COHOMOLOGY OF SYMPLECTIC GROUPS AND MEYER’S SIGNATURE THEOREM

DAVE BENSON, CATERINA CAMPAGNOLO, ANDREW RANICKI, AND CARMEN ROVI

Abstract. Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of 4, and can be computed using an element of H^{2}(Sp(2g, ℤ), ℤ). Denoting by 1 → ℤ → { \atop Sp(2g, ℤ)} → Sp(2g, ℤ) → 1 the pullback of the universal cover of Sp(2g, ℝ), Deligne proved that every finite index subgroup of { \atop Sp(2g, ℤ)} contains 2ℤ. As a consequence, a class in the second cohomology of any finite quotient of Sp(2g, ℤ) can at most enable us to compute the signature of a surface bundle modulo 8. We show that this is in fact possible and investigate the smallest quotient of Sp(2g, ℤ) that contains this information. This quotient ℌ is a non-split extension of Sp(2g, 2) by an elementary abelian group of order 2^{2g+1}. There is a central extension 1 → ℤ/2 → ℌ̃ → ℌ → 1, and ℌ̃ appears as a quotient of the metaplectic double cover Mp(2g, ℤ) = { \atop Sp(2g, ℤ)}/2ℤ. It is an extension of Sp(2g, 2) by an almost extraspecial group of order 2^{2g+2}, and has a faithful irreducible complex representation of dimension 2^{g}. Provided g ⩾ 4, ℌ̃ is the universal central extension of ℌ. Putting all this together, we provide a recipe for computing the signature modulo 8, and indicate some consequences.

1. Introduction

Let Σ_{g} → M → Σ_{h} be an oriented surface bundle over a surface. This is determined by a homotopy class of maps Σ_{h} → BAut^{+}(Σ_{g}). If g ⩾ 2 then the connected components of Aut^{+}(Σ_{g}) are contractible (Hamstrom [17]), and π_{0} Aut^{+}(Σ_{g}) = Γ_{g} is the (orientation preserving) mapping class group of Σ_{g}. So BAut^{+}(Σ_{g}) ≃ BΓ_{g}, and the bundle is classified by a homotopy class of maps Σ_{h} → BΓ_{g}, or equivalently by the monodromy homomorphism

π_{1}(Σ_{h}) = ⟨a_{1}, b_{1}, …, a_{h}, b_{h} | [a_{1}, b_{1}] … [a_{h}, b_{h}] = 1⟩ → Γ_{g}.

Now Γ_{g} acts on H^{1}(Σ_{g}, ℤ) ≅ ℤ^{2g} preserving the symplectic form given by cup product into H^{2}(Σ_{g}, ℤ) ≅ ℤ. So we have a map Γ_{g} → Sp(2g, ℤ), which is surjective. Composing, we obtain a map

χ: π_{1}(Σ_{h}) → Γ_{g} → Sp(2g, ℤ),

and an induced map in cohomology

χ^{∗}: H^{2}(Sp(2g, ℤ), ℤ) → H^{2}(π_{1}(Σ_{h}), ℤ).

Meyer [22] constructed a 2-cocycle τ on Sp(2g, ℤ) and proved that

signature(M) = ⟨χ^{∗}[τ], [Σ_{h}]⟩,

2010 Mathematics Subject Classification. 20J06 (primary); 55R10, 20C33 (secondary). Monday 9^{th} October, 2017

and that this is divisible by 4. In fact, for g ⩾ 3, we have

H^{2}(Sp(2g, ℤ), ℤ) ≅ ℤ

and [τ] corresponds to 4 under a suitably chosen isomorphism. For g ⩾ 3, Sp(2g, ℤ) is perfect, so has a universal central extension.

Denote by { \atop Sp(2g, ℤ)} the central extension obtained by pulling back the universal cover of Sp(2g, ℝ):

[figure: 1; ℤ; { \atop Sp(2g, ℤ)}; Sp(2g, ℤ); 1; 1; ℤ; { \atop Sp(2g, ℝ)}; Sp(2g, ℝ); 1]

Then for g ⩾ 4 the group { \atop Sp(2g, ℤ)} is the universal central extension of Sp(2g, ℤ), while for g = 3 there is an extra copy of ℤ/2 coming from the fact that Sp(6, 2) has an exceptional double cover (see Lemma 6.11). Note also that the centre of Sp(2g, ℤ) has order two. The centre of { \atop Sp(2g, ℤ)} is twice as big as the subgroup ℤ displayed above; it is isomorphic to ℤ × ℤ/2 if g is even, and ℤ if g is odd.

Theorem 1.1 (Deligne [5]). The group { \atop Sp(2g, ℤ)} is not residually finite. Every subgroup of finite index contains the subgroup 2ℤ.

To rephrase Deligne’s theorem, every finite quotient of { \atop Sp(2g, ℤ)} is in fact a finite quotient of the metaplectic double cover Mp(2g, ℤ) of Sp(2g, ℤ) defined by

[figure: 1; 1; 2ℤ; 2ℤ; 1; ℤ; { \atop Sp(2g, ℤ)}; Sp(2g, ℤ); 1; 1; ℤ/2; Mp(2g, ℤ); Sp(2g, ℤ); 1; 1; 1]

A consequence of the theorem of Deligne is that if we compose χ with the map to a finite quotient of Sp(2g, ℤ), we lose information about the signature; the best we can hope to do is compute the signature modulo eight.

Our purpose in this paper is to produce a normal subgroup 𝔎 ⊴ Sp(2g, ℤ) with finite quotient ℌ = Sp(2g, ℤ)/𝔎 of shape 2^{2g+1} ^{·}Sp(2g, 2) (see §5.2 of the Introduction to the Atlas [4] for notation describing group extensions), and a double cover ℌ̃ of ℌ which inflates to the metaplectic double cover Mp(2g, ℤ) of Sp(2g, ℤ). The group ℌ̃ has a 2^{g} dimensional faithful irreducible representation over ℚ[i] which we shall investigate in a subsequent paper [1].

