The University of Chicago

Eilenberg-Mac Lane Algebras and Their Computation

An Invariant Description of H(Π, 1)

a dissertation submitted to

the faculty of the division of the physical sciences

in candidacy for the degree of doctor of philosophy department of mathematics

by ross m. hamsher chicago, illinois

august, 1973

Document prepared by Dave Benson and Sharon Southwick, Sept 2000

Contents

1. Algebraic Techniques for Computing H(Π) 2 1A. The Normalized Bar Construction 2 1B. The Theory of Cross Effects 3 2. Homology Operations on H(Π) 5 2A. Torsion-free DG-rings with Divided Powers 5 2B. Bockstein Homomorphisms 7 2C. Caps 8 2D. Torsion Generators and Their Relations 10 2E. Auxiliary Relations in H(Π) 12 3. A Description of the Ring H(Π) by Generators and Relations 13 3A. Statement of the Main Result: The Functor R(Π) 13 3B. A Reduction to Consideration of Finite Groups 14 3C. Cross Effects of R(Π) for Torsion Groups Π 15 3D. Ring Generators for R(Π) When Π is Finite 17 3E. Group Generators for R(Π) When Π is Finite 18 3F. Calculation of H(Π) for Finite Groups Π 19 4. Generators and Relations for the Groups H_{k}(Π) 23 4A. Relations in the Groups H_{k}(Π) 23 4B. A Presentation of H_{k}(Π): The Functor S_{k}(Π) 26

1. Algebraic Techniques for Computing H(Π)

1A. The Normalized Bar Construction.

We now very briefly introduce the normalized bar construction, which we dnote by B(Π), on the integral group ring of Π, where Π is an abelian group written additively. A very complete account of these complexes is to be found in (Eilenberg and Mac Lane 1953), where the complexes are denoted by A_{N}(Π, 1). B(Π) is the free nonnegative complex with generators all cells:

[x_{1}, …, x_{n}], x_{i} ∈ Π, x_{i} ≠ 0. (1.1)

The dimension of the cell (1.1) in n. There is only one zero-dimensional cell, the empty cell, written [ ].

The boundary ∂ for B(Π) is defined by the formula

(1.2) ∂[x_{1}, …, x_{n}] = [x_{2} …, x_{n}] + ∑_{i=1}^{n−1}(−1)^{i}[x_{1}, …, x_{i} + x_{i+1}, …, x_{n}]

+ (−1)^{n}[x_{1}, …, x_{n−1}],

where in interpreting this formula a cell with one entry zero is to be zero.

A (p, q) shuffle of the set of integers {1, …, p+q} is a permutation σ such that:

σi < σj if 1 ≤ i < j ≤ p or p + 1 ≤ i < j ≤ p + q (1.3)

Then if [x_{1}, …, x_{p}] and [x_{p+1}, …, x_{p+1}] are any two cells of B(Π), their (shuffle) product is defined by:

[x_{1}, …, x_{p}] ∗ [x_{p+1}, …, x_{p+q}] = ∑_{σ}(−1)^{ε(σ)}[x_{σ(1)}, …, x_{σ(p+q)}], (1.4)

where σ ranges over all (p, q) shuffles of {1, …, p + q}, and ε(σ) is the sign of the permutation σ.

With these definitions, (B(Π), ∂, ∗) is a free DG-ring, and [ ] is the identity for multiplication. B(Π) is a functor, where a homomorphism f : Π → Π′ yields a homomorphism of DG-rings, B(f) : B(Π) → B(Π′), which sends a cell (1.1) in B(Π) to the cell [fx_{1}, …, fx_{n}] in B(Π′). Eilenberg and Mac Lane (1953) show that the homology ring functor H(B(Π)) is H(Π), and we adopt this as our definition of H(Π).

The remaining fact that we shall need about B(Π) is the direct sum (or product) theorem of Eilenberg and Mac Lane (1954, Th. 6.1a). Let Π_{1}, …, Π_{n} be abelian groups. Then there is a system of DG-ring homomorphisms (which for fixed n yields a transformation natural in Π_{1}, …, Π_{n})

f(Π_{1}, …, Π_{n}) : B(Π_{1} + ⋯ + Π_{n}) → B(Π_{1}) ⊗ ⋯ ⊗ B(Π_{n}), (1.5)

with the properties that it induces an isomorphism of homology rings, and, for c_{i} ∈ B(Π_{i}), 1 ≤ i ≤ n:

f(c_{1} ∗ ⋯ ∗ c_{n}) = c_{1} ⊗ ⋯ ⊗ c_{n}. (1.6)

an invariant description of H(Π, 1) 3

Note that in (1.5) and elsewhere in this paper, the symbol + written between two abelian groups stands for their direct sum.

1B. The Theory of Cross Effects.

Here we shall give the basic definitions and state the relevant facts on cross effects; a complete description of this notion is given in the original account of Eilenberg and Mac Lane (1954, §§8-10). Cross effects are useful in considering an arbitrary functor U from an abelian category A to an abelian category B and especially in establishing that a transformation from one such functor to another is a natural isomorphism (or epimorphism or monomorphism).

We first have the preliminary notions of deviation operations and of deviations of a function from one abelian group to another. Let A be an abelian group (written multiplicatively here) and ZA, its integral group ring. We define a binary operation ⊤ (called the deviation operation) on ZA. For a_{1}, a_{2} ∈ A, we have:

a_{1}⊤a_{2} = a_{1}a_{2} − a_{1} − a_{2}. (1.7)

The operation is then extended to make it a biadditive operation on ZA. The operation ⊤ is commutative and also associative; in view of the latter, we may drop parentheses in formulas involving iterations of the operation ⊤.

Let f be a function from A to an abelian group B, where B is written additively. Extend this map f to a homomorphism (also denoted by f) from ZA to B. Then the deviation functions are the functions,

f(a_{1}⊤ … ⊤a_{n}) : A × ⋯ × A (n factors) → B.

For such functions one has the expansion:

(1.8) f(a_{1}⊤ … ⊤a_{n}) = f(a_{1} … a_{n}) − f(a_{2} … a_{n}) − f(a_{1}a_{3} … a_{n}) − ⋯ − f(a_{1} … a_{n+1}) + ⋯ + (−1)^{n}f(a_{1}) + ⋯ + (−1)^{n}f(a_{n}).

It would be useful in proving the statements to be made below to note that if f(1) = 0, then f(a_{1}⊤ … ⊤a_{n}) = 0 when at least one a_{i} is the identity element of A. This follows, since 1⊤x = −1 for x ∈ ZA. The deviations of f measure the departure of f from being a homomorphism. For instance, f(a_{1}⊤a_{2}) vanishes identically if, and only if, f is a homomorphism; f(a_{1}⊤a_{2}⊤a_{3}) vanishes identically if, and only if, f(a_{1}⊤a_{2}) is a biadditive; and so on.

Let U: A → B be a functor, where A and B are abelian categories. U is uniquely up to natural isomorphism the direct sum of subfunctors U_{0} and Ū where U_{0} is naturally isomorphic to a constant functor, and Ū is zero preserving. For A ∈ A one has:

(1.9)  U_{0}(A) = Im[U(0) → U(A)], and
(1.10)  Ū(A) = Ker[U(A) → U(0)].

U_{0} is the zeroth cross effect of U. The nth cross effects of U, n > 0, are all derived from Ū, Ū itself is defined to be the first cross effect of U. The nth cross effect of U is

Ū(A_{1}, …, A_{n}) : A × ⋯ × A (n factors) → B.

To define it, let A = A_{1} + ⋯ + A_{n}, and let ψ_{i} ∈ Hom(A, A) be projection on the ith coordinate, 1 ≤ i ≤ n. Ū gives a function from Hom(A, A) to Hom(Ū(A), Ū(A)) to which the previously defined notion of deviations may be applied. We make the definition:

Ū(A_{1}, …, A_{n}) = Im[Ū(ψ_{1}⊤ … ⊤ψ_{n})]. (1.11)

Given n morphisms, f_{1} : A_{i} → A′_{i} one defines the morphism

Ū(f_{1}, …, f_{n}) : Ū(A_{1}, …, A_{n}) → Ū(A′_{1}, …, A′_{n}) (1.12)

to be the map induced by:

Ū(f_{1} + ⋯ + f_{n}) : Ū(A_{1} + ⋯ + A_{n}) → Ū(A′_{1} + ⋯ + A′_{n}).

(1.11) and (1.12) complete the definition of the higher cross-effect functors of U. Since Ū is zero preserving, the cross-effect object (1.11) is zero if any A_{i} is zero, and the cross-effect morphism (1.12) is zero if any f_{1} is zero.

We should note that, for the functor H(Π), the seemingly ambiguous notation H_{0}(Π) stands either for the zero-dimensional elements or for the zeroth cross-effect functor; however, these two subfunctors are the same. H̄(Π) consists of the elements of positive dimension in H(Π). The higher cross-effect functors are naturally graded; indeed, for n > 0,

H̄(Π_{1}, …, Π_{n}) = ∑_{k=1}^{∞} H_{k}(Π_{1}, …, Π_{n}). (1.13)

(In writing the righthand side of (1.13), we have adopted the convention that when a functor is zero preserving, we omit the bars above cross-effect functors.)

The basic facts on cross effects is that, with notations as above, there are isomorphisms (natural in A_{1}, …, A_{n}):

Ū(A) ≅ ∑_{σ} Ū(A_{σ_{1}}, …, A_{σ_{k}}) (1.14)

where σ = {σ_{1}, …, σ_{k}} ranges over all nonempty subsets of {1, …, n}. We shall briefly describe the proof. If μ_{σ} : A_{σ_{1}} + ⋯ + A_{σ_{k}} → A is the natural injection, then Ū(μ_{σ}) is a monomorphism. The injection of Ū(A_{σ_{1}}, …, A_{σ_{k}}) into Ū(A) provided by the isomorphism (1.14) is to be that induced by Ū(μ_{σ}). The image of Ū(A_{σ_{1}}, …, A_{σ_{k}}) in Ū(A) is easily seen to be the image of the endomorphism of Ū(A) : E_{σ} = Ū(ψ_{σ_{1}}⊤ … ⊤ψ_{σ_{k}}). One shows by direct calculation that the E_{σ}’s, σ a nonempty subset of {1, …, n}, form a set of orthogonal idempotents, with sum 1, in the ring Hom(Ū(A), Ū(A)); (1.14) follows. Also, when B is a category of sets with structure, an element of Ū(A) lies in Ū(A_{σ_{1}}, …, A_{σ_{k}}) if, and only if, it is left fixed by E_{σ}.

Let U and V be functors from A to B, and let α be a natural transformation from U to V. Then, by restriction, α induces natural transformations:

α_{0} : U_{0}(A) → V_{0}(A) and (1.15)

ᾱ(A_{1}, …, A_{n}) : Ū(A_{1}, …, A_{n}) → V̄ (A_{1}, …, A_{n}). (1.16)

Because of (1.14) α is a natural isomorphism (or epimorphism or monomorphism) if, and only if, the transformation (1.15) and all the transformations (1.16) are natrual isomorphisms (or epimorphisms or monomorphisms).

To illustrate a similar use of (1.14), let A be the category of finite abelian groups. Since a finite group Π may be written as a direct sum of cyclic groups so arranged that the order of one divides the order of the next, to show that a natural transformation α is a natural isomorphism (or epimorphism or monomorphism), it would suffice to show that α_{0}(0) and ᾱ(Π_{1}, …, Π_{n}) (where each Π_{i} is cyclic with order dividing that of Π_{i+1}) are isomorphisms (or epimorphisms or monomorphisms).

