Cohomology of Groups

D. J. Benson Department of Mathematics

University of Georgia Athens GA 30602, USA djb@sloth.math.uga.edu

P. H. Kropholler School of Mathematical Sciences Queen Mary & Westfield College London E1 4NS, England P.H.Kropholler@qmw.ac.uk

1 Introduction

The cohomology of groups is one of the crossroads of mathematics. It has its origins in representation theory, class field theory and algebraic topology, and thanks to the extraordinary work of Quillen, it has led to the modern development of algebraic Ktheory. It is therefore not surprising that wherever one looks in pure mathematics and mathematical physics, one cannot escape its influence. In this survey, we can only hope to give a brief glimpse of the power and beauty of the subject.

For an account of the origins of the subject, we cannot do better than refer the reader to the excellent 1978 survey article of Mac Lane [88]. If we fall short of a full historical account, it is because we cannot hope to compete. We feel obliged, nonetheless, to begin with a few words on the beginnings of the cohomology of groups.

The first appearances only feature the low degree cohomology. Schur [102, 103] (1904, 1907), in his studies of projective representations (i.e., homomorphisms to a projective general linear group, not to be confused with the theory of projective modules), studied what we now write as H^{2}(G, C^{×}). This parametrizes the central extensions of the group by C^{×}. For a finite group, the dual object H_{2}(G, Z) is called the Schur multiplier. In the case of a perfect group, this appears as the extending central subgroup of the universal central extension of G. Schreier [101] (1926) and Baer [12] (1934) generalized this notion to group extensions which are not necessarily central. The same idea appears in the theory of crossed product algebras, as developed by Brauer, Hasse and Noether [37] (1932). Here, the appropriate group is H^{2}(Gal(L/K), L^{×}). These authors proved that all central simple algebras can be described as crossed products. This fact is central to the development of class field theory.

2 Algebraic topology

The theory of cohomology of groups in degrees higher than two really begins with a theorem in algebraic topology. Hurewicz [75] (1936), having just defined the higher homotopy groups of a topological space, proved that for an aspherical space (namely, a path connected space X for which π_{n}(X) = 0 for all n ≥ 2), the fundamental group determines all the homology groups. Although Hurewicz works with homology with integer coefficients, the same methods work in homology or cohomology, and with arbitrary non-twisted or twisted coefficients. One possible approach to group cohomology is to define H_{n}(π_{1}(X), A) = H_{n}(X, A) and H^{n}(π_{1}(X), A) = H^{n}(X, A), for coefficients A.

Hopf [73] (1942) found an algebraic description of the second homology group of an aspherical space as

H_{2}(X, Z) = R ∩ [F, F]/[F, R]

where F is a free group mapping onto π_{1}(X) and R is the subgroup of relations. This is exactly the same formula as Schur had written down for his multiplier nearly forty years earlier.

Based on the work of Hurewicz and Hopf, Eilenberg and Mac Lane [56] (1943) constructed an algebraically defined chain complex K(G) whose homology groups are exactly the homology groups of an aspherical space X with π_{1}(X) = G. The degree n term in this complex is the free Z-module with symbols [g_{1}|g_{2}| ⋯ |g_{n}] with g_{i} ∈ G, and the boundary homomorphism is given by

∂[g_{1}|g_{2}| ⋯ |g_{n}] = [g_{2}| ⋯ |g_{n}] + ∑_{i=1}^{n−1}(−1)^{i}[g_{1}| ⋯ |g_{i}g_{i+1}| ⋯ |g_{n}]
+ (−1)^{n}[g_{1}| ⋯ |g_{n−1}].

This complex may be viewed as the simplicial chains on an “Eilenberg–Mac Lane space” K(G, 1) whose simplices correspond to these symbols, and where the ith face of such a simplex is given by the ith term in the above formula. The universal cover of this space has simplices corresponding to symbols g_{0}[g_{1}|g_{2}| ⋯ |g_{n}], and with boundary homomorphism given by

∂g_{0}[g_{1}|g_{2}| ⋯ |g_{n}] = g_{0}g_{1}[g_{2}| ⋯ |g_{n}] + ∑_{i=1}^{n−1}(−1)^{i}g_{0}[g_{1}| ⋯ |g_{i}g_{i+1}| ⋯ |g_{n}]
+ (−1)^{n}g_{0}[g_{1}| ⋯ |g_{n−1}].

This complex forms a free resolution of Z as a ZG-module. Namely, it is an exact sequence

⋯ → F_{n} → ⋯ → F_{1} → F_{0} → Z → 0

with each F_{i} a free ZG-module.

Hopf [74] (1945) noticed the algebraic analogue of the theorem of Hurewicz, and used it to give the definition of homology of a group G with coefficients in an arbitrary module, by taking a free resolution and forming the quotient by the augmentation ideal of the group algebra. He proved a comparison theorem which showed that the resulting definition is independent of the choice of resolution. In modern language, he described group homology in terms of Tor. Cohomology is similarly described in terms of Ext. The appropriate definitions are as follows. If A is a ring and N and M are A-modules, we form a projective resolution of M as an A-module

⋯ → P_{n} → ⋯ → P_{1} → P_{0} → M → 0.

A projective module is simply a direct summand of a free module, and it is an easy step to generalize from free to projective modules. The characterization in terms of the usual lifting property is all that is needed for the comparison theorem to work; this theorem states that any map of modules extends to a chain map between projective resolutions. Furthermore, any two such chain maps are chain homotopic. Then Tor_{n}^{A}(N, M) is defined to be the nth homology group of the chain complex

⋯ → N ⊗_{A} P_{n} → ⋯ → N ⊗_{A} P_{1} → N ⊗_{A} P_{0} → 0.

In particular, we have Tor_{0}^{A}(N, M) = Hom_{A}(N, M). Applying the comparison theorem to the identity map on M shows that these groups Tor_{n}^{A}(N, M) are independent (up to natural isomorphism) of the choice of projective resolution. It is this statement which may be interpreted as the algebraic analogue of Hurewicz’s statement that the homology groups of an aspherical space are determined by the fundamental group. In these terms, Hopf’s definition amounts to

H_{n}(G, M) = Tor_{n}^{ZG}(Z, M).

In particular, we have H_{0}(G, M) = M^{G}, the fixed points of G on M.

The definition of Tor is symmetric, in the sense that we get the same answer if we form a projective resolution of N:

⋯ → P′_{n} → ⋯ → P′_{1} → P′_{0} → N → 0

and take the homology of the chain complex

⋯ → P′_{n} ⊗_{A} M → ⋯ → P′_{1} ⊗_{A} M → P′_{0} ⊗ M → 0.

The definition of Ext is similar. We may either form an injective resolution of M:

0 → M → I_{0} → I_{1} → ⋯ → I_{n} → ⋯

and define Ext_{A}^{n}(N, M) to be the nth cohomology group of the cochain complex

0 → Hom_{A}(N, I_{0}) → Hom_{A}(N, I_{1}) → ⋯ → Hom_{A}(N, I_{n}) → ⋯

or we may form a projective resolution of N as above and define Ext_{A}^{n}(N, M) to be the nth cohomology group of the cochain complex

0 → Hom_{A}(P′_{0}, M) → Hom_{A}(P′_{1}, M) → ⋯ → Hom_{A}(P′_{n}, M) → ⋯

In these terms, group cohomology is defined by

H^{n}(G, M) = Ext_{ZG}^{n}(Z, M).

Since a short exact sequence of chain complexes induces a long exact sequence of (co)homology groups, we get long exact sequences in (co)homology from short exact sequences of modules.

It is also worth remarking at this stage that if R is a ring of coefficients, then a projective resolution of R as an RG-module may be obtained by tensoring R with a projective resolution of Z as a ZG-module. It follows that if M is an RG-module then

Ext_{RG}^{n}(R, M) ≅ Ext_{ZG}^{n}(Z, M).

Returning to the algebraic topology, we say that a space X is an Eilenberg– Mac Lane space K(G, 1) if X is an aspherical space with the homotopy type of a connected CW-complex with π_{1}(X) ≅ G. Such a space is unique up to homotopy equivalence, and the simplicial chains C_{∗}(K̃(G, 1); R) on its universal cover K̃(G, 1) form a free resolution of the coefficient ring R over the group ring RG. Since

C^{∗}(K(G, 1); R) = Hom_{R}(C_{∗}(K(G, 1); R), R) ≅ Hom_{RG}(C_{∗}(K̃(G, 1); R), R),

we have H^{∗}(K(G, 1); R) ≅ Ext_{RG}^{∗}(R, R) = H^{∗}(G, R).

There are two useful generalizations of this. The first is that if n ≥ 1 and A is an abelian group then a space X is an Eilenberg–Mac Lane space K(A, n) if X has the homotopy type of a connected CW-complex with π_{n}(X) ≅ A and π_{i}(X) = 0 for i ≠ n. It turns out that such a space “represents” cohomology in the sense that there is a natural one-one correspondence between homotopy classes of maps from a CW-complex Y to X and elements of H^{n}(Y; A).

Secondly, if G is a topological group, one can form a classifying space BG for principal G-bundles over a paracompact base space. The total space of the universal principal G-bundle over BG is written EG. It is characterized by the property that it is a contractible space with a free G-action. Thus the loop space ΩBG is homotopy equivalent to the group G. In case G happens to be discrete, BG is an Eilenberg– Mac Lane space K(G, 1), and EG is its universal cover.

