The University of Chicago

THE INTEGRAL HOMOLOGY ALGEBRA OF AN EILENBERG-MAC LANE SPACE

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

gerald john decker

chicago, illinois

June, 1974

in memory of alfred daniel decker

Document prepared by Dave Benson and Sharon Southwick, Nov 2000 Acknowledgments. While this dissertation has been an attempt, in a small way, to add to the body of mathematical knowledge, it could not have been done without the help of several people. I wish to thank my wife, Barbara, for her patience and perseverance during the writing of this paper, and for her typing of the final manuscript. I am grateful to Saunders Mac Lane for our numerous discussions, for his patience in reading and correcting the several drafts of this work, and, finally, for the many things he has taught me of the beauty and nature of mathematics.

Contents

Introduction 1 1. The Torsion Algebra 2 2. Homology is a Tor quotient 10 3. H_{∗}(π, n; ℤ) 20 4. Cohomology 33 Appendix: Elementary Equivalences in Tor(H𝒳(π, n)) 39 References 42

The Integral Homology Algebra of a K(π, n) 1

Introduction

The principal result of this paper is a natural description by generators and relations of the integral homology algebra of an Eilenberg-Mac Lane space X of type K(π, n), where we assume that π is abelian and n ≥ 1 (this is a restriction only for n = 1). Such a space (CW-complex) X is characterized by the property that π_{n}(X) = π and π_{i}(X) = 0 for i ≠ n, where π_{i}( ) denotes the ith homotopy group. It is well known that H_{∗}(X) depends only upon π and n, hence we identify H_{∗}(π, n; R) ≅ H_{∗}(X; R) for R a coefficient ring (assumed commutative). While the homotopy groups of a K(π, n) are quite simple, the homology is in general far from it. Our interest here is the case R = ℤ, however, the techniques and results given below are valid fo R an arbitrary principal ideal domain.

The associated algebraic problem is to determine the homology of the reduced normalized bar construction B̄(A_{n}) of a graded, connected, strictly commutative differential algebra A_{n} defined inductively by A_{1} = Rπ, the group ring of π over R, and A_{n} = B̄(A_{n−1}). Hence H_{∗}(π, n; R) ≅ H_{∗}(B̄(A_{n}); R). We will denote B̄(A_{n}) by B̄^{(n)}(Rπ) in future references.

Under the natural pairing R ⊗ R ≅ R (⊗ over R), and the homomorphism π + π → π given by addition, the graded group H_{∗}(π, n; R) is a strictly commutative, connected R-algebra which admits a structure of divided powers [1]. If R is a field, H_{∗}(π, n; R) is a Hopf algebra. For R = ℤ_{p}, p a prime, Cartan [3] has completely determined this algebra, and André [1] has shown that such a connected, commutative Hopf algebra with divided powers is the universal enveloping coalgebra of a graded Lie coalgebra, thus placing Cartan’s original result in a general algebraic setting.

The starting point for the present work is the analysis of the case R = ℤ given by Cartan [3]. To a group π and n ≥ 1, Cartan associates a family {A_{α}} of connected, commutative, differential algebras (“elementary” complexes) determined, essentially, by the various admissible sequences of homology operations, α, arising for each prime p and fixed n. For each A_{α}, there is a chain map g_{α}: A_{α} → B̄^{(n)}(ℤπ) which is not itself unique, but is unique up to homotopy and so induces a unique map in homology. If X(π, n) = ⨂_{α} A_{α}, and g: X(π, n) → B̄^{(n)}(ℤπ) is the map induced by the g_{α}, the following result gives a means of determining the integral homology.

Theorem: The algebra homomorphism g_{∗}: H_{∗}(X(π, n)) → H_{∗}(π, n; ℤ) is an epimorphism with kernel N(π, n) generated in homology by specified identifications among the generators of the A_{α}. The algebra X(π, n) and the map g_{∗} are natural for maps π → π′ of abelian groups.

The problem thus becomes one of calculating the homology algebra of a tensor product of differential algebras. In particular, restricting ourselves to the finite case, the question is H_{∗}(A_{1} ⊗⋯⊗A_{r}) for r ≥ 1. For the case r = 2, and by induction for larger r, the classic Künneth formula (assuming each algebra is torsion free) gives

0 → Σ H_{∗}(A_{1}) ⊗ H_{∗}(A_{2}) → H_{∗}(A_{1} ⊗ A_{2}) → Σ Tor^{R}(H_{∗}(A_{1}), H_{∗}(A_{2})) → 0

and so the homology additively up to a group extension. This does not suffice to determine the algebraic structure in homology. For this, a more detailed analysis is required.

