COHOMOLOGY OF SPLIT GROUP EXTENSIONS AND CHARACTERISTIC CLASSES

NANSEN PETROSYAN

Abstract. There are characteristic classes that are the obstructions to the vanishing of the differentials in the Lyndon-Hochschild-Serre spectral sequence of a split extension of an integral lattice L by a group G. These characteristic classes exist in the r-th page of the spectral sequence provided that the differentials d_{i} = 0 for all i < r. When L decomposes into a sum of G-sublattices, we show that there are defining relations between the characteristic classes of L and the characteristic classes of its summands.

1. Introduction

Suppose G is a group and L is a finite rank integral ℤG-lattice. Let Γ = L ⋊ G be the semidirect product group induced by the action of G on L. Our objective is to analyze the split group extension 0 → L → Γ → G → 1 and its associated Lyndon-Hochschild-Serre spectral sequence {E_{∗}, d_{∗}}. In [1] we showed that when G is a cyclic group of prime order, the spectral sequence with integral coefficients collapses at E_{2} without extension problems. In fact, our proof stated in [1] applies not only to integral coefficients, but to all coefficient modules in a certain category which we denote by M_{F} (L, G). This category is essential in the definition of characteristic classes (see Theorem 2.2). It consists of all finite rank F Γ-lattices M on which L acts trivially, where F is a given principal ideal domain. In particular, F as a trivial F Γ-module is an object of M_{F} (L, G).

Let us now suppose that L decomposes into a sum of ℤG-sublattices; L = L′ ⊕ L′′. Let {E∗^{′}, d∗^{′}} and {E∗^{′′}, d∗^{′′}} be the Lyndon-Hochschild-Serre spectral sequences associated to the respective semidirect product groups L′ ⋊ G and L′′ ⋊ G. The goal of the present paper is to find necessary and sufficient conditions on the latter defined spectral sequences such that {E_{∗}, d_{∗}} has no nonzero differentials up to a given page.

In section 2, we define characteristic classes v_{r}^{t}(L), introduced by Charlap-Vasquez and Sah (see [2] and [3]), that are the obstructions to the vanishing of the differentials in {E_{∗}, d_{∗}}. These classes exist and lie in the images of the differentials d_{r}^{0,t} :

Date: June 22, 2012.

Key words and phrases. spectral sequence, group cohomology.

E_{r}^{0,t}(H_{t}(L, F)) → E_{r}^{r,t−r+1}(H_{t}(L, F)) provided that the differentials d_{i}^{∗,t} vanish for all i < r and for all coefficient modules in M_{F} (L, G). Our main result is the following decomposition theorem.

Theorem 1.1. Let G be any group. Assume L′ and L′′ are ℤG-lattices of finite rank and L = L′ ⊕ L′′. Let r ≥ 2 and t ≥ 0.

(a) Suppose v_{k}^{i}(L′) = v_{k}^{j}(L′′) = 0 for all i, j ≤ t and for all k < r. Then d_{2}^{s,m} = ⋯ = d_{r−1}^{s,m} = 0 for all s ≥ 0, all m ≤ t, and all coefficient modules in M_{F} (L, G). Additionally,

v_{r}^{t}(L) = ∑_{i+j=t} ((C_{j}^{′r})_{∗}(v_{r}^{i}(L′)) + (−1)^{i}(C_{i}^{′′r})_{∗}(v_{r}^{j}(L′′))).

(b) Suppose v_{k}^{t}(L) = 0 for all k < r. Then d_{k}^{′s,t} = d_{k}^{′′s,t} = 0 for all s ≥ 0, all k < r, and all coefficient modules in M_{F} (L, G). Additionally,

v_{r}^{t}(L′) = D_{∗}^{′r}(v_{r}^{t}(L)) and v_{r}^{t}(L′′) = D_{∗}^{′′r}(v_{r}^{t}(L)).

In section 3, we define the F G-equivariant homomorphisms C_{j}^{′r}, C_{i}^{′′r}, D^{′r}, and D^{′′r} that induce the maps (C_{j}^{′r})_{∗} : E_{2}^{′s,i−r+1} → E_{2}^{s,t−r+1}, (C_{i}^{′′r})_{∗} : E_{2}^{′′s,j−r+1} → E_{2}^{s,t−r+1}, D_{∗}^{′r} : E_{2}^{s,t−r+1} → E_{2}^{′s,t−r+1}, and D_{∗}^{′′r} : E_{2}^{s,t−r+1} → E_{2}^{′′s,t−r+1}, respectively. The homomorphisms C_{j}^{′2} and C_{i}^{′′2} were first considered by Charlap and Vasquez in [2]. They showed that the sum formula holds when r = 2. We use a different approach to generalize this result to all pages of the spectral sequence and also to prove a converse.

The problem of establishing the collapse of {E_{∗}, d_{∗}} in general can be a difficult one. The spectral sequence can have nonzero differentials even when G is abelian. For instance, Totaro proved in [4] that for any prime number p and G = C_{p}^{2}, there is a semidirect product group Γ = L ⋊ C_{p}^{2} such that in the associated Lyndon-Hochschild-Serre spectral sequence with ℤ_{p} coefficients there always exist nonzero differentials at E_{p} or later.