3 Cohomological finiteness conditions

The links between algebraic topology and group theory lead naturally to the idea of cohomological finiteness conditions. It is useful to compare this with the idea of a finiteness condition in abstract group theory. The latter notion, very prevalent in the work of Philip Hall and other influential groups theorists some thirty years ago, has had a powerful influence on the study of abstract infinite groups. As a definition, we say that a finiteness condition is any group theoretic property which holds for all finite groups. For example, the properties of being finitely generated or finitely presented are finiteness conditions. Further important examples include residual finiteness, Hopficity, and linearity.

By analogy with this, we can define a cohomological finiteness condition to be any group theoretic property which holds for all those groups G which admit a finite Eilenberg–Mac Lane space, K(G, 1). Finite generation and finite presentation are both examples of cohomological finiteness conditions, but residual finiteness, Hopficity and linearity are not cohomological finiteness conditions.

A group is said to be of type (F) if it satisfies the strongest possible cohomological finiteness condition, namely that there is a finite Eilenberg–Mac Lane space. Examples of such groups include torsion-free arithmetic groups and torsion-free polycyclic-by-finite groups. The case of arithmetic groups was studied in detail by Borel and Serre, [35]. A further very important source of examples is given by torsion-free subgroups of finite index in Coxeter groups. All these examples are also residually finite, Hopfian and linear. But in addition, all finitely generated torsion-free one-relator groups are of type (F), having an Eilenberg–Mac Lane space formed by adjoining a single 2-cell to a finite bouquet of circles, and amongst these there are examples of groups which are non-Hopfian, and in particular not residually finite, and far from linear.

On the other hand there are some cohomological finiteness conditions which are also abstract finiteness conditions. Perhaps the most important of these is the property that the cohomology ring H^{∗}(G, Z) is finitely generated. This plays a major role in the study of cohomology of finite groups, as we shall see subsequently.

There is an important family of cohomological finiteness conditions which are best defined in terms of projective resolutions, and which make sense for modules over any associative ring. Let R be a ring and let M be an R-module. We say that M is of type (FP)_{n} if there is a projective resolution

⋯ → P_{i} → P_{i−1} → ⋯ → P_{1} → P_{0} → M → 0,

in which the P_{i} are finitely generated for all i ≤ n. The module M is said to be of type (FP)_{∞} if the projective resolution can be chosen so that every P_{i} is finitely generated, or in other words, if there is a projective resolution of finite type. The module M is said to have finite projective dimension if the projective resolution can be chosen of finite length, that is, with P_{i} = 0 for all sufficiently large i. In this case the projective dimension of a non-zero module M is defined to be the least n such that there is a projective resolution terminating with P_{n}. The module M is said to be of type (FP) if there is a finite projective resolution, that is a resolution which is simultaneously of finite type and of finite length; and finally, M is said to be of type (FL) if there is a finite free resolution, that is a finite projective resolution in which each P_{i} is free.

It is natural to ask how sensitive these definitions are to the choice of projective resolution, because one imagines choosing a projective resolution step by step. For example, if a module is of type (FP)_{∞} and one has made a choice of the first few steps in a projective resolution so that the projectives so far chosen are finitely generated, is it possible to continue the resolution so that it realizes the (FP)_{∞} property, or is it conceivable that some early choice that was made renders it impossible to maintain finitely generated projectives indefinitely even in the presence of the property (FP)_{∞}? The answer in all cases is that projective resolutions behave as well as one could wish them to. With the exception of type (FL), where the story involves an extra twist, there are two important ways of understanding the stability of these definitions.

The first of these depends on Schanuel’s lemma which states that given two short exact sequences of R-modules

0 → K → P → M → 0

and

0 → L → Q → M → 0

which both end with the same module M, and in which P and Q are projective, then P ⊕ L is isomorphic o Q ⊕ K. An easy induction yields the following embellishment: if

0 → K → P_{i} → P_{i−1} → ⋯ → P_{0} → M → 0

and

0 → L → Q_{i} → Q_{i−1} → ⋯ → Q_{0} → M → 0

are two partial projective resolutions of M then

K ⊕ Q_{i} ⊕ P_{i−1} ⊕ Q_{i−2} ⊕ P_{i−3} ⊕ ⋯ ≅ L ⊕ P_{i} ⊕ Q_{i−1} ⊕ P_{i−2} ⊕ Q_{i−3} ⊕ · · · .

From this it is easy to see that our homological finiteness conditions for projective resolutions behave well. For example, if M is of type (FP)_{∞} and the first partial resolution is taken from a projective resolution of finite type, then any partial resolution 0 → L → Q_{i} → Q_{i−1} → ⋯ → Q_{0} → M → 0 in which the Q_{i} are finitely generated automatically has L finitely generated, and hence any partial projective resolution by finitely generated projectives can always be continued so that all projectives are finitely generated. Except for type (FL), similar remarks can be made of the other finiteness conditions. The embellished Schanuel’s Lemma also shows that M has type (FP)_{∞} if and only if it has type (FP)_{n} for all n, and that M has type (FP) if and only if has type (FP)_{∞} and finite projective dimension.

The second way of understanding the invariance of these finiteness conditions is to interpret them as properties of the cohomology functors Ext_{R}^{∗}(M, −). Since these functors can be defined by using projective resolutions of M, but are at the same time independent of the particular choice of resolution, their properties on the one hand reflect the nature of projective resolutions of M but are, on the other hand, invariant. Projective dimension is the simplest to interpret this way: M has finite projective dimension if and only if the functors Ext_{R}^{n}(M, −) are zero for all sufficiently large n, and if M is a non-zero module of finite projective dimension then its projective dimension is the largest n for which the functor Ext_{R}^{n}(M, −) is non-zero.

To interpret other properties through Ext one needs to consider filtered colimit systems. A filtered colimit system (N_{λ} | λ ∈ Λ) consists of a directed partially ordered set Λ together with a family of R-modules N_{λ} and a compatible system of maps N_{λ} → N_{μ} for each pair λ ≤ μ. We write lim_{⟶} N_{λ} for the colimit of such a system. Now it can be shown that an R-module M is of type (FP)_{n} (resp. (FP)_{∞}) if and only if

lim_{⟶} Ext_{R}^{i}(M, N_{λ}) = 0

for all i ≤ n (resp. for all i) and all (N_{λ}) filtered colimit systems satisfying lim_{⟶} N_{λ} = 0. Notice that in view of this it becomes transparent that M is of type (FP)_{∞} if and only if it is of type (FP)_{n} for all n. Another very important consequence is that for M of type (FP)_{n}, the functors Ext_{R}^{i}(M, −) commute with arbitrary direct sums when i ≤ n.

The story for the property (FL) is more subtle and we postpone this to the next section.

Now suppose that G is a group and that X is the universal cover of a cellular K(G, 1). Then G acts freely on X by covering transformations, and X is contractible, so that the associated augmented chain complex

→ C_{i}(X) → ⋯ → C_{1}(X) → C_{0}(X) → Z → 0

is a free resolution of the trivial module Z over the group ring ZG. Finiteness properties of the chosen K(G, 1) are reflected in this chain complex, so that for example, if one begins with a finite K(G, 1) then the chain complex is a finite free resolution of Z and hence Z is a ZG-module of type (FL). In general, one says that the group G is of type (FL) when this conclusion holds. If one begins with any finite dimensional K(G, 1) then the chain complex has finite length and Z has finite projective dimension. In this case one says that G has finite cohomological dimension, and its dimension can be defined to be the projective dimension of Z over ZG. It is now natural to define the properties (FP)_{n}, (FP)_{∞} and (FP) for groups simply in terms of the corresponding property for the trivial module Z over the group ring.

For groups G these properties are interrelated in the following way:

(F) ⇒ (FL) ⇒ (FP) ⇒ (FP)_{∞} ⇒ ⋯ ⇒ (FP)_{2} ⇒ (FP)_{1}.

Moreover, a group is of type (FP)_{1} if and only if it is finitely generated, and every finitely presented group is of type (FP)_{2}. There is no known example of a group of type (FP)_{2} which is not finitely presented. In general, a group is finitely presented if and only if there is a cellular K(G, 1) with finite 2-skeleton, and so in particular, every group of type (F) is finitely presented as well as being of type (FL). One of the most important facts here is the theorem of Wall [122] which states that the converse holds: every finitely presented group of type (FL) has a finite K(G, 1). This allows one to translate topological questions into algebra, although there remains one major difficulty. From the algebraic point of view, the property (FP) is much easier to grasp than (FL), being equivalent to type (FP)_{∞} together with finite cohomological dimension. However there are few theorems in algebra which enable one to establish that a group has type (FL), and usually this requires powerful geometric methods. The work of Borel and Serre on arithmetic groups is a good example of this: in [34] they use the natural action of the arithmetic group on the symmetric space for the ambient Lie group, and although this provides a fast proof that arithmetic groups have finite cohomological dimension, it takes considerably more work to establish that they have type (F). At present there is no known example of a group of type (FP) which is not of type (FL).

