_{p,q}^{1}, p+q = 2n+1, in the E^{1} term of the augmented spectral sequence associated to the the action of Γ_{3n+1,r+1} on BX_{2}. The map d_{0,2n+1}^{1} is the map (ϕ_{r})_{∗} (all other cases are the same). Near p = 1, the sequence is:

H_{2n+1}Γ_{3n+1,r+1} H_{2n+1}Γ_{3n+1,r}
H_{2n}Γ_{3n+1,r} (H_{2n}Γ_{3n,r+1})^{2}

⋯

[1]  [12]

[21]

(H_{2n}Γ_{3n−1,r+2})^{3} ⊕ (H_{2n}Γ_{3n,r})^{3}
⋯
[123]  [132]
[231]  [321]
[312]  [213]

where the labels below the horizontal line describe the cell orbits in terms of their permutations P (α).

Consider the map d_{2,2n}^{1}. Let α = < α_{1}, α_{2}, α_{3} > with P (α) = [132]. Then

d^{1}(α ⊗ ω) =< α_{2}, α_{3} > ⊗ ω− < α_{1}, α_{3} > ⊗ ω+ < α_{1}, α_{2} > ⊗ ω

for every ω in H_{2n}(Γ_{α}) ≅ H_{2n}(Γ_{3n,r}). Here

P (< α_{2}, α_{3} >) = [21]

and

P (< α_{1}, α_{3} >) = P (< α_{1}, α_{2} >) = [12].

There exists f in Γ_{3n+1,r+1} such that f(α_{3}) = α_{2}, f(α_{1}) = α_{1}, and f|_{Γ_{α}} = 1. (In fact, f is the Dehn twist on a simple closed curve homotopic to α_{2} ∪ α_{3}.) It follows that the last two terms above cancel so that

d^{1}(α ⊗ ω) =< α_{2}, α_{3} > ⊗ ω.

This map may be identified with ϕ_{i}; since 3n > 3(2n)/2 − 1, inductively (3) tells us it is an isomorphism.

Next look at β = < β_{1}, β_{2}, β_{3} > with P (β) = [231].

d^{1}(β ⊗ ω) = < β_{2}, β_{3} > ⊗ ω − < β_{1}, β_{3} > ⊗ ω + < β_{1}, β_{2} > ⊗ ω

for every ω in H_{2n}(Γ_{β}) ≅ H_{2n}(Γ_{3n−1,r+2}). This time

P (< β_{2}, β_{3} >) = P (< β_{1}, β_{3} >) = [21]

and

P (< β_{1}, β_{2} >) = [12].

Since the [21] part has been accounted for, we look at the [12] part. This map may be identified with

(ψ_{i,j})_{∗} : H_{2n}(Γ_{3n−1,r+2}) → H_{2n}(Γ_{3n,r+1}).

Inductively (by(2)), this has cokernel of rank 1 generated by κ_{1}^{n}. Therefore E_{1,2n}^{2} has rank at most 1 generated by [12] ⊗ κ_{1}^{n}. This creates the the class λ_{n,r} described in the previous section. Since λ_{n,r} will appear in the creation of the class κ_{1}^{n+1} in H_{2n+2}(Γ_{3n+3,∗}) it is non-zero here.

Next consider E_{p,q}^{1}, p + q = 2n + 1, for p fixed, p > 1. Recall that

C_{k}(BX_{2}/Γ_{g,r}) = ⨁_{s=max(k−g,0)}^{[k/2]} C_{2,k}^{s}

where C_{2,k}^{s} is the free abelian group on the configurations in BX_{2}/Γ_{g,r} of species s. Let δ_{s}(n) denote the number of orbits under Γ_{g,r} of configurations of species s which involve n arcs in the arcsystem.