2. The subgroup 𝔎 ⩽ Sp(2g, ℤ) and the main theorem

Denote by J the 2g × 2g matrix

(\begin{matrix}0 & I \\ −I & 0 \end{matrix}) .

Regarding J as a symplectic form, we have

Sp(2g, ℤ) = \{(\begin{matrix}A & B \\ C & D \end{matrix}) = X | X^{t}JX = J\} .

Since J^{−1} = −J, a matrix is symplectic if and only if its transpose is symplectic. Writing out the above condition explicitly, a matrix is symplectic if and only if

(i) AB^{t} and CD^{t} are symmetric, and AD^{t} − BC^{t} = I, or equivalently

(ii) A^{t}C and B^{t}D are symmetric, and A^{t}D − C^{t}B = I. We write Sp(2g, 2) for the matrices satisfying the same conditions over 𝔽_{2}, and note that reduction modulo two Sp(2g, ℤ) → Sp(2g, 2) is surjective (Newman and Smart [24]).

We write Γ(2g, N) ⩽ Sp(2g, ℤ) for the principal congruence subgroup consisting of symplectic matrices which are congruent to the identity modulo N. We write Γ(2g, N, 2N) for the Igusa subgroup [20] of Γ(2g, N) consisting of the matrices ( \begin{matrix}A & B \\ C & D \end{matrix}) where the entries of Diag(AB^{t}) and Diag(CD^{t}) are divisible by 2N, or equivalently where the entries of Diag(A^{t}C) and Diag(B^{t}D) are divisible by 2N. If N = 1, this is the theta subgroup, also known as the symplectic quadratic group, and denoted Sp^{q}(2g, ℤ). It is the inverse image in Sp(2g, ℤ) of the orthogonal subgroup O^{+}(2g, 2) ⩽ Sp(2g, 2).

Definition 2.1. We write 𝔎 for the subgroup of Sp(2g, ℤ) consisting of matrices

(\begin{matrix}I + 2a & 2b \\ 2c & I + 2d \end{matrix}) ∈ Sp(2g, ℤ)

satisfying:

(i) The vectors of diagonal entries Diag(b) and Diag(c) are even, and

(ii) the trace Tr(a) is even.

Thus we have Γ(2g, 4) ⩽ 𝔎 ⩽ Γ(2g, 2) and |Γ(2g, 2) : 𝔎| = 2^{2g+1}. The interpretation of the subgroup 𝔎 is that it is the inverse image in Sp(2g, ℤ) of the largest subspace of Γ(2g, 2)/Γ(2g, 4) on which the quadratic form in part (iv) of the theorem below is identically zero.

Our main theorem is as follows. We assume that g ⩾ 4 for the purpose of simplifying the statements. In an appendix we include statements for all values of g. The main difference for low values of g is that the cohomology of Sp(2g, 2) in degrees one and two contributes some further annoying complications.

Theorem 2.2. Let g ⩾ 4.

(i) 𝔎 is a normal subgroup of Sp(2g, ℤ). We write ℌ for the quotient Sp(2g, ℤ)/𝔎.

(ii) The quotient Γ(2g, 2)/𝔎 ⩽ ℌ is an elementary abelian 2-group (ℤ/2)^{2g+1}.

(iii) The extension

1 → (ℤ/2)^{2g+1} → ℌ → Sp(2g, 2) → 1

does not split. (iv) The group (ℤ/2)^{2g+1} supports an invariant quadratic form 𝔮 given by

𝔮 (\begin{matrix}I + 2a & 2b \\ 2c & I + 2d \end{matrix}) = Tr(a) + ⟨Diag(b), Diag(c)⟩

(see Remark 5.3 for definition of the pointy brackets here).

(v) The action of Sp(2g, 2) on (ℤ/2)^{2g+1} described by the extension in (iii) gives the exceptional isomorphism Sp(2g, 2) ≅ O(2g + 1, 2), the orthogonal group of the quadratic form 𝔮.

(vi) We have H^{2}(ℌ, ℤ/2) ≅ ℤ/2, and an associated central extension

1 → ℤ/2 → ℌ̃ → ℌ → 1.

(vii) For n ⩾ 2, the inflation map H^{2}(ℌ, ℤ/2) → H^{2}(Sp(2g, ℤ/2^{n}), ℤ/2) is an isomor-

phism. (viii) The non-zero element of H^{2}(Sp(2g, ℤ)/𝔎, ℤ/2) inflates to the reduction modulo two

of \frac{1}{4}[τ] as an element of H^{2}(Sp(2g, ℤ), ℤ/2). (ix) Restricting the central extension of ℌ to the subgroup Γ(2g, 2)/𝔎 gives an almost

extraspecial group 2^{1+(2g+1)} ⩽ ℌ̃.

The proof of this theorem occupies the rest of the paper.

3. Extraspecial and almost extraspecial groups

For background on extraspecial and almost extraspecial groups, we refer to §I.5.5 of Gorenstein [13] and §III.13 of Huppert [19], as well as the papers of Bouc and Mazza [2], Carlson and Thévenaz [3], Glasby [10], Griess [14], Hall and Higman [15], Lam and Smith [21], Quillen [26], Schmid [31], Stancu [32], and the letter from Isaacs to Diaconis reproduced in the appendix of Diaconis [7].

The cohomology ring H^{∗}((ℤ/2)^{n}, ℤ/2) is a polynomial ring in generators z_{1}, …, z_{n} of degree one. Thus

H^{1}((ℤ/2)^{n}, ℤ/2) = Hom((ℤ/2)^{n}, ℤ/2)

is an n dimensional vector space spanned by the linear forms z_{1}, …, z_{n}. An element of degree two is therefore a quadratic form 𝔮 on (ℤ/2)^{n}. Letting 𝔟 be the associated bilinear form (ℤ/2)^{n} × (ℤ/2)^{n} → ℤ/2, we have

𝔮(x + y) = 𝔮(x) + 𝔮(y) + 𝔟(x, y).

In the corresponding central extension

1 → ℤ/2 → E → (ℤ/2)^{n} → 1

the role played by 𝔮 and 𝔟 is as follows. If x and y are elements of (ℤ/2)^{n}, choose preimages x̂ and ŷ in E. Then as elements of the central ℤ/2, we have x̂^{2} = 𝔮(x) and [x̂, ŷ] = 𝔟(x, y).