The crucial observation is that, for a large class of torsion-free differential graded R-algebras, it is possible to describe H_{∗}(A_{1} ⊗ A_{2}) directly in terms of H_{∗}A_{1}, H_{∗}A_{2} and a certain algebraic construction, the torsion algebra. The appropriate algebras are those which possess capping operations, in effect, those algebras which permit a suitably natural representation of their R-torsion homology classes. The elementary complexes A_{α} of Cartan are shown to possess such operations [Chapter 2].

For a graded R-module M, we define the torsion algebra of M, Tor(M), as a graded, commutative R-algebra given by generators and relations. The generators are symbols τ_{h}(m_{1}, …, m_{r}) where the m_{i} are homogeneous elements of M, h ∈ R (R − {0} if r > 1) such that hm_{1} = ⋯ = hm_{r} = 0 in M. The relations are given in Chapter 1, Section 3. The algebra Tor(M) in effect enlarges the more standard construction of the symmetric algebra of M, S(M), by taking into account the presence of non-trivial R-torsion in M. In particular, we have the inclusion S(M) → Tor(M) induced by the homomorphism ι: M → Tor(M) given by ι(m) = τ_{0}(m). If M is torsion-free (for example, when R is a field) Tor(M) collapses to the symmetric algebra, S(M).

As a convenience in what follows, we define the torsion algebra of an indexed family M_{I} = {M_{i} | i ∈ I} of graded R-modules, Tor(M_{I}), and observe that for M = ⨁_{i∈I} M_{i}, we have an isomorphism Tor(M_{I}) ≅ Tor(M).

Combining these observations, we utilize the Künneth formula and a suitable filtration of the algebra Tor( ) to prove

Theorem: If A_{1}, A_{2} are torsion-free, graded, connected, commutative R-algebras with capping operations, there is a natural (with respect to maps preserving these operations) R-algebra homomorphism Φ: Tor({HA_{1}, HA_{2}}) → H_{∗}(A_{1}⊗A_{2}) that is an epimorphism with kernel K(A_{1}, A_{2}) given by specific relations among generators in Tor( ) [Theorem 2.4.8].

Passing to the case of an infinite number of such algebras {A_{α}} by appropriate colimit arguments and naturality, we have a computation of the homology algebra H_{∗}(⨂_{α} A_{α}). Applying this to the elementary complexes, A_{α}, of Cartan gives our determination of H_{∗}(π, n; R) as a quotient of the torsion algebra of the family {H_{∗}A_{α}}. We identify, then, the generators and relations by identifying the respective kernels K({A_{α}}) and N(π, n) for R = ℤ.

As an example of the usefulness of this presentation of H_{∗}(π, n; ℤ), we calculate the groups H_{n+k}(π, n; ℤ) for 0 ≤ k ≤ 2n; we give a complete description for n = 1, which was done independently by Hamsher [5]; and we determine the groups H_{2+k}(π, 2; ℤ) for 0 ≤ k ≤ 7, the instances for k ≤ 4 having been obtained by Eilenberg and Mac Lane [4]. We observe that H_{∗}(π, 2; ℤ) ≅ Γ(π), for π torsion free, where Γ(π) is the divided power algebra on π.

Finally, using the results above, we give a description of the cohomology groups H^{n+k}(π, n; G) in the range 0 ≤ k ≤ n, with G an arbitrary coefficient group.

1. The Torsion Algebra

1.1. Preliminaries, The Category of Families. Throughout this paper R is a fixed commutative principal ideal domain with unit. All modules are unitary R-modules; differential graded modules are ℤ-graded, ℤ the integers, and the differentials have degree −1. All R-algebras are associative and augmented.

Our work shall take place within sub-categories (not necessarily full) of the following basic categories:

ℰns – the category of sets with morphisms set maps,

Rℳ – the category of graded R-modules with gradation bounded below and morphisms degree-preserving R-module homomorphisms,

R𝒜 – the category of graded, commutative, augmented R-algebras with gradation bounded below and morphisms degree-preserving augmented R-algebra homomorphisms,

R𝒟𝒜 – the category of graded, augmented, strictly commutative differential R-algebras A such that H_{0}(A) ≃ R (i.e., A is connected), with gradation bounded below, and morphisms all degree-preserving differential R-algebra homomorphisms,

R𝒟ℱ𝒜 – the full sub-category of _{R}𝒟𝒜 whose objects are torsion free as Rmodules.