To make the argument work we will need to ignore the configurations of species 0 which occur in E_{p+1,2n+1−p}^{1}. Therefore the portion of the E^{1} term we need to consider may be written as:

H_{2n+1−p}(Γ_{3n+2−p,r−1+p})^{δ_{0}(p)}  H_{2n+1−p}(Γ_{3n+1−p,r+p})^{δ_{0}(p+1)}
⊕  d_{p,2n+1−p}^{1}  ⊕
H_{2n+1−p}(Γ_{3n+3−p,r−3+p})^{δ_{1}(p)}  ⟵  H_{2n+1−p}(Γ_{3n+2−p,r−2+p})^{δ_{1}(p+1)} ←
⊕  ⊕
H_{2n+1−p}(Γ_{3n+4−p,r−5+p})^{δ_{2}(p)}  H_{2n+1−p}(Γ_{3n+3−p,r−4+p})^{δ_{2}(p+1)}
⊕  ⊕
⋮  ⋮

(∗)

d_{p+1,2n+1−p}^{1}
⟵  H_{2n+1−p}(Γ_{3n+1−p,r−1+p})^{δ_{1}(p+2)}
⊕
H_{2n+1−p}(Γ_{3n+2−p,r−3+p})^{δ_{2}(p+2)}
⊕
⋮

The smallest genus of any stabilizer involved is 3n + 1 − p. Split F_{3n+1,r+1} into two surfaces glued along a single boundary component:

F_{3n+1,r+1} = F_{3n+1−p,r} ∪ F_{p,3}

where ∂_{r}F and ∂_{r+1}F are the non-glued components of ∂F_{p,3} and ∂_{1}F, …, ∂_{r−1}F are those of ∂F_{3n+1−p,r}. All the curves which are required to make the configurations in (∗) may be taken on F_{p,3} in non-separating configurations. Now we want to combine the inductive assumptions (3) and (5) to conclude that

H_{2n+1−p}(Γ_{3n+1−p,r}) → H_{2n+1−p}(Γ_{3n+1−p+t,r+u})

is an isomorphism for every t ≥ 0, u ≥ −t. The key cases to check are:

(i) t = 0, u = 1 which is (3) and requires 3n+1−p ≥ a_{2n+1−p} +3(2n+1−p)/2 and

(ii) t = 1, u = −1 which is (5) and requires 3n+2−p ≥ c_{2n+1−p}+3(2n+1−p)/2.

Both of these are easy to verify using the fact that p ≥ 2. This means that the sequence (∗) may be identified with the tensor product of H_{2n+1−p}(Γ_{3n+1−p,r}) with

⨁_{s=1}^{[(p+1)/2]} C_{2,p+1}^{s} → ⨁_{s=0}^{[p/2]} C_{2,p}^{s} → ⨁_{s=0}^{[(p−1)/2]} C_{2,p−1}^{s}.

This sequence is exact by Lemma 1.5, so E_{p,2n+1−p}^{2} = 0 for p ≥ 2. Since the maps (ϕ_{i})_{∗} are always injective (see [Harer3], Lemma 3.1), part 1 follows.

Step 2: Next we show that

(ψ_{i,j})_{∗} : H_{2n}(Γ_{3n−1,r+1}) → H_{2n}(Γ_{3n,r})

has cokernel of rank 1 with generator κ_{1}^{n}. Look at E_{p,q}^{1}, p + q = 2n, in the E^{1} term of the augmented spectral sequence associated to the the action of Γ_{3n,r} on BX_{1}. As before, the map d_{0,2n}^{1} is equivalent to the map (ψ_{i,j})_{∗} which we are studying. Near p = 1, the sequence is:

H_{2n}Γ_{3n,r} H_{2n}Γ_{3n−1,r+1}
H_{2n−1}Γ_{3n−1,r+1} H_{2n−1}Γ_{3n−2,r+2} ⊕ H_{2n−1}Γ_{3n−1,r}
⋯
[12]  [21]
H_{2n−1}Γ_{3n−3,r+3} ⊕ (H_{2n−1}Γ_{3n−2,r+1})^{5}
⋯
[123]  [132]

[213] [231] [312]

Consider the map d_{2,2n−1}^{1}. Let α = < α_{1}, α_{2}, α_{3} > with P (α) = [132]. The same arguments as in step 1 show that

d^{1}(α ⊗ ω) =< α_{2}, α_{3} > ⊗ ω

with P (< α_{2}, α_{3} >) = [21]. This map may be identified with ψ_{i,j}, so inductively (by (4)) it is surjective.