We conclude with a discussion of cohomological dimension, which has proved to be a remarkably subtle group theoretic invariant. Of all the cohomological finiteness conditions introduced in this section, this is the only one which is always inherited by subgroups. Indeed, if H is a subgroup of G then

cd(H) ≤ cd(G)

because the trivial module Z has a projective resolution of length cd(G) as a ZGmodule and this serves equally as a projective resolution over ZH. From another point of view, a group G has finite cohomological dimension if and only if it admits a finite dimensional K(G, 1). Non-trivial finite groups have infinite cohomological dimension, and consequently all groups of finite cohomological dimension are torsion-free. There is a beautiful and fundamental theorem of Serre which states that if H is a subgroup of finite index in a torsion-free group G and H has finite cohomological dimension then so does G, and moreover, in these circumstances cd(H) = cd(G). A good acount of this result can be found in Brown’s book, [38]. We can give only a brief outline of the proof of this result here. First we may assume that H is normal in G. Now G can be identified with a subgroup of the wreath product W of H by the finite quotient Q := G/H. Since H has finite cohomological dimension there is a finite dimensional K(H, 1), and its universal cover X is a space on which H acts freely by covering translations. The wreath product W now acts naturally on the cartesian product Y := X × ⋯ × X of |Q| copies of X with finite isotropy, and through its identification as a subgroup of W, G also acts on Y. The fact that W acts with finite isotropy means that G acts freely, because G is torsion-free. The conclusion is that Y/G is a finite dimensional K(G, 1) and so G has finite cohomological dimension. The fact that cd(G) = cd(H) can deduced with techniques of homological algebra.

Since many groups which arise in nature, such as arithmetic groups and Coxeter groups, are not torsion-free, it may seem unlikely that cohomological dimension could be a useful invariant. However, Serre’s theorem provides a way around this because it leads to the notion of virtual cohomological dimension. A group G is said to have finite virtual cohomological dimension if it has a subgroup of finite index which has finite cohomological dimension. If G is such a group, then vcd(G) is defined to be the cohomological dimension of a torsion-free subgroup of finite index. The point here is that Serre’s theorem guarantees vcd to be well defined: it makes no difference which torsion-free subgroup of finite index is chosen. Arithmetic groups, Coxeter groups, and polycyclic-by-finite groups are amongst many naturally occurring families of groups of finite vcd.

4 Euler characteristics and K-theory

If G is a group of type (F) then its Euler characteristic χ(G) can be defined as the Euler characteristic of any finite K(G, 1). This is an invariant of the group because one has the formula

χ(G) = ∑_{i}(−1)^{i} dim H_{i}(G, Q),

in terms of the rational homology groups. This formula makes sense more generally, provided the rational homology groups are finite dimensional and only finitely many are non-zero. Based on this, K. S. Brown has developed a theory of Euler characteristics for a much larger class of groups. Care has to be taken: it is not simply a matter of adopting the alternating sum formula whenever it makes sense because one wants to preserve the following property which holds for groups of type (F).

If G is a group of type (F) and H is a subgroup of finite index then H is also of type (F) and

χ(H) = |G : H|χ(G).

In view of this, one can define χ(G) for any group G which is virtually of type (F) by the formula

χ(G) := \frac{1}{|G : H|}χ(H),

where H is any subgroup of finite index which is of type (F). In general, this formula yields a rational number, not necessarily an integer. For example, when G is finite, χ(G) = \frac{1}{|G|}. Since arithmetic groups are virtually of type (F), these have well defined Euler characteristics and many interesting calculations have been carried out.

Using Swan’s work on the ranks of projective modules over integral group rings of finite groups, Brown showed that if G is torsion-free and virtually of type (F) then χ(G) is an integer. Now a torsion-free group which is virtually of type (F) is of type (FP), and looking at this from a different point of view, one can use the alternating sum formula to define an integral Euler characteristic for any group of type (FP) so that, because of Swan’s theorem, the finite index formula holds. Hence there is a well behaved rational Euler characteristic which can be defined for any group which is virtually of type (FP). Brown’s theory is more general even than this, although at present the Euler characteristic remains of most interest for groups which are virtually of type (F). We refer the reader to Brown’s book, [38], for a more detailed account of this topic.

Here, we turn briefly to K-theory. For any ring R, let K_{0}(R) denote the Grothendieck group of finitely generated projective R-modules. We write [P] for the class in K_{0}(R) of a projective module P. If M is any R-module of type (FP) then we write [M] for the class

[M] := ∑_{i}(−1)^{i}[P_{i}],

where P_{∗} → M is a finite projective resolution of M. The generalized form of Schanuel’s Lemma shows at once that this does not depend on the choice of finite resolution, and hence every module of type (FP) has a well defined Euler class in K_{0}(R). Given an additive homomorphism, or rank function ρ : K_{0}(R) → Z, we can associate the integer ρ([M]) to M, which can be viewed as a kind of Euler characteristic.

Suppose that G is a group of type (FP). Then the trivial module Z is of type (FP) over ZG, and the Euler class E(G) can be defined to be the class [Z] in K_{0}(ZG). Over the integral group ring, the simplest rank function on projective modules is given by ρ([P]) := rank_{Z}(P ⊗_{ZG} Z), and using this one obtains the formula

ρ(E(G)) = χ(G).

If G is of type (F), or more generally of type (FL) then the Euler class is imply a multiple of the class [ZG]:

E(G) = χ(G)[ZG].

This formula is very significant, because it is not hard to show that when G is of type (FP) then this formula holds if and only if G is of type (FL). In this way the Euler class of a group of type (FP) carries crucial information.

At present, in all known cases, the Grothendieck group K_{0}(ZG) is generated by the class [ZG] when G is torsion-free. If K_{0}(ZG) were always trivial for torsion-free group then it would follow that every group of type (FP) is in fact of type (FL). Thus the question of whether there are groups of type (FP) which are not of type (FL) is really K-theoretic. A great deal of research has been carried out into K-theory. The work of Adem, [1, 2, 3] is especially relevant to our discussion here.

5 Ends and cohomological dimension one

There is an attractive quality to any result in abstract group theory whose proof depends on a cohomological insight. One such result, suggested by Serre, and later proved by Stallings and Swan [112, 117] states that if G is a torsion-free group which has a free subgroup of finite index then G is free. Serre proved an analogous result for pro-p-groups, but using very different techniques.

To understand how Stallings and Swan proved this result, and the way in which cohomological techniques are involved, we need to briefly recount the basis of the Bass-Serre theory of group actions on trees. At a combinatorial level a graph Γ is a quadruple (V, E, ι, τ) comprising a set of vertices V, a set of edges E and two functions ι, τ : E → V which indicate the initial and terminal vertices of each edge. Bass and Serre prefer to work with unoriented graphs, but for this account it seems convenient to work with directed graphs as we have defined them. Any graph Γ = (V, E, ι, τ) can be realized geometrically as a 1-dimensional CW -complex: if I denotes the unit interval then one first forms the disjoint union of V and E × I, and then one identifies each (e, 0) with ιe and each (e, 1) with τe. The graph has an associated chain complex with the group of 1-chains being ZE, the free abelian group on E and the group of 0-chains being ZV, the free abelian group on V. It is useful to consider the augmented chain complex:

0 → ZE → ZV → Z → 0,

because this carries the crucial homological data for the graph. In particular, this sequence is exact at Z if and only if the graph is non-empty, it is exact at ZV if and only if the graph, or rather its realization as a space, is connected , and it is exact at ZE if and only if the graph has no loops, which is to say that the graph is a forest. Thus the chain complex is a short exact sequence if and only if the graph is a non-empty connected forest, or more simply, a tree.

An action of a group G on a graph comprises an action on the vertex and edge sets which is compatible with the initial and terminal vertex maps. If Γ is a G-graph then the associated chain complex is naturally a chain complex of G-modules. One of the most important examples of a G-graph is the Cayley graph of G with respect to a subset X of G. This is defined by setting V := G, E := X × G, ι(x, g) := g, and τ(x, g) := xg, and G acts by right multiplication. It is connected if and only if G is generated by X, and one of the fundamental cornerstones of the Bass-Serre theory states that it is a tree if and only if G is the free group on X. Since the group G always acts freely on its Cayley graphs, then in the case when G is the free group on X, the chain complex is a free resolution of Z over ZG of length one, and this gives an algebraic proof that free groups have cohomological dimension one. Geometrically, one sees this, because a free group has a bouquet of circles as an Eilenberg–Mac Lane space.

A remarkable theorem proved by Stallings [112] for finitely generated groups and generalized by Swan [117] to arbitrary groups states that the converse holds: every group of cohomological dimension one is free. This is a very deep result, but notice that the group theoretic application with which we opened this section follows immediately from it. If G is torsion-free and has a free subgroup of finite index then Serre’s theorem on cohomological dimension together with the basic results from Bass-Serre theory, shows that G has cohomological dimension one, and hence G is free by Stallings-Swan.

In the remainder of this section we discuss Stallings contribution to this area and some of the ideas it has led to. Let G be a finitely generated group and let Γ be the Cayley graph of G with respect to some finite generating set. Then Γ is connected, but it is possible that, if a finite set of edges of Γ are removed then Γ will become disconnected and even that it may have more than one infinite component. The number of ends of G is defined to be the supremum of the number of infinite components of Γ \ F as F runs through all finite sets of edges. It can be shown that this number e(G) is an invariant independent of the choice of finite generating set for G. Moreover, it is known that e(G) can take only one of four possible values: 0, 1, 2 or ∞. It is very easy to see from the definition that e(G) = 0 if and only if G is finite. If G is infinite then there is the classical formula of Hopf:

e(G) = 1 + dim_{F} H^{1}(G, FG),

where F denotes the field of two elements.