Using the cross-effect functors, one may define the (possibly infinite) degree of a functor U. The degree of U is n if the (n + 1)st, but not the nth, cross-effect functor vanishes identically. The zero functor has degree minus one, while if no cross effect of U vanishes, the degree is infinity. A functor of degree zero is naturally isomorphic to a constant functor; a zero-preserving functor of degree one is simply an additive functor, while the general functor of degree one is naturally isomorphic to the direct sum of a constant functor and an additive functor; and so on. Clearly, the degree of a functor is analogous to the degree of a polynomial. Also, functors of all degrees exist. Pertinent to our discussion is the fact — easily deduced from the Künneth formula and the direct sum theorem (section 1A) — that H_{k}(Π) has degree k; hence, H(Π) has degree infinity.

2. Homology Operations on H(Π)

2A. Torsion-free DG-rings with Divided Powers.

Let (G, ∗) be a graded ring. Then a system of divided powers, {γ_{i}}_{i=0}^{∞}, for G (cf. Cartan 1967, Exp. 7) is a sequence of functions, with values in G, satisfying the following properties.

(2.1) γ_{i}(x) is defined for x of positive, even dimension in G. γ_{0}(x) = 1,
γ_{1}(x) = x, and deg γ_{i}(x) = i deg(x).

γ_{p}(x) ∗ γ_{q}(x) = (p, q)γ_{p+q}(x), where (p, q) = \frac{(p + q)!}{p!q!}. (2.2)

γ_{r}(x + y) = ∑_{p+q=r} γ_{p}(x)γ_{q}(y). (2.3)

(2.4) For i ≥ 2, γ_{i}(x ∗ y) = 0 if deg x and deg y are odd.

For all i, γ_{i}(x ∗ y) = x^{i}γ_{i}(y) if deg x and deg y are even, and deg y > 0.

If G_{1} and G_{2} are graded rings with divided powers, a homomorphism f, of degree zero, preserves divided powers if fγ_{k}(x) = γ_{k}(f(x)) for all x of positive, even dimension in G_{1}.

If (G, ∗, ∂) is a DG-ring, we say G is a DG-ring with divided powers if a sequence of functions {γ_{i}}_{i=0}^{∞}, as above, is given for G, which satisfy (2.1)–(2.4), and in addition:

∂γ_{i}(x) = ∂x ∗ γ_{i−1}(x), i ≥ 1; and (2.5)

(2.6) for i ≥ 1, γ_{i}(b) is a boundary if b is a boundary of positive, even dimension

By (2.5) the divided power of a cycle in G is a cycle. By (2.6) and (2.4), if c_{1} and c_{2} ∈ G_{2n}, n > 0, are homologous cycles, then γ_{i}(c_{1}) ∼ γ_{i}(c_{2}) for all i. Thus, the homology ring H(G) inherits a system of divided powers satisfying (2.1)–(2.4).

From now on let (G, ∂, ∗) be a torsion-free DG-ring. Since G is torsion free, in order to have (2.1) and (2.2), divided powers must be defined by the formula:

γ_{k}(x) = \frac{x^{k}}{k!} (2.7)

for x of positive, even dimension. When x^{k} is divisible by k! for each x of positive, even dimension, G has a system of divided powers, which automatically satisfy (2.1)–(2.5); however, (2.6) may or may not hold.

When G is a torsion-free DG-ring with divided powers, and h is a positive integer, then hB(Π) is a differential ideal in B(Π), which by (2.4) is closed under the taking of positive divided powers. Thus G/hG is naturally a DG-ring with divided powers in such a way that the projection G → G/hG preserves divided powers. From the divided power structure in G/hG, H(G; Z_{h}) is an algebra with divided powers such that the following homomorphisms preserve divided powers:

φ^{h} : H(G) → H(G; Z_{h}) and
λ^{hk,h} : H(G; Z_{hk}) → H(G; Z_{h}),

where φ^{h} and λ^{hk,h} are the coefficient homomorphisms induced by the natural ring homomorphisms Z → Z_{h} and Z_{hk} → Z_{h}, respectively.

The notion of divided powers has proved useful in analyses of the algebras H(Π, n) and in many other contexts as well (e.g., Cartan 1967; André 1971a; André 1971b). Cartan (1967, Exp. 7, Th. 1) showed that the normalized bar construction on a strictly commutative DG-ring has a system of divided powers satisfying (2.1)–(2.6). In particular, by this result, B(Π) has a system of divided powers defined by (2.7), which system satisfies (2.1)–(2.6); thus, B(Π) is a torsion-free DG-ring with divided powers.

2B. Bockstein Homomorphisms.

Let G be a torsion-free DG-ring. For a positive integer h, multiplication by h, G ⟶^{h} G, gives an exact sequence,

0 → G ⟶^{h} G → G/hG → 0, (2.8)

which yields a long exact sequence of homology,

⋯ → H_{q}(G) ⟶^{h} H_{q}(G) ⟶^{φ_{q}^{h}} H_{q}(G : Z_{h}) ⟶^{β_{q}^{h}} H_{q−1}(G) → … (2.9)

where the connecting homomorphism, β^{h}, is called the Bockstein homomorphism, and φ^{h} is the coefficient homomorphism from the ring homomorphism Z → Z_{h}. A deeper analysis of the Bockstein homomorphism is to be found in (Browder 1961). For now, we only note several additional facts.

We have a commutative diagram with exact rows,

[figure: 0; G; hk; k; G; G/hkG; 0; 0; G; h; G; G/hG; 0] (2.10)

which, upon the taking of homology, yields a commutative ladder, part of which gives the commutative squares:

\xymatrix{H_{q}(G; Z_{hk}) \ar[r]^{β_{q}^{hk}} \ar[d]^{λ_{q}} & H_{q−1}(G) \ar[d]^{k} \\ H_{q}(G; Z_{h}) \ar[r]^{β_{q}^{h}} & H_{q−1}(G)} (2.11)

where λ_{q} = λ_{q}^{hk,h} is the coefficient homomorphism induced by the natural ring homomorphism Z_{hk} → Z_{h}. Thus, we have derived the formula

kβ_{q}^{hk} = β_{q}^{h}λ_{q}^{hk,h}. (2.12)

Let x ∈ H_{p}(G; Z_{h}), and let a ∈ G_{p} be a cycle mod h that represents x. Then β_{q}^{h}(x) is represented by the cycle \frac{∂}{h}(a) ∈ G_{p−1}. Suppose that y ∈ H_{q}(G; Z_{hk}) and z ∈ H_{r}(G; Z_{hk}), and that b and c are cycles mod hk in G that represent y and z, respectively. By the product rule for the differential ∂, we have that

∂(a) ∗ ∂(b ∗ c) = (−1)^{q}∂(a ∗ b) ∗ ∂(c) + (−1)^{qr+r}∂(a ∗ c) ∗ ∂(b).

Hence,

\frac{∂}{h}(a) ∗ \frac{∂}{hk}(b ∗ c) = (−1)^{q} \frac{∂}{h}(a ∗ b) ∗ \frac{∂}{hk}(c) + (−1)^{qr+r} \frac{∂}{h}(a ∗ c) ∗ \frac{∂}{hk}(b). (2.13)

This proves

(2.14)  β_{p}^{h}(x) ∗ β_{q+r}^{hk}(y ∗ z) = (−1)^{q}β_{p+q}^{h}(x ∗ λ(y)) ∗ β_{r}^{hk}(z)
+ (−1)^{qr+r}β_{p+r}^{h}(x ∗ λ(z)) ∗ β_{q}^{hk}(z),

where λ = λ^{hk,h} : H(G; Z_{hk}) → H(G; Z_{h}). When k = 1, (2.14) resembles the Jacobi identity for a graded Lie algebra.

Our final observation depends on a special property of the functor H(Π). We shall need the well-known fact (cf. section 3F) that if Π is cyclic, then H_{2}(Π) = 0. For a fixed positive integer h, the Bockstein homomorphism β_{2}^{h} : H_{2}(Π; Z_{h}) → Π is natural in Π. From the exact sequence (2.9) the image of β_{2}^{h} is _{h}Π.

Proposition 2.1. For each positive integer h, there is a unique transformation α^{h}: _{h}Π → H_{2}(Π; Z_{h}) that is natural in Π, and such that β_{2}^{h}α^{h}(x) = x for x ∈ _{h}Π.

Proof. Let Z_{h}(c) be a cyclic group of order h, with generator c. Since H_{2}(Z_{h}(c)) = 0, and hZ_{h}(c) = Z_{h}(c), by (2.9) β_{2}^{h} is an isomorphism in this case. Thus, α^{h} is completely determined on Z_{h}(c). If Π is any abelian group, and x ∈ _{h}Π, to make α^{h} natural, we must have:

α^{h}(x) = H_{2}(f; Z_{h})(β_{2}^{h})^{−1}(c), (2.15)

where f : Z_{h}(c) → Π sends c to x. This shows the uniqueness of α^{h}, and the transformation α^{h} defined by (2.15) is clearly natural in Π and satisfies β_{2}^{h}α^{h}(x) = x for x ∈ _{h}Π, which proves the proposition.

The natural transformation α^{h} was discovered by Cartan (1967, Exp. 11), who formulated its description differently than we have done here. We note the key properties of α^{h} to be used in the sequel (cf. Cartan 1967, Exp. 11, Props. 1 and 2).

(2.16)  β_{2}^{h}α^{h}(x) = x, x ∈ _{h}Π
(2.17)  α^{h}(ky) = λα^{hk}(y) for y ∈ _{hk}Π, λ = λ_{2}^{hk,h}.
(2.18)  α^{h}(x⊤y) = \frac{h(h−1)}{2}φ_{2}^{h}(x ∗ y), for x, y ∈ _{h}Π
(2.19)  α^{h}(kx) = kα^{h}(x), x ∈ _{h}Π

Indeed, (2.16) is the defining property of α^{h}, and when Π = Z_{hk}(y), the commutative square (2.11) with q = 2 proves (2.17), which follows for any element in _{hk}Π by the naturality of α^{h} and α^{hk}. (2.19) follows, since α^{h} is a homomorphism on cyclic groups of order h, but may also be deduced from (2.18) by induction on k. (2.18) will be proved in the next section.

2C. Caps.

If z is a cyclic in a torsion-free complex, whose homology class is annihilated by h, a chain c with ∂c = hz is called an h-cap for z. Note that an h-cap (for some element) is simply a cycle mod h. For x ∈ _{h}Π, define V^{h}(x) ∈ B_{2}(Π) by the formula:

V^{h}(x) = ∑_{i=1}^{h−1}[x, ix]. (2.20)

Then, the boundary formula (1.2) readily yields

∂V^{h}(x) = h[x], (2.21)

and, thus, V^{h}(x) is an h-cap for [x]. The transformation V^{h} : _{h}Π → B_{2}(Π) is clearly natural in Π. The homolgy class in H_{2}(Π; Z_{h}) that is represented by V^{h}(x) is mapped by β_{2}^{h} to x, by virtue of (2.21). Thus, by Proposition 2.1, the natural transformation from _{h}Π to H_{2}(Π; Z_{h}) that sends x ∈ _{h}Π to the homology class of V^{h}(x) in H(Π; Z_{h}) is α^{h}. We have proved:

Lemma 2.2. For x ∈ _{h}Π, V^{h}(x) ∈ B_{2}(Π) represents α^{h}(x).