Definition 3.1. We say that a quadratic form 𝔮 is non-singular if the radical 𝔟^{⊥} of the associated bilinear form 𝔟 is {0}, and non-degenerate if 𝔟^{⊥} ∩ 𝔮^{−1}(0) = {0}.

If 𝔮 is non-singular then n = 2g is even; in this case there are two isomorphism classes of quadratic forms, distinguished by the Arf invariant. The corresponding groups E defined by the central extension

1 → ℤ/2 → E → (ℤ/2)^{2g} → 1

are called extraspecial 2-groups, and are characterised by the properties

Φ(E) = [E, E] = Z(E) ≅ ℤ/2.

The two isomorphism classes of extraspecial groups are denoted 2_{+}^{1+2g} (Arf invariant zero) and 2_{−}^{1+2g} (Arf invariant one).

If 𝔮 is singular but non-degenerate then n = 2g + 1 is odd; in this case there is one isomorphism class of quadratic forms. The corresponding groups E defined by the central extension

1 → ℤ/2 → E → (ℤ/2)^{2g+1} → 1

are called almost extraspecial groups. The central product of ℤ/4 with an extraspecial group of either isomorphism type of order 2^{1+2g} gives the almost extraspecial group of order 2^{1+(2g+1)}.

If G is a group, we write Aut(G) for the group of automorphisms of G, Out(G) for the group of outer automorphisms, and Inn(G) for the group of inner automorphisms. These fit into short exact sequences

1 → Z(G) → G → Inn(G) → 1
1 → Inn(G) → Aut(G) → Out(G) → 1.

Writing the automorphism groups of the extraspecial and almost extraspecial groups as extensions of the outer by the inner automorphisms in this way, we have sequences

1 → (ℤ/2)^{2g} → Aut(2_{+}^{1+2g}) → O^{+}(2g, 2) → 1
1 → (ℤ/2)^{2g} → Aut(2_{−}^{1+2g}) → O^{−}(2g, 2) → 1

1 → (ℤ/2)^{2g} → Aut(2^{1+(2g+1)}) → Sp(2g, 2) × ℤ/2 → 1. (3.2)

which do not split provided g ⩾ 4. It is the last case that is of interest to us: in this case the extra factor of ℤ/2 in the outer automorphism group Out(E) acts by inverting the central element of order four, and for g ⩾ 3 the derived subgroup Out(E)′ is Sp(2g, 2).

It was proved by Griess [14] using representation theory, that in each case, there is an extension of the extraspecial group by its outer automorphism group, and of the almost extraspecial group by the subgroup of index two in its outer automorphism group.

We are interested in the almost extraspecial case. In this case, what Griess proved (part (b) of Theorem 5 of [14]) is that there is a group which he denotes H_{0} of shape 2^{1+(2g+1)}Sp(2g, 2), with the following properties. The normal 2-subgroup O_{2}(H_{0}) is the almost extraspecial group 2^{1+(2g+1)}, and the quotient H_{0}/Z(H_{0}) is isomorphic to the subgroup of index two in Aut(2^{1+(2g+1)}).

Dempwolff [6] proved that for g ⩾ 2 there is a unique isomorphism class of non-split extensions of Sp(2g, 2) by an elementary abelian group (ℤ/2)^{2g} with non-trivial action. We shall combine the results of Griess and Dempwolff to show that the group ℌ of Theorem 2.2 is isomorphic to the quotient of Griess’ group H_{0} by the central subgroup of order two. This in turn allows us to compute H^{2}(ℌ, ℤ/2) and relate it to H^{2}(Sp(2g, ℤ), ℤ).

There is another approach to this, which we describe in a separate paper [1]. This avoids the use of the theorems of Griess and Dempwolff, replacing them with a computation showing that the group ℌ̃ has a Curtis–Tits–Steinberg type presentation. This approach is closely related to the action of ℌ̃ on a certain 2^{g} dimensional space of theta functions, and shows that the following defines a projective representation σ: Sp(2g, ℤ) → U(2^{g}, ℚ[i])/{±I} with kernel 𝔎, and then induces a 2^{g} dimensional representation ℌ̃ → U(2^{g}, ℚ[i]).

The underlying vector space for the representation has as a basis the vectors e_{w} for w ∈ {0, 1}^{n}. In the following matrices, we regard det A, which is really an element of (ℤ/4)^{×} = \{1, −1\}, as being either +1 or −1 in ℂ, and \sqrt{det A} is either 1 or i.

σ (\begin{matrix}I & B \\ 0 & I \end{matrix}): e_{w} ↦ i^{w^{t}Bw}e_{w}
σ (\begin{matrix}A & 0 \\ 0 & (A^{t})^{−1} \end{matrix}): e_{w} ↦ \sqrt{det A} e_{(A^{t})^{−1}w}
σ (\begin{matrix}0 & I \\ −I & 0 \end{matrix}): e_{w} ↦ \frac{1}{(1 − i)^{g}} ∑_{w′}(−1)^{w^{t}w′}e_{w′}.

Note that Sp(2g, ℤ) is generated by these elements, but it is not at all obvious that the relations in Sp(2g, ℤ) hold up to sign for the linear transformations listed here; this is proved in [1]. Note also that in the first formula above, the matrix B may be interpreted as having diagonal entries in ℤ/4 and off-diagonal entries in ℤ/2, so that it represents a quadratic form on (ℤ/2)^{g}, taking values in ℤ/4.

Further references for the representation described here include Funar and Pitsch [9], Glasby [10], Gocho [11, 12], Nebe, Rains and Sloane [23], Runge [27, 28, 29], and Tsushima [35].

4. Signature modulo eight

Given an oriented surface bundle over a surface Σ_{g} → M → Σ_{h}, recall that we had an associated map χ: π_{1}(Σ_{h}) → Sp(2g, ℤ). Composing with σ: Sp(2g, ℤ) → U(2^{g}, ℚ[i])/{±I}, we obtain a map

ϕ: π_{1}(Σ_{h}) = ⟨a_{1}, b_{1}, …, a_{h}, b_{h} | [a_{1}, b_{1}] … [a_{h}, b_{h}] = 1⟩ → U(2^{g}, ℚ[i])/{±I}.