A question, first posed by Adem (see [1]), that is still open is whether the spectral sequence collapses integrally at E_{2} without extension problems when G is an arbitrary finite cyclic group. In view of our results, we can make the following

Conjecture 1.2. Let C_{n} be a cyclic group of order n and let L be a finite rank integral lattice. The Lyndon-Hochschild-Serre spectral sequence {E_{∗}, d_{∗}} of any split group extension 0 → L → Γ → C_{n} → 1 collapses at E_{2} for all coefficient modules in M_{F} (L, C_{n}).

Note that part (a) of Theorem 1.1 reduces this conjecture to the case where L is an indecomposable ℤC_{n}-lattice (see Corollary 4.2).

COHOMOLOGY OF SPLIT GROUP EXTENSIONS AND CHARACTERISTIC CLASSES 3

2. Preliminary Results

Henceforth, let G be any group and let L be any finite rank integral ℤG-lattice. Denote by Γ the associated semidirect product L ⋊ G. Suppose F is a principal ideal domain. For each F Γ-module M we have the Lyndon-Hochschild-Serre spectral sequence

E_{2}^{p,q}(M) = H^{p}(G, H^{q}(L, M)) ⟹ H^{p+q}(Γ, M).

In the proof of our main result, we make use of the multiplicative structure of this spectral sequence. Namely, any Γ-pairing of F Γ-modules A, B, and C

· : A ⊗_{F} B → C,

determines an F -pairing

E_{r}^{p,q}(A) ⊗_{F} E_{r}^{s,t}(B) → E_{r}^{p+s,q+t}(C).

In addition, for the differential d_{r} and for each a ∈ E_{r}^{p,q}(A) and b ∈ E_{r}^{s,t}(B) we have the product formula

d_{r}^{p+s,q+t}(a · b) = d_{r}^{p,q}(a) · b + (−1)^{p+q}a · d_{r}^{s,t}(b).

We denote by M_{F} (L, G) the category of all F Γ-modules M, such that M is a finitely generated free F -module and L acts trivially on M. In particular, F with a trivial Γaction is a module in this category. M_{F} (L, G) has the property that if M ∈ M_{F} (L, G), then H_{i}(L, M) and H^{i}(L, M) ≅ Hom_{F} (H_{i}(L, F), M) are also in M_{F} (L, G). Observe that the projection of Γ onto G induces an equivalence between this category and the category of all finite rank F G-lattices.

Let t ≥ 0 and r ≥ 2. Set M = H_{t}(L, F) ≅ ∧^{t}(L) ⊗ F. Assume d_{k}^{0,t} = 0 for all 2 ≤ k ≤ r − 1. By applying the Universal Coefficient Theorem it follows

E_{r}^{0,t}(H_{t}(L, F)) = E_{2}^{0,t}(H_{t}(L, F))
= H^{0}(G, H^{t}(L, H_{t}(L, F)))
≅ H^{0}(G, Hom_{F} (H_{t}(L, F), H_{t}(L, F))) (by UCT)
= Hom_{F G}(H_{t}(L, F), H_{t}(L, F))
≅ Hom_{F G}( ∧^{t} (L) ⊗ F, ∧^{t}(L) ⊗ F ).

Definition 2.1. With the preceding assumptions, let id^{t} : ∧^{t}(L) ⊗ F → ∧^{t}(L) ⊗ F be the identity homomorphism. Identifying along the above isomorphisms, denote by [f] the class in E_{r}^{0,t}(H_{t}(L, F)) corresponding to a map f ∈ Hom_{F G}(∧^{t}(L) ⊗ F, ∧^{t}(L) ⊗ F). Then v_{r}^{t}(L) := d_{r}^{0,t}([id^{t}]) ∈ E_{r}^{r,t−r+1}(H_{t}(L, F)) is said to be a characteristic class of the spectral sequence {E_{r}, d_{r}}.

Characteristic classes were first considered by Charlap and Vasquez in [2], but only in the case when r = 2. Sah in [3] extended their definition to all r ≥ 2 by proving the following key theorem.

Theorem 2.2. (Sah, 1972, [3]) Let {E_{∗}, d_{∗}} be the Lyndon-Hochschild-Serre spectral sequence of the extension 0 → L → Γ → G → 1. Suppose there exist integers r ≥ 2 and t ≥ 0 such that d_{2}^{s,t} = ⋯ = d_{r−1}^{s,t} = 0 for all s ≥ 0 and for all coefficient modules in M_{F} (L, G).

(a) There is a canonical epimorphism θ : E_{r}^{s,0}(H^{t}(L, M)) → E_{r}^{s,t}(M) for all s ≥ 0 and for all M ∈ M_{F} (L, G).

(b) We have

d_{r}^{s,t}(x) = (−1)^{s}y · v_{r}^{t}(L),

for all x ∈ E_{r}^{s,t}(M), all M ∈ M_{F} (L, G), and all y ∈ E_{r}^{s,0}(H^{t}(L, M)) with θ(y) = x.