Next, look at β = < β_{1}, β_{2}, β_{3} > with P (β) = [231]. As before, only the [12] part of d^{1}(β) matters to us. This map may be identified with

(ϕ_{i})_{∗} : H_{2n−1}(Γ_{3n−2,r+1}) → H_{2n−1}(Γ_{3n−2,r+2}).

Inductively (by(1)), it has cokernel of rank 1 generated by

λ_{n−1,r+1} = σ_{0} ⊗ κ_{1}^{n−1}.

Therefore E_{1,2n−1}^{2} has rank at most 1 generated by [12] ⊗ λ_{n−1,r+1}. This will then create the class κ_{1}^{n} as in the previous section.

Next consider E_{p,q}^{1}, p + q = 2n, for p fixed, p > 1. Again recall that

C_{k}(BX_{1}/Γ_{g,r}) = ⨁_{s=max(k−g+1,0)}^{[(k+1)/2]} C_{1,k}^{s}

where C_{1,k}^{s} is the free abelian group on the configurations in BX_{1}/Γ_{g,r} of species s. Let ϵ_{s}(n) denote the number of orbits under Γ_{g,r} of configurations of species s which involve n arcs in the arcsystem.

Again we will ignore the configurations of species 0 which occur in E_{p+1,2n−p}^{1}. Therefore the portion of the E^{1} term we need to consider is:

H_{2n−p}(Γ_{3n−p,r+p})^{ϵ_{0}(p)}  H_{2n−p}(Γ_{3n−1−p,r+1+p})^{ϵ_{0}(p+1)}
⊕ d_{p,2n−p}^{1} ⊕
H_{2n−p}(Γ_{3n+1−p,r−2+p})^{ϵ_{1}(p)} ⟵  H_{2n−p}(Γ_{3n−p,r−1+p})^{ϵ_{1}(p+1)} ⟵
⊕  ⊕
H_{2n−p}(Γ_{3n+2−p,r−4+p})^{ϵ_{2}(p)}  H_{2n−p}(Γ_{3n+1−p,r−3+p})^{ϵ_{2}(p+1)}
⊕  ⊕
⋮  ⋮

(∗∗)

d_{p+1,2n−p}^{1}
⟵  H_{2n−p}(Γ_{3n−1−p,r+p})^{ϵ_{1}(p+2)}
⊕
H_{2n−p}(Γ_{3n−p,r−2+p})^{ϵ_{2}(p+2)}
⊕
⋮

The smallest genus of any stabilizer involved is 3n − 1 − p. Split F_{3n,r} into two surfaces, glued along a single boundary component:

F_{3n,r} = F_{3n−1−p,r} ∪ F_{p+1,2}

where ∂_{r}F is the non-glued component of ∂F_{p+1,2} and ∂_{1}F, …, ∂_{r−1}F are those of ∂F_{3n−1−p,r}. All the curves which are required to make the configurations in (∗∗) may be taken on F_{p+1,2} in non-separating configurations. As before we combine the inductive assumptions (3) and (5) to conclude that

H_{2n−p}(Γ_{3n−1−p,r}) → H_{2n−p}(Γ_{3n−1−p+t,r+u})

is an isomorphism for every t ≥ 0, u ≥ −t. The key cases are:

(i) t = 0, u = 1 which is (3) and requires 3n − 1 − p ≥ a_{2n−p} + 3(2n − p)/2 and

(ii) t = 1, u = −1 which is (5) and requires 3n − p ≥ c_{2n−p} + 3(2n − p)/2.

Again these are easy to verify using the fact that p ≥ 2.

This means that the sequence (∗∗) may be identified with the tensor product of H_{2n−p}(Γ_{3n−1−p,r}) with

⨁_{s=1}^{[(p+2)/2]} C_{1,p+1}^{s} → ⨁_{s=0}^{[(p+1)/2]} C_{1,p}^{s} → ⨁_{s=0}^{[p/2]} C_{1,p−1}^{s}.

This sequence is exact by Lemma 1.5. Part 2 follows.

Step 3: Next we show that

(ϕ_{i})_{∗} : H_{k}(Γ_{g,r}) → H_{k}(Γ_{g,r+1})

is an isomorphism for g ≥ 3k/2 + a_{k} where a_{k} = −1 when k is even and a_{k} = 1/2 when k is odd. Injectivity of this map is automatic (see, e.g. [Harer3], Lemma 3.1) so we only need to verify surjectivity.