Each category is naturally a sub-category of the one preceding it in the list above. We write C ∈ 𝒞 to mean C is an object of 𝒞 and f ∈ 𝒞(C, C′) to mean f is a morphism between the objects C and C′ in 𝒞.

In what follows, we assume that the elements of objects I of ℰns are given an order, subject only to the condition that I ⊆ J (I a subset of J) requires that the order on the set I agree with that imposed upon I by the ordering in J.

Definition 1.1.1. The category of families, ℱ(𝒞), of 𝒞 is the category whose objects are indexed sets C_{I} = {C_{i} | i ∈ I} of objects of 𝒞, where I ∈ ℰns is non-empty. A morphism f_{I}: C_{I} → C_{J} is a pair (m, {f_{i} | i ∈ I}) where m: I → J is an injective morphism of sets and {f_{i} | i ∈ I} is an indexed set of morphisms f_{i} ∈ 𝒞(C_{i}, C_{m(i)}).

Given objects and morphisms f_{I}: C_{I} → C_{J} and g_{J}: C_{J} → C_{K} with f_{I} = (m, {f_{i}}) and g_{J} = (m′, {g_{j}}), define the composite g_{J} ∘ f_{I}: C_{I} → C_{J} by the pair (m′ ∘ m, {g_{m′∘m(i)} ∘ f_{i}}) where m′ ∘ m: I → J is injective in ℰns and g_{m′∘m(i)} ∘ f_{i}: C_{i} → C_{m′∘m(i)} is a morphism in 𝒞.

Clearly the pair (id_{I}, {id_{C_{i}} | i ∈ I}): C_{I} → C_{I}, where id_{I} is the identity on I and id_{C_{i}} is the identity on C_{i}, gives the identity morphism in ℱ(𝒞). This composition is clearly associative and preserves identity, so we have a category.

Definition 1.1.2. A filtered set 𝒮 is a small category such that 𝒮(s, s′) has at most one element for s, s′ ∈ 𝒮, and such that for any s, s′, there is an object s′′ ∈ 𝒮 with 𝒮(s, s′′) and 𝒮(s′, s′′) both non-empty. A filtered system F over 𝒮 in a category 𝒞 is a functor F: 𝒮 → 𝒞. A filtered colimit over F, lim_{𝒮^{→}}F, is an initial object in the

category of F -cones on 𝒞 [10]. 𝒞 has filtered colimits if every filtered system in 𝒞 has a colimit in 𝒞.

We define a functor E: ℱ(𝒞) → ℰns by E(C_{I}) = I, and E(f_{I}) = m for f_{I} = (m, {f_{i}}). Note that E is not full since E(f_{I}) is always injective.

Proposition 1.1.3. If the category 𝒞 has filtered colimits, the category ℱ(𝒞) also has filtered colimits.

Proof. Given a filtered system F: 𝒮 → ℱ(𝒞), let E ∘ F: 𝒮 → ℰns be the filtered system on the index sets, and let I = lim_{S^{→}} E ∘ F be the filtered colimit (regarded as

a set), and let m_{s}: E ∘ F (s) → I for s ∈ 𝒮 be the associated colimit map in ℰns. Note that each m_{s} is injective since each map F (s → s′) is. For i ∈ I, let 𝒮(i) be the full subcatgory of 𝒮 with objects s such that m_{s}(x) = i for some x ∈ E ∘ F (s). Define a functor F_{i}: 𝒮(i) → 𝒞 by F_{i}(s) = C_{x} ∈ F (s) (the member of the family with index x), where m_{s}(x) = i, and define the morphism F_{i}(s → s′) to be the morphism in 𝒞 corresponding to the index x in ℱ(𝒞)(F (s), F (s′)). This functor is well defined since the m_{s} are all injective. Let C_{I} = { lim_{𝒮(i)^{→}} F_{i} | i ∈ I} ∈ ℱ(𝒞), then

it is straightforward to verify, using the universal properties of the colimits in ℰns and 𝒞, that the family C_{I} is the filtered colimit of the system defined by F.

Given two categories 𝒞 and 𝒟 having filtered colimits, and a functor H: 𝒞 → 𝒟, H preserves filtered colimits if it carries colimits over filtered systems in 𝒞 to colimits over filtered systems in 𝒟. Furthermore, H induces a functor ℱ(H): ℱ(𝒞) → ℱ(𝒟) by ℱ(H)(C_{I}) = ℱ(H)({C_{i}|i ∈ I}) = {H(C_{i}) | i ∈ I} and ℱ(H)(f_{I}) = ℱ(H)(m, {f_{i} | i ∈ I}) = (m, {H(f_{i}) | i ∈ I}).