(c) Let σ : H → G be a group homomorphism which converts L into a ℤH-module. Assume d_{2}^{s,t} = ⋯ = d_{r−1}^{s,t} = 0 holds for all s ≥ 0 and for all objects in M_{F} (L, H). The characteristic class w_{r}^{t}(L) for the category M_{F} (L, H) is then the image of v_{r}^{t}(L) under the map induced on the spectral sequences.

Note that, with the assumptions of the theorem, characteristic classes v_{r}^{t}(L) are obstructions to the vanishing of the differentials d_{r}^{s,t} for all integers s ≥ 0. The following corollary is an immediate consequence of Sah’s theorem.

Corollary 2.3. Let t ≥ 0. The Lyndon-Hochschild-Serre spectral sequence {E_{∗}, d_{∗}} has d_{r}^{s,t} = 0 for all s ≥ 0, all r < n, and all M ∈ M_{F} (L, G) if and only if the edge differentials d_{r}^{0,t} : E_{r}^{0,t}(H_{t}(L, F)) → E_{r}^{r,t−r+1}(H_{t}(L, F)) are zero for all r < n.

Corollary 2.4. Suppose φ : L → Γ is the natural inclusion. Given an integer t ≥ 0, the following statements are equivalent.

(a) In {E_{∗}, d_{∗}}, the differentials d_{r}^{s,t} vanish for all r, s ≥ 0 and all M ∈ M_{F} (L, G).

(b) φ^{∗} : H^{t}(Γ, M) → H^{t}(L, M) maps onto H^{t}(L, M)^{G} for all M ∈ M_{F} (L, G).

(c) φ^{∗} : H^{t}(Γ, H_{t}(L, F)) → H^{t}(L, H_{t}(L, F)) maps onto the G-invariants H^{t}(L, H_{t}(L, F))^{G}.

Proof. Clearly (b) implies (c). Note that the map φ^{∗} is given by the composition

H^{t}(Γ, M) ↠ E_{∞}^{0,t}(M) = E_{t+2}^{0,t}(M) ⊂ ⋯ ⊂ E_{2}^{0,t}(M) = H^{t}(L, M)^{G} ⊂ H^{t}(L, M).

If d_{r}^{s,t} = 0 for all r, s ≥ 0 and all M ∈ M_{F} (L, G), then E_{∞}^{0,t}(M) = E_{t+2}^{0,t}(M) = ⋯ = E_{2}^{0,t}(M) for all M ∈ M_{F} (L, G). Hence, (a) implies (b). To prove that (a) follows from (c), assume φ^{∗} : H^{t}(Γ, H_{t}(L, F)) → H^{t}(L, H_{t}(L, F))^{G} is onto. Then by the above composition we have E_{∞}^{0,t}(H_{t}(L, F)) = E_{t+2}^{0,t}(H_{t}(L, F)) = ⋯ = E_{2}^{0,t}(H_{t}(L, F)). This shows that d_{r}^{0,t} : E_{r}^{0,t}(H_{t}(L, F)) → E_{r}^{r,t−r+1}(H_{t}(L, F)) is zero for all r ≥ 2. By the previous corollary d_{r}^{s,t} = 0 for all s ≥ 0, r ≥ 2, and M ∈ M_{F} (L, G). □

3. Decomposition Theorem

Suppose L is a direct sum of ℤG-sublattices L′ and L′′. In this section we derive relations between the characteristic classes of L and the characteristic classes of L′ and L′′.

For convenience, we use ∧^{∗}(L) to denote ∧^{∗}(L) ⊗ F. Recall that there is a standard F G-module decomposition

∧^{n}(L) ≅ ⨁_{i+j=n} ∧^{i}(L′) ⊗ ∧^{j}(L′′).

Definition 3.1. Given integers i, j ≥ 0 and r ≥ 1, define F G-equivariant homomorphisms

C_{j}^{′r} : Hom_{F} (∧^{i}(L′), ∧^{i+r−1}(L′)) → Hom_{F} (∧^{i+j}(L), ∧^{i+j+r−1}(L))

by

C_{j}^{′r}(f)(x ⊗ y) = \{ \begin{matrix}f(x) ⊗ y & if x ∈ ∧^{i}(L′) and y ∈ ∧^{j}(L′′) \\ 0 & otherwise \end{matrix}

and

C_{i}^{′′r} : Hom_{F} (∧^{j}(L′′), ∧^{j+r−1}(L′′)) → Hom_{F} (∧^{i+j}(L), ∧^{i+j+r−1}(L))

by

C_{i}^{′′r}(f)(x ⊗ y) = \{ \begin{matrix}x ⊗ f(y) & if x ∈ ∧^{i}(L′) and y ∈ ∧^{j}(L′′) \\ 0 & otherwise. \end{matrix}

Let Γ′ = L′ ⋊ G and Γ′′ = L′′ ⋊ G. There are associated split exact sequences of groups

0 → L′ → Γ′ → G → 1 and 0 → L′′ → Γ′′ → G → 1.