As in steps 1 and 2, look at E_{p,q}^{1}, p + q = k, in the E^{1} term of the augmented spectral sequence associated to the the action of Γ_{g,r} on BX_{2}. The map d_{0,k}^{1} is equivalent to the map (ϕ_{i})_{∗}. Near p = 1, the sequence is:

H_{k}Γ_{g,r+1} H_{k}Γ_{g,r}
H_{k−1}Γ_{g,r} (H_{k−1}Γ_{g−1,r+1})^{2} (H_{k−1}Γ_{g−2,r+2})^{3} ⊕ (H_{k−1}Γ_{g−1,r})^{3}

⋯

[12]  [123]  [132]
[21]  [231]  [321]
[312]  [213]

Consider the map d_{2,k−1}^{1}. Let α = < α_{1}, α_{2}, α_{3} > with P (α) = [132]. Then as before two boundary terms cancel so that

d^{1}(α ⊗ ω) =< α_{2}, α_{3} > ⊗ ω

with P (< α_{2}, α_{3} >) = [21]. This map may be identified with ϕ_{i}. When k is odd, this is an isomorphism (by(3)), since then a_{k} = 1/2 and a_{k−1} = −1, so that g ≥ 3k/2 + a_{k} implies g − 1 ≥ 3(k − 1)/2 + a_{k−1}. When k is even, say k = 2n, and g = 3n − 1 this map has cokernel λ_{n−1,r} by (1). When g > 3n − 1 it is an isomorphism. In the case g = 3n − 1, if this class weren’t killed by a higher differential, it would create a class in H_{2n}(Γ_{3n−1,r+1}) which would lie in the image of H_{2}(Γ_{2,r+2}) ⊗ H_{2n−2}(Γ_{3n−3,1}) (since κ_{1}^{n−1} lies in H_{2n−2}(Γ_{3n−3,1}) and [12] ⊗ σ_{0} lies in F_{2,r+2} where [12] is nonseparating). But H_{2}(Γ_{2,r+2}) is 0 ([Harer1]) so indeed it must be killed.

Next look at β = < β_{1}, β_{2}, β_{3} > with P (β) = [231]. Again the [12] part of d^{1}(β) is the part that interests us; it may be identified with

(ψ_{i,j})_{∗} : H_{k−1}(Γ_{g−2,r+2}) → H_{k−1}(Γ_{g−1,r+1}).

Inductively (by(4)), this map is surjective whenever g − 1 ≥ 3(k − 1)/2 + b_{k−1}, i.e. when g ≥ 3k/2 − 1/2 + b_{k−1}. Since it is easily verified that b_{k−1} = a_{k} + 1/2 for all k, surjectivity follows from the assumption g ≥ 3k/2 + a_{k}.

Next consider E_{p,q}^{1}, p + q = k, for p fixed, p > 1. As in step 1 the portion of the E^{1} term we need to consider may be written as:

H_{k−p}(Γ_{g+1−p,r+1+p})^{δ_{0}(p)}  H_{k−p}(Γ_{g−p,r+2+p})^{δ_{0}(p+1)}
⊕ d_{p,k−p}^{1} ⊕
H_{k−p}(Γ_{g+2−p,r−1+p})^{δ_{1}(p)} ⟵  H_{k−p}(Γ_{g+1−p,r+p})^{δ_{1}(p+1)} ⟵
⊕  ⊕
H_{k−p}(Γ_{g+3−p,r−3+p})^{δ_{2}(p)}  H_{k−p}(Γ_{g+2−p,r−2+p})^{δ_{2}(p+1)}
⊕  ⊕
⋮  ⋮

(∗ ∗ ∗)

d_{p+1,k−p}^{1}
⟵  H_{k−p}(Γ_{g−p,r+1+p})^{δ_{1}(p+2)}
⊕
H_{k−p}(Γ_{g+1−p,r−1+p})^{δ_{2}(p+2)}
⊕
⋮

The smallest genus of any stabilizer involved is g − p. Split F_{g,r+1} into two surfaces, glued along a single boundary component:

F_{g,r+1} = F_{g−p,r} ∪ F_{p,3}

where ∂_{r}F and ∂_{r+1}F are the non-glued components of ∂F_{p,3} and ∂_{1}F, …, ∂_{r−1}F are those of ∂F_{g−p,r}. All the curves which are required to make the configurations in (∗ ∗ ∗) may be taken on F_{p,3} in non-separating configurations.