Now the commutators [ϕ(a_{i}), ϕ(b_{i})] are well defined in U(2^{g}, ℚ[i]), since changing the sign on ϕ(a_{i}) or ϕ(b_{i}) changes the sign twice in the commutator. Since the product of the commutators is in the kernel of ϕ, we have

[ϕ(a_{1}), ϕ(b_{1})] … [ϕ(a_{h}), ϕ(b_{h})] = ±I ∈ U(2^{g}, ℚ[i]).

Theorem 4.1. We have

[ϕ(a_{1}), ϕ(b_{1})] … [ϕ(a_{h}), ϕ(b_{h})] = \begin{cases}I & iff signature(M) ≡ 0 (mod 8) \\ −I & iff signature(M) ≡ 4 (mod 8). \end{cases}

Remarks 4.2. (1) As a method of computation, this theorem is not very useful, because of the large size of the matrices involved. Endo [8] provided a much more efficient and purely algebraic method for computing the signature, and not just modulo eight. On the other hand, there are consequences of the theorem that are not very apparent from the point of view of Endo’s method.

(2) The following is a consequence of the theta function point of view, and will be discussed in a separate paper [1]. Let Sp^{q}(2g, ℤ) be the theta subgroup of Sp(2g, ℤ). If the image of χ lies in Sp^{q}(2g, ℤ) then we have signature(M) ≡ 0 (mod 8). In particular, this holds if the action of π_{1}(Σ_{h}) on H^{1}(Σ_{g}, ℤ/2) is trivial. This proves a special case of the Klaus–Teichner conjecture; see the introduction to [16] for details.

(3) Consider next the subgroup consisting of the matrices ( \begin{matrix}A & B \\ C & D \end{matrix}) ∈ Sp(2g, ℤ) such that the entries of C are even, and of Diag(C) are divisible by four. If the image of χ lies in this subgroup then again we have signature(M) ≡ 0 (mod 8). This will be proved in [1].

5. Symplectic groups and their Lie algebras

Let R be a commutative ring, and Sp(2g, R) be the symplectic group of dimension 2g over R. Explicitly, this consists of matrices X with entries in R, and satisfying X^{t}JX = J, where J is the symplectic form

J = (\begin{matrix}0 & I \\ −I & 0 \end{matrix})

and I is a g × g identity matrix. Denoting by V_{R} a free R-module of rank g, and setting

W_{R} = V_{R}^{∗} = Hom_{R}(V_{R}, R),

the matrices X act on U_{R} = V_{R} ⊕ W_{R}, preserving the skew-symmetric bilinear form

⟨ , ⟩: U_{R} × U_{R} → R

given by

⟨(v, w), (v′, w′)⟩ = w′(v) − w(v′).

For the action of matrices in Sp(2g, R), we regard (v, w) as a column vector of length 2g with entries in R. The skew-symmetric bilinear form induces an isomorphism from U_{R} to U_{R}^{∗} sending u to ⟨u, ⟩. If R = 𝔽_{2}, we shall write U, V and W instead of U_{𝔽_{2}}, V_{𝔽_{2}} and W_{𝔽_{2}}.

The Lie algebra 𝔰𝔭(2g, R) consists of matrices Y with entries in R, and satisfying

JY + Y^{t}J = 0.

Thus

Y = (\begin{matrix}a & b \\ c & −a^{t} \end{matrix})

where b and c are symmetric. To say that b is symmetric is to say that as an element of

Hom_{R}(W_{R}, V_{R}) ≅ V_{R} ⊗_{R} V_{R}

it is invariant under the transposition swapping the two tensor factors. Thus b is an element of the divided square D^{2}(V_{R}) (which may not be identified with the symmetric square S^{2}(U_{R}) unless 2 happens to be invertible in R, which will not be the case for us). Similarly, we have c ∈ D^{2}(W_{R}) and

a ∈ Hom_{R}(V_{R}, V_{R}) ≅ V_{R} ⊗_{R} W_{R}.

Putting this together, we see that

Y ∈ D^{2}(V_{R}) ⊕ D^{2}(W_{R}) ⊕ (V_{R} ⊗_{R} W_{R}) ≅ D^{2}(U_{R}).

Thus, as a module for Sp(2g, R), we have identified the Lie algebra 𝔰𝔭(2g, R) with the divided square of the natural module. More abstractly, if u ∈ U_{R} then the symmetric tensor u ⊗ u is identified with the endomorphism sending x to ⟨u, x⟩u. Polarising, this identifies u⊗u′+u′⊗u with the endomorphism of U_{R} sending x to ⟨u, x⟩u′ + ⟨u′, x⟩u. We have therefore proved the following.

Theorem 5.1. For any commutative ring R, we have isomorphisms

𝔰𝔭(2g, R) ≅ D^{2}(U_{R}) ≅ R^{g(2g+1)}.

The first isomorphism is an isomorphism of Sp(2g, R)-modules, while the second is an isomorphism of R-modules.

We are interested in the group Sp(2g, ℤ/4). This sits in a short exact sequence

1 → 𝔰𝔭(2g, 𝔽_{2}) → Sp(2g, ℤ/4) → Sp(2g, 2) → 1.

The elementary abelian 2-subgroup is identified with 𝔰𝔭(2g, 𝔽_{2}), and consists of the matrices I + 2Y with Y ∈ 𝔰𝔭(2g, 𝔽_{2}). These have the form

(\begin{matrix}I + 2a & 2b \\ 2c & I − 2a^{t} \end{matrix})

with b and c symmetric. We have a short exact sequence

\xymatrix{0 \ar[r] & Λ^{2}(U) \ar[r] & D^{2}(U) \ar[r] \ar[d]^{≅} & U \ar[r] & 0 \\  &  & 𝔰𝔭(2g, 𝔽_{2})}

where Λ^{2}(U) is spanned by elements of the form u⊗u′ +u′ ⊗u. As a submodule of 𝔰𝔭(2g, 𝔽_{2}), this consists of the matrices where Diag(b) = Diag(c) = 0. The quotient U corresponds to the diagonal entries in b and c. Thus the above short exact sequence can be thought of as a short exact sequence of groups

1 → Γ(2g, 2, 4)/Γ(2g, 4) → Γ(2g, 2)/Γ(2g, 4) → Γ(2g, 2)/Γ(2g, 2, 4) → 1.