Suppose v_{∗}^{∗}(L′) and v_{∗}^{∗}(L′′) are the respective characteristic classes of the Lyndon-Hochschild-Serre spectral sequences {E′_{∗}, d∗^{′}} and {E′′_{∗}, d∗^{′′}} corresponding to these extensions. Let ι′ : L′ → L be the natural inclusion and let p′ : L → L′ be the natural projection. Similarly, define ι′′ : L′′ → L and p′′ : L → L′′. As a straightforward application of the definitions, we obtain the following lemma.

Lemma 3.2. The map C_{0}^{′r} : Hom_{F} (∧^{i}(L′), ∧^{i+r−1}(L′)) → Hom_{F} (∧^{i}(L), ∧^{i+r−1}(L)) is given by the composition p^{′∗} ∘ ι′_{∗}, where

Hom_{F} (∧^{i}(L′), ∧^{i+r−1}(L′)) ⟶^{ι′_{∗}} Hom_{F} (∧^{i}(L′), ∧^{i+r−1}(L)) ⟶^{p^{′∗}} Hom_{F} (∧^{i}(L), ∧^{i+r−1}(L)),

and C_{0}^{′′r} is the composition p^{′′∗} ∘ ι′′_{∗}, where

Hom_{F} (∧^{j}(L′′), ∧^{j+r−1}(L′′)) ⟶^{ι′′_{∗}} Hom_{F} (∧^{j}(L′′), ∧^{j+r−1}(L)) ⟶^{p^{′′∗}} Hom_{F} (∧^{j}(L), ∧^{j+r−1}(L)).

Proof. Suppose f ∈ Hom_{F} (∧^{i}(L′), ∧^{i+r−1}(L′)). For any x ∈ ∧^{i}(L′) and y ∈ ∧^{0}(L′′), (p^{′∗} ∘ ι′_{∗})(f)(x ⊗ y) = y(ι′_{∗}(f)(x)) = y(f(x) ⊗ 1) = f(x) ⊗ y = C_{0}^{′r}(f)(x ⊗ y). The second assertion of the lemma follows analogously. □

Definition 3.3. Given integers i, j ≥ 0 and r ≥ 1, let ∧^{i+r−1}p′ : ∧^{i+r−1}(L) → ∧^{i+r−1}(L′) and ∧^{j+r−1}p′′ : ∧^{j+r−1}(L) → ∧^{j+r−1}(L′′) be the maps induced by the projections p′ and p′′, respectively. Define F G-equivariant homomorphisms

D^{′r} : Hom_{F} (∧^{i}(L), ∧^{i+r−1}(L)) → Hom_{F} (∧^{i}(L′), ∧^{i+r−1}(L′))

by

D^{′r}(f)(x) = ∧^{i+r−1}p′(f(x ⊗ 1)) for all x ∈ ∧^{i}(L′),

and

D^{′′r} : Hom_{F} (∧^{j}(L), ∧^{j+r−1}(L)) → Hom_{F} (∧^{j}(L′′), ∧^{j+r−1}(L′′))

by

D^{′′r}(f)(y) = ∧^{j+r−1}p′′(f(1 ⊗ y)) for all y ∈ ∧^{j}(L′′).

Lemma 3.4. The map D^{′r} is given by the composition p′_{∗} ∘ ι^{′∗}, where

Hom(∧^{i}(L), ∧^{i+r−1}(L)) ⟶^{ι^{′∗}} Hom(∧^{i}(L′), ∧^{i+r−1}(L)) ⟶^{p′_{∗}} Hom(∧^{i}(L′), ∧^{i+r−1}(L′)),

and D^{′′r} is the composition p′′_{∗} ∘ ι^{′′∗}, where

Hom(∧^{j}(L), ∧^{j+r−1}(L)) ⟶^{ι^{′′∗}} Hom(∧^{j}(L′′), ∧^{j+r−1}(L)) ⟶^{p′′_{∗}} Hom(∧^{j}(L′′), ∧^{j+r−1}(L′′)).

Proof. This is an immediate consequence of the definitions of D^{′r} and D^{′′r}. □

Proposition 3.5. Let i ≥ 0 and r ≥ 2. Suppose E_{r}^{′s,t}(M) = E_{2}^{′s,t}(M) and E_{r}^{s,t}(H_{i}(L, F)) = E_{2}^{s,t}(H_{i}(L, F)) when (s, t) = (0, i) and when (s, t) = (r, i − r + 1), for M = H_{i}(L′, F) and for M = H_{i}(L, F). Then d_{r}^{0,i} ∘ (C_{0}^{′1})_{∗} = (C_{0}^{′r})_{∗} ∘ d_{r}^{′0,i} and d_{r}^{′0,i} ∘ D_{∗}^{′1} = D_{∗}^{′r} ∘ d_{r}^{0,i}.

Proof. The first claim asserts that the following diagram commutes.