For a and b in a torsion-free complex, let a ∼_{h} b signify that a and b are

homologous cycles mod h. Then (2.17) and (2.19) become:

V^{h}(ky) ∼_{h} V^{hk}(y) for y ∈ _{hk}Π; and (2.17)′

V^{h}(kx) ∼_{h} V^{h}(x) for x ∈ _{h}Π. (2.19)′

When x ∈ _{h}Π, one easily sees that we have an equality:

V^{hk}(x) = kV^{h}(x). (2.22)

Finally, direct calculation shows that, for x, y ∈ _{h}Π,

V^{h}(x⊤y) − \frac{h(h−1)}{2}[x] ∗ [y] = ∂(R − S) − h[y, x]

where

R = ∑_{0≤j<i≤h−1}[x, y, ix + jy], and
S = ∑_{0≤j≤i<h−1}[y, x, ix + jy].

Thus,

V^{h}(x⊤y) ∼_{h} \frac{h(h−1)}{2}[x] ∗ [y], (2.18)′

which proves (2.18)

Since in (2.18) φ_{2}^{h}(x ∗ y) has order dividing h, \frac{h(h−1)}{2}φ_{2}^{h}(x ∗ y) has order dividing 2. When h is odd, α^{h}(x⊤y) = 0; that is, α^{h} is a homomorphism when h is odd. When h is even, the deviation in (2.18) may be nonzero; viz., if Π = Z_{2}(x) + Z_{2}(y), α^{2}(x⊤y) = φ^{2}(x ∗ y) ≠ 0.

For later use we note:

Lemma 2.3. \frac{∂}{h}γ_{k}V^{h}(x) ∗ \frac{∂}{h}γ_{m}V^{h}(x) = 0 for x ∈ _{h}Π and k, m ≥ 1.

Proof. The formula is immediate for k = m = 1. Using (2.5), one then proceeds by an easy induction on k + m.

2D. Torsion Generators and Their Relations.

Using the operations defined in the preceding sections, consider the following elements in H(Π):

C_{i_{1},...,i_{n}}^{h}(x_{1}, …, x_{n}) = β_{q+1}^{h}(γ_{i_{1}}α^{h}(x_{1}) ∗ ⋯ ∗ γ_{i_{n}}α^{h}(x_{n})) ∈ H_{q}(Π), (2.23)

where h is a positive integer, the x’s are in_{h}Π, the i’s are nonnegative integers, and q = 2(i_{1} + ⋯ + i_{n}) − 1. By Lemma 2.2 the homology class (2.23) is represented by the cycle:

\frac{∂}{h}(γ_{i_{1}}V^{h}(x_{1}) ∗ ⋯ ∗ γ_{i_{n}}V^{h}(x_{n})) ∈ B_{q}(Π). (2.24)

We shall often abbreviate the symbol (2.23) by writing C_{L}^{h}(X) for it; L = (i_{1}, …, i_{n}) stands for the ordered list of nonnegative integers, and x = (x_{1}, …, x_{n}) stands for the matching ordered list of elements in _{h}Π. We shall, for notational convenience, allow empty lists with the conventions that: C_{∅}^{h}(∅) = 0 and C_{L,∅}^{h}(X, ∅) = C_{L}^{h}(X). We note that by (2.1), C_{0,L}^{h}(y, X) = C_{L}^{h}(X) and C_{0}^{h}(y) = 0 for y ∈ _{h}Π. Any element in H(Π) of the form (2.23) will be referred to as a torsion generator in H(Π). A torsion generator C_{L}^{h}(X) is not defined unless the entries of X are all in _{h}Π.

We shall now state and prove some relations among the torsion generators and their products that hold generally in H(Π) for any abelian group Π. Let X, Y and Z be possibly empty lists of elements in the torsion subgroup of Π, and let L, M and N be lists of nonnegative integers that match the lists X, Y, and Z, respectively. The list kX is obtained from the list X by multiplying the entries of X by the integer k. Below, u, v, w, x_{1}, …, and x_{n} denote elements in the torsion subgroup of Π; p, q, r, i_{1}, …, and i_{n} denote nonnegative integers; and h and k stand for positive integers. With these conventions we have

Proposition 2.4. The following relations hold in H(Π), whenever the torsion generators in them are defined.

A. C_{1}^{h}(u) = u.

B. C_{i_{σ1},...,i_{σn}}^{h}(x_{σ1}, …, x_{σn}) = C_{i_{1},...,i_{n}}^{h}(x_{1}, …, x_{n}), where σ permutes {1, …, n}.

C1. C_{i,M}^{hk}(u, Y) = k^{i−1}C_{i,M}^{h}(u, kY), i > 0.

C2. C_{M}^{h}(kY) = kC_{M}^{hk}(Y).

D. C_{r,L}^{h}(u+v, X) = ∑_{p+q=r}C_{p,q,L}^{h}(u, v, X)+ \frac{h(h−1)}{2} ∑_{p+q=r−1}C_{p+1}^{h}(u)C_{q+1}^{h}(v)C_{L}^{h}(X).

E1. C_{(p,q,L)}^{h}(u, u, X) = (p, q)C_{p+q,L}^{h}(u, X).

E2. C_{L,p}^{h}(X, ku)C_{q}^{hk}(u) = (p, q − 1)C_{L}^{h}(X)C_{p+q}^{hk}(u), q > 0.

F. C_{L}^{h}(X)C_{M,N}^{hk}(Y, Z) = C_{L,M}^{h}(X, kY)C_{N}^{hk}(Z) + C_{L,N}^{h}(X, kZ)C_{M}^{hk}(Y).

Proof. Relations A, B, C2, E1, and F may be easily deduced directly from the definition (2.23). Relation A restates the defining property (2.16) of α^{h}, while relation B is true since in H(Π; Z_{n}), even-dimensional elements commute. C2 follows from (2.12) and (2.17), and E1 is a consequence of the axiom (2.2) for divided powers. Finally, F follows from (2.14) and (2.17).

Owing to (2.24), the following slightly more general homologies in B(Π) prove C1, D, and E2, respectively.

\frac{∂}{hk}(γ_{i}V^{hk}(u) ∗ a) ∼ k^{i−1} \frac{∂}{h}(γ_{i}V^{h}(u) ∗ a). (2.25)

\frac{∂}{h}(γ_{r}V^{h}(u + v) ∗ c) ∼ ∑_{p+q=r} \frac{∂}{h}(γ_{p}V^{h}(u) ∗ γ_{q}V^{h}(v) ∗ c) (2.26)

+\frac{h(h−1)}{2} ∑_{p+q=r−1} \frac{∂}{h}(γ_{p+1}V^{h}(u)) ∗ \frac{∂}{h}(γ_{q+1}V^{h}(v)) ∗ \frac{∂c}{h}.

\frac{∂}{h}(c ∗ γ_{p}V^{h}(ku)) ∗ \frac{∂}{hk}(γ_{q}V^{hk}(u)) ∼ (p, q − 1)\frac{∂c}{h} ∗ \frac{∂}{hk}(γ_{p+q}V^{hk}(u)). (2.27)

Here a is any cycle mod hk, while c is any cycle mod h, in B(Π), and the other letters have the same meanings as in C1, D, and E2.

To prove (2.25), let u ∈ _{hk}Π. Then by (2.22) and (2.4),

γ_{i}V^{hk}(u) = k^{i}γ_{i}V^{h}(u).

Thus,

\frac{∂}{hk}(γ_{i}V^{hk}(u) ∗ a) = k^{i−1} \frac{∂}{h}(γ_{i}V^{h}(u) ∗ a).

In proving (2.26) and (2.27), we shall make use of the easy fact that if x, y, and z are cycles mod h in a torsion-free DG-ring, with x ∼_{h} y, then:

\frac{∂}{h}(x ∗ z) ∼ \frac{∂}{h}(y ∗ z). (2.28)

Let u ∈ _{hk}Π. Then, by (2.17)′,

γ_{p}V^{hk}(u) ∼_{h} γ_{p}V^{h}(ku).

Then we have, by (2.28):

\frac{∂}{h}(c ∗ γ_{p}V^{h}(ku)) ∗ \frac{∂}{hk}(γ_{q}V^{hk}(u)) ∼ \frac{1}{h^{2}k}∂(c ∗ γ_{p}V^{hk}(u)) ∗ ∂(γ_{q}V^{hk}(u)). (2.29)

Using Lemma 2.3 and (2.5), we see that the righthand side of (2.29) is:

\frac{1}{h^{2}k}∂(c) ∗ γ_{p}V^{hk}(u) ∗ ∂γ_{q}V^{hk}(u). (2.30)

But, by (2.5) and (2.2),

γ_{p}V^{hk}(u) ∗ ∂γ_{q}V^{hk}(u) = (p, q − 1)∂γ_{p+q}V^{hk}(u).

Putting this into (2.30), we prove (2.27).

Finally, let us prove (2.26). By (2.4), γ_{r} (\frac{h(h−1)}{2}u ∗ v) = 0, r ≥ 2. Thus, by (2.3),

γ_{r} ([V^{h}(u) + V^{h}(u)] + \frac{h(h−1)}{2}u ∗ v) = γ_{r}(V^{h}(u) + V^{h}(v))

+ \frac{h(h−1)}{2}γ_{r−1}(V^{h}(u) + V^{h}(v)) ∗ u ∗ v.

Thus, for any h-cap c, by (2.18)′ and the identity above:

(2.31) \frac{∂}{h}(γ_{r}V^{h}(u + v) ∗ c) ∼ \frac{∂}{h}(γ_{r}[V^{h}(u) + V^{h}(v)] ∗ c)

+ \frac{h(h−1)}{2} \frac{∂}{h}(γ_{r−1}[V^{h}(u) + V^{h}(v)] ∗ u ∗ v ∗ c).

By (2.3) the first term, RHS, of (2.31) is the same as the first term, RHS, of (2.26). Using (2.3) and writing \frac{∂}{h}V^{h}(x) for x, x = u or v, we have:

\frac{∂}{h}(γ_{r−1}[V^{h}(u) + V^{h}(v)] ∗ u ∗ v ∗ c)

= ∑_{p+q=r−1} \frac{∂}{h^{3}} [γ_{p}V^{h}(u) ∗ ∂V^{h}(u) ∗ γ_{q}V^{h}(v) ∗ ∂V^{h}(v) ∗ c]

= ∑_{p+q=r−1} \frac{∂}{h}(γ_{p+1}V^{h}(u)) ∗ \frac{∂}{h}(γ_{q+1}V^{h}(v)) ∗ \frac{∂c}{h},

where we have used (2.5) in the process. Thus, the second term, RHS, of (2.31) is the same as the second summation in (2.26). This proves (2.26) and completes the proof of Proposition 2.4.

2E. Auxiliary Relations in H(Π).

Let us note some consequences of relations A–F. Relation B implies that C1, D, E1 and E2 may be applied to torsion generators with the significant entry in any argument. As previously noted, any or all of the lists in relations A–F may be empty. The consequences D′ and D with X = ∅ and the consequence E2′ of E2 with X = ∅ and k = 1 are especially useful.

D′. C_{r}^{h}(u + v) = ∑_{p+q=r} C_{p,q}^{h}(u, v).

E2′. C_{p}^{h}(u)C_{q}^{h}(u) = 0.

Lemma 2.5. The following relations are consequences of relations A–F.

D1. C_{r,L}^{h}(ku, X) = k^{r}C_{r,L}^{h}(u, X) where k is any integer.