More generally, we have short exact sequences

1 → 𝔰𝔭(2g, 𝔽_{2}) → Sp(2g, ℤ/2^{n+1}) → Sp(2g, ℤ/2^{n}) → 1

and

1 → Γ(2g, 2^{n}, 2^{n+1})/Γ(2g, 2^{n+1}) → Γ(2g, 2^{n})/Γ(2g, 2^{n+1}) → Γ(2g, 2^{n})/Γ(2g, 2^{n}, 2^{n+1}) → 1.

Proposition 5.2. As modules over Sp(2g, 2), for g ⩾ 1 and n ⩾ 1 we have

Γ(2g, 2^{n})/Γ(2g, 2^{n}, 2^{n+1}) ≅ U, Γ(2g, 2^{n}, 2^{n+1})/Γ(2g, 2^{n+1}) ≅ Λ^{2}(U).

Now the symplectic form on U gives us a map Λ^{2}(U) → 𝔽_{2}, which sends u ⊗ u′ + u′ ⊗ u to ⟨u, u′⟩. We write Y for the kernel of this map, and we write Z for D^{2}(U)/Y, an 𝔽_{2}-vector space of dimension 2g+1. Putting these together, we have the following diagram of modules.

[figure: 0; 0; Y; Y; 0; Λ^{2}(U); D^{2}(U); U; 0; 0; 𝔽_{2}; Z; U; 0; 0; 0]

We claim that the symplectic form on U lifts to a non-degenerate orthogonal form on Z, invariant under Sp(2g, 2). The quadratic form Z → 𝔽_{2} is given by

𝔮(u ⊗ u) = 0, 𝔮(u ⊗ u′ + u′ ⊗ u) = ⟨u, u′⟩,

and the associated symmetric bilinear form is

𝔟(u ⊗ u, u′ ⊗ u′) = ⟨u, u′⟩,
𝔟(u ⊗ u′ + u′ ⊗ u, u′′ ⊗ u′′) = 0,
𝔟(u ⊗ u′ + u′ ⊗ u, u′′ ⊗ u′′′ + u′′′ ⊗ u′′) = 0.

A priori, these are a quadratic form and associated bilinear form on D^{2}(U). But they clearly vanish identically on Y, and define a non-degenerate but singular quadratic form and associated bilinear form on Z. These are invariant under Sp(2g, 2), which is therefore the orthogonal group on Z ≅ 𝔽_{2}^{2g+1}, displaying the isomorphism

Sp(2g, 2) ≅ O(2g + 1, 2).

Remark 5.3. Translating back from D^{2}(U) to 𝔰𝔭(2g, 𝔽_{2}), the quadratic and bilinear form are given as follows:

𝔮 (\begin{matrix}a & b \\ c & −a^{t} \end{matrix}) = Tr(a) + ⟨Diag(b), Diag(c)⟩
𝔟 ((\begin{matrix}a & b \\ c & −a^{t} \end{matrix}) , (\begin{matrix}a′ & b′ \\ c′ & −a^{′t} \end{matrix})) = ⟨Diag(b), Diag(c′)⟩ + ⟨Diag(b′), Diag(c)⟩.

Here, the pointy brackets denote the standard inner product on 𝔽_{2}^{g} given by multiplying corresponding coordinates and summing.

The normal subgroup 𝔎 described in Section 2 is the inverse image of

Y ⩽ 𝔰𝔭(2g, 𝔽_{2}) ⩽ Sp(2g, ℤ/4)

under the quotient map Sp(2g, ℤ) → Sp(2g, ℤ/4). Thus there is a short exact sequence

1 → Z → ℌ → Sp(2g, 2) → 1

and the subgroup Z ≅ (ℤ/2)^{2g+1} may be viewed as the orthogonal module 𝔽_{2}^{2g+1} for Sp(2g, 2) via conjugation.

Remark 5.4. The submodule structure of the Sp(2g, 2)-modules Λ^{2}(U) of dimension g(2g − 1) and D^{2}(U) ≅ 𝔰𝔭(2g, 𝔽_{2}) of dimension g(2g + 1) can be described explicitly as follows (see also Hiss [18]). There is a map Λ^{2}(U) → 𝔽_{2} corresponding to the symplectic form, given by

u ⊗ u′ + u′ ⊗ u ↦ ⟨u, u′⟩.

There is a dual map 𝔽_{2} → Λ^{2}(U) coming from the fact that the representation Λ^{2}(U) is self-dual. In terms of the natural bases v_{1}, …, v_{g} of V and w_{1}, …, w_{g} of W, this is given by

1 ↦ ∑_{i}(v_{i} ⊗ w_{i} + w_{i} ⊗ v_{i}).

If g = 1 then Λ^{2}(U) ≅ 𝔽_{2} is one dimensional, Y = 0, and Z = D^{2}(U) decomposes as a direct sum 𝔽_{2} ⊕ U.

If g ⩾ 2 is even then the composite 𝔽_{2} → Λ^{2}(U) → 𝔽_{2} is zero, and the quotient of the kernel by the image is a simple module S of dimension g(2g −1)−2. Thus Λ^{2}(U) is uniserial (i.e., it has a unique composition series) with composition factors 𝔽_{2}, S, 𝔽_{2}.

If g ⩾ 3 is odd, then the composite is non-zero, and Λ^{2}(U) decomposes as a direct sum of a trivial module 𝔽_{2} and a simple module S of dimension g(2g − 1) − 1.