Proposition 1.1.4. If H: 𝒞 → 𝒟 preserves filtered colimits, the induced functor ℱ(H): ℱ(𝒞) → ℱ(𝒟) preserves filtered colimits.

Proof. Using the notation of Proposition 1.1.3, we have C_{I} = {C_{i} | i ∈ I}, where C_{i} = lim_{𝒮(i)^{→}} F_{i} and I = lim_{𝒮^{→}}E ∘ F. In E(𝒟), the filtered system is G = ℱ(H) ∘ F.

Let D_{J} = lim_{𝒮^{→}}G, then, observing that E = E ∘ ℱ(H), it follows that J = lim_{𝒮^{→}}E ∘ G = lim_{𝒮^{→}}E ∘ F = I, so the index sets are the same, and G_{i} = H ∘ F_{i} for i ∈ I.

Consequently, lim_{𝒮(i)^{→}} G_{i} = lim_{𝒮(i)^{→}} H ∘ F_{i} = H( lim_{𝒮(i)^{→}} F_{i}) = H(C_{i}) since H preserves

filtered colimits. It follows that D_{J} = D_{I} = ℱ(H)(C_{I}).

1.2. The Functors T and ℱ(H_{∗}). Let us assume now that the category 𝒞 is a subcategory of _{R}𝒜 which has filtered colimits and is closed under tensor product.

Define a functor T: ℱ(𝒞) → 𝒞 as follows: If C_{I} ∈ ℱ(𝒞) is a finite family, set T (C_{I}) = ⨂_{i∈I} C_{i}, the tensor product taken according to the order imposed on I. That ⨂_{i∈I} C_{i} ∈ 𝒞 follows from the standard fact that a finite tensor product of graded, augmented, strictly commutative R-algebras is again such an algebra, multiplication being given by signed transposition and the multiplication of each algebra.

If C_{I} ∈ ℱ(𝒞) is an infinite family, set T (C_{I}) = ⨂_{i∈I} C_{i}, the infinite tensor product in 𝒞, defined as the colimit, over the set of finite subsets of I directed by inclusion, of all finite tensor products with inclusion (under the natural coherence of ⊗_{R}). It is again standard that this colimit is a graded, augmented, strictly commutative R-algebra in 𝒞.

If f_{I}: C_{I} → C_{J} is a morphism in ℱ(𝒞) with f_{I} = (m, {f_{i} | i ∈ I}), I finite, set f_{I} = σ(⨂_{i} f_{i}) where σ: (⨂_{i∈I} C_{m(i)}) ⊗ (⨂_{j∈J′} C_{J}) → ⨂_{j∈J} C_{j} is the signed transposition induced by permuting the order on the index set m(I) ∪ J′ = m(I) ∪ J − m(I) = J. In the infinite case, let f_{I} be the filtered colimit of the morphisms induced by all the finite subsets of I directed by inclusion (note that σ is the identity).

Proposition 1.2.1. For 𝒞 as above, the functor T: ℱ(𝒞) → 𝒞 preserves filtered colimits.

Proof. This follows from the fact that ⊗_{R} preserves filtered colimits over a finite family of systems, and from the definition of the infinite tensor product as a colimit of finite tensor products.

If we restrict the category further, and require that 𝒞 be a subcategory of _{R}𝒟𝒜 (having filtered colimits), there is the homology functor H_{∗}: 𝒞 → _{R}𝒜, which assigns to each differential R-algebra A ∈ 𝒞 its homology algebra, H_{∗}(A; R). It is well known that H_{∗} preserves filtered colimits, so that ℱ(H_{∗}): ℱ(𝒞) → ℱ(_{R}𝒜) also preserves filtered colimits by Propositon 1.1.4.

1.3. The Torsion Algebra of a Module. Let M ∈ _{R}ℳ, then we associate to M a graded, commutative, augmented R-algebra, the torsion algebra, Tor(M), defined by generators and relations.

For n ≥ 0, let A = (m_{1}, …, m_{n}) be a string (possibly empty) of homogeneous elements of M. Define a symbol τ_{h}(A) for such a string whenever h ∈ R − {0} such that hm_{1} = ⋯ = hm_{n} = 0 in M. Such a symbol will be called an h-torsion generator. If A is empty, define the “empty symbol” τ_{h}( ) for all h ∈ R − {0}. For n = 1, define special symbols, τ_{0}(m), for m a homogeneous element of M.