Motivated by his work on 3-dimensional manifolds, Stallings proved a remarkable theorem about groups G with e(G) > 1. The theorem concerns splittings of groups. Here, we say that a group G splits over a subgroup H if and only if there is an action of G on a tree with one orbit of edges so that H is an edge stabilizer and so that all vertex stabilizers are proper subgroups of G. Let T be a G-tree with one orbit of edges, and let e be an edge with stabilizer H := G_{e}. There can be at most two orbits of vertices. If ιe and τe are in different orbits then G is isomorphic to the free product of the vertex stabilizers G_{ιe} and G_{τe}, amalgamated over H, and if ιe and τe belong to the same orbit, so that there is a g ∈ G with (τe)g = ιe, then there is only one orbit of vertices and G is isomorphic to the HNN-extension G_{ιe}∗_{H,g}. Stallings theorem [112] asserts that if G is a finitely generated group then e(G) > 1 if and only if G splits over a finite subgroup. One direction of this theorem is not hard. If G does split over a finite subgroup then one can use the associated G-tree to construct almost invariant subsets of G in the following way: let e be an edge of T and let B be the set of g ∈ G such that e points towards (ιe)g. It turns out that B is almost invariant in the sense that for all g ∈ G, the symmetric difference B + Bg is finite. Finite subsets of G naturally correspond to elements of FG, and so the function g ↦ B + Bg can be regarded as a derivation from G to FG which in turn gives rise to a non-trivial element of H^{1}(G, FG). Hopf’s formula now shows that e(G) > 1. The other direction of Stallings theorem is a real tour de force and we cannot give a detailed account here.

Stallings theorem can be used to show that finitely generated groups of cohomological dimension one are free. If G is such a group, then one shows first that e(G) > 1 by means of Hopf’s formula. Therefore G splits over a finite subgroup. Being torsion-free, G is therefore a non-trivial free product of two subgroups. Since these subgroups still have cohomological dimension one, the argument can be repeated, until one concludes that G is free. For infinitely generated groups, additional arguments are needed. Swan solved this problem [117], showing that all groups of cohomological dimension one are free.

Subsequently much further work has been done on the theory of ends. Holt’s paper [72] in which he showed that uncountable locally finite groups have one end is an important landmark. Much more has been discovered by Dunwoody [51, 52]. Dunwoody’s work has led to a theory of accessibility for groups. A group G is called accessible if there is a G-tree with finite edge stabilizers, finitely many orbits of edges, and with all vertex stabilizers having at most one end. Dunwoody proved that all finitely presented groups are accessible, and more recently he has shown that there exist finitely generated groups which are not accessible and that such groups always contain an infinite locally finite subgroup. This last result is a far reaching generalization of a theorem of Linnell asserting that a finitely generated group is accessible if there is a bound on the orders of the finite subgroups.

6 Duality groups

If M is a closed orientable manifold of dimension n then there are isomorphisms between the ith homology and the (n−i)th cohomology for each i. This is the familiar Poincaré duality for manifolds and the isomorphisms can be defined as cap products with the fundamental class of the manifold. In this article we discuss products in cohomology in the next section, and the reader may ask why it is possible to discuss duality between homology and cohomology of groups before products are introduced. While it is true that duality in group cohomology can be described in terms of cap products, it is possible to develop a workable theory which avoids any discussion of products. This approach was first taken by Bieri and Eckmann, [19, 22, 27, 29, 28].

A group G is called an n-dimensional duality group if there is a module D such that for each i there is a natural isomorphism

H^{i}(G, M) ≅ H_{n−i}(G, D ⊗ M),

where D ⊗ M is made into a G-module with the diagonal action.

Let G be such a group. Bieri and Eckmann reformulated the definition of duality in the following way First, it is an immediate consequence of the definition that the cohomology of G vanishes in dimensions greater than n, and that if M is a free ZGmodule, then the cohomology groups H^{i}(G, M) are zero except possibly in dimension n. Hence G is a group of cohomological dimension n. Secondly, since homology always commutes with direct limits, it follows that the cohomology functors also commute with direct limits and hence that G is of type (FP)_{∞}. Thus G is a group of type (FP). Thirdly, it can be shown that D is necessarily torsion-free as an additive group and that it is isomorphic to H^{n}(G, ZG)^{op}. (Here we are viewing cohomology and homology as functors from right ZG-modules to abelian groups, but since ZG is a bimodule, the cohomology group H^{n}(G, ZG) inherits a left module structure. If L is any left ZG-module we write L^{op} for the right module with the same additive group and with G action defined by ℓ · g := g^{−1}ℓ.) Thus an n-dimensional duality group G satisfies three properties:

• G is of type (FP);

• H^{i}(G, ZG) = 0 for i ≠ n; and

• D := H^{n}(G, ZG)^{op} is torsion-free as additive group.

Bieri and Eckmann then showed that these three properties actually imply that G is a duality group with dualizing module D. The Bieri-Eckmann approach avoids using cup or cap products, and has proved invaluable for further study of duality groups.

In the special case when G is a duality group with dualizing module D being infinite cyclic, one says that G is a Poincaré duality group. Here there is a very close link with manifold theory. A Poincaré duality group is called orientable if its action on the dualizing module is trivial, and it is called non-orientable if the action is non-trivial. Moreover, no example is known of a Poincaré duality group which is not the fundamental group of a closed aspherical manifold. In a series of papers, Bieri, Eckmann, Müller and Linnell proved that every 2-dimensional Poincaré duality group is the fundamental group of a closed 2-manifold. Subsequently Thomas, Hillman, Kropholler and Roller [120, 70, 71, 80, 82, 83, 84, 85] have developed the theory of 3-dimensional Poincaré duality groups and there are indications of a close link with Thurston’s geometrization programme for 3-manifolds.

The key idea behind the study of Poincaré duality groups is a generalization of the theory of ends. Scott introduced a notion of ends of a pair (G, H), where G is a group and H is a subgroup. If G is finitely generated then e(G, H) can be defined to be the number of ends of the quotient graph Γ/H where Γ is a Cayley graph for G. He showed that if G splits over H then e(G, H) > 1. However, the converse does not hold. For example, the group G generated by x, y, z subject to relations

x^{2} = y^{3} = z^{7} = xyz

has many free abelian subgroups A such that e(G, A) = 2 but it admits no splitting over any subgroup. This is an interesting example because it is the fundamental group of a closed Seifert fibred 3-manifold, and in particular, it is a Poincaré duality group. Nevertheless, some splitting theorems have been proved in terms of the new end invariant. The first of these is Scott’s theorem that if H ≤ G are finitely generated groups and H is an intersection of subgroups of finite index in G then e(G, H) > 1 if and only if some subgroup of finite index in G splits over H. A second theorem of this kind, due to Kropholler [80] states that if H ≤ G are groups which both have one end [e(G) = e(H) = 1] and if H ∩ H^{g} = 1 for all g ∉ H, then e(G, H) > 1 if and only if G splits over H. There are more delicate versions of these results, and these play a role in understanding 3-dimensional Poincaré duality groups.

A further key ingredient is the beautiful theorem of Strebel [115] which asserts that if G is an n-dimensional Poincaré duality group then every subgroup of infinite index in G has cohomological dimension strictly less than n. This is an important ingredient in the study of 2-dimensional Poincaré duality groups where it shows that every subgroup of infinite index has cohomological dimension ≤ 1 and hence is free by the Stallings-Swan theorem.

Finally we should mention Davis’ paper [50]. Davis provides a way of constructing many examples of closed aspherical manifolds using Coxeter groups. Not only has this led to examples of aspherical manifolds not covered by Euclidean space, but it has also provided many other interesting examples of Poincaré duality groups. Mess has shown in [89] how one can use Davis’ construction to embed certain groups into Poincaré duality groups. In particular, every group of type (F) and of dimension n can be embedded into a (2n+1)-dimensional Poincaré duality group.

7 Products

One advantage of cohomology over homology is the existence of cup products and Yoneda products. Yoneda products are defined in terms of composition of maps between projective resolutions. This gives a map

Ext_{A}^{m}(M_{2}, M_{3}) ⊗ Ext_{A}^{n}(M_{1}, M_{2}) → Ext_{R}^{m+n}(M_{1}, M_{3}).

Cup products are defined in terms of tensor products of resolutions. They are not defined in the same generality as Yoneda products, but they are defined for modules over group rings, or more generally over Hopf algebras. The cup product is a map

Ext_{RG}^{m}(N_{1}, M_{1}) ⊗ Ext_{RG}^{n}(N_{2}, M_{2}) → Ext_{RG}^{m+n}(N_{1} ⊗ N_{2}, M_{1} ⊗ M_{2}),

and hence in particular

H^{m}(G, M_{1}) ⊗ H^{n}(G, M_{2}) → H^{m+n}(G, M_{1} ⊗ M_{2}),

so that H^{∗}(G, R) becomes a ring.^{1} This ring is graded commutative, in the sense that xy = (−1)^{|x||y|}yx, where |x| denotes the degree of an element x.

1Here and elsewhere, unless otherwise stated, tensor products of modules are taken to be over the coefficient ring, with diagonal G-action.

Any cup product may be written as a Yoneda product as follows:

Ext_{RG}^{m}(N_{1}, M_{1}) ⊗ Ext_{RG}^{m}(N_{2}, M_{2}) →^{∪} Ext_{RG}^{n+m}(N_{1} ⊗ N_{2}, M_{1} ⊗ M_{2})

↓ (−⊗M_{2},N_{1}⊗−) ↗ Yoneda
Ext_{RG}^{m}(N_{1} ⊗ M_{2}, M_{1} ⊗ M_{2}) ⊗ Ext_{RG}^{n}(N_{1} ⊗ N_{2}, N_{1} ⊗ M_{2})

However, not every Yoneda product may be written as a cup product. In particular, Yoneda and cup products agree on Ext_{RG}^{∗}(R, R), but even if M is a simple RG-module, Ext_{RG}^{∗}(M, M) need not be graded commutative. Carlson [43] has constructed examples with G finite, R a field of characteristic p and M a simple RG-module, where the rings Ext_{RG}^{∗}(M, M) have complete matrix rings as quotients.