H^{0}(G, H^{i}(L′, H_{i}(L′, F))) ⟶^{d_{r}^{′0,i}} H^{r}(G, H^{i−r+1}(L′, H_{i}(L′, F)))
( C_{0}^{′1})_{∗}↓  _{( C_{0}^{′r})_{∗}}↓
H^{0}(G, H^{i}(L, H_{i}(L, F))) ⟶^{d_{r}^{0,i}} H^{r}(G, H^{i−r+1}(L, H_{i}(L, F)))

By Lemma 3.2, we know that the map (C_{0}^{′∗})_{∗} is induced by the inclusion ι′ : L′ → L and the projection p′ : L → L′. Hence, this diagram is the outer square of the commutative diagram below.

H^{0}(G, H^{i}(L′, H_{i}(L′, F))) ⟶^{d_{r}^{′0,i}} H^{r}(G, H^{i−r+1}(L′, H_{i}(L′, F)))
ι′_{∗}↓  ^{ι}′_{∗}↓
H^{0}(G, H^{i}(L′, H_{i}(L, F))) ⟶^{d_{r}^{′0,i}} H^{r}(G, H^{i−r+1}(L′, H_{i}(L, F)))
p^{′∗}↓  ^{p^{′∗}}↓
H^{0}(G, H^{i}(L, H_{i}(L, F))) ⟶^{d_{r}^{0,i}} H^{r}(G, H^{i−r+1}(L, H_{i}(L, F)))

The second claim follows by a similar argument using Lemma 3.4. □

As previously noted, the category M_{F} (L, G) can be identified with the category of all finite rank F G-lattices. In view of this, we will not distinguish between M_{F} (L′, G), M_{F} (L′′, G), and M_{F} (L, G).

Lemma 3.6. There are natural morphisms of spectral sequences π^{′∗} : Er^{′} → E_{r}, ϕ^{′∗} : E_{r} → Er^{′} and π^{′′∗} : Er^{′′} → E_{r}, ϕ^{′′∗} : E_{r} → Er^{′′}, such that for all r ≥ 2, all s ≥ 0, and all coefficient modules in M_{F} (L, G),

d_{r}^{′s,t} = ϕ^{′∗} ∘ d_{r}^{s,t} ∘ π^{′∗} and d_{r}^{′′s,t} = ϕ^{′′∗} ∘ d_{r}^{s,t} ∘ π^{′′∗}.

Proof. We observe that the map ι′ induces a natural inclusion ϕ′ : Γ′ → Γ and the map p′ induces a natural projection π′ : Γ → Γ′ such that the composition π′ ∘ ϕ′ is the identity map on Γ′. Let φ : L → Γ and φ′ : L′ → Γ′ be the canonical inclusions. It follows that ϕ′ ∘ φ′ = φ ∘ ι′ and π′ ∘ φ = φ′ ∘ p′. Hence, the homomorphisms ϕ′ and π′ give rise to spectral sequence morphisms ϕ^{′∗} : E_{∗} → E∗^{′} and π^{′∗} : E∗^{′} → E_{∗}, such that ϕ^{′∗} ∘ π^{′∗} is the

identity morphism on {E∗^{′}, d∗^{′}}. Then, d_{r}^{′s,t} = ϕ^{′∗} ∘ π^{′∗} ∘ d_{r}^{′s,t} = ϕ^{′∗} ∘ d_{r}^{s,t} ∘ π^{′∗}. Analogously,

ι′′ and p′′ induce an inclusion ϕ′′ : Γ′′ → Γ and a projection π′′ : Γ → Γ′′, respectively, such that the composition π′′ ∘ ϕ′′ is the identity map on Γ′′ and d_{r}^{′′s,t} = ϕ^{′′∗} ∘ d_{r}^{s,t} ∘ π^{′′∗}. □

We are now ready to prove our main result.

Theorem 3.7. Recall that G is an arbitrary group, and that L is a ℤG-lattice of finite rank that decomposes into a direct sum of the ℤG-lattices L′ and L′′. Let r ≥ 2 and t ≥ 0.

(a) Suppose v_{k}^{i}(L′) = v_{k}^{j}(L′′) = 0 for all i, j ≤ t and for all k < r. Then d_{2}^{s,m} = ⋯ = d_{r−1}^{s,m} = 0 for all s ≥ 0, all m ≤ t, and all coefficient modules in M_{F} (L, G). Additionally,

v_{r}^{t}(L) = ∑_{i+j=t} ((C_{j}^{′r})_{∗}(v_{r}^{i}(L′)) + (−1)^{i}(C_{i}^{′′r})_{∗}(v_{r}^{j}(L′′))).

(b) Suppose v_{k}^{t}(L) = 0 for all k < r. Then d_{k}^{′s,t} = d_{k}^{′′s,t} = 0 for all s ≥ 0, all k < r, and all coefficient modules in M_{F} (L, G). Additionally,

v_{r}^{t}(L′) = D_{∗}^{′r}(v_{r}^{t}(L)) and v_{r}^{t}(L′′) = D_{∗}^{′′r}(v_{r}^{t}(L)).