Assign to each symbol τ_{h}(A), A = (m_{1}, …, m_{n}) a length ℓ(τ_{h}(A)) = n, and a degree |τ_{h}(A)| = ∑_{k=1}^{n} |m_{k}| + n − 1, where |m_{k}| is the homogeneous degree of m_{k} in M. Define ℓ(τ_{h}( )) = 0 and |τ_{h}( )| = −1. Since the length and degree are clearly independent of h, it will sometimes be convenient to abbreviate to the symbols ℓ(A), |A|, without ambiguity.

If A = (A_{1}, …, A_{p}), p > 0 where A_{1}, …, A_{p} are strings of homogeneous elements as above, and τ_{h}(A_{j}) is defined for j = 1, …, p and h ∈ R − {0}, we will write τ_{h}(A) = τ_{h}(A_{1}, …, A_{p}). Note that ℓ(A) = ∑_{j=1}^{p} ℓ(A_{j}) and |A| = ∑_{j=1}^{p} |A_{i}|+p−1.

Given a string A = (m_{1}, …, m_{n}), let hA = (hm_{1}, …, hm_{n}), then the symbol τ_{h}(A) is defined whenever hA = 0, 0 denoting the n-tuple of zeros (0, …, 0).

Definition 1.3.1. Let Tor(M) ∈ _{R}𝒜 be the associative, commutative, R-algebra on generators 1 (the unit of R with degree 0) and the symbols τ_{h}(A), h ∈ R − {0} (R if n = 1), with their assigned degrees subject to the following relations (where we assume all symbols defined as appropriate)

(i)  τ_{h}( ) = 0 for h ∈ R;
(ii) τ_{0}(m) = τ_{h}(m) for m ∈ M, h ∈ R such that hm = 0;
(iii) kτ_{hk}(A) = τ_{h}(kA) for hkA = 0, with h, k ∈ R − {0};
(iv) τ_{hk}(A, B) = k^{ℓ(B)−1}τ_{h}(kA, B) for hkA = hB = 0, ℓ(B) > 0,
and h, k ∈ R − {0};  (1.3.2)
(v) τ_{h}(A_{σ_{1}}, …, A_{σp}) = (−1)^{sgn σ}τ_{h}(A_{1}, …, A_{p}) for hA_{1} = ⋯ = 0,

where σ is a permutation on 1, …, p and sgn σ = ∑_{\begin{matrix}i<j \\ σj<σi \end{matrix}}(|A_{i}| + 1)(|A_{j}| + 1);

(vi) τ_{h}(0, A) = 0 for hA = 0;
(vii) [Jacobi relation] (−1)^{a}τ_{hk}(A, B)τ_{h}(C) + (−1)^{b}τ_{h}(kB, C)τ_{hk}(A) +

(−1)^{c}τ_{h}(C, kA)τ_{hk}(B) = 0 for strings such that hkA = hkB = hC = 0 for

h, k ∈ R − {0} where a = |A|(|C| + 1), b = |B|(|A| + 1), c = |C|(|B| + 1);

(viii) τ_{0}(m_{1} + m_{2}) = τ_{0}(m_{1}) + τ_{0}(m_{2}) for m_{1}, m_{2} ∈ M
τ_{0}(km) = kτ_{0}(m) for k ∈ R, m ∈ M.

Define the augmentation by ε(1) = 1, and ε(τ_{h}(A)) = 0.

We observe that an immediate consequence of (ii) and (iv) is that τ_{h}(m) = τ_{h′}(m) for any h, h′ ∈ R − {0} such that hm = 0 = h′m, so that all length one h-torsion generators of a fixed m are identified. Relations (ii), (iii), and (viii) imply that the length one h-torsion generators are linear over R.

Proposition 1.3.3. The Jacobi relation (1.3.2, vii) “generalizes” commutativity in Tor(M) in the sense that for any string A, B, or C empty, the relation reduces to one expressing commutativity among the non-trivial torsion generators.