We want to combine the inductive assumptions (3) and (5) to conclude that

H_{k−p}(Γ_{g−p,r}) → H_{k−p}(Γ_{g−p+t,r+u})

is an isomorphism for every t ≥ 0, u ≥ −t. Then we may proceed as before to finish the proof of (3). The primary cases to check are

(i) t = 0, u = 1 which is (3) and requires g − p ≥ 3(k − p)/2 + a_{k−p} and

(ii) t = 1, u = −1 which is (5) and requires g − p + 1 ≥ 3(k − p)/2 + c_{k−p}.

For (i) notice that a_{k} ≥ a_{k−p} − p/2 holds as long as p ≥ 2 (but not for p = 1 which is why we did that case separately). For (ii) notice that a_{k} ≥ c_{k−p} − p/2 − 1 holds for p ≥ 2. This finishes step 3.

Step 4: Next we show that

(ψ_{i,j})_{∗} : H_{k}(Γ_{g−1,r+1}) → H_{k}(Γ_{g,r})

is surjective for g ≥ 3k/2 + b_{k}. Look at E_{p,q}^{1}, p + q = k, in the E^{1} term of the augmented spectral sequence associated to the the action of Γ_{g,r} on BX_{1}. As before, d_{0,k}^{1} is equivalent to the map (ψ_{i,j})_{∗}. Near p = 1, the sequence is:

H_{k}Γ_{g,r} H_{k}Γ_{g−1,r+1}
H_{k−1}Γ_{g−1,r+1} H_{k−1}Γ_{g−2,r+2} ⊕ H_{k−1}Γ_{g−1,r}
⋯
[12]  [21]

H_{k−1}Γ_{g−3,r+3} ⊕ (H_{k−1}Γ_{g−2,r+1})^{5}
⋯
[123]  [132]

[213] [231] [312] [321]

Consider d_{2,k−1}^{1}. Let α = < α_{1}, α_{2}, α_{3} > with P (α) = [132]. The same arguments as in step 1 show that

d^{1}(α ⊗ ω) =< α_{2}, α_{3} > ⊗ ω

with P (< α_{2}, α_{3} >) = [21]. This map may be identified with ψ_{i,j}. When k is even, this is inductively surjective by (4) since g ≥ 3k/2+1 implies g−1 ≥ 3(k−1)/2−1/2. When k is odd, say k = 2n + 1, and g > 3n + 1, g − 1 ≥ 3(2n)/2 + 1 so again it’s surjective. Finally, when k = 2n + 1 and g = 3n + 1, the cokernel of this map is generated by κ_{1}^{n}. We claim that this class is killed off in the spectral sequence.

To see this, notice that it is of the form x ⊗ κ_{1}^{n} where x is the Dehn twist on a non-separating simple closed curve. But the class x ⊗ κ_{1} is 0 in H_{3}(Γ_{4,r}) by results of [Harer2].

Next, look at β = < β_{1}, β_{2}, β_{3} > with P (β) = [231]. The [12] part of d^{1}(β) may be identified with

(ϕ_{i})_{∗} : H_{k−1}(Γ_{g−2,r+1}) → H_{k−1}(Γ_{g−2,r+2}).

By (3) this is an isomorphism as long as g − 2 ≥ 3(k − 1)/2 + a_{k−1}. This follows from the fact that b_{k} = a_{k−1} + 1/2 for every k.