8 Tate cohomology

Motivated by the development of class field theory, Tate introduced a variation on the cohomology of finite groups, Ĥ^{n}(G, M) where n runs over the positive and negative integers, which agrees with ordinary cohomology for n ≥ 1 and is reindexed ordinary homology for n ≤ −2. Long exact sequences for this theory extend in both directions.

If G is a finite group, we define a complete resolution of Z as a ZG-module to be a doubly infinite exact sequence of free modules

⋯ → F_{−n} → ⋯ → F_{−1} → F_{0} → F_{1} → ⋯ → F_{n} → ⋯

with the property that the non-negative part forms a free resolution of Z as a ZGmodule. For example, one may form the Z-dual of a resolution of Z by finitely generated free ZG-modules, shift in degree by one, and splice to the original resolution:

⋯ → Hom_{Z}(F_{n−1}, Z) → ⋯ → Hom_{Z}(F_{0}, Z) ⟶ F_{0} → F_{1} → ⋯ → F_{n} → ⋯

↘ ↗ Z ↗ ↘

0  0

Tate cohomology Ĥ^{n}(G, M) (−∞ ≤ n ≤ ∞) with coefficients in a ZG-module M is then defined to be the cohomology of the doubly infinite complex Hom_{ZG}(F_{n}, M). So for n > 0 this agrees with ordinary group cohomology, while for n < −1, we have Ĥ^{n}(G, M) ≅ H_{−n−1}(G, M).

If H is a subgroup of G, then the inclusion of H in G induces a restriction map in (ordinary or) Tate cohomology

res_{G,H} : Ĥ^{∗}(G, M) → Ĥ^{∗}(H, M)

and a corestriction map

cor_{H,G} : Ĥ^{∗}(H, M) → Ĥ^{∗}(G, M).

In positive degrees, these are the usual restriction and transfer maps, while in negative degrees (regarded as homology groups) the roles are reversed. This has the interesting consequence that transfer may be defined in terms of restriction by dimension shifting past zero.

If H has index n in G, then cor_{H,G} ∘ res_{G,H} is equal to multiplication by n. In particular, this implies that Ĥ^{∗}(G, M) is killed by multiplication by |G|, so that it is entirely torsion. Furthermore, it implies that if P is a Sylow p-subgroup of G, then res_{G,P} injects the p-primary component of Ĥ^{∗}(G, M) into Ĥ^{∗}(P, M). If M is killed by multiplication by p, and M is not a projective F_{p}P -module, then for every value of n ∈ Z, we have Ĥ^{n}(P, M) ≠ 0. This follows from the fact that F_{p}P is self-injective and the fact that every non-trivial F_{p}P -module has a non-zero fixed point. If, on the other hand, M is Z-torsion-free, then the long exact sequence in Tate cohomology arising from the short exact sequence

0 → M ⟶^{p} M → M/pM → 0

shows that if Ĥ^{n}(P, M) is zero for two consecutive values of n then M/pM is a projective F_{p}P -module, hence a projective F_{p}G-module, and so the p-primary part of Ĥ^{∗}(H, M) is zero for every subgroup H of G. If this holds for all primes p, we say that M is cohomologically trivial. By dimension shifting, the assumption that M is Z-torsion-free may be dropped.

9 Class field theory

In Helmut Hasse’s article [69] on the history of class field theory, he states the view that “the sharply profiled lines and individual features of this magnificent edifice seem to me to have lost somewhat of their original splendour and plasticity by the penetration of class field theory with cohomological concepts and methods, which set in so powerfully after the [second world] war”. It is our hope to convince the reader that in fact the introduction of methods from the cohomology of groups only served to enhance the splendor of this subject.

The principal goal of class field theory is to describe the abelian extensions of a number field in terms of the internal structure of the field. This is usually described in terms of reciprocity maps. We shall limit ourselves to discussing that part of class field theory which is most easily dealt with in cohomological terms. For a fuller account of the local case see Serre [110, 111], and for the global theory see Tate [119].

The subject begins with the computation of the Brauer group of a local field. For any field K, the Brauer group Br(K) has as its elements the isomorphism classes of division rings Δ which are finite dimensional over K and which have K as their center. If Δ_{1} and Δ_{2} are two such, then Δ_{1}⊗_{K}Δ_{2} is a full matrix ring over another such division algebra Δ_{3}. This gives Br(K) its group structure, with identity element represented by K itself and with Δ^{op} playing the role of the inverse of Δ. If L is a finite extension of K, we obtain a map Br(K) → Br(L) by tensoring over K with L, and the kernel of this map is written Br(L/K). It consists of the division rings over K split by L. Every such division ring can be written as a crossed product algebra, so that there is an isomorphism between Br(L/K) and H^{2}(G, L^{×}), where G = Gal(L/K).

In the case where K is a local field, it turns out that H^{2}(G, L^{×}) ≅ Z/|G|. Moreover, generators for these cyclic groups can be chosen consistently, in the sense that if H ≤ G then the restriction of the generator of H^{2}(G, L^{×}) to H^{2}(H, L^{×}) is the generator for the latter. We write u ∈ H^{2}(G, L^{×}) for the generator.

Now write Ω^{2}Z for the second kernel in a projective resolution of Z as a ZG-module. This is not uniquely defined, but by Schanuel’s lemma it is uniquely defined modulo adding projective summands. Then we can represent u ∈ H^{2}(G, L^{×}) by a ZG-module homomorphism û : Ω^{2}Z → L^{×}. Adding projective summands to Ω^{2}Z if necessary, we may assume that this map is surjective, say with kernel M. For any H ≤ G, the short exact sequence

0 → M → Ω^{2}Z ⟶^{û} L^{×} → 0

of ZH-modules gives rise to a long exact sequence in Tate cohomology, a portion of which is as follows:

Ĥ^{1}(H, L^{×}) →Ĥ^{2}(H, M) →Ĥ^{2}(H, Ω^{2}Z) →^{u_{∗}} Ĥ^{2}(H, L^{×}) →Ĥ^{3}(H, M) →Ĥ^{3}(H, Ω^{2}Z).

Now by Hilbert’s Theorem 90, Ĥ^{1}(H, L^{×}) = 0. By the above calculation of the Brauer group, the map u_{∗} from Ĥ^{2}(H, Ω^{2}Z) ≅ Ĥ^{0}(H, Z) ≅ Z/|H| to Ĥ^{2}(H, L^{×}) is an isomorphism. Also, Ĥ^{3}(H, Ω^{2}Z) ≅ Ĥ^{1}(H, Z) ≅ Hom(H, Z) = 0 since H is finite. So by the above long exact sequence, we deduce that for all subgroups H of G, Ĥ^{2}(H, M) and Ĥ^{3}(H, M) are zero. Thus M is cohomologically trivial, and so the map u_{∗} from Ĥ^{n}(G, Ω^{2}Z) ≅ Ĥ^{n−2}(G, Z) to Ĥ^{n}(G, L^{×}) (i.e., cup product with u) is an isomorphism for all n ∈ Z.

Let us apply this in the case n = 0. The group Ĥ^{−2}(G, Z) = H_{1}(G, Z) is just the abelianization G_{ab}, while Ĥ^{0}(G, L^{×}) is fixed points modulo transfers. Since we are working multiplicatively, transfer here really means the usual number theoretic norm map, and so we have

G_{ab} ≅ K^{×}/N_{L/K}(L^{×}).

The inverse map to this cup product isomorphism is called the local reciprocity map, or norm residue symbol, and it has an explicit character theoretic characterization which may be found in Serre [110]. This allows the explicit construction of all abelian extensions of K.

In the global case, we must replace the multiplicative group L^{×} by the idèle class group C_{L} = J_{L}/L^{×}. The properties of this module are analogous to the properties of L^{×} in the local case. Namely Ĥ^{1}(G, C_{L}) = 0, and there are compatible isomorphisms Ĥ^{2}(G, C_{L}) ≅ Z/|G|. Thus the generator u ∈ Ĥ^{2}(G, C_{L}) gives rise in the same way as before to a map û : Ω^{2}Z → C_{L} which induces isomorphisms

Ĥ^{n−2}(G, Z) → Ĥ^{n}(G, C_{L}).

In particular, the corresponding reciprocity law is the inverse of the isomorphism

G_{ab} ≅ C_{K}/N_{L/K}(C_{L})

arising from the case n = 0. Details may be found in Tate [119].

Finally, we mention that the notion of a class formation is designed to abstract what makes the above arguments work. See the notes of Artin and Tate [8].

10 The complete cohomology of Mislin and Vogel

Although Tate cohomology was introduced only for finite groups, it was subsequently generalized by Farrell G with vcdG < ∞. As well as Farrell’s paper [61], there is also a useful account in [38]. Farrell’s approach was similar to Tate’s and involved the use of complete projective resolutions.