E3. C_{L,p}^{h}(X, ku)C_{M,q}^{hk}(Y, u) = (p, q − 1)C_{L,M}^{h}(X, kY)C_{p+q}^{hk}(u)

+ (p, q)C_{L,p+q}^{h}(X, ku)C_{M}^{hk}(Y), q > 0.

G1. hC_{L}^{h}(X) = 0

G2. C_{i,L}^{h}(0, X) = 0, i > 0.

Proof. We start proving E3. By relation F,

C_{L,p}^{h}(X, ku)C_{M,q}^{hk}(Y, u) = C_{L,M,p}^{h}(X, kY, ku)C_{q}^{hk}(u)

+ C_{L,p,q}^{h}(X, ku, ku)C_{M}^{hk}(Y).

One then applies E2 to the first term, and E1 to the second term, of the righthand side to obtain E3.

Next let us prove G2. By relation A, C_{1}^{h}(0) = 0. Then, by relation D′, one goes by induction on r to obtain that C_{i}^{h}(0) = 0, i ≥ 1. The case of relation D with u = v = 0, r = 1, yields: C_{2}^{h}(0, X) = 0. One then uses these results, relation D with u = v = 0, and induction on r to obtain G2. One then uses G2 and the case h = 1 of C2 to obtain G1.

Finally, we prove D1. For k = 0, this is simply a consequence of G2, while the case k = 1 of D1 is obvious. One proceeds by induction on k to obtain D1 for nonnegative integers. Indeed, D1 being true for integers i, 0 ≤ i ≤ k, we use D with v = ku. Then the second summation drops out by the induction hypothesis and E2′. Thus, again using the induction hypothesis and relation E1, we have that

C_{r,L}^{h}((k + 1)u, x) = ∑_{p+q=r} C_{p,q,L}^{h}(u, ku, X)

= ∑_{p+q=r}(p, q)k^{q}C_{r,L}^{h}(u, X) = (k + 1)^{r}C_{r,L}^{h}(u, X).

To prove D1 for k negative, one replaces k by a positive integer in its congruence class mod h and then uses G1. This proves all the auxiliary relations.

3. A Description of the Ring H(Π) by Generators and Relations

3A. Statement of the Main Result: The Functor R(Π).

We intend to show that the elements in Π together with the torsion generators (2.23) generate the ring H(Π), and that relations A–F and the fact that multiplication is strictly commutative imply all other relations among the generators for H(Π) and their products. To facilitate the proof, we introduce the graded ring R(Π) defined by generators and relations as follows. R_{0}(Π) = Z; R_{1}(Π) = Π; and the remaining generators are all symbols R_{i_{1},...,i_{n}}^{h}(x_{1}, …, x_{n}) subject to the same conventions as the torsion generators in H(Π). The only relations we require are that 1 ∈ Z act as an identity for multiplication, that multiplication be strictly commutative, and that relations A–F of section 2D, with the letter C be replaced by the letter R, hold in R(Π). R(Π) is made a functor in the obvious way: for a homomorphism f of abelian groups, R(f) is defined on generators by applying f to their entries. R(f) is then a homomorphism of graded rings.

A transformation (natural in Π) τ : R(Π) → H(Π) is defined by: τ(1) = 1; τ(x) = x for x ∈ R_{1}(Π); and τR_{L}^{h}(X) = C_{L}^{h}(X). The rest of this chapter is devoted to proving our main result:

Theorem I. The natural transformation τ : R(Π) → H(Π) is a natural isomorphism.

3B. A Reduction to Consideration of Finite Groups.

The functor R(Π), being defined by generators and relations, commutes with direct limits. Since the functor H(Π) also commutes with direct limits, it follows that if τ : R(Π) → H(Π) is a natural homomorphism such that τ_{Π} is an isomoprhism when Π is finitely generated, then τ is a natural isomorphism.

Let us next consider a property of the functor H(Π). Let A be a torsionfree abelian group, let Π be any abelian group, and let i and j be the injections of A and Π, respectively, into the direct sum A + Π. Since H(A) is torsion free, by the Künneth formula, the map

H(i) ⊗ H(j) : H(A) ⊗ H(Π) → H(A + Π)

is an isomorphism of graded rings. A similar fact for R(Π) is useful and easily proved. This is

Lemma 3.1. Let A be a torsion-free group, let Π be any group, and let i and j be the injections of A and Π into the direct sum A + Π. Then the map

R(i) ⊗ R(j) : R(A) ⊗ R(Π) → R(A + Π)

is an isomorphism of graded rings.

Proof. It suffices to define the inverse map, f : R(A+Π) → R(A)⊗R(Π). The map f is defined on generators of R(A + Π) by the formulas:

f(1) = 1 ⊗ 1;
f(a + b) = a ⊗ 1 + 1 ⊗ b, a ∈ A, b ∈ Π; and
f(R_{L}^{h}(X)) = 1 ⊗ R_{L}^{h}(X),

where R_{L}^{h}(X) has entries in Π. Since the entries in the generators in relations A–F must be elements of Π, f respects these relations, and since both algebras are strictly commutative, f is a well-defined ring homomorphism. The composite maps f ∘ R(i) ⊗ R(j) and R(i) ⊗ R(j) ∘ f are the identity maps (as one readily verifies on a set of generators in each case), and this proves the lemma.

Proposition 3.2. If τ : R(Π) → H(Π) is a natural homomorphism such that τ_{Π} is an isomorphism when Π is either finite or finitely generated and free, then τ is a natural isomorphism.

Proof. Let G be a finitely generated abelian group. We may write G as a direct sum, G = F + Π, where F is finitely generated and free, and Π is finite. Let i and j be the injections of F and Π into the direct sum F + Π.

We then have a commutative diagram:

[figure: R(F) ⊗ R(Π); τ_{F} ⊗τ_{Π}; R(i)⊗R(j); H(F) ⊗ H(Π); H(i)⊗H(j); R(F + Π); H(F + Π); R(G); τ_{G}; H(G) .]

Since the vertical map are isomorphisms, τ_{G} is an isomorphism if τ_{F} and τ_{Π} are isomorphisms. The proposition follows, since, under the hypotheses, τ_{G} is an isomorphism for finitely generated groups, and the functors R(Π) and H(Π) commute with direct limits.

When A is torsion free, R(A) is obviously the exterior algebra on A (with A in dimension one), and, clearly, τ : R(A) → H(A) is an isomorphism. This, coupled with the proposition, yields:

Corollary 3.3. τ_{Π} : R(Π) → H(Π) is an isomorphism for every abelian group Π if such is the case for every finite abelian group.

In view of this corollary, we henceforth restrict our attention to the functors R(Π) and H(Π) for finite groups Π.

3C. Cross Effects of R(Π) for Torsion Groups Π.

Before proceeding with the results of this section, we shall make a convention to be in force in this section and in 3D and 3E as well: we shall (tacitly) assume that all torsion generators have positive subscripts. This causes no problem, since an arbitrary torsion generator, R_{L}^{h}(X), is the same as the torsion generator Rh ; L̂(X̂) where L̂ is the list with the zero entries deleted, and X̂ is the list X with deleted entries corresponding to those deleted from L. If all entries in L are zero, then R_{L}^{h}(X) = 0. Call a torsion generator with no subscript zero normalized. Also, we shall freely use the fact that the auxiliary relations (section 2E) hold in R(Π).

We shall describe the cross-effect functors R_{k}(Π_{1}| ⋯ |Π_{n}), k > 0, when the arguments Π_{i} are torsion groups. (Following similar ideas, it is not difficult to obtain a description for arguments Π_{i} any groups, but we have no need of this result.) We fix a direct sum decomposition, Π = Π_{1} + ⋯ + Π_{n}, of a torsion group Π, and we view the various cross-effect groups R_{k}(Π_{i_{1}}| ⋯ |Π_{i_{q}}), with {i_{1} < ⋯ < i_{q}} a nonempty subset of {1, …, n}, as subgroups of R(Π). Since Π is a torsion group, and element x of order h in Π = R_{1}(Π) is also represented by the symbol R_{1}^{h}(X) (by relation A). Thus, 1 and the torsion generators R_{L}^{h}(X) alone, generate R(Π). Call a generator R_{L}^{h}(X) unmixed (relative to the direct sum decomposition of Π) if each of its argument x_{i} lies in one of the direct factors Π_{ℓ}, 1 ≤ ℓ ≤ n, of Π. If a generator R_{L}^{h}(x_{1}, …, x_{p}) is mixed, then one writes each of its arguments x_{i} as a sum, x_{i} = x_{i,1} +⋯+x_{i,n}, where x_{i,j} ∈ Π_{j}. Then, going by a finite (downward) induction on the number of nonzero x_{i,j}’s, one uses relation D (which may be applied to a sum of elements in any argument of a torsion generator, owing to relation B) to prove that a generator R_{L}^{h}(X) of R(Π) is a sum of products of unmixed generators. Thus we have:

Lemma 3.4. R_{k}(Π), k > 0, is generated as a group by all products of (normalized) unmixed torsion generators in R_{k}(Π).

Let σ be a nonempty subset of {1, …, n}. An element P in R(Π) that is a product of unmixed generators will be said to have type σ if: (1) for each i ∈ σ, there is at least one argument, in one of the torsion generators of which P is a product, that lies in Π_{i}; and (2) for each j ∉ σ, no entry in any of the generators, of which P is a product, lies in Π_{j}. Now, if an element P in R_{k}(Π) is a product of unmixed generators, the representation of P as such is not necessarily unique; consequently, P might have more than one type. The following lemma, however, shows that a nonzero product of unmixed generators has only one type.

Lemma 3.5. Let σ = {i_{1} < ⋯ < i_{q}} be a nonempty subset of {1, …, n}. Then, an element in R_{k}(Π), k > 0, that is a product of unmixed generators and has type σ lies in R_{k}(Π_{i_{1}}| ⋯ |Π_{i_{q}}).

Proof. Let ψ_{i} : Π → Π be projection on the ith coordinate (with respect to the given direct sum decomposition, Π = Π_{1} + ⋯ + Π_{n}). Write R_{k}(ψ_{i_{1}}⊤ … ⊤ψ_{i_{q}}), according to its definition, as a sum:

(3.1) R_{k}(ψ_{i_{1}}⊤ … ⊤ψ_{i_{q}}) = R_{k}(ψ_{i_{1}} + ⋯ + ψ_{i_{q}}) − R_{k}(ψ_{i_{2}} + ⋯ + ψ_{i_{q}}) − ⋯
− R_{k}(ψ_{i_{1}} + ⋯ + ψ_{i_{q−1}}) + ⋯ + (−1)^{q}R_{k}(ψ_{i_{1}})
+ ⋯ + (−1)^{q}R_{k}(ψ_{i_{q}}).

The map ψ_{i_{1}} + ⋯ + ψ_{i_{q}} applied to any argument of P leaves the entry there fixed, so R_{k}(ψ_{i_{1}}+⋯+ψ_{i_{q}})P = P. Let ψ be a sum of certain, but not all, of the maps ψ_{i_{1}}, …, ψ_{i_{q}}, and suppose that ψ_{i_{r}} does not occur in ψ. Then ψ(x_{i_{r}}) = 0 for any x_{i_{r}} ∈ Π_{i_{r}}. Thus, R_{k}(ψ)P = 0, since one of its entries is zero. These observations show that each of the maps, except R_{k}(ψ_{i_{1}} + ⋯ + ψ_{i_{q}}), on the righthandside of (3.1) annihilates P. Thus, R_{k}(ψ_{i_{1}}⊤ ⋯ ⊤ψ_{i_{q}}) leaves P fixed, and P ∈ R_{k}(Π_{i_{1}}| ⋯ |Π_{i_{q}}), which proves the lemma.