Next consider E_{p,q}^{1}, p + q = k, for p fixed, p > 1. Once again we will ignore the configurations of species 0 which occur in E_{p+1,k−p}^{1}, so the portion of the E^{1} term under consideration is:

H_{k−p}(Γ_{g−p,r+p})^{ϵ_{0}(p)}  H_{k−p}(Γ_{g−1−p,r+1+p})^{ϵ_{0}(p+1)}
⊕  d_{p,k−p+1}^{1}  ⊕
H_{k−p}(Γ_{g+1−p,r−2+p})^{ϵ_{1}(p)}  ⟵  H_{k−p}(Γ_{g−p,r−1+p})^{ϵ_{1}(p+1)} ⟵
⊕  ⊕
H_{k−p}(Γ_{g+2−p,r−4+p})^{ϵ_{2}(p)}  H_{k−p}(Γ_{g+1−p,r−3+p})^{ϵ_{2}(p+1)}
⊕  ⊕
⋮  ⋮

(∗ ∗ ∗∗)

d_{p+1,2n−p}^{1}
⟵  H_{k−p}(Γ_{g−1−p,r+p})^{ϵ_{1}(p+2)}
⊕
H_{k−p}(Γ_{g−p,r−2+p})^{ϵ_{2}(p+2)}
⊕
⋮

The smallest genus of any stabilizer involved is g − 1 − p. Split F_{g,r} into two surfaces, glued along a single boundary component:

F_{g,r} = F_{g−1−p,r} ∪ F_{p+1,2}

where ∂_{r}F is the non-glued component of ∂F_{p+1,2} and ∂_{1}F, …, ∂_{r−1}F are those of ∂F_{g−1−p,r}. All the curves which are required to make the configurations in (∗ ∗ ∗∗) may be taken on F_{p+1,2} in non-separating configurations. As before we need to combine the inductive assumptions (3) and (5) to conclude that

H_{k−p}(Γ_{g−1−p,r}) → H_{k−p}(Γ_{g−1−p+t,r+u})

is an isomorphism for every t ≥ 0, u ≥ −t. The key cases are

(i) t = 0, u = 1 which is (3) and requires g − p − 1 ≥ 3(k − p)/2 + a_{k−p} and

(ii) t = 1, u = −1 which is (5) and requires g − p ≥ 3(k − p)/2 + c_{k−p}.

For (i) notice that b_{k} ≥ a_{k−p} − p/2 + 1 holds as long as k is even, or k = 2n + 1, p ≥ 2 and g ≥ 3n + 2. It also holds when p ≥ 3, g = 3n + 1, k = 2n + 1, but the case k = 2n + 1, g = 3n + 1, p = 2, must be treated separately. This is done below. For (ii) b_{k} ≥ c_{k−p} − p/2 holds for all p ≥ 2 when k is even. When k is odd, say k = 2n + 1, it holds as long as g ≥ 3n + 2 or g = 3n + 1 and p ≥ 3. Again the special case k = 2n + 1, g = 3n + 1, p = 2 must be treated separately.

We are reduced therefore to the case where k = 2n + 1, g = 3n + 1 and p = 2. The portion of the spectral sequence we are looking at is:

H_{2n−1}Γ_{3n−1,r+2} ⊕ H_{2n−1}Γ_{3n,r} H_{2n−1}Γ_{3n−2,r+3} ⊕ (H_{2n−1}Γ_{3n−1,r+1})^{5}
⋯  ⋯
[12] [21] [123] [132]

[213] [231] [312] [321]

H_{2n−1}Γ_{3n−3,r+4} ⊕ (H_{2n−1}Γ_{3n−2,r+2})^{15} ⊕ (H_{2n−1}Γ_{3n−1,r})^{8}
⋯
[1234]  [1243]  [2143]
⋯  ⋯

As before

[132] ↦ [21]

and

[231] ↦ [12]

are isomorphisms on H_{2n−1} by (5) (since 3n ≥ 3(2n − 1)/2 + 3/2) and (3) (since 3n − 1 ≥ 3(2n − 1)/2 + 1/2). Next,

[2341] ⊗ H_{2n−1} → [123] ⊗ H_{2n−1}

has cokernel equal to

[123] ⊗ σ_{0} ⊗ κ^{n−1}

by (1). This extra class is killed by

[1342] ⊗ σ_{0} ⊗ κ^{n−1}.

Finally,

[1324] → [213] − [123] + [123] − [132]
[1432] → [321] − [132] + [132] − [132]
[1423] → [312] − [123] + [132] − [132]

and all are surjective on H_{2n−1} of the first summand since 3n−1 ≥ 3(2n−1)/2 − 1/2.