More recently, Vogel and Mislin have independently discovered a generalized Tate cohomology theory, [67, 90] which works for any group. This theory does not depend on complete resolutions, and has many exciting applications. Mislin was strongly influenced by the paper [63] of Gedrich and Gruenberg in which the authors develop a theory of terminal completions of cohomological functors for certain classes of groups. A different approach to generalized Tate cohomology was discovered by Vogel [67] and by Benson and Carlson, [16]. The paper by Benson and Carlson is ostensibly about Tate cohomology (of finite groups), but they work with definitions which make sense for arbitrary groups and which turn out to yield a theory isomorphic to that of Vogel and Mislin. This theory, which we shall call the complete cohomology, turns out to be ideally suited for proving that certain groups have finite cohomological dimension. We shall denote the cohomology groups by Ĥ^{j}(G, M). Like ordinary cohomology, one can have coefficients in any G-module M, and each cohomology group is functorial in M. The theory shares basic properties with ordinary cohomology, but it also enjoys some distinctive features. One point to emphasize at once is that Ĥ^{j}(G, M) is defined and can be non-zero for all integers j, positive and negative, as indeed must be the case because complete cohomology coincides with Tate cohomology for finite groups.

Some properties of complete cohomology are just as for ordinary cohomology. First there are natural long exact sequences of complete cohomology associated to short exact sequences of coefficient modules.

Thus, for any short exact sequence 0 → A → B → C → 0 of G-modules, there are natural connecting homomorphisms δ: Ĥ^{j}(G, C) → Ĥ^{j+1}(G, A) which, together with functorially induced maps, give rise to a long exact sequence:

⋯ → Ĥ^{j}(G, A) → Ĥ^{j}(G, B) → Ĥ^{j}(G, C) → Ĥ^{j+1}(G, A) → ⋯

Secondly, if G is of type (FP)_{∞} then the functors Ĥ^{j}(G, ) commute with filtered colimits, just as the ordinary cohomology functors do.

There are, however, two properties which distinguish complete cohomology from ordinary cohomology. The first of these is very striking, and of great importance:

Ĥ^{0}(G, Z) = 0 if and only if cdG < ∞.

This stands out because it ensures that complete cohomology need not be identically zero. It is wonderfully convenient that this zeroeth cohomology group with trivial coefficients carries so much information. In general, not surprisingly, Ĥ^{0}(G, Z) is very hard to compute for groups of infinite cohomological dimension. If G is finite then we know that

Ĥ^{0}(G, Z) = Z/|G|Z

but even for polycyclic-by-finite groups, no general formula is known.

The last property we mention, like the first, is really axiomatic. This is built in to complete cohomology when it is defined, and the theory satisfies a universal property in relation to ordinary cohomology subject to this condition: For all integers j, and all projective modules P, the complete cohomology groups Ĥ^{j}(G, P) are zero.

This property does not usually hold for ordinary cohomology. There are exceptions: for example if G is a free abelian group of infinite rank then H^{j}(G, P) is zero for all j (including j = 0) and all projective modules P. For such groups, the complete cohomology and the ordinary cohomology coincide. More generally, if there is an integer j such that the ordinary cohomology vanishes on projectives from dimension j onwards, then the complete cohomology and the ordinary cohomology coincide from that point on.

At the time of writing, there have been only a few applications for complete cohomology, but those that have been discovered strike us as very exciting. One of the best results to emerge is Kropholler’s theorem [81] that soluble and linear groups of type (FP)_{∞} are virtually of type (FP). The proof uses the fact that complete cohomology provides a criterion for a group to have finite cohomological dimension. But in order to prove such a result, one also needs a method of computing Ĥ^{0}(G, Z). The method depends on the fact that both soluble and linear groups admit useful actions on finite dimensional contractible complexes. For linear groups, one needs the methods introduced by Alperin and Shalen when they established their criterion for finite dimensionality. For soluble groups one uses the simpler fact that every infinite finitely generated soluble group has an infinite quotient which is a crystallographic group, and which thus acts on a Euclidean space.

These results can be generalized in various ways. For example, in [81] Kropoller also proves that if G is a torsion-free soluble or linear group then every ZG-module of type (FP)_{∞} has finite projective dimension. It seems likely that complete cohomology will have many further applications in the near future.

On the other hand, these results do not hold for arbitrary groups. Brown and Geoghegan [42, 41] have shown that there exist torsion-free groups of type (FP)_{∞} which have infinite cohomological dimension.

11 Spectral sequences

There are several spectral sequences of importance in group cohomology. Perhaps the most commonly used is the Lyndon–Hochschild–Serre spectral sequence of a group extension. If

1 → N → G → G/N → 1

is a short exact sequence of groups and group homomorphisms, and R is a commutative ring of coefficients, then for any RG-module M there are spectral sequences in homology and cohomology:

E_{pq}^{2} = H_{p}(G/N, H_{q}(N, M)) ⇒ H_{p+q}(G, M),
E_{2}^{pq} = H^{p}(G/N, H^{q}(N, M)) ⇒ H^{p+q}(G, M).

To construct these spectral sequences, we take a projective resolution of R as an R(G/N)-module

⋯ → P_{1} → P_{0} → R → 0

and as an RG-module

⋯ → Q_{1} → Q_{0} → R → 0

and we form the double complexes

E_{pq}^{0} = P_{p} ⊗_{R(G/N)} (Q_{q} ⊗_{R} NM) ≅ (P_{p} ⊗ Q_{q}) ⊗_{RG} M,
E_{0}^{pq} = Hom_{R(G/N)}(P_{p}, Hom_{RN}(Q_{q}, M)) ≅ Hom_{RG}(P_{p} ⊗_{R} Q_{q}, M).

In the spectral sequences of these double complexes, we have

E_{pq}^{1} = P_{p} ⊗_{R(G/N)} H_{q}(N, M),
E_{1}^{pq} = Hom_{R(G/N)}(P_{p}, H^{q}(N, M)),

and so the E_{pq}^{2}, respectively E_{2}^{pq} terms are as given above. Since P_{∗} ⊗_{R} Q_{∗} is a projective resolution of R as an RG-module, the spectral sequences converge to H_{∗}(G, M), respectively H^{∗}(G, M).

Calculations involving the Lyndon–Hochschild–Serre spectral sequence tend to be intricate, and to depend on having available as much extra structure as possible. This involves products, Steenrod operations, restrictions, transfers, and so on. Particularly useful is the fact (Serre [109]) that the Steenrod operations commute with transgressions (i.e., the differentials going from the vertical edge to the horizontal edge) in the cohomology spectral sequence with F_{p} coefficients.

In the case of a central extension of groups, or more generally a central product of groups, there is the Eilenberg–Moore spectral sequence. Namely, if G_{1} and G_{2} are groups and ϕ_{1} : Z ↪ Z(G_{1}), ϕ_{2} : Z ↪ Z(G_{2}) are inclusions of a common central subgroup, then we may form the central product

G_{1} ∘ G_{2} = G_{1} × G_{2}/{(ϕ_{1}(z), ϕ_{2}(z)), z ∈ Z}.

In this situation, there is a pullback diagram of fibrations of Eilenberg–Mac Lane spaces

K(G_{1} ∘ G_{2}, 1) → K(G_{1}/ϕ_{1}(Z), 1)
↓  ↓
K(G_{2}/ϕ_{2}(Z), 1) →  K(Z, 2).

The Eilenberg–Moore spectral sequence of this pullback square is

Tor_{H^{∗}(K(Z,2);R)}^{∗∗}(H^{∗}(G_{1}/ϕ_{1}(Z), R), H^{∗}(G_{2}/ϕ_{2}(Z), R)) ⇒ H^{∗}(G_{1} ∘ G_{2}, R).

The case where G_{1} = ϕ_{1}(Z) applies to any central extension, and is useful for example in calculating the cohomology of finite p-groups. Rusin [99] calculated the mod two cohomology of the groups of order 32 this way.

Another spectral sequence which is of occasional use in finite group cohomology is the Atiyah spectral sequence [9]. This has as its E_{2} term the cohomology ring H^{∗}(G, Z), and converges to the completion R(CG)^{∧} of the complex representation ring R(CG) at the augmentation ideal.

12 Cohomological dimension

We have already discussed groups of low cohomological dimension, and especially aspects of the theory of cohomological dimension one. In higher dimensions many interesting calculations have been carried out, but calculations are often very hard. Infinite soluble groups provide a good source of examples. If G is a soluble group, one says that it has finite Hirsch length if and only if there is a subnormal series 1 = G_{0} ≤ G_{1} ≤ ⋯ ≤ G_{n} = G in which the factors G_{i}/G_{i−1} are either infinite cyclic or locally finite. For such groups, the Hirsch length h(G) is defined to be the number of infinite cyclic factors in any such series. This is easily seen to be an invariant of the group. Stammbach, [113] established that a soluble group has finite cohomological dimension if and only if it is torsion-free of finite Hirsch length. The simplest instance of this is a free abelian group of finite rank, in which case the cohomological dimension is equal to the rank. Gruenberg took this further by showing that if G is a torsion free nilpotent group with finite Hirsch length then cdG is equal to either h(G) or h(G) + 1, and that cdG = h(G) if and only if G is finitely generated. Bieri [20] studied this question for soluble groups in [20], and it became clear that the cohomological dimension of a soluble group was an interesting invariant. In general, for any torsion-free soluble group G of finite Hirsch length, it is the case that either cdG = h(G) or cdG = h(G)+1. After this had been established, it was some years before progress was made. Then Gildenhuys [65] solved the question in the case of Hirsch length 2, coming up with a complete list of the soluble groups G with cdG = h(G) = 2. These are precisely the groups with a presentation on two generators x and y subject to a single relation

y^{−1}xy = x^{m},

where m is a non-zero integer. Gildenhuys and Strebel studied the question further in [66], and finally a complete story for soluble groups emerged in Kropholler’s paper [79]. The result is that if G is a soluble group then the following statements are equivalent

• cdG = h(G) < ∞;

• G is of type (FP);

• G is a duality group;

• G is torsion-free and can be built up from the trivial group by a finite sequence of HNN-extensions and finite extensions.