Proof. If A (or equivalently, B) is empty, the relation becomes

(−1)^{−(|C|+1)}τ_{hk}(B)τ_{h}(C) + (−1)^{|C|(|B|+1)}τ_{h}(C)τ_{hk}(B) = 0

and on combining signs becomes

τ_{h}(C)τ_{hk}(B) = (−1)^{|C||B|}τ_{hk}(B)τ_{h}(C)  (1.3.4)

which expresses commutativity. If C is empty, the relation becomes

(−1)^{|B|(|A|+1)}τ_{h}(kB)τ_{hk}(A) + (−1)^{−(|B|+1)}τ_{h}(kA)τ_{hk}(B) = 0

or τ_{h}(kB)τ_{hk}(A) = (−1)^{|A||B|}τ_{h}(kA)τ_{hk}(B) on combining signs. By 1.3.2(iii), the equality becomes

kτ_{hk}(B)τ_{hk}(A) = (−1)^{|A||B|}kτ_{hk}(A)τ_{hk}(B)

again expressing a commutativity relation in Tor(M).

Finally, if any two or more A, B, C are empty, the products in the Jacobi identity are all zero by 1.3.2(i) so the identity is trivial.

The following propositions record some properties of the R-algebra Tor(M) that will be useful in the sequel. We first observe that the h-torsion generators are themselves torsion in Tor(M).

Proposition 1.3.5. If τ_{h}(A) is a generator in Tor(M), then hτ_{h}(A) = 0.

Proof. hτ_{h}(A) = τ_{1}(hA) = τ_{1}(0) = 0 by 1.3.2(iii) and (vi).

The following result is immediate.

Proposition 1.3.6. Tor( ): _{R}ℳ → _{R}𝒜 is a functor.

Since Tor(M) is defined as an R-algebra by generators and relations, we have

Proposition 1.3.7. If 𝒞 is a subcategory of _{R}ℳ that has filtered colimits, then Tor( ): 𝒞 → _{R}𝒜 preserves filtered colimits.

The Jacobi relations allows us to determine an R-module generating set for Tor(M) which will be required in the following chapters.

Lemma 1.3.8. Let A_{1}, A_{2} and B be non-empty strings of homogeneous elements of M such that q = τ_{hk}(A_{1}, A_{2})τ_{h}(B) is defined. Then q can be written as

q = ±τ_{h}(B_{1})τ_{hk}(A_{1}) ± τ_{h}(B_{2})τ_{hk}(A_{2})

where B_{1} = (kA_{2}, B) and B_{2} = (B, kA_{1}).

Proof. Apply the Jacobi relation to the three strings A_{1}, A_{2}, B.

As a consequence of this observation and commutativity, it follows that any product of the form τ_{hk}(A)τ_{h}(B) can be reduced to a linear combination of products where at most one of the factors has length greater than one (this decomposition is not unique).

Proposition 1.3.9. Tor(M) is generated as an R-algebra by 1 and the torsion generators τ_{h}(A); and as an R-module by products of the form

τ_{h}(m_{1}, …, m_{n})τ_{hk_{1}}(m_{n+1}) ⋯ τ_{hk_{ℓ}}(m_{n+ℓ}) · Q.  (1.3.10)

for n, ℓ ≥ 0 where m_{i} ∈ M, h, k_{1}, …, k_{ℓ} ∈ R − {0} and Q is a product of special torsion generators, τ_{0}(m).

Proof. The first assertion is trivial from the definitions. For the second, consider a product q = τ_{h}(A)τ_{h′}(A′). If either is a special torsion generator we are done, otherwise h, h′ ∈ R − {0}, and hA = 0 = h′A′. Since h and h′ both annihilate q by Proposition 1.3.5, their greatest common divisor h̄ also annihilates q (R is a PID). If h = h̄s and h′ = h̄s′, s, s′ ∈ R − {0}, then there are u, u′ ∈ R − {0} such that us+u′s′ = 1 in R. Hence q = 1·q = usq+u′s′q = uτh̄(sA)τ_{h′}(A′)+u′τ_{h}(A)τh̄(s′A′) by 1.3.2(iii). Consequently, any product of two h-torsion generators can be reduced to a linear combination of products of the form τ_{h}(A)τ_{hk}(B). By repeating this argument, any product or torsion generators in Tor(M) can be reduced to a product of the form

τ_{h}(A)τ_{hk_{1}}(B_{1}) ⋯ τ_{hk_{ℓ}}(B_{ℓ}) · Q  (1.3.11)

for ℓ ≥ 0, where Q is a product of special torsion generators or 1. By Lemma 1.3.8 and the remark following, a product of the form 1.3.11 can be reduced to a linear combination of products again of the form of 1.3.11 but with at least one factor of lower length in each term (everything in Q is already of length one). Hence by downward induction, any non-trivial product can be reduced to a linear combination of products of the form 1.3.10.

A symmetry relation in Tor(M), which is a direct consequence of the permutation and Jacobi relations, will be useful.