Since R_{k}(Π) is the direct sum of all the cross-effect groups R_{k}(Π_{i_{1}}| ⋯ |Π_{i_{q}}), as above, Lemma 3.4 and 3.5 give us a description of the cross-effect subgroups of R_{k}(Π) as follows:

Proposition 3.6. For a nonempty subset σ = {i_{1} < ⋯ < i_{q}} of {1, …, n}, the subgroup R_{k}(Π_{i_{1}}| ⋯ |Π_{i_{q}}) of R_{k}(Π) is the subgroup generated by all type σ products of (normalized) unmixed generators in R_{k}(Π).

3D. Ring Generators for R(Π) When Π is Finite.

Let Π be a finite group. We may write Π as a direct sum, Π = Π_{1} +… Π_{n}, where Π_{i} = Z_{h_{i}}(X_{i}), and h_{1}|h_{2}| ⋯ |h_{n}. For 1 ≤ i ≤ j, we may write h_{i}ℓ_{i,j} = h_{j}, where ℓ_{i,j} is a positive integer. These notations are kept throughout this section and the next.

We shall call a torsion generator for R(Π) a standard generator (relative to the given direct sum decomposition of Π and choices of generators x_{i} for the cyclic factors of Π) if it is one of the symbols

R_{(e_{i_{1}},...,e_{iq})}^{h_{i_{1}}}(x_{i_{1}}, ℓ_{i_{1},i_{2}}x_{i_{2}}, …, ℓ_{i_{1},i_{q}}x_{i_{q}}),

where 1 ≤ i_{1} < i_{2} < ⋯ < i_{q} ≤ n. Then we have

Proposition 3.7. R(Π) is generated as a ring by 1 and the (normalized) standard generators.

We first need another lemma, which states a consequence of the relation that is valid in R(A) for any group A.

Lemma 3.8. Let A be any abelian group. Let y_{1}, …, y_{p} ∈ A, where y_{1} has order r, and y_{i} has order s_{i}r. Let L = (i_{1}, …, i_{p}) be a list of positive integers. Then, any generator of the form R_{L}^{t}(a_{1}y_{1}, …, a_{p}y_{p}) is an integral multiple of R_{L}^{r}(y_{1}, s_{2}y_{2}, …, s_{p}y_{p}).

Proof. We may assume the a_{i}’s to be positive, since they may be replaced by positive integers, if needed. We have the divisibility conditions: r|ta_{1}, and s_{i}r|ta_{i}. Write t = kg and a_{1} = b_{1}h, where r = gh. Also, write ka_{i} = hs_{i}b_{i}, which is possible since s_{1}gh|kga_{i}. Then we have:

R_{L}^{t}(a_{1}y_{1}, …, a_{p}y_{p}) = R_{L}^{kg}(b_{1}hy_{1}, a_{2}y_{2}, …, a_{p}y_{p})
= b_{1}^{i_{1}}R_{L}^{kg}(hy_{1}, a_{2}y_{2}, …, a_{p}y_{p}) (by D1)
= b_{1}^{i_{1}}k^{i_{1}−1}R_{L}^{g}(hy_{1}, ka_{2}y_{2}, …, ka_{p}y_{p}) (by C1)
= b_{1}^{i_{1}}k^{i_{1}−1}R_{L}^{g}(hy_{1}, hs_{2}b_{2}y_{2}, …, hs_{p}b_{p}y_{p})
= b_{1}^{i_{1}}k^{i_{1}−1}hR_{L}^{gh}(y_{1}, s_{2}b_{2}y_{2}, …, s_{p}b_{p}y_{p}) (by C2)
= b_{1}^{i_{1}}k^{i_{1}−1}(b_{2}^{i_{2}} … b_{p}^{i_{p}})R_{L}^{r}(y_{1}, s_{2}y_{2}, …, s_{p}y_{p}) (by D1).

Proof of Propositon 3.7. To prove the proposition, it suffices, by Lemma 3.4, to show that any unmixed generator in R(Π) (relative to the decomposition Π = Π_{1} + ⋯ + Π_{n}) is an integral multiple of a standard generator. Any unmixed generator in R(Π) is represented, owing to relation B, by a symbol of the form:

R_{L}^{t}(a_{1,i_{1}}x_{i_{1}}, …, a_{n_{1},i_{1}}x_{i_{1}}, …, a_{1,i_{k}}x_{i_{k}}, …, a_{n_{k},i_{k}}x_{i_{k}})

where 1 ≤ i_{1} < ⋯ < i_{k} ≤ n. By Lemma 3.8 this generator is an integral multiple of:

R_{L}^{h_{i_{1}}} (x_{i_{1}}, …, x_{i_{1}}, …, ℓ_{i_{1},i_{k}}x_{i_{k}}, …, ℓ_{i_{1},i_{k}}x_{i_{k}}).

Apply relation E1, as many times as necessary, to write the generator as an integral multiple of the standard generator

R_{M}^{h_{1}}(x_{i_{1}}, …, ℓ_{i_{1},i_{k}}x_{i_{k}}),

which has no repeated arguments.

3E. Group Generators for R(Π) When Π is Finite.

Keeping the notations and terminology of section 3D, we fix a finite group Π and a direct sum decomposition, Π = Π_{1} + ⋯ + Π_{n}, Π_{i} = Z_{h_{i}}(x_{i}), with h_{1}| ⋯ |h_{n}. We write ℓ_{j} for ℓ_{1,j} of the last section. An element of R(Π_{1}| ⋯ |Π_{n}) is defined to be a basic product (with respect to the direct sum decomposition given and choice of generators for the factors) if it has the form:

R_{e_{1},e_{i_{1}}}^{h_{1}} , …, e_{i_{p}}(x_{1}, ℓ_{i_{1}}x_{i_{1}}, …, ℓ_{i_{p}}x_{i_{p}})R_{e_{j_{1}}}^{h_{j_{1}}} (x_{j_{1}}) … R_{e_{jq}}^{h_{jq}} (x_{j_{q}}), (3.2)

where {1 < i_{1} < ⋯ < i_{p}} and {j_{1} < ⋯ < j_{q}} partition {1, …, n} into disjoint subsets, and, of course, the e’s are positive integers. We single these elements out for the following reason:

Proposition 3.9. The basic products in R_{k}(Π) generate the nth crosseffect group, R_{k}(Π_{1}| ⋯ |Π_{n}).

Proof. As a consequence of Propositions 3.7 and 3.6, the group R_{k}(Π_{1}| ⋯ |Π_{n}) is generated by all type {1, …, n} elements that are products of standard generators. Let P be such an element. If in two of the standard generators in P, there is an entry in the same cyclic factor of Π, we may use E3 to eliminate the duplication. For, take two standard generators

S = R_{L_{0}}^{h_{i_{1}}} (x_{i_{1}}, ℓ_{i_{1},i_{2}}x_{i_{2}}, …, ℓ_{i_{1},i_{s}}x_{i_{s}}) and
T = R_{M_{0}}^{h_{j_{1}}}(x_{j_{1}}, ℓ_{j_{1},j_{2}}x_{j_{2}}, …, ℓ_{j_{1},j_{r}}x_{j_{r}}),

where L_{0} = (e_{i_{1}}, …, e_{i_{s}}) and M_{0} = (f_{j_{1}}, …, f_{j_{r}}) are lists of positive integers. Suppose that two indices are equal, say i_{a} = j_{b}. Now h_{i_{1}}|h_{j_{1}}, or h_{j_{1}}|h_{i_{1}} (i.e., i_{1} ≤ j_{1} or j_{1} ≤ i_{1}); say i_{1} ≤ j_{1}. Then, we have i_{1} ≤ j_{1} ≤ i_{a} = j_{a}. Hence:

h_{j_{1}} = h_{i_{1}}ℓ_{i_{1},j_{1}}; h_{j_{b}} = h_{j_{1},j_{b}}h_{j_{1}} = ℓ_{i_{1},j_{1}}ℓ_{j_{1},j_{b}}h_{i_{1}};

and

ℓ_{i_{1},i_{a}}(= ℓ_{i_{1},j_{b}}) = ℓ_{i_{1},j_{1}}ℓ_{j_{1},j_{b}}.

Using relation B, we now apply relation E3 with p = e_{i_{a}}, q = f_{j_{b}}, h = h_{i_{1}}, k = ℓ_{i_{1},j_{1}}, and u = ℓ_{j_{1},j_{b}}x_{j_{b}} to write ST as a linear combination of two products of standard generators in which the index i_{a} = j_{b} is not repeated. We may continue elimination of repeated indices until no repeats occur.

Thus, to prove the proposition, it suffices to express an element P as a linear combination of basic products where P is a type {1, …, n} element that is a product of standard generators, and the number of arguments in all generators combined is n. To prove this, we go by a finite (downward) induction on the number r of arguments in the generator in which x_{1} appears. If r = n, P is a basic product.

If r < n, and P is not a basic product, we use relation F as follows. Let the factor of P in which x_{1} appears be:

R_{L}^{h_{1}}(x_{1}, ℓ_{i_{2}}x_{i_{2}}, …, ℓ_{i_{r}}x_{i_{r}}).

Write X for (x_{1}, ℓ_{i_{2}}x_{i_{2}}, …, ℓ_{i_{r}}x_{i_{r}}). One of the standard generators that is a factor of P has two or more arguments. Write this standard generator as

R_{(m,N)}^{h_{j_{1}}}(x_{j_{1}}, Z)

where

Z = (ℓ_{j_{1},j_{2}}x_{j_{2}}, ….ℓ_{j_{1},j_{s}}x_{j_{s}}).

Then, by relation F with h = h_{1} and k = ℓ_{j_{1}}, we have:

C_{L}^{h_{1}}(X)C_{(m,N)}^{h_{j_{1}}}(x_{j_{1}}, Z) = C_{L,m}^{h_{1}}(X, ℓ_{j_{1}}x_{j_{1}})C_{N}^{h_{j_{1}}} (Z) + C_{L,N}^{h_{1}}(X, ℓ_{j_{1}}Z)C_{m}^{h_{j_{1}}} (x_{j_{1}}).

All the generators occurring on the righthand side, upon permutation of their arguments (using relation B), are standard generators except C_{N}^{h_{j_{1}}} (Z), which is an integral multiple of the standard generator

C_{N}^{h_{j_{2}}} (x_{j_{2}}, ℓ_{j_{2},j_{3}}x_{j_{3}}, …, ℓ_{j_{2},j_{s}}x_{j_{s}})

by Lemma 3.8.

We may thus express P as a sum of two elements of type {1, …, n} where each element is a multiple of a product of standard generators, with exactly n arguments occurring in all the generators combined, and where the number of arguments in the generator having x_{1} as an entry is at least r + 1. By induction, P is a linear combination of basic products. This completes the proof of Proposition 3.9.

3F. Calculation of H(Π) for Finite Groups Π.

Eilenberg and Mac Lane (1954, §§14 and 15) have introduced certain complexes to facilitate computing the homology ring H(Π) when Π is a finitely generated abelian group. We begin this section by giving a brief accout of these complexes for the case of a finite group. We refer the reader to the aforementioned paper of Eilenberg and Mac Lane for complete details on what we shall describe below.