As in the case of the Stallings-Swan theorem, one sees that the cohomological dimension is intimately related to the structure of the group, and it remains mysterious that this single invariant should carry so much information.

Another important theorem concerns the cohomological dimension of linear groups. Here, in general there is no exact formula for the dimension. Alperin and Shalen [7] proved that a finitely generated linear group (meaning a subgroup of GL_{n}(C)) has finite cohomological dimension if and only if there is a bound on the ranks of the unipotent subgroups. To prove this, they depend on two techniques. The first is a generalized version of the method Borel and Serre used to study arithmetic groups. Secondly, they develop the theory of valuations and use the fact that if A is a finitely generated commutative ring then, associated to each discrete valuation on A one can construct an (n − 1)-dimensional affine building on which SL_{n}(A) acts. Their method draws attention to an important principle in studying cohomological dimension. In general, to show that a group G has finite cohomological dimension it suffices to find a free action of G on a finite dimensional contractible cell complex. In practice, one cannot always easily find free actions, but if G acts on an r-dimensional contractible complex in such a way that all isotropy groups have dimension ≤ n, then cdG ≤ n + r.

13 Elementary abelian subgroups

Evens [58] and Venkov [121] (1961) proved independently that the cohomology ring H^{∗}(G, R) of a finite group G with coefficients in a commutative Noetherian ring R is finitely generated. The two proofs are quite different in nature. Venkov’s is topological, and uses the theory of Chern classes. Evens’ is purely algebraic, and in fact he proves the stronger theorem that for any commutative coefficient ring R, if M is an RGmodule which is Noetherian over R, then H^{∗}(G, M) is Noetherian as a module over H^{∗}(G, R). The ideas in Evens’ proof gave rise to his formulation of the Evens norm map [59] (of course, he didn’t call it that), which is a multiplicative analogue of the usual additive transfer map.

In the case where R = k is a field of characteristic p dividing |G|, Quillen [93] (1971) gave an explicit description of the homogeneous affine variety V_{G}(k) obtained by looking at the maximal ideal spectrum of H^{∗}(G, k). His description involves the structure of the set of elementary abelian subgroups of G.

We begin by explaining that since H^{∗}(G, k) is graded commutative, if p is odd then elements of odd degree square to zero and elements of even degree commute, while if p = 2 then all elements commute. In either case, H^{∗}(G, k) modulo its nil radical is commutative. We write H^{·}(G, k) to denote the even cohomology ring if p is odd, and the whole of H^{∗}(G, k) if p = 2. Then H^{∗}(G, k) and H^{·}(G, k) have “the same” maximal ideals.

If ζ_{1}, …, ζ_{s} are homogeneous elements generating H^{·}(G, k) then the map

k[ζ_{1}, …, ζ_{s}] → H^{·}(G, k)

gives rise to a map of maximal ideal spectra (by the weak Nullstellensatz)

max H^{·}(G, k) → A^{s}(k)

which embeds max H^{·}(G, k) as a closed homogeneous subvariety of affine space A^{s}. Of course, this embedding depends on the choice of generators, but the abstract homogeneous affine variety max H^{·}(G, k) does not.

If ϕ : A → B is a morphism of finitely generated commutative k-algebras, with the property that the kernel is nilpotent and for some value of t, the p^{t}th power of every element of B lies in Im(ϕ), then ϕ^{∗} : max B → max A is bijective. Such a map is called an “inseparable isogeny” or an “F -isomorphism” (F for Frobenius). Quillen’s theorem states that the restriction maps give rise to an inseparable isogeny

H^{∗}(G, k) → lim_{\overleftarrow{E}} H^{∗}(E, k),

so that

lim_{\overrightarrow{E}} V_{E}(k) → V_{G}(k)

is bijective. Here, the limit is taken over the category whose objects are the elementary abelian p-subgroups E of G, and whose morphisms are generated by the conjugations and inclusions in G. This theorem says that an element of H^{∗}(G, k) is nilpotent if and only if its restriction to every elementary abelian subgroup is nilpotent; and moreover, given elements x_{E} ∈ H^{n}(E, k) for each elementary abelian subgroup E, consistent under conjugations and restrictions, there is an element x ∈ H^{np^{a}}(G, k) for some a ≥ 0 such that for each E we have res_{G,E}(x) = X_{E}^{p^{a}}.

Five years later, Chouinard [48] (1976) proved that a kG-module is projective if and only if its restriction to every elementary abelian p-subgroup E of G is projective. The connection between this and Quillen’s theorem was not apparent until Alperin and Evens [6] (1981) formulated the notion of the complexity c_{G}(M) of a finitely generated kG-module M, and proved that it is equal to the maximal complexity of a restriction of M to an elementary abelian p-subgroup E of G:

c_{G}(M) = max_{E≤G} c_{E}(M ↓_{E}).

The definition of complexity is that if ⋯ → P_{1} → P_{0} → M → 0 is a minimal projective resolution of M, then c_{G}(M) is the smallest integer c ≥ 0 such that

lim_{n→∞} dim_{k} P_{n}/n^{c} = 0.

The fact that there is such a value of c follows from Evens’ finite generation theorem. It turns out that M is projective if and only if c_{G}(M) = 0, and M is a direct sum of a projective and a periodic module (a module is said to be periodic if its minimal resolution repeats) if and only if c_{G}(M) = 1 (Eisenbud [57]; see also the beginning of the next section).

Another way to interpret the complexity is as follows. By Evens’ finite generation theorem, Ext_{kG}^{∗}(M, M) is a finitely generated module over the image of the map

H^{∗}(G, k) = Ext_{kG}^{∗}(k, k) ⟶^{⊗M} Ext_{kG}^{∗}(M, M)

given by tensoring exact sequences with M. We write I_{G}(M) for the kernel of this map. Then since Ext_{kG}^{∗}(M, S) is finitely generated as a module over Ext_{kG}^{∗}(M, M) for each simple module S, it follows that c_{G}(M) is the least integer c such that

lim_{n→∞} dim_{k} Ext_{kG}^{n}(M, M)/n^{c} = 0,

and is therefore equal to the Krull dimension of Ext_{kG}^{∗}(M, M), or equivalently the Krull dimension of H^{∗}(G, k)/I_{G}(M). So the case M = k of the Alperin–Evens theorem may be interpreted as Quillen’s statement that the Krull dimension of H^{∗}(G, k) is equal to r_{p}(G).

The work of Alperin and Evens, together with some work of Carlson [44, 45] on rank varieties of modules for elementary abelian groups, led to the formulation of the notion of varieties for modules. If M is a finitely generated kG-module, we define V_{G}(M) to be the closed homogeneous subvariety of V_{G}(k) determined by the ideal I_{G}(M); namely the subset consisting of the maximal ideals containing I_{G}(M). The following is a list of properties of the varieties V_{G}(M):

(i) The dimension of V_{G}(M) is equal to the complexity of M. In particular, V_{G}(M) = {0} if and only if M is projective, and V_{G}(M) is a finite union of lines through the origin if and only if M is a direct sum of a projective and a periodic module.

(ii) V_{G}(M_{1} ⊕ M_{2}) = V_{G}(M_{1}) ∪ V_{G}(M_{2}).

(iii) V_{G}(M_{1} ⊗ M_{2}) = V_{G}(M_{1}) ∩ V_{G}(M_{2}).

(iv) If 0 → M_{1} → M_{2} → M_{3} → 0 is a short exact sequence of finitely generated kG-modules, then V_{G}(M_{i}) ⊆ V_{G}(M_{j}) ∪ V_{G}(M_{k}) whenever {i, j, k} = {1, 2, 3}.

(v) Write Ω(M) for the kernel of the projective cover of M. Then V_{G}(Ω(M)) = V_{G}(M).

(vi) (Avrunin–Scott [11]) The map lim_{\overrightarrow{E}} V_{E}(M) → V_{G}(M) induced by Quillen’s map

lim_{\overrightarrow{E}} V_{E}(k) → V_{G}(k) is an inseparable isogeny.

(vii) If 0 ≠ ζ ∈ H^{n}(G, k) is represented by a cocycle ζ̂ : Ω^{n}(k) → k, write L_{ζ} for the kernel of ζ̂. Then V_{G}(L_{ζ}) is the closed homogeneous hypersurface V_{G}⟨ζ⟩ given by regarding ζ as a polynomial function on V_{G}(k).

(viii) (Carlson’s connectedness theorem [46]) If V_{G}(M) = V_{1} ∪ V_{2}, where V_{1} and V_{2} are closed homogeneous subvarieties intersecting in the origin, then M decomposes as M_{1} ⊕ M_{2} with V_{G}(M_{1}) = V_{1} and V_{G}(M_{2}) = V_{2}. In particular, if M is indecomposable then V_{G}(M) is (projectively) connected.

It is worth remarking that it follows from (ii) and (vii) that every closed homogeneous subvariety of V_{G}(k) is the variety of some finitely generated module; namely a tensor product of suitable L_{ζ}’s.

Recently (Benson, Carlson and Rickard [17, 18]), it has been realized that much of what is described in this section can be extended to infinitely generated modules. A kGmodule M is said to have complexity at most c if every map from a finitely generated kG-module to M factors through a finitely generated module of complexity at most c.