This finishes part 4.

Step 5: Finally we come to the proof that

(ψ_{i,j})_{∗} : H_{k}(Γ_{g−1,r+1}) → H_{k}(Γ_{g,r})

is an isomorphism for g ≥ 3k/2 + c_{k}. To prove this we will show first that d_{1,k}^{1} = 0. Then we will verify that E_{p,q}^{2} = 0 for p ≥ 2, p + q = k + 1. Since c_{k} ≥ b_{k} this will suffice as we already know from step 4 that E_{p,q}^{2} = 0 for p ≥ 1, p + q = k.

Look at the expanded part of the spectral sequence consisting of the terms E_{p,q}^{1} with p + q = k − 1, k, k + 1 and k + 2. As before, d_{0,k}^{1} is equivalent to the map (ψ_{i,j})_{∗}. For q = k we have:

H_{k}Γ_{g,r} H_{k}Γ_{g−1,r+1} H_{k}Γ_{g−2,r+2} ⊕ H_{k}Γ_{g−1,r}
⋯  ⋯
[1]  [12]  [21]

H_{k}Γ_{g−3,r+3} ⊕ (H_{k}Γ_{g−2,r+1})^{5}
⋯
[123]  [132]

[213] [231] [312] [321]

The map d_{1,k}^{1} restricts to 0 on the [21] summand by the usual argument. For the other term, consider d_{2,k}^{1} applied to H_{k} of the stabilizer of [312]. The part which goes to [12] may be identified with

(ϕ_{i})_{∗} : H_{k}(Γ_{g−2,r+1}) → H_{k}(Γ_{g−2,r+2}).

This is an isomorphism as long as g − 2 ≥ 3k/2 + a_{k}. When k is even, c_{k} = a_{k} + 2, so this follows for all g ≥ 3k/2 + c_{k}. When k = 2n + 1 this holds for g ≥ 3n + 4, but for g = 3n + 3 this is the map

(ϕ_{i})_{∗} : H_{2n+1}(Γ_{3n+1,r+1}) → H_{2n+1}(Γ_{3n+1,r+2})

of part (1), which has cokernel generated by λ_{n,r}, occuring as σ_{0} ⊗ κ_{1}^{n}. This creates κ_{1}^{n+1} in H_{2n+2}(Γ_{3n+3,r}), so it must map to 0 under d^{1}. Thus we have that d_{1,k}^{1} = 0.

Next look at the case where q = k − 1. The sequence is:

H_{k−1}Γ_{g−1,r+1} H_{k−1}Γ_{g−2,r+2} ⊕ H_{k−1}Γ_{g−1,r}

⋯

[1]  [12]  [21]
H_{k−1}Γ_{g−3,r+3} ⊕ (H_{k−1}Γ_{g−2,r+1})^{5}
⋯
[123]  [132]

⋯

H_{k−1}Γ_{g−4,r+4} ⊕ (H_{k−1}Γ_{g−3,r+2})^{15} ⊕ (H_{k−1}Γ_{g−2,r})^{8}
⋯  ⋯
[1234]  [1243]  [2143]
⋯  ⋯

The map

[132] ⊗ H_{k−1}(Γ_{g−2,r+1}) → [21] ⊗ H_{k−1}(Γ_{g−1,r})

is an isomorphism for g − 1 ≥ 3(k − 1)/2 + c_{k−1}. This follows from g ≥ 3k/2 + c_{k} for all k. Similarly the map

[312] ⊗ H_{k−1}(Γ_{g−2,r+1}) → [12] ⊗ H_{k−1}(Γ_{g−2,r+2})

is an isomorphism for g − 2 ≥ 3(k − 1)/2 + a_{k−1} which follows similarly for all k. Therefore d_{2,k−1}^{1} is surjective and has kernel equivalent to

[123] ⊗ H_{k−1}(Γ_{g−3,r+3}) ⊕ {[213], [231], [321]} ⊗ H_{k−1}(Γ_{g−2,r+1}).