For a positive integer h, let M(h) be the nonnegative complex of free groups described as follows. M_{0}(h) is the free abelian group generated by the cell 1. In odd dimensions 2k+1, M_{2k+1}(h) is the infinite cyclic group generated by the cell w_{k}^{h}; in even dimensions 2k, M_{2k}(h) is the infinite cyclic group generated by the cell v_{k}^{h}. We shall refer to elements in the set {1, w_{0}, v_{1}, … } as the cells of M(h) and to other elements as chains. The boundary ∂ in M(h) is given by the formulas:

∂1 = 0; ∂w_{k}^{h} = 0; ∂v_{k}^{h} = hw_{k−1}^{h}.

The multiplication in M(h), which makes M(h) into a DG ring, is given by decreeing that 1 is the identity for multiplication and:

w_{k}^{h}w_{ℓ}^{h} = 0, (3.3)

w_{k}^{h}v_{ℓ}^{h} = v_{ℓ}^{h}w_{k}^{h} = (k, ℓ)w_{k+ℓ}^{h}, and (3.4)

v_{k}^{h}v_{ℓ}^{h} = (k, ℓ)v_{k+ℓ}, (3.5)

where (k, ℓ) = \frac{(k+ℓ)!}{k!ℓ!}. M(h) is easily shown to be a torsion-free DG-ring with divided powers; moreover, by (3.5)

γ_{k}v_{1}^{h} = v_{k}^{h}, k ≥ 1. (3.6)

Next, define two integer-valued functions s_{h}^{1}(i) and s_{h}^{2}(i, j) for integers i and j as follows. s_{h}^{1}(i) = r where 0 ≤ r ≤ h − 1, and i ≡ r (mod h).

If s_{h}^{1}(i) = r, and s_{h}^{1}(j) = s, then s_{h}^{2}(i, j) = s_{h}^{2}(r, s) where

s_{h}^{2}(r, s) = 0 if r + s < h, and
s_{h}^{2}(r, s) = 1 if r + s ≥ h.

Note that the functions s_{h}^{1} and s_{h}^{2} are zero when one of their arguments is divisible by h.

Define a map f_{x} : B(Z_{h}(x)) → M(h) by defining f_{x} on a cell in B(Π) by the formulas (ibid., (15.6), p. 97):

f_{x}[ ] = 1; (3.7)

f_{x}[a_{1}x, b_{1}x, …, a_{ℓ}xb_{ℓ}x] = {∏_{i=1}^{ℓ} s_{h}^{2}(a_{i}, b_{i})} v_{ℓ}^{h}; and (3.8)

f_{x}[a_{1}x, b_{1}x, …, a_{ℓ}x, b_{ℓ}x, cx] = {s_{h}^{1}(c) ∏_{i=1}^{ℓ} s_{h}^{2}(a_{i}, b_{i})} w_{ℓ}^{h}. (3.9)

Eilenberg and Mac Lane (ibid., Th. 15.1) prove that f_{x} is a homomorphism of DG-rings that induces an isomorphism of homology rings.

Next, let Π be a finite group, and write Π = Z_{h_{1}}(x_{1}) + ⋯ + Z_{h_{n}}(x_{n}) where h_{1}|h_{2}| ⋯ |h_{n}. Let M = M(h_{1}) ⊗ ⋯ ⊗ M(h_{n}). One combines the DGhomomorphism in the direct sum theorem (section 1A) and the maps above to obtain a map

f_{x_{1},...,x_{n}} : B(Π) → M,

which is a homomorphism of DG-rings, and which induces an isomorphism of the homology rings.

One easily shows that M is a torsion-free DG-ring with divided powers. It is clear from the definition (2.7) of divided powers in a torsion-free graded ring that any homomorphism from one torsion-free graded ring to another (in particular, f_{x_{1},...,x_{n}}) preserves divided powers.

Let Π = Π_{1} +⋯+Π_{n} be a direct sum decomposition of the abelian group Π. For a subset σ of {1, …, n}, we define a subcomplex

K^{σ} ⊆ B(Π_{1}) ⊗ ⋯ ⊗ B(Π_{n})

as follows. K^{∅} = (B(Π_{1}) ⊗ ⋯ ⊗ B(Π_{n}))_{0} ≅ Z. If σ = {i_{1} < ⋯ < i_{k}} is nonempty, K^{σ} = B^{+}(Π_{i_{1}}) ⊗ ⋯ ⊗ B^{+}(Π_{i_{k}}), where B^{+}(Π) is the subcomplex of elements of positive dimension in B(Π). Then,

B(Π_{1}) ⊗ ⋯ ⊗ B(Π_{n}) = ∑_{σ} K^{σ},

where σ ranges over all subsets of {1, …, n}. Eilenberg and Mac Lane (ibid., Th. 20.1) show that for a nonempty subset σ of {1, …, n}, the map

f : B(Π) → B(Π_{1}) ⊗ ⋯ ⊗ B(Π_{n})

induces an isomorphism in homology:

H̄(Π_{i_{1}}| ⋯ |Π_{i_{k}}) ⟶^{≅} H(K^{σ}).

Let M^{+}(h) denote the subcomplex of elements of positive dimension in M(h), and set L = M^{+}(h_{1})⊗⋯⊗M^{+}(h_{n}), which we view as a subcomplex of M = M(h_{1})⊗⋯⊗M(h_{n}). L has a basis consisting of all cells c_{1}⊗⋯⊗c_{n} ∈ M with deg c_{i} > 0, 1 ≤ i ≤ n. (By a cell in M, we mean any one of the elements c_{1} ⊗ ⋯ ⊗ c_{n} where c_{i} is a cell of M(h_{i}), 1 ≤ i ≤ n.) Let Π = Z_{h_{1}}(x_{1}) + ⋯ + Z_{h_{n}}(x_{n}). By the remarks in the preceding paragraph, it is clear that the isomorphism (f_{x_{1}...,x_{n}})_{∗} : H(Π) ⟶^{≅} H(M) yields, by restriction, an isomorphism

H(Z_{h}(x_{1})| ⋯ |Z_{h}(x_{n})) ⟶^{≅} H(L).

Thus, we call L the nth cross-effect subcomplex of M.

By Corollary 3.3 and the theory of cross effects, we conclude the proof of our main result by proving

Proposition 3.10. Let Π_{i} = Z_{h_{i}}(x_{i}), 1 ≤ i ≤ n, with h_{1}|h_{2}| ⋯ |h_{n}, and let Π = Π_{1} + ⋯ + Π_{n}. Then, by restriction, τ_{Π} : R(Π) → H(Π) induces an isomorphism from R̄(Π_{1}| ⋯ |Π_{n}) onto H̄(Π_{1}| ⋯ |Π_{n}). Moreover, the basic products (relative to the given direct sum decomposition of Π and choices of generators for the cyclic factors of Π) constitute a basis for the free Z_{h_{1}}-module R̄(Π_{1}| ⋯ |Π_{n}).

Proof. Hereafter, let us refer to a basic product relative to the given decomposition and choices of generators of the cyclic factors of Π as simply a basic product. These elements generate R̄(Π_{1}| ⋯ |Π_{n}) by Proposition 3.9. By relation G1, any basic product is annihilated by h_{1}. Thus R̄(Π_{1}| ⋯ |Π_{n}) is a Z_{h_{1}}-module. We shall show that τ_{Π} maps the basic products to a Z_{h_{1}}-basis for the group H̄(Π_{1}| ⋯ |Π_{n}), which proves the proposition.

Let M = M(h_{1})⊗⋯⊗M(h_{n}). To establish our previous claims, it suffices to show that the images of the basic products under the map

(f_{x_{1},...,x_{n}})_{∗}τ_{Π} : R(Π) → H(M)

constitute a basis for the free Z_{h_{1}}-module H(L), where L is the nth cross-effect subcomplex of M, as defined above.

Let us first compute the image in H(L) of the basic products in R̄(Π_{1}| ⋯ |Π_{n}). By (3.8), for a finite cyclic group Z_{h}(x), f_{x}(V^{h}(x)) = v_{1}^{h}. By (3.6), f_{x}(γ_{k}V^{h}(x)) = v_{k}^{h}, k ≥ 1, and f_{x} (\frac{∂}{h}(γ_{k}V^{h}(x))) = \frac{∂}{h}(v_{k}^{h}) = w_{k−1}^{h}. In view of (2.24), (2.17)′, and these observations, the image of a basic product (3.2) is represented in L by the cycle:

(3.10) \frac{∂}{h_{1}} (v_{e_{1}}^{h_{1}} ∗ v_{e_{i_{1}}}^{h_{i_{1}}} ∗ ⋯ ∗ v_{e_{ip}}^{h_{ip}}) ∗ w_{e_{j{−1 \atop 1}}}^{h_{j_{1}}} ∗ ⋯ ∗ w_{e_{j{−1 \atop q}}}^{h_{jq}}

= \frac{∂}{h_{1}} (v_{e_{1}}^{h_{1}} ∗ v_{e_{i_{1}}}^{h_{i_{1}}} ∗ ⋯ ∗ v_{e_{ip}}^{h_{ip}} ∗ w_{e_{j{−1 \atop p}}}^{h_{j_{1}}} ∗ ⋯ ∗ w_{e_{j{−1 \atop q}}}^{h_{jp}}),

where ∗ denotes the product in M. In tensor notation, the element (3.10) is:

±\frac{∂}{h_{1}} (v_{e_{1}}^{h_{1}} ⊗ c_{2} ⊗ ⋯ ⊗ c_{n}) (3.11)

where c_{i_{ℓ}}^{h_{i_{ℓ}}} = v_{i_{ℓ}} and c_{j_{ℓ}} = w_{j_{ℓ}^{−1}}^{h_{j_{ℓ}}}. In view of this, we establish Proposition 3.10 by proving

Proposition 3.11. H(L) is a free Z_{h_{1}}-module with basis formed by the homology classes of elements in the set

S = \{\frac{∂}{h_{1}} (v_{e}^{h_{1}} ⊗ c_{2} ⊗ ⋯ ⊗ c_{n}) : e > 0 and c_{i} is a cell of M^{+}(h_{i}), 2 ≤ i ≤ n\} .

Proof. Let us write h for h_{1}. By the Künneth formula, if K and L are torsion-free complexes with h · H(K) = 0, then h · H(K ⊗ L) = 0. Thus, since h · H(M^{+}(h)) = 0 (as is easily verified),

h · H(L) = h · H(M^{+}(h) ⊗ [M^{+}(h_{2}) ⊗ ⋯ ⊗ M^{+}(h_{n})]) = 0.

Let C and B denote the groups of cycles and boundaries, respectively, in L. Thus, hC ⊆ B. Since the boundary of a chain in L is divisible by h, it is clear that h · C = B, and, hence, that H(L) = C/hC. Thus, to prove the proposition, it suffices to show that the cycles in S constitute a Z-basis for C.