The complexity of a module is equal to the maximal complexity of a restriction to an elementary abelian subgroup, just as in the finitely generated case.

Instead of a single variety, an infinitely generated module has a collection of varieties. If M itself is finitely generated, then V_{G}(M) consists of the closed homogeneous subvarieties of V_{G}(M), but for an infinitely generated module, V_{G}(M) need not have a unique maximal element. Nonetheless, elements of V_{G}(M) certainly have a maximal dimension, and this is equal to the complexity of M.

14 Multiple complexes

If M is a finitely generated kG-module with the property that V_{G}(M) is one dimensional, then we may choose an element 0 ≠ ζ ∈ H^{n}(G, k) (for some n > 0) satisfying V_{G}⟨ζ⟩ ∩ V_{G}(M) = {0}. Then L_{ζ} ⊗ M is projective, and so the exact sequence

0 → L_{ζ} ⊗ M → Ω^{n}(k) ⊗ M → M → 0

shows that Ω^{n}(k) ⊗ M is isomorphic to a direct sum of M with a projective module. It follows easily from Schanuel’s lemma that Ω^{n}(k) ⊗ M is also isomorphic to a direct sum of Ω^{n}(M) with a projective module, and so M is a direct sum of a periodic and a projective module. It should be mentioned that this is not the original proof of Eisenbud’s theorem [57], but it is the one which generalizes to a “multiple periodicity” theorem for arbitrary finitely generated modules (Benson and Carlson [14]).

We begin with a construction. If ζ ∈ H^{n}(G, k), we form the pushout

0  0
↓  ↓
L_{ζ} =  L_{ζ}
↓  ↓
0 → Ω^{n}(k) → P_{n−1} → ⋯ → P_{0} → k → 0
↓  ↓  ↓  ↓
0 → k → P_{n−1}/L_{ζ} → ⋯ → P_{0} → k → 0
↓  ↓
0  0

Truncate the bottom row of this diagram by removing the copies of k at the beginning and the end. The resulting complex C_{ζ} has length n − 1. It has one dimension of homology in degrees zero and n − 1, and is otherwise exact. This complex may be spliced to itself infinitely often to form a periodic complex

⋯ → P_{n−1}/L_{ζ} → ⋯ → P_{0} → P_{n−1}/L_{ζ} → ⋯ → P_{0} → 0

resolving the trivial module. All the modules except the copies of P_{n−1}/L_{ζ} are projective. We write C_{ζ}^{(∞)} for this complex.

If M is a finitely generated kG-module of complexity c, then the Noether normalization lemma implies that we can find a polynomial subring k[ζ_{1}, …, ζ_{c}] ⊆ H^{∗}(G, k) generated by homogeneous elements ζ_{i} ∈ H^{n_{i}}(G, k), which injects into Ext_{kG}^{∗}(M, M), and over which the latter is finitely generated as a module. Under these conditions, the module

P_{n_{1}−1}/L_{ζ_{1}} ⊗ ⋯ ⊗ P_{n_{c}−1}/L_{ζ_{c}} ⊗ M

is projective, and so the complex

C_{ζ_{1}}^{(∞)} ⊗ ⋯ ⊗ C_{ζ_{c}}^{(∞)} ⊗ M

is a “c-fold multiply periodic” projective resolution of M. It has the same polynomial rate of growth as the minimal resolution, though it is not in general minimal.

In case M = k, the trivial module, the complexity is r = r_{p}(G), the maximal rank of an elementary abelian p-subgroup of G. In this case, the elements ζ_{1}, …, ζ_{r} are a homogeneous system of parameters for the cohomology ring. The complex

C = C_{ζ_{1}} ⊗ ⋯ ⊗ C_{ζ_{r}}

may be thought of as a period hypercube for the above constructed resolution. It is a finite Poincaré duality complex of projective modules, in the sense that it is homotopy equivalent to its graded dual, shifted in degree by s = ∑_{i=i}^{r}(n_{i} − 1).

A great deal of information about group cohomology may be obtained from the equivariant cohomology spectral sequence of the complex C:

E_{2}^{pq} = Ext_{kG}^{p}(H_{q}(C), k) ⇒ H^{p+q}(Hom_{kG}(C, k)).

For example, if the cohomology ring happens to be Cohen–Macaulay, then this spectral sequence is effectively the Koszul complex, and it converges to H^{∗}(G, k)/(ζ_{1}, …, ζ_{r}). It follows that this finite quotient ring satisfies Poincaré duality with dualizing degree s, which means that the Poincaré series p_{G}(t) = ∑_{j=0}^{∞} dim_{k} H^{j}(G, k) satisfies the functional equation

p_{G}(1/t) = (−t)^{r_{p}(G)}p_{G}(t).

15 Calculations

In this section, we describe some of the calculations which have been carried out in finite group cohomology, and their theoretical impact. We begin with Quillen’s calculation [95] of the cohomology of the finite general linear groups H^{∗}(GL(n, F_{q}), F_{l}), where q = p^{a} is a prime power, l ≠ p is a prime. He showed that there is a map from BGL(n, F_{q}) to the homotopy fixed point set F Ψ^{q} of the Adams operation Ψ^{q} : BU → BU which induces an epimorphism in cohomology away from the prime p, and this enabled him to give explicit generators and relations for the mod l cohomology. He was not able to calculate the cohomology at the prime p (nobody since has managed this either), but he was able to show that as n gets larger, the cohomology in any particular degree is eventually zero.

This calculation led him to the definition of the plus construction (see for example Gersten [64]), which is a procedure for killing a perfect normal subgroup of the fundamental group of a space by adding 2-cells and 3-cells, without altering the cohomology. He observed that there is a homotopy equivalence between BGL(∞, F_{q})^{+} (the perfect normal subgroup in this case is the derived subgroup of GL(∞, F_{q}) = lim_{n→∞} GL(n, F_{q})) and the homotopy fixed point set of Ψ^{q} on BU. He used this as one of the motivations for his definition of algebraic K-theory: if Λ is a ring then the algebraic K-groups of Λ are defined by

K_{i}(Λ) = π_{i}(BGL(∞, Λ)^{+}) (i ≥ 1).

Again, the plus construction is with respect to the derived subgroup of GL(∞, Λ), which is the perfect subgroup E(Λ) generated by the elementary matrices, which differ from the identity matrix in a single off-diagonal entry. The above calculation of the cohomology of the finite general linear groups then proves that

K_{2j−1}(F_{q}) = Z/(q^{2j} − 1) K_{2j}(F_{q}) = 0.

In general, the group K_{1}(Λ) is isomorphic to GL(∞, Λ)/E(Λ), and K_{2}(Λ) is isomorphic to H_{2}(E(Λ), Z).

Analogous calculations have been made for the finite unitary, orthogonal and symplectic groups by Fiedorowicz and Priddy [62], and for the Chevalley groups of exceptional Lie type by Kleinerman [78].

Another example of the plus construction comes from the cohomology of the symmetric groups. The homology of the infinite symmetric group Σ_{∞} = lim_{→} Σ_{n} was calculated by Nakaoka [91]. Based on this, together with work of Dyer and Lashof [53], Priddy proved [92] that there is a canonical map BΣ_{∞}×Z → Ω^{∞}S^{∞} = lim_{→} Ω^{n}S^{n}, inducing isomorphisms in mod p and integral homology. The induced map (BΣ_{∞})^{+} × Z → Ω^{∞}S^{∞} is then a homotopy equivalence. Here, the plus construction is performed with respect to the infinite alternating group, which is a perfect subgroup of index two.

The example of Quillen above splits off this one, essentially as the image of the Jhomomorphism (away from p).

In the case of a group G with perfect derived group, the plus construction BG^{+} with respect to the derived group is homotopy equivalent to the Bousfield–Kan Zcompletion [36] Z_{∞}BG.

If G is a finite group, then the Bousfield–Kan Z-completion of BG is homotopy equivalent to the product of the F_{p}-completions, as p runs over the prime divisors of |G|. The fundamental group π_{1}(F_{p})_{∞}BG is equal to G/O^{p}(G), the largest p-factor group of G, so π_{1}Z_{∞}BG is the largest nilpotent quotient of G. It is an interesting question in general to ask what sort of spaces one obtains by looking at the loop spaces Ω(F_{p})_{∞}BG. For example, when G is a perfect group with a cyclic Sylow p-subgroup P of order p^{r}, then this space is the homotopy fiber of a self-map of degree p^{r} of a 2e − 1-sphere (Cohen [49]). Here, e = |N_{G}(P) : C_{G}(P)|.

A theorem of Kan and Thurston [77] says that given any topological space X, one can find a group G and a homology equivalence ϕ : BG → X (with arbitrary coefficients). The proof is constructive, but produces very large groups in general. Letting N denote the kernel of the epimorphism ϕ_{∗} : G → π_{1}(X), it follows that N is a perfect normal subgroup of G, and there is an induced weak homotopy equivalence from BG^{+} (with respect to N) to X.

Various other calculations should also be mentioned. Quillen [94] calculated the mod two cohomology of the extraspecial 2-groups. The analogous computation for odd primes still eludes us, though some progress has been made. A survey of work on the cohomology of extraspecial groups can be found in [15]. Rusin [99] has computed the mod two cohomology of all the groups of order 32. The cohomology of a number of the sporadic simple Mathieu groups at the prime two has been calculated: that of M_{11} by Benson and Carlson [13], of M_{12} by Adem, Maginnis and Milgram [4], and of M_{22} by Adem and Milgram [5].

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