Next look at the cells corresponding to the permutations [2413], [2431], [3241] and [4123]. Since g ≥ 3k/2 + c_{k}, the map

H_{k−1}(Γ_{g−2,r}) → H_{k−1}(Γ_{g−2,r+1})

is an isomorphism by (3). Also,

[2413] → [312] − [213] + [132] − [231]
[2431] → [321] − [231] + [231] − [132]

(with the middle two terms cancelling on H_{k−1} by the usual argument) and

[3241] → [231] − [231] + [321] − [213]

(with the first two terms cancelling on H_{k−1} by the usual argument). Together these imply that H_{k−1} is killed for the stabilizers of [213], [231] and [321]. Finally,

[4123] → [123] − [312] + [312] − [312].

The first term of this map is

H_{k−1}(Γ_{g−3,r+2}) → H_{k−1}(Γ_{g−3,r+3})

which is an isomorphism in all cases except when k = 2n, g = 3n + 1. The latter situation is case (1) of the main theorem, so it has cokernel generated by

[123] ⊗ σ_{0} ⊗ κ_{1}^{n−1}

which is killed by

[1423] ⊗ σ_{0} ⊗ κ_{1}^{n−1}.

This implies that E_{2,k−1}^{2} = 0.

To complete the proof suppose q < k − 1. The part of the E^{1} term that we look at is equivalent to the sequence:

H_{k−p}(Γ_{g−p,r+p})^{ϵ_{0}(p)}  H_{k−p}(Γ_{g−1−p,r+1+p})^{ϵ_{0}(p+1)}
⊕  d_{p,k−p+1}^{1}  ⊕  d_{p+1,2n−p}^{1}
H_{k−p}(Γ_{g+1−p,r−2+p})^{ϵ_{1}(p)}  ⟵  H_{k−p}(Γ_{g−p,r−1+p})^{ϵ_{1}(p+1)}  ⟵
⊕  ⊕
H_{k−p}(Γ_{g+2−p,r−4+p})^{ϵ_{2}(p)}  H_{k−p}(Γ_{g+1−p,r−3+p})^{ϵ_{2}(p+1)}
⊕  ⊕
⋮  ⋮

H_{k−p}(Γ_{g−2−p,r+2+p})^{ϵ_{0}(p+2)}
⊕  d_{p+2,2n−p}^{1}
⟵ H_{k−p}(Γ_{g−1−p,r+p})^{ϵ_{1}(p+2)} ⟵ H_{k−p}(Γ_{g−2−p,r+1+p})^{ϵ_{1}(p+3)}
⊕  ⊕
H_{k−p}(Γ_{g−p,r−2+p})^{ϵ_{2}(p+2)}  H_{k−p}(Γ_{g−1−p,r−1+p})^{ϵ_{2}(p+3)}
⊕  ⊕
⋮  ⋮

As usual we reduce to showing that

H_{k−p}(Γ_{g−2−p,r}) → H_{k−p}(Γ_{g−2−p+t,r+u})

is an isomorphism for t ≥ 0 and u ≥ −t. The key cases are:

(i) t = 0, u = 1 which is (3) and requires g − p − 2 ≥ 3(k − p)/2 + a_{k−p} and (ii) t = 1, u = −1 which is (5) and requires g − p − 1 ≥ 3(k − p)/2 + c_{k−p}.

Then, for p ≥ 2, (i) holds because c_{k} ≥ a_{k−p} + 2 − p/2 and (ii) holds because c_{k} ≥ c_{k−p} + 1 − p/2.

This completes the proof of the main theorem.

References

[Harer1] Harer, J. , The third homology group of the moduli space of curves, Duke Math. Jl. 63 #1 (1991), 25–55.

[Harer2] Harer, J., The fourth homology group of the moduli space of curves, (to appear).

[Harer3] Harer, J., Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. 121 (1985), 215–249.

[Harer4] Harer J., Stability of the homology of the moduli spaces of Riemann surfaces with spin structure, Math. Ann. 287 (1990), 323–334.

[Ivanov] Ivanov, N., To appear in Contemporary Math.

[Miller] Miller, E., The homology of the mapping class group, J. Differential Geometry, 24 (1986), 1–14.

[Morita] Morita, S., Characteristic Classes of Surface Bundles , Inventiones Math., 90 (1987), 551–577.

Department of Mathematics, Duke University, Durham, NC 27708–0320 E-mail address: harer@math.duke.edu