Proof. To show part (a), we will use induction on k ≥ 2 to prove that v_{k}^{m}(L) satisfies the sum formula for all m ≤ t and all k ≤ r. For each k < r, since v_{k}^{i}(L′) = v_{k}^{j}(L′′) = 0 for all i, j ≤ t, this will imply v_{k}^{m}(L) = 0 and hence, by Theorem 2.2(b), d_{k}^{s,m} = 0 for all s ≥ 0, all m ≤ t, and all M ∈ M_{F} (L, G).

Suppose d_{2}^{s,m} = ⋯ = d_{k−1}^{s,m} = 0 for all s ≥ 0, all m ≤ t, and all M ∈ M_{F} (L, G). Recall that v_{k}^{m}(L) = d_{k}^{0,m}([id^{m}]), where id^{m} ∈ Hom_{F G}(∧^{m}(L), ∧^{m}(L)) is the identity map. Similarly, v_{k}^{i}(L′) = d_{k}^{′0,i}([id^{′i}]) and v_{k}^{j}(L′′) = d_{k}^{′′0,j}([id^{′′j}]). Consider the decomposition id^{m} = ∑_{i+j=m}id_{ij}, where id_{ij} : ∧^{i+j}(L) → ∧^{i+j}(L) is the F -linear map given by

id_{ij}(x ⊗ y) = \{ \begin{matrix}x ⊗ y & if x ∈ ∧^{i}(L′) and y ∈ ∧^{j}(L′′) \\ 0 & otherwise. \end{matrix}

Given i, j ≥ 0, let x ∈ ∧^{i}(L′) and y ∈ ∧^{j}(L′′). Then, id_{ij}(x⊗y) = x⊗y = (x⊗1)∧(1⊗y) = C_{0}^{′1}(id^{′i})(x ⊗ 1) ∧ C_{0}^{′′1}(id^{′′j})(1 ⊗ y). This implies [id_{ij}] = (C_{0}^{′1})_{∗}([id^{′i}]) · (C_{0}^{′′1})_{∗}([id^{′′j}]) ∈ H^{0}(G, H^{i+j}(L, H_{i+j}(L, F))), and thus

[id^{m}] = ∑_{i+j=m}(C_{0}^{′1})_{∗}([id^{′i}]) · (C_{0}^{′′1})_{∗}([id^{′′j}]).

By applying the product formula for the differentials, we compute

v_{k}^{m}(L) = d_{k}^{0,m}( ∑_{i+j=m}(C_{0}^{′1})_{∗}([id^{′i}]) · (C_{0}^{′′1})_{∗}([id^{′′j}]))

= ∑_{i+j=m} d_{k}^{0,m}((C_{0}^{′1})_{∗}([id^{′i}]) · (C_{0}^{′′1})_{∗}([id^{′′j}]))

= ∑_{i+j=m} (d_{k}^{0,i}((C_{0}^{′1})_{∗}([id^{′i}])) · (C_{0}^{′′1})_{∗}([id^{′′j}]) + (−1)^{i}(C_{0}^{′1})_{∗}([id^{′i}]) · d_{k}^{0,j}((C_{0}^{′′1})_{∗}([id^{′′j}])))

=^{3.5} ∑_{i+j=m} ((C_{0}^{′k})_{∗}(d_{k}^{′0,i}([id^{′i}])) · (C_{0}^{′′1})_{∗}([id^{′′j}]) + (−1)^{i}(C_{0}^{′1})_{∗}([id^{′i}]) · (C_{0}^{′′k})_{∗}(d_{k}^{′′0,j}([id^{′′j}])))

= ∑_{i+j=m} ((C_{0}^{′k})_{∗}(v_{k}^{i}(L′)) · (C_{0}^{′′1})_{∗}([id^{′′j}]) + (−1)^{i}(C_{0}^{′1})_{∗}([id^{′i}]) · (C_{0}^{′′k})_{∗}(v_{k}^{j}(L′′)))

= ∑_{i+j=m} ((C_{j}^{′k})_{∗}(v_{k}^{i}(L′)) + (−1)^{i}(C_{i}^{′′k})_{∗}(v_{k}^{j}(L′′))).

The last equality follows from the definitions of C_{∗}^{′∗} and C_{∗}^{′′∗} and the fact that id^{′i} and id^{′′j} are identity maps.

To prove (b), we will use induction on k ≥ 2 to show that v_{k}^{t}(L′) = D_{∗}^{′r}(v_{k}^{t}(L)) and v_{k}^{t}(L′′) = D_{∗}^{′′r}(v_{k}^{t}(L)) for all k ≤ r. By Theorem 2.2, for each k < r and for all s ≥ 0 this will imply that d_{k}^{′s,t} = d_{k}^{′′s,t} = 0 for all coefficient modules in M_{F} (L, G). Note that this assertion also follows from Lemma 3.6.