Proposition 1.3.12. In Tor(M) let A_{1}, …, A_{n}, n ≥ 2, be strings of homogeneous elements of M so that τ_{h}(A_{1}), …, τ_{h}(A_{n}), h ∈ R − {0} are defined. Then

∑_{j=1}^{n}(−1)^{ε(j)}τ_{h}(A_{j+1}, …, A_{n}, A_{1}, …, A_{j−1})τ_{h}(A_{j}) = 0  (1.3.13)

where ε(j) = |(A_{1}, …, A_{j})| |(A_{j+1}, …, A_{n})| + |(A_{1}, …, Â_{j}, …, A_{n})| ( ˆ denotes omission).

Proof. To simplify calculations of signs, we compute mod-2 on the exponents of −1. ε(1) = |A_{1}||A_{2}|+|A_{2}| and ε(2) = |(A_{1}, A_{2})|+|A_{1}| = (|A_{1}|+|A_{2}|+1)+|A_{1}| = |A_{2}|+1 so 1.3.13 becomes

(−1)^{|A_{2}|}[(−1)^{|A_{1}||A_{2}|}τ_{h}(A_{2})τ_{h}(A_{1}) − τ_{h}(A_{1})τ_{h}(A_{2})] = 0

by commutativity in Tor(M).

Assume the result for n and take new strings

B_{1} = A_{1}, …, B_{n−1} = A_{n−1}, B_{n} = (A_{n}, A_{n+1}).

Then 1.3.13 for n strings becomes

∑_{j=1}^{n−1}(−1)^{e(j)}τ_{h}(A_{j+1}, …, A_{n+1}, A_{1}, …, A_{j−1})τ_{h}(A_{j})

+ (−1)^{e(n)}τ_{h}(A_{1}, …, A_{n−1})τ_{h}(A_{n}, A_{n+1}) = 0 (1.3.14) where e(j) = |B_{1}, …, B_{j}||B_{j+1}, …, B_{n}| + |B_{1}, …, B̂_{j}, …, B_{n}|. By definition of the B_{j}’s, ε(j) = e(j) for j = 1, …, n, and

e(n) = |B_{1}, …, B_{n}| + |B_{1}, …, B_{n−1}|
= (|B_{1}, …, B_{n+1}| + |B_{n}| + 1) + |B_{1}, …, B_{n−1}|
= |B_{n}| + 1 = |A_{n}, A_{n+1}| + 1.

Applying the Jacobi relation to the strings A = A_{n}, B = A_{n+1}, and C = (A_{1}, …, A_{n−1}) gives

τ_{h}(A_{n}, A_{n+1})τ_{h}(A_{1}, …, A_{n−1}) = (−1)^{a}τ_{h}(A_{n+1}, A_{1}, …, A_{n−1})τ_{h}(A_{n})

+ (−1)^{b}τ_{h}(A_{1}, …, A_{n})τ_{h}(A_{n+1}) (1.3.15)

where a = |A_{n}|(|C|+1)+1+|A_{n+1}|(|A_{n}|+1), b = |A_{n}|(|C|+1)+1+|C|(|A_{n+1}|+1). Since Tor(M) is commutative,

(−1)^{e(n)}τ_{h}(A_{1}, …, A_{n−1})τ_{h}(A_{n}, A_{n+1}) = (−1)^{d}τ_{h}(A_{n}, A_{n+1})τ_{h}(A_{1}, …, A_{n−1}) (1.3.16)

where

d = e(n) + |A_{n}, A_{n+1}||A_{1}, …, A_{n−1}| = |A_{n}, A_{n+1}|(|A_{1}, …, A_{n−1}| + 1) + 1
= |A_{n}, A_{n+1}|(|C| + 1) + 1
= |A_{n}|(|C| + 1) + 1 + (|A_{n+1}| + 1)(|C| + 1).

Multiplying 1.3.15 by (−1)^{d} and substituting 1.3.16 gives

(−1)^{e(n)}τ_{h}(A_{1}, …, A_{n−1})τ_{h}(A_{n}, A_{n+1}) = (−1)^{d_{l}}τ_{h}(A_{n+1}, A_{1}, …, A_{n−1})τ_{h}(A_{n})

+ (−1)^{d_{2}}τ_{h}(A_{1}, …, A_{n})τ_{h}(A_{n+1}) (1.3.17)

where

d_{1} = d + a = (|A_{n+1}| + 1)(|C| + 1) + |A_{n+1}|(|A_{n}| + 1)
= |A_{n+1}|(|C| + |A_{n}| + 1) + |C| + |A_{n+1}| + 1
= |A_{n+1}|(|A_{1}, …, A_{n}|) + |A_{1}, …, Â_{n}, A_{n+1}|
= ε(n)

and

d_{2} = d + b = |C|(|A_{n+1}| + 1) + (|A_{n+1}| + 1)(|C| + 1)
= (|A_{n+1}| + 1) = ε(n + 1).