For one of the cycles in S, we have the expansion

\frac{∂}{h}(v_{e_{1}}^{h} ⊗ c_{2} ⊗ ⋯ ⊗ c_{n}) = w_{e−1}^{h} ⊗ c_{2} ⊗ ⋯ ⊗ c_{n} + v_{e}^{h} ⊗ \frac{∂}{h}(c_{2} ⊗ ⋯ ⊗ c_{n}). (3.12)

Suppose that

∑_{j=1}^{k} n_{j} \frac{∂}{h}(v_{e_{j}}^{h} ⊗ c_{2}^{j} ⊗ ⋯ ⊗ c_{n}^{j}) = 0, (3.13)

where c_{i}^{j} is a cell of M^{+}(h_{i}), and where v_{e_{j}}^{h} ⊗c_{2}^{j} ⊗⋯⊗c_{n}^{j} ≠ v_{e_{j′}}^{h} ⊗c_{2}^{j′} ⊗⋯⊗c_{n}^{j′} for j ≠ j′. Consider the expansion of (3.13) as a linear combination of cells. From (3.12) we see that for a particular superscript j′, the term

n_{j′}(w_{e_{j′}^{−1}}^{h} ⊗ c_{2}^{j′} ⊗ ⋯ ⊗ c_{n}^{j′})

occurs only in the expansion of the term with superscript j′, and that no other term in the sum (3.13) has the cell w_{e_{j′}^{−1}}^{h} ⊗c_{2}^{j′} ⊗⋯⊗c_{n}^{j′} in its expansion.

Thus, n_{j′} = 0. This proves that the elements of S are lineary independent.

Since C = \frac{1}{h}B, in order to show that the elements in S generate C, it suffices to express an element of the form:

\frac{∂}{h}(w_{f}^{h} ⊗ c_{2} ⊗ ⋯ ⊗ c_{n}) = −\frac{∂}{h}(v_{f+1}^{h}) ⊗ \frac{∂}{h}(c_{2} ⊗ ⋯ ⊗ c_{2}) (3.14)

(where f ≥ 0 and c_{i} ∈ M^{+}(h_{i})) as a linear combination of elements in S. We shall now use the product ∗ in M. By an easy consequence of (2.13) with k = 1, we have for the cycle in (3.14) the expansion:

−\frac{∂}{h}(v_{f+1}^{h}) ∗ \frac{∂}{h}(c_{2} ∗ ⋯ ∗ c_{n}) = ∑_{j=2}^{n} ± \frac{∂}{h}(v_{f+1}^{h} ∗ c_{2} ∗ ⋯ ĉ_{j} ⋯ ∗ c_{n}) ∗ \frac{∂}{h}(c_{j}). (3.15)

(The exact signs, which are easily computed, do not concern us here.) Consider one of the terms

±\frac{∂}{h}(v_{f+1}^{h} ∗ c_{2} ∗ ⋯ ĉ_{j} ⋯ ∗ c_{n}) ∗ \frac{∂}{h}(c_{j}) (3.16)

arising in the summation (3.15). If c_{j} has odd dimension, this term is zero, whereas, if c_{j} is v_{g}^{h_{j}}, the element (3.16) is:

±ℓ_{j} \frac{∂}{h}(v_{f+1}^{h} ∗ c_{2} ∗ ⋯ ∗ c_{j−1} ∗ w_{g−1}^{h_{j}} ∗ c_{j+1} ∗ ⋯ ∗ c_{n}), (3.17)

where h_{j} = ℓ_{j}h. Combining formulas (3.15), (3.16), and (3.17), one sees that the cycle (3.14) is a linear combination of cycles in S. This establishes Propositions 3.10 and 3.11 and, thus, establishes our main result.

4. Generators and Relations for the Groups H_{k}(Π)

This chapter presumes a knowledge of the results, terminology, and methods of proof in the preceding chapter.

4A. Relations in the Groups H_{k}(Π).

One can, of course, get a description of the functor H_{k}(Π) by taking as group generators all k-dimensional elements of R(Π) that are products of elements of Π and of torsion generators in R(Π), subject to the relations implied by relations A–F and the (strict) commutative law in R(Π). The purpose of the present chapter is to sharpen this description somewhat. Observe that one may take as group generators for H_{k}(Π) all k-dimensional elements of the form

[C_{L}^{h}(X)C_{e_{1}}^{h_{1}}(x_{1}) ⋯ C_{e_{p}}^{h_{p}}(x_{p})][y_{1} … y_{q}], (4.1)

where p ≥ 0; q ≥ 0; y_{1}, …, y_{q} ∈ Π; and the term in either bracket is allowed to be 1. Call a torsion generator wide if it has two or more arguments. The difference between (4.1) and the arbitrary product of generators is that only one wide torsion generator is allowed. To show that elements of the form (4.1) generate H(Π) as a group, it suffices to show that a product of torsion generators is a linear combination of products of torsion generators with only one wide generators present; thus, it suffices to prove that the k-dimensional elements of the form (4.1) generate H_{k}(Π) when Π is a finite group. In this case we may write Π = Z_{h_{1}}(x_{1}) + ⋯ + Z_{h_{n}}(x_{n}), where h_{1}| ⋯ |h_{n}. By the basic fact on cross effects (1.14), one may write H_{k}(Π) as the direct sum of the cross-effect subgroups (derived from the given decomposition of Π). By Proposition 3.9, each of these cross-effect subgroups is generated by elements of the form (4.1). This proves that the k-dimensional elements of the form (4.1) generate the group H_{k}(Π) for Π finite and, hence, for any group Π. A conceptual proof of this result may also be given using the multiple Künneth formulas (Mac Lane 1960; Hungerford 1965).

We now develop relations among generators of the form (4.1), sufficient to imply all other relations. A permissible transposition of the torsion generators in (4.1) is a transposition of two adjacent, one-argument torsion generators. The effect of a permissible transposition is to change the sign of the generator (4.1). For elements in the second brackets in (4.1), one has the two exterior product rules:

[ ][⋯ y · y ⋯] = 0, and (4.2)

[ ][⋯ (y + y′) ⋯] = [ ][⋯ y ⋯] + [ ][⋯ y′ ⋯]. (4.3)

The empty brackets represent either 1 or a product of torsion generators (with only one wide generator present). By (4.1), if σ permutes {1, …, q}, then

[ ][y_{σ1} … y_{σq}] = (−1)^{ε(σ)}[ ][y_{1} … y_{q}],

where ε(σ) is the sign of the permutation σ.

The remaining relations are as follows. (Each relation below, except relation F^{♯}, is directly analogous to, and follows immediately from the similarly lettered relation in section 2D or section 2E.)

A^{♯}. [ ][x ⋯][ C_{1}^{h}(x)][⋯], where hx = 0.

B^{♯}. [C_{i_{1},...,i_{n}}^{h}(x_{1}, …, x_{n}) …][ ] = [C_{i_{σ1},...,i_{σn}}^{h}(x_{σ1}, …, x_{σn}) …][ ], where σ permutes {1, …, n}.

C1^{♯}. [… C_{i,M}^{hk}(u, Y) …][ ] = k^{i−1}[… C_{i,M}^{h}(u, kY) …][ ], i > 0.

C2^{♯}. [… C_{M}^{h}(kY) …][ ] = k[… C_{M}^{hk}(Y) …][ ], k > 0.

D1^{♯}. [… C_{r,L}^{h}(ku, X) …][ ] = k^{r}[… C_{r,L}^{h}(u, X) …][ ].

D^{♯}. [C_{L,r}^{h}(X, u + v) …][ ] = ∑_{p+q=r}[C_{L,p,q}^{h}(X, u, v) …][ ]

+ \frac{h(h−1)}{2} ∑_{p+q=r−1}[C_{L}^{h}(X)C_{p+1}^{h}(u)C_{q+1}^{h}(v) …][ ].

E1^{♯}. [C_{p,q,L}^{h}(u, u, X) …][ ] = (p, q)[C_{p+q,L}^{h}(u, X) …][ ].

E2^{♯}. [… C_{L,p}^{h}(X, ku)C_{q}^{hk}(u) …][ ] = (p, q − 1)[… C_{L}^{h}(X)C_{p+q}^{hk}(u) …][ ].

F^{♯}. [C_{L}^{h}(X)C_{r}^{m}(u + v) …][ ] = s ∑_{p+q=r}{[C_{L,p}^{g}(eX, fu)C_{q}^{m}(v) …][ ]

+ [C_{L,q}^{g}(eX, fv)C_{p}^{m}(u) …][ ]}

− t ∑_{p+q=r} {∑_{j=1}^{n}[C_{L̂_{j},p,q}^{g}(eX̂_{j}, fu, fv)C_{i_{j}}^{h}(x_{j}) …][ ]},

where: g is the greatest common divisor of h and m; sh + tm = g; h = ge; m = gf; and, if X = (x_{1}, …, x_{n}), and L = (i_{1}, …, i_{n}), then

X̂_{j} = (x_{1}, …, x_{j−1}, x_{j+1}, …, x_{n}), and
L̂_{j} = (i_{1}, …, i_{j−1}, i_{j+1}, …, i_{n}).

In relations A^{♯}–F^{♯}, the superscripts, subscripts, and entries in the torsion generators are subject to exactly the same conventions as given in section 2D. The ellipsis marks represent other torsion generators, while the symbol [ ] represents either 1 or an element y_{1} ⋯ y_{q}, y_{i} ∈ Π. It should also be noted that: relations B^{♯} and E1^{♯} apply to a wide generator only; relation D^{♯} applies to the first torsion generator (wide or not); relations C1^{♯}, C2^{♯}, and D1^{♯} apply to any torsion generator; relation E2^{♯} applies to any two adjacent torsion generators; and relation F^{♯} applies to the first two torsion generators.

Let us now prove relation F^{♯}. To begin, we note a relation valid in any torsion-free DG-ring G. Let h, m, g, s, and t be integers as in relation F^{♯}. Let a be an h-cap, and b, an m-cap, in G. Then, since 1/hm = s/gm + t/hg, we have

\frac{∂}{h}(a)\frac{∂}{m}(b) = s\frac{∂}{g} (a)\frac{∂}{m}(b) + t\frac{∂}{h}(a)\frac{∂}{g} (b). (4.4)

Note that both a and b are g-caps, since g divides both h and m. Now write h = ge, and m = gf, as in relation F^{♯}. Then, by (4.4), using the representing cycles for torsion generators (2.24) and also (2.17)′, one easily obtains a homology in B(Π) that proves

C_{L}^{h}(X)C_{r}^{m}(u + v) = sC_{L}^{g}(eX)C_{r}^{m}(u + v) + tC_{L}^{h}(X)C_{r}^{g}(fu + fv). (4.5)

To prove relation F^{♯}, we shall show that

(4.6) C_{L}^{h}(X)C_{r}^{m}(u + v) = s ∑_{p+q=r}{C_{L̂_{j},p,q}^{g}(eX̂_{j}, fu, fv)C_{q}^{m}(v)

+ C_{L,q}^{g}(eX, fv)C_{p}^{m}(u)} − t ∑_{p+q=r} {∑_{j=1}^{n} C_{L̂_{j},p,q}^{g}(eX̂_{j}, fu, fv)C_{i_{j}}^{h}(x_{i_{j}})} .

Indeed, to the first term, RHS, of (4.5), one first applies relation D1 to C_{r}^{m}(u+ v) and then applies relation F to each of the resulting terms in the sum (which possible, since g divides m) to obtain the first summation, RHS, of (4.6). One uses a similar procedure on the second term, RHS, of (4.5). Again apply relation D1 to the term C_{r}^{g}(fu + fv). For a term in the resulting sum, one has the following expansion (an easy inductive consequence of relation F).

C_{L}^{h}(X)C_{p,q}^{g}(fu, fv) = − ∑_{j=1}^{n} C_{L̂_{j},p,q}^{g}(eX̂_{j}, fu, fv)C_{i}^{h}(x_{j}). (4.7)

The form that the expansion takes the time is due to the manner in which relation F may be applied, since, in this case, g divides h; the minus sign is due to an interchange of torsion generators. This procedure shows that the second term, RHS, of (4.5) is the same as the second summation, RHS, of (4.6), which prove (4.6) and, thereby, proves relation F^{♯}.