In both cases with g ⩾ 2, D^{2}(U) has Λ^{2}(U) as its unique maximal submodule. We can therefore draw diagrams for the structure of D^{2}(U) ≅ 𝔰𝔭(2g, 𝔽_{2}) as follows.

g = 1: 𝔽_{2} ⊕ U  g ⩾ 2 even : \begin{matrix}U \\ 𝔽_{2} \\ S \\ 𝔽_{2} \end{matrix}  g ⩾ 3 odd : [figure: U; 𝔽_{2}; S]

For g ⩾ 2, the quotient Z of D^{2}(U) has structure

\begin{matrix}U \\ 𝔽_{2} \end{matrix}

and this is the orthogonal module for Sp(2g, 2) ≅ O(2g + 1, 2). The submodule Y is S for g ⩾ 3 odd, it is a non-split extension

0 → 𝔽_{2} → Y → S → 0

for g ⩾ 2 even, and Y = 0 for g = 1.

Lemma 5.5. (i) For g ⩾ 1 we have H_{0}(Sp(2g, 2), Y) = 0 and H_{0}(Sp(2g, 2), U) = 0. (ii) For g ⩾ 2 we have H_{0}(Sp(2g, 2), Z) = 0 and H_{0}(Sp(2g, 2), 𝔰𝔭(2g, 𝔽_{2})) = 0.

Proof. This follows immediately from the structure of Y, U, Z, and 𝔰𝔭(2g, 𝔽_{2}) as Sp(2g, 2)- modules given in the above remark, since these modules admit no non-trivial homomorphisms to 𝔽_{2} with trivial action. □

6. Computations in degree two homology and cohomology

Lemma 6.1. (i) We have H_{2}(Sp(2g, 2)) = 0 for g ⩾ 4, and H_{2}(Sp(6, 2)) ≅ ℤ/2. (ii) We have H^{2}(Sp(2g, 2), ℤ/2) = 0 for g ⩾ 4 and H^{2}(Sp(6, 2), ℤ/2) ≅ ℤ/2.

Proof. (i) This is computed in the paper of Steinberg [34].

(ii) This follows from the universal coefficient theorem, since Sp(2g, 2) is perfect for g ⩾ 3. □

Lemma 6.2. For n ⩾ 1 and g ⩾ 2 we have H_{0}(Sp(2g, ℤ/2^{n}), H_{1}(Γ(2g, 2^{n}))) = 0.

Proof. Proposition 10.1 of Sato [30] computes H_{1}(Γ(2g, N)), the abelianisation of Γ(2g, N). Namely, the derived subgroup is Γ(2g, N^{2}) if N is odd, and Γ(2g, N^{2}, 2N^{2}) if N is even.

Taking N = 2^{n}, it gives

H_{1}(Γ(2g, 2^{n})) ≅ Γ(2g, 2^{n})/Γ(2g, 2^{2n}, 2^{2n+1}).

As modules over Sp(2g, ℤ/2^{n}) we have

Γ(2g, 2^{n})/Γ(2g, 2^{2n}) ≅ 𝔰𝔭(2g, ℤ/2^{n})
Γ(2g, 2^{2n})/Γ(2g, 2^{2n}, 2^{2n+1}) ≅ U

(cf. Section 5). This gives us a short exact sequence

(6.3)  0 → U → H_{1}(Γ(2g, 2^{n})) → 𝔰𝔭(2g, ℤ/2^{n}) → 0.

We also have short exact sequences

0 → 𝔰𝔭(2g, 𝔽_{2}) → 𝔰𝔭(2g, ℤ/2^{n}) → 𝔰𝔭(2g, ℤ/2^{n−1}) → 0.

By Lemma 5.5 (ii), for g ⩾ 2 we have

H_{0}(Sp(2g, ℤ/2^{n}), 𝔰𝔭(2g, 𝔽_{2})) = H_{0}(Sp(2g, 2), 𝔰𝔭(2g, 𝔽_{2})) = 0

and so by induction on n and right exactness of H_{0}, we have

H_{0}(Sp(2g, ℤ/2^{n}), 𝔰𝔭(2g, ℤ/2^{n})) = 0.

Finally, by Lemma 5.5 (i) we have

H_{0}(Sp(2g, ℤ/2^{n}), U) = H_{0}(Sp(2g, 2), U) = 0.

Therefore, using right exactness of H_{0} on the sequence (6.3), the lemma is proved. □

Proposition 6.4. For n ⩾ 1 and g ⩾ 2, (i) the map H_{2}(Sp(2g, ℤ)) → H_{2}(Sp(2g, ℤ/2^{n})) is surjective, and (ii) the map H_{2}(Sp(2g, ℤ/2^{n+1})) → H_{2}(Sp(2g, ℤ/2^{n})) is surjective.

Proof. (i) The short exact sequence

1 → Γ(2g, 2^{n}) → Sp(2g, ℤ) → Sp(2g, ℤ/2^{n}) → 1

gives rise to a five term sequence in homology

H_{2}(Sp(2g, ℤ)) → H_{2}(Sp(2g, ℤ/2^{n})) → H_{0}(Sp(2g, ℤ/2^{n}), H_{1}(Γ(2g, 2^{n})) → H_{1}(Sp(2g, ℤ)) → H_{1}(Sp(2g, ℤ/2^{n})) → 0.

The proposition therefore follows immediately from Lemma 6.2.

(ii) This is similar, using the observation that H_{1}(Γ(2g, 2^{n})/Γ(2g, 2^{n+1})) ≅ 𝔰𝔭(2g, 𝔽_{2}), so that by Lemma 5.5 (ii) we have

H_{0}(Sp(2g, ℤ/2^{n}), H_{1}(Γ(2g, 2^{n})/Γ(2g, 2^{n+1}))) = 0.  □

Corollary 6.5. For n ⩾ 1 and g ⩾ 3, the map

H^{2}(Sp(2g, ℤ/2^{n}), A) → H^{2}(Sp(2g, ℤ), A)

is injective for any abelian group of coefficients A with trivial action.