For the identity map id^{′t} : ∧^{t}(L′) → ∧^{t}(L′), we have id^{′t}(x) = x = ∧^{t}p′(x ⊗ 1) = ∧^{t}p′(id^{t}(x ⊗ 1)) = D^{′1}(id^{t})(x) for all x ∈ ∧^{t}(L′). This implies [id^{′t}] = D_{∗}^{′1}([id^{t}]) ∈ E_{k}^{′0,t}(H_{t}(L′, F)). Thus, it follows

v_{k}^{t}(L′) = d_{k}^{′0,t}([id^{′t}]) = d_{k}^{′0,t} ∘ D_{∗}^{′1}([id^{t}])=^{3.5}D_{∗}^{′k} ∘ d_{k}^{0,t}([id^{t}]) = D_{∗}^{′k}(v_{k}^{t}(L)).

By an analogous argument, v_{k}^{t}(L′′) = D_{∗}^{′′k}(v_{k}^{t}(L)). □

Remark 3.8. Using the same assumptions as in Theorem 3.7(a), the sum formula can be simplified. Since E_{r}^{′r,i−r+1} = E_{r}^{′′r,j−r+1} = 0, v_{r}^{i}(L′) = v_{r}^{j}(L′′) = 0 holds a priori for all i, j < r − 1. It is an easy exercise to check that d_{r}^{′0,r−1} = 0 and d_{r}^{′′0,r−1} = 0 for all M ∈ M_{F} (L, G), since they are differentials with target in the 0th row (see Prop.1, [3]).

COHOMOLOGY OF SPLIT GROUP EXTENSIONS AND CHARACTERISTIC CLASSES 11

Therefore, v_{r}^{r−1}(L′) = v_{r}^{r−1}(L′′) = 0 and we have

v_{r}^{t}(L) = ∑_{i+j=t} ((C_{j}^{′r})_{∗}(v_{r}^{i}(L′)) + (−1)^{i}(C_{i}^{′′r})_{∗}(v_{r}^{j}(L′′)))

= ∑_{i+j=t−r}(C_{j}^{′r})_{∗}(v_{r}^{i+r}(L′)) + ∑_{i+j=t−r}(−1)^{i}(C_{i}^{′′r})_{∗}(v_{r}^{j+r}(L′′))

= ∑_{i+j=t−r} ((C_{j}^{′r})_{∗}(v_{r}^{i+r}(L′)) + (−1)^{i}(C_{i}^{′′r})_{∗}(v_{r}^{j+r}(L′′))).

4. Some Corollaries

Theorem 3.7 has particularly interesting applications if characteristic classes of the ℤGlattices L′, L′′, and L are viewed as obstructions to the vanishing of the differentials in the associated Lyndon-Hochschild-Serre spectral sequences. We use the same notation as before.

Corollary 4.1. Let r ≥ 2. If v_{k}^{i}(L′) = v_{k}^{j}(L′′) = 0 for all i, j ∈ [k, r] and for all k < r, then d_{2}^{s,m} = ⋯ = d_{r−1}^{s,m} = 0 for all s ≥ 0, all m ≤ r, and all M ∈ M_{F} (L, G). Moreover,

v_{r}^{r}(L) = (C_{0}^{′r})_{∗}(v_{r}^{r}(L′)) + (C_{0}^{′′r})_{∗}(v_{r}^{r}(L′′)).

Proof. This is a direct consequence of Theorem 3.7 and the preceding remark.

□

A consequence of Lemma 3.6 is the fact that when the differentials in the Lyndon-Hochschild-Serre spectral sequence {E_{∗}, d_{∗}} are all zero, the same is true for the spectral sequences corresponding to the ℤG-sublattices L′ and L′′. The next corollary gives us a converse.

Corollary 4.2. Suppose v_{p}^{t}(L′) = v_{q}^{t}(L′′) = 0 for all p ≤ dim(L′), all q ≤ dim(L′′), and all t ≥ 0. Then d_{k}^{s,t} = 0 for all M ∈ M_{F} (L, G), all s, t ≥ 0, and all k ≥ 2. Moreover, if in addition {E_{∗}, d_{∗}} has no extension problems, then for every n ≥ 0 and for all M ∈ M_{F} (L, G) we have

H^{n}(Γ, M) = ⨁_{i+j=n} H^{i}(G, H^{j}(L, M)).

Proof. Since v_{p}^{t}(L′) = d_{p}^{′0,t}([id^{′t}]) ∈ E_{p}^{p,t−p+1}(H_{t}(L′, F)), this class lies in the image of the map

d_{p}^{′0,t} : H^{0}(G, H^{t}(L′, H_{t}(L′, F))) → H^{p}(G, H^{t−p+1}(L′, H_{t}(L′, F))).

Note that d_{p}^{′0,t} = 0 when t > dim(L′) or p > t + 1. If p = t + 1, then d_{t+1}^{′0,t} = 0, since it is a differential with a target in the 0th row (see Prop.1, [3]). Therefore, if v_{p}^{∗}(L′) = 0 for all p ≤ dim(L′), then all characteristic classes of the spectral sequence {E′_{∗}, d′_{∗}} are zero. A similar argument shows that all characteristic classes of {E′′_{∗}, d′′_{∗}} are zero when v_{q}^{∗}(L′′) = 0 for all q ≤ dim(L′′). □

Corollary 4.3. Let t ≥ 0. Set Γ′ = L′ ⋊ G and Γ′′ = L′′ ⋊ G. Let φ′ : L′ → Γ′, φ′′ : L′′ → Γ′′, and φ : L → Γ be the natural inclusions.