We conclude this section by proving two auxiliary relations.

Lemma 4.1. The following two relations in H_{k}(Π) are consequences of relations A^{♯}–F^{♯}.

G1^{♯}. h[… C_{L}^{h}(X) …][ ] = 0.

G2^{♯}. [… C_{i,L}^{h}(0, X) …][ ] = 0, i > 0.

(Both of these relations apply to any torsion generator in the generator (4.1).)

Proof. G1^{♯} follows easily from G2^{♯} and C2^{♯}, so it suffices to prove G2^{♯}. One proves G2^{♯} in the case when the 0 occurs in the first torsion generator by an argument analogous to the argument in Lemma 2.4. It suffices to prove:

[C_{L}^{h}(X)C_{r}^{m}(0) …][ ] = 0, r > 0. (4.8)

We may reduce to the case m = 1, owing to relation C1^{♯}. Then, relation F^{♯} (with m = 1, g = 1, s = 0, t = 1, e = h, and f = 0) shows that the generator (4.8) is:

− ∑_{p+q=r} {∑_{j=1}^{n}[C_{L̂_{j},p,q}^{1}(hX̂_{j}, 0, 0)C_{i_{j}}^{h}(x_{j}) …][ ]} ,

which is zero by the preceding remarks.

4B. A Presentation of H_{k}(Π): The Functor S_{k}(Π).

We contend that relations A^{♯}–F^{♯} together with the permissible transposition law and the exterior product rules suffice to imply all possible relations among the generators of the form (4.1) in H_{k}(Π). To express this result more precisely, introduce the functor S_{k}(Π), which has as generators all kdimensional elements of the form (4.1) with the letter C replaced by the letter S. The relations that we impose are the analogues, with the letter C replaced by letter S, of relations A^{♯}–F^{♯}, the permissible transposition law, and the exterior product rules. One makes S_{k}(Π) a functor in the obvious way so that one has a transformation, natural in Π,

σ_{k} : S_{k}(Π) → H_{k}(Π),

which sends a generator in S_{k}(Π) to the generator in H_{k}(Π) with S’s replaced by C’s. The main result of this chapter is

Theorem II. The natural transformation σ_{k} : S_{k}(Π) → H_{k}(Π) is a natural isomorphism.

Proof. It is clear that, when A is a torsion-free group, S_{k}(A) = Λ^{k}A, the kth exterior product of A, and that σ_{k}(A) is an isomorphism when A is torison free.

Next, we prove that σ_{k}(Π) is an isomorphism when Π is a torsion group. Since the functors S_{k} and H_{k} commute with direct limits, it suffices to prove this when Π is a finite group. By the theory of cross effects, it suffices to prove that, by restriction, σ_{k} induces isomorphisms

S_{k}(Z_{h_{1}}(x_{1})| ⋯ |Z_{h_{n}}(x_{n})) → H_{k}(Z_{h_{1}}(x_{1})| ⋯ |Z_{h_{n}}(x_{n})), (4.9)

where h_{1}| ⋯ |h_{n}. Set Π = Z_{h_{1}}(x_{1}) + ⋯ + Z_{h_{n}}(x_{n}). Since all the elements in S_{k}(Π) have finite order, we have, by one or more applications of relation A^{♯}, that S_{k}(Π) is generated by all k-dimensional elements of the form

S_{L}^{h}(X)S_{i_{1}}^{h_{1}}(y_{1}) …, S_{i_{r}}^{h_{r}}(y_{r}). (4.10)

Furthermore, al these symbols may be assumed normalized (that is, all subscripts are positive). We shall now adopt the analogues in the present context of the terminology in sections 3C, 3D and 3E (relative to the given decomposition of Π and choices of generators for the cyclic factors of Π).

To prove that the homomorphism (4.9) is an isomorphism, it remains to show that the basic products — elements in S_{k}(Π) of the form (3.2) with C’s there replaced by S’s — generate S_{k}(Z_{h_{1}}(x_{1})| ⋯ |Z_{h_{n}}(x_{n})), since, by relation G1^{♯}, a basic product is annihilated by h_{1}. It is manifestly obvious that relations D^{♯} and F^{♯} imply that S_{k}(Π) is generated by unmixed elements of the form (4.10). Since we have relation G2^{♯} at hand, it is clear (cf. the proof of Lemma 3.5) that S_{k}(Z_{h_{1}}(x_{1})| ⋯ |Z_{h_{n}}(x_{n})) is generated as a group by all type {1, …, n} elements of the form (4.10). Any of the torsion generators in such an element may be brought to standard form — except for possible repeated entries in the first torsion generator, if it is wide — by applying the technique in Lemma 3.8, which clearly uses relation C1^{♯}, C2^{♯}, and D1^{♯} in the present context. If there are repeated entries in the wide generator, they may be eliminated by one or more applications of relation E1^{♯}. If an element has any entry in the wide generator, and an entry in one of the other torsion generators, in the same cyclic factor of Π, the superscript on the wide torsion generator divides the superscript on the other torsion generator. Furtherore, the latter torsion generator may be brought next to the wide torsion generator by applications of the permissible transposition law, so relation E2^{♯} applies to eliminate the duplicated entry. Thus, we may assume that there is no entry common to the wide torsion generator and another torsion generator. If two one-argument torsion generators have the same entry, the permissible transposition law and relation E2^{♯} with X = ∅ show that the element is zero. Then, with possible rearrangement of entries in the wide torsion generator (using relation B^{♯}) and with possible permutations of the torsion generators (using the permissible transposition law), we see that S_{k}(Z_{h_{1}}(x_{1})| ⋯ |Z_{h_{n}}(x_{n})) is generated by the k-dimensional elements of the form

S_{L}^{h_{i_{1}}} (x_{i_{l}}, ℓ_{i_{1},i_{2}}x_{i_{2}}, …, ℓ_{i_{1},i_{p}}x_{i_{p}})S_{e_{j_{1}}}^{h_{j_{1}}} (x_{j_{1}}) … S_{e_{jq}}^{h_{jq}} (x_{j_{q}}), (4.11)

where {i_{1} < ⋯ < i_{p}} and {j_{1} < ⋯ < j_{q}} partition {1, …, n} into disjoint subsets, and, for i ≤ j, h_{j} = ℓ_{i,j}h_{i}. If i_{1} = 1, the element in (4.11) is a basic product. If j_{1} = 1, use relation F^{♯} — with h = h_{i_{1}}; m = h_{j_{1}} = h_{1}; g = h_{1}; s = 0; t = 1; u = x_{1}; N = 0; X = (x_{i_{1}}, ℓ_{i_{1},i_{2}}x_{i_{2}}, …, ℓ_{i_{1},i_{p}}x_{i_{p}}); and L = (f_{i_{1}}, …, f_{i_{p}}) — relation B^{♯} to write (4.11) in the form

− ∑_{s=1}^{p} S_{i_{1},L̂_{s}}^{h}(x_{1}, X̂_{s})S_{f_{js}}^{h_{js}} (ℓ_{i_{1},i_{s}}x_{i_{s}})S_{e_{j_{2}}}^{h_{j_{2}}} (x_{j_{2}}) … S_{e_{jq}}^{h_{jq}} (x_{j_{q}}). (4.12)

One then applies the analogue of Lemma 3.8 to either of the first two torsion generators in a term of the sum (4.12) and applies the permissible transposition law to each term to write each term (4.12) as a multiple of a basic product. Thus, the basic products generate S_{k}(Z_{h_{1}}(x_{1})| ⋯ |Z_{h_{n}}(x_{n})); this proves that σ_{k}(Π) is an isomorphism when Π is a finite abelian group, and, hence, that σ_{k}(Π) is an isomorphism when Π is a torsion abelian group.

Finally, to prove Theorem II, it suffices to prove that σ_{k}(G) is an isomorphism if G is a finitely generated abelian group. We shall use the preceding results to prove somewhat more generally that σ_{k}(G) is an isomorphism when G = Π + A where Π is a torsion group and A is a torsion-free group. We have, by the Künneth formula, the decomposition:

H_{k}(Π + A) = ∑_{i+j=k} H_{i}(Π) ⊗ H_{j}(A). (4.13)

We shall define an inverse map for the map

σ_{k}(Π + A) : S_{k}(Π + A) → H_{k}(Π + A).

Define σ̃_{k}(Π+A) : H_{k}(Π+A) → S_{k}(Π+A) as follows. Let i+j = k. One has the homomorphism from S_{i}(Π) ⊗ S_{j}(A) to S_{k}(Π + A) which sends an element x ⊗ y ∈ S_{i}(Π) ⊗ S_{j}(A) — with x an element of the form (4.1) in S_{i}(Π) and a = a_{1} ⋯ a_{q}, a_{ℓ} ∈ A, 1 ≤ ℓ ≤ q — to the element xa in S_{k}(Π + A). One defines σ̃_{k}(Π + A) by defining σ̃_{k}(Π + A) on the summand H_{i}(Π) ⊗ H_{j}(A) of H_{k}(Π + A) to be the composite

H_{i}(Π) ⊗ H_{j}(A) ⟶^{[σ_{i}(Π)⊗σ_{j}(A)]^{−1}} S_{i}(Π) ⊗ S_{j}(A) → S_{k}(Π + A).

It is immediate that σ(Π+A)σ̃(Π+A) is the identity map of H_{k}(Π+A). It is an easy consequence of the exterior product rules that S_{k}(Π+A) is generated by elements of the form

[ ]a_{1} ⋯ a_{q} (4.14)

where [ ] is a generator in S_{k−q}(Π); a_{i} ∈ A, and 0 ≤ q ≤ k. Plainly, σ̃_{k}(Π + A)σ_{k}(Π + A) leaves any element in S_{k}(Π + A) of the form (4.14) fixed; hence, σ̃_{k}(Π+A)σ_{k}(Π+A) is the identity map on S_{k}(Π+A). Thus, σ_{k}(Π+A) is an isomorphism, which concludes the proof of Theorem II.

List of References

1. André, M. 1971a. On the Structure of Hopf Algebras with Divided Powers. J. of Alg. 18:19–50.

2. —-. 1971b. Hopf and Eilenberg-Mac Lane Algebras. In Reports of the Midwest Category Seminar V, Lecture Notes in Math. 195, pp. 1–28. Berlin, Heidelberg, New York: Springer-Verlag.

3. Browder, W. 1961. Torsion in H-spaces. Ann. of Math. 74:24–51.

4. Cartan, H. 1967. Algèbres d’Eilenberg-Mac Lane et Homotopie. In Séminaire Henri Cartan, vol. 3: 1954–57. New York: Benjamin.

5. Eilenberg, S., and Mac Lane, S. 1945. Relations Between the Homology and Homotopy Groups of Spaces. Ann. of Math. 46:480–509.

6. —–. 1953. On the Groups H(Π, n), I. Ann. of Math. 58:55–106.

7. —–. 1954. On the Groups H(Π, n), II: Methods of Computation. Ann. of Math. 60:49–139.

8. Hungerford, T. W. 1965. Multiple Künneth Formulas for Abelian Groups. Trans. Amer. Math. Soc. 118:257–76.

9. Mac Lane, S. 1954. The Homology Products in K(Π, n). Proc. Amer. Math. Soc. 5:642–51.

10. —–. 1960. Triple Torsion Products and Multiple Künneth Formulas. Math. Annalen. 140:51–64.