Proof. This follows directly from Proposition 6.4 together with the universal coefficient theorem for cohomology, as the groups Sp(2g, ℤ) and Sp(2g, ℤ/2^{n}) are perfect for g ⩾ 3. □

Proposition 6.6. For g ⩾ 2, the maps H_{2}(Sp(2g, ℤ/4)) → H_{2}(ℌ) → H_{2}(Sp(2g, 2)) are surjective.

Proof. For the first map, we use the five term sequence for the short exact sequence

1 → Y → Sp(2g, ℤ/4) → ℌ → 1,

and the computation

H_{0}(ℌ, H_{1}(Y)) = H_{0}(Sp(2g, 2), Y) = 0

given in Lemma 5.5. Note that Y is an elementary abelian 2-group, so H_{1}(Y) ≅ Y. The computation for the second map is similar, using the short exact sequence

1 → Z → ℌ → Sp(2g, 2) → 1

and the computation H_{0}(Sp(2g, 2), Z) = 0 given in Lemma 5.5. □

Corollary 6.7. For g ⩾ 3, the inflation map H^{2}(ℌ, A) → H^{2}(Sp(2g, ℤ/4), A) is injective for any abelian group of coefficients A with trivial action.

Proof. This follows directly from Proposition 6.6 and the universal coefficient theorem for cohomology, as the groups Sp(2g, ℤ/4) are perfect for g ⩾ 3, hence all their quotients are perfect as well. □

Proposition 6.8. For g ⩾ 4 the group ℌ = Sp(2g, ℤ)/𝔎 ≅ Sp(2g, ℤ/4)/Y is isomorphic to the quotient H̄_{0} of the group H_{0} of Griess, described in Section 3, by its central subgroup of order two.

Proof. Examine the extension

(6.9)  1 → Γ(2g, 2)/Γ(2g, 2, 4) → Sp(2g, ℤ/4)/Γ(2g, 2, 4) → Sp(2g, 2) → 1.

This is non-split, since the element of order two in Sp(2g, 2) which swaps the first basis vectors of L and L^{∗} and fixes the remaining basis vectors does not lift to an element of order two in Sp(2g, ℤ/4)/Γ(2g, 2, 4).

Let E be the almost extraspecial group O_{2}(H_{0}) of shape 2^{1+(2g+1)}. The action of Sp(2g, 2) on Γ(2g, 2)/Γ(2g, 2, 4) ≅ U is the same as the action of Out(E)′ on Inn(E) (see (3.2)), namely the natural symplectic module. It follows from the main theorem of Dempwolff [6] that

H^{2}(Sp(2g, 2), Γ(2g, 2)/Γ(2g, 2, 4))

is one dimensional. Therefore Sp(2g, ℤ/4)/Γ(2g, 2, 4) is isomorphic to the group Aut(E)′. Since Γ(2g, 4) ⊆ 𝔎 ⊆ Γ(2g, 2, 4) it follows that the short exact sequence

1 → Z → ℌ → Sp(2g, 2) → 1

also does not split. We have Γ(2g, 2, 4)/𝔎 ≅ ℤ/2, and since g ⩾ 4, by Lemma 6.1 we have H^{2}(Sp(2g, 2), ℤ/2) = 0. So H^{2}(Sp(2g, 2), Γ(2g, 2, 4)/Y) = 0, and hence H^{2}(Sp(2g, 2), Z) is at most one dimensional. Since we have a non-split extension (6.9), it is exactly one dimensional. The modules E/[E, E] and Z for Sp(2g, 2) are both isomorphic to the natural orthogonal module of dimension 2g + 1, so it follows that ℌ is isomorphic to H̄_{0}. □

Remark 6.10. In the case g = 3, Proposition 6.8 is still true, but needs a bit more work. The group H^{2}(Sp(6, 2), ℤ/2) is one dimensional by Lemma 6.1, and we are left with the nasty possibility that ℌ = Sp(6, ℤ/4)/Y is isomorphic to a quotient of the pullback of H̄_{0} → Sp(6, 2) and { \atop Sp(6, 2)} → Sp(6, 2) by the diagonal central element of order two. In order to prove that ℌ is really isomorphic to H̄_{0} and not this other group, it suffices to construct a matrix representation of a double cover of ℌ of dimension eight. Explicit matrices for this representation were given in Section 3. On the other hand, the smallest faithful irreducible complex representation in the case of the other possibility has dimension 64. It is worth noticing, though, that it does not matter which possibility is true, if we just wish to prove the next theorem.

Lemma 6.11. We have H_{2}(Sp(2g, ℤ)) = ℤ for g ⩾ 4 and ℤ ⊕ ℤ/2 for g = 3.

Proof. See for example Stein [33], Theorem 2.2 for g = 3 and Theorem 5.3 and Remark 5 following Corollary 5.5 for g ⩾ 4. See also Theorem 5.1 of Putman [25] for g ⩾ 4. □

Theorem 6.12. For g ⩾ 3 we have H_{1}(ℌ) = 0. For g ⩾ 4 we have H_{2}(ℌ) = ℤ/2, and for g = 3 we have H_{2}(ℌ) = ℤ/2 ⊕ ℤ/2. The map

H_{2}(Sp(2g, ℤ/2^{n})) → H_{2}(ℌ)

is an isomorphism for n ⩾ 2.

Proof. The computation of the abelianisation H_{1}(ℌ) is straightforward. It follows from Propositions 6.4 and 6.6 that for n ⩾ 2 the maps

(6.13) H_{2}(Sp(2g, ℤ)) → H_{2}(Sp(2g, ℤ/2^{n})) → H_{2}(Sp(2g, ℤ/4)) → H_{2}(ℌ) → H_{2}(Sp(2g, 2))

are surjective, and from Theorem 1.1 that the kernel of the first map contains every element divisible by two. Consulting Lemma 6.11, we see that H_{2}(Sp(2g, ℤ/2^{n})) and H_{2}(ℌ) are quotients of the groups given.

By Proposition 6.8, there is a non-trivial element of H_{2}(ℌ, ℤ/2) which is killed by the map to H_{2}(Sp(2g, 2)). Namely, the central extension ℌ̃ → ℌ is not inflated from Sp(2g, 2) because the kernel of ℌ̃ → Sp(2g, 2) is the non-abelian group E.