(a) If φ^{′∗} : H^{m}(Γ′, H_{m}(L′, F)) → H^{m}(L′, H_{m}(L′, F))^{G} and φ^{′′∗} : H^{m}(Γ′′, H_{m}(L′′, F)) → H^{m}(L′′, H_{m}(L′′, F))^{G} are surjective for all m ≤ t, then φ^{∗} : H^{m}(Γ, M) → H^{m}(L, M)^{G} is surjective for all m ≤ t and for all M ∈ M_{F} (L, G).

(b) If φ^{∗} : H^{t}(Γ, M) → H^{t}(L, M)^{G} is surjective, then φ^{′∗} : H^{t}(Γ′, M) → H^{t}(L′, M)^{G} and φ^{′′∗} : H^{t}(Γ′′, M) → H^{t}(L′′, M)^{G} are surjective for all M ∈ M_{F} (L, G).

Proof. To prove (a), we observe that Corollary 2.4 implies v_{r}^{i}(L′) = v_{r}^{j}(L′′) = 0 for all r ≥ 0 and for all i, j ≤ t. Then, by Theorem 3.7, d_{r}^{s,m} = 0 for all r, s ≥ 0, all m ≤ t, and all coefficient modules in M_{F} (L, G). Applying again Corollary 2.4 finishes the proof.

For part (b), let ι^{′∗} : H^{t}(L, M)^{G} → H^{t}(L′, M)^{G}, ϕ^{′∗} : H^{t}(Γ, M) → H^{t}(Γ′, M), and p^{′∗} : H^{t}(L′, M)^{G} → H^{t}(L, M)^{G} be the induced maps of the inclusions ι′ : L′ → L, ϕ′ : Γ′ → Γ, and the projection p′ : L → L′, respectively. Then, we have φ^{′∗} ∘ϕ^{′∗} = ι^{′∗} ∘φ^{∗}. Since ι^{′∗} ∘ p^{′∗} is the identity map on H^{t}(L′, M)^{G}, ι^{′∗} is surjective and hence ι^{′∗} ∘ φ^{∗} is surjective. The previous equality shows that φ^{′∗} is also surjective. Similarly, it follows that φ^{′′∗} is surjective. □

Given an arbitrary finite rank integral ℤG-lattice L, Lieberman’s result (Thm.4, [3]) states that in the associated Lyndon-Hochschild-Serre spectral sequence {E_{∗}, d_{∗}}, for any s, t ≥ 0 and r ≥ 2, the image of the differential d_{r}^{s,t} is a torsion group annihilated by the integers m^{t−r+1}(m^{r−1} − 1) for all m ∈ ℤ. Using this fact, it was proved in [3] that if F is a field of nonzero characteristic p, then d_{r}^{s,t} = 0 for all s, t ≥ 0 and all r < p. The next corollary follows from combining this result with the sum formula of Remark 3.8.

Corollary 4.4. Let F be a field of nonzero characteristic p. Assume L′ and L′′ are ℤG-lattices of finite rank and L = L′ ⊕ L′′.

(a) d_{r}^{s,t} = 0 for all s, t ≥ 0, all r < p, and all M ∈ M_{F} (L, G).

(b) d_{p}^{s,t}(x) = (−1)^{s}y · ∑_{i+j=t−p}((C_{j}^{′p})_{∗}(v_{p}^{i+p}(L′)) + (−1)^{i}(C_{i}^{′′p})_{∗}(v_{p}^{j+p}(L′′)))

for all M ∈ M_{F} (L, G), for all x ∈ E_{p}^{s,t}(M), and for all y ∈ E_{p}^{s,0}(H^{t}(L, M)) such that θ(y) = x.

□

Acknowledgments

I am thankful to Alejandro Adem for suggesting to me to study the cohomology of semidirect product groups and for always giving good advice. I also thank Jim Davis and Karel Dekimpe for their many conversations. Lastly, I would like to thank the referee for many helpful comments and suggestions which led to a better exposition of the results in the article.

References

[1] Adem, A., Ge, J., Pan, J., Petrosyan, N. Compatible Actions and Cohomology of Crystallographic Groups, J. of Algebra 320 (2008), 341–353.

[2] Charlap, L. Vasquez, A., Characteristic classes for modules over groups I, Trans. Amer. Math. Soc. 137 (1969), 533–549.

[3] Sah, C.-H., Cohomology of split group extensions, J. of Algebra 29 (1974), 255–302.

[4] Totaro, B., Cohomology of Semidirect Product Groups, J. of Algebra 182 (1996), 469–475.

Department of Mathematics, Catholic University of Leuven, Kortrijk, Belgium

E-mail address: Nansen.Petrosyan@kuleuven-kortrijk.be