Comparing the value of H_{2}(Sp(2g, ℤ)) given in Lemma 6.11 with the value of H_{2}(Sp(2g, 2)) given in Lemma 6.1, the theorem follows. □

Corollary 6.14. The map

H^{2}(ℌ, A) → H^{2}(Sp(2g, ℤ/2^{n}), A)

is an isomorphism for g ⩾ 3, n ⩾ 2, and any abelian group of coefficients A with trivial action.

Proof. This follows from Theorem 6.12 and the universal coefficient theorem. □

Summary of homology and cohomology groups

Values for g ⩾ 4 \begin{cases}even \\ odd \end{cases} are as follows:

Group  H_{1}(−)  H_{2}(−)  H^{2}(−, ℤ) H^{2}(−, ℤ/8)  H^{2}(−, ℤ/2)
Γ_{g}  0  ℤ  ℤ  ℤ/8  ℤ/2
Sp(2g, ℤ)  0  ℤ  ℤ  ℤ/8  ℤ/2
P Sp(2g, ℤ)  0  \begin{cases}ℤ ⊕ ℤ/2 \\ ℤ \end{cases}  ℤ  \begin{cases}ℤ/8 ⊕ ℤ/2 \\ ℤ/8 \end{cases} \begin{cases}ℤ/2 ⊕ ℤ/2 \\ ℤ/2 \end{cases}
Sp(2g, ℤ/4)  0  ℤ/2  0  ℤ/2  ℤ/2
P Sp(2g, ℤ/4)  0 \begin{cases}ℤ/2 ⊕ ℤ/2 \\ ℤ/4 \end{cases}  0  \begin{cases}ℤ/2 ⊕ ℤ/2 \\ ℤ/4 \end{cases} \begin{cases}ℤ/2 ⊕ ℤ/2 \\ ℤ/2 \end{cases}
ℌ = Sp(2g, ℤ/4)/Y  0  ℤ/2  0  ℤ/2  ℤ/2
Sp(2g, 2)  0  0  0  0  0

Values for g = 3:

Group  H_{1}(−) H_{2}(−) H^{2}(−, ℤ) H^{2}(−, ℤ/8) H^{2}(−, ℤ/2)
Γ_{3}  0  ℤ ⊕ ℤ/2  ℤ  ℤ/8 ⊕ ℤ/2 ℤ/2 ⊕ ℤ/2
Sp(6, ℤ)  0  ℤ ⊕ ℤ/2  ℤ  ℤ/8 ⊕ ℤ/2 ℤ/2 ⊕ ℤ/2
P Sp(6, ℤ)  0  ℤ ⊕ ℤ/2  ℤ  ℤ/8 ⊕ ℤ/2  ℤ/2
Sp(6, ℤ/4)  0 ℤ/2 ⊕ ℤ/2  0  ℤ/2 ⊕ ℤ/2 ℤ/2 ⊕ ℤ/2
P Sp(6, ℤ/4)  0 ℤ/4 ⊕ ℤ/2  0  ℤ/4 ⊕ ℤ/2 ℤ/2 ⊕ ℤ/2
ℌ = Sp(6, ℤ/4)/Y  0 ℤ/2 ⊕ ℤ/2  0  ℤ/2 ⊕ ℤ/2 ℤ/2 ⊕ ℤ/2
Sp(6, 2)  0  ℤ/2  0  ℤ/2  ℤ/2

Values for g = 2:

Group  H_{1}(−) H_{2}(−) H^{2}(−, ℤ) H^{2}(−, ℤ/8) H^{2}(−, ℤ/2)
Γ_{2}  ℤ/10  ℤ/2  ℤ/10  (ℤ/2)^{2}  (ℤ/2)^{2}
Sp(4, ℤ)  ℤ/2 ℤ ⊕ ℤ/2 ℤ ⊕ ℤ/2 ℤ/8 ⊕ (ℤ/2)^{2} (ℤ/2)^{3}
P Sp(4, ℤ)  ℤ/2 ℤ ⊕ (ℤ/2)^{2} ℤ ⊕ ℤ/2 ℤ/8 ⊕ (ℤ/2)^{3} (ℤ/2)^{4}
Sp(4, ℤ/4)  ℤ/2  (ℤ/2)^{2}  ℤ/2  (ℤ/2)^{3}  (ℤ/2)^{3}
P Sp(4, ℤ/4)  ℤ/2  (ℤ/2)^{3}  ℤ/2  (ℤ/2)^{4}  (ℤ/2)^{4}
ℌ = Sp(4, ℤ/4)/Y ℤ/2  (ℤ/2)^{2}  ℤ/2  (ℤ/2)^{3}  (ℤ/2)^{3}
Sp(4, 2)  ℤ/2  ℤ/2  ℤ/2  (ℤ/2)^{2}  (ℤ/2)^{2}

Values for g = 1:

Group  H_{1}(−) H_{2}(−) H^{2}(−, ℤ) H^{2}(−, ℤ/8) H^{2}(−, ℤ/2)
Γ_{1}  ℤ/12  0  ℤ/12  ℤ/4  ℤ/2
Sp(2, ℤ)  ℤ/12  0  ℤ/12  ℤ/4  ℤ/2
P Sp(2, ℤ)  ℤ/6  0  ℤ/6  ℤ/2  ℤ/2
Sp(2, ℤ/4)  ℤ/4 ℤ/2  ℤ/4  (ℤ/2)^{2}  ℤ/4 ⊕ ℤ/2
P Sp(2, ℤ/4)  ℤ/2 ℤ/2  ℤ/2  (ℤ/2)^{2}  (ℤ/2)^{2}
ℌ = Sp(2, ℤ/4)/Y ℤ/4 ℤ/2  ℤ/4  (ℤ/2)^{2}  ℤ/4 ⊕ ℤ/2
Sp(2, 2)  ℤ/2  0  ℤ/2  ℤ/2  ℤ/2

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Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland, United Kingdom

Department of Mathematics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany

School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, Scotland, United Kingdom

Department of Mathematics, Indiana University, Bloomington IN 47405, USA