This establishes the validity of signs as given by ε for n + 1, and proves 1.3.13.

Definition 1.3.18. For 𝒞 a subcategory of _{R}ℳ and M_{I} ∈ ℱ(𝒞), define Tor(M_{I}), the torsion algebra of a family, to be the graded, augmented, commutative Ralgebra generated by 1 and torsion generators τ_{h}(m_{1}, …, m_{n}) where m_{j} ∈ M_{i_{j}} and hm_{j} = 0 in M_{i_{j}}, for some i_{j} ∈ I and M_{i_{j}} ∈ M_{I}. The m_{j} must be homogeneous and h ∈ R − {0} (R if n = 1). Define length, degree as before, and require that the τ_{h}(A) satisfy relations 1.3.2, for A = (m_{1}, …, m_{n}).

Clearly, Tor(M_{I}) satisfies the previous propositions stated for Tor(M).

We relate the torsion algebra of a module, Tor(M), to the more familiar symmetric algebra, S(M), by observing that the map ι_{0}(m) = τ_{0}(m) of M → Tor(M) is a homomorphism of R-modules by relation 1.3.2(viii). By universality, ι_{0} extends to an algebra homomorphism, ι: S(M) → Tor(M). Since the only relations on length one generators are those expressing linearity, the map g is an inclusion of R-algebras. Hence the torsion algebra extends the symmetric algebra by taking into account the torsion properties of M. If M is torsion free over R, we recover the symmetric algebra over M.

1.4. The Multilinear Torsion Algebra. With the notation of §1.3, we define the multilinear form of the torsion algebra of a family, M_{I}.

Definition 1.4.1. Let Torl(M_{I}) be the graded, connected, commutative R-algebra obtained from Tor(M_{I}) by imposing the additional relations

τ_{h}(m_{1}, …, km_{ℓ}, …, m_{n}) = kτ_{h}(m_{1}, …, m_{ℓ}, …, m_{n}) for k ∈ R (1.4.2)

τ_{h}(m_{1}, …, m_{ℓ} + m′_{ℓ}, …, m_{n}) = τ_{h}(m_{1}, …, m_{ℓ}, …, m_{n}) + τ_{h}(m_{1}, …, m′_{ℓ}, …, m_{n}) where all symbols are defined, m_{ℓ}, m′_{ℓ} are homogeneous elements of M_{i_{ℓ}} ∈ M_{I} of the same degree, ℓ = 1, …, n, and h ∈ R − {0}.

Definition 1.4.3. M_{I} ∈ ℱ(_{R}ℳ) is of finite type if each M_{i} is of finite type in _{R}ℳ. If M ∈ _{R}ℳ and m ∈ M is a torsion element (hm = 0 for some h ∈ R − {0}) define o(m) ∈ R as the unique (up to unit) generator of the ideal ann_{R}(m) = {k ∈ R | km = 0}.

Then for any h ∈ R with hm = 0, it follows that o(m)|h in R. With this we can give a restricted set of torsion generators for Torl(M_{I}).

Proposition 1.4.4. If M_{I} ∈ ℱ(_{R}ℳ) is a family of finite type, then Torl(M_{I}) is generated as an R-algebra by 1, the special symbols τ_{0}(m), and h-torsion generators of the form τ_{h}(m_{1}, …, m_{n}), where m_{j} ∈ M_{i_{j}} ∈ M_{I} are homogeneous elements such that o(m_{1})|o(m_{j}) for j = 1, …, n.

Proof. For n ≤ 1 the assertion is trivial. If n ≥ 1, let τ_{h}(A) = τ_{h}(m_{1}, …, m_{n}) be an arbitrary h-torsion generator with m_{j} of degree d_{j}. Let (M_{i_{j}})_{d_{j}} denote the submodule of degree d_{j}. By the structure theorem for finitely generated Rmodules, each (M_{i_{j}})_{d_{j}} can be written as a direct sum of cyclic R-submodules with generators of prime power order (a PID is a unique factorization domain). By linearity, τ_{h}(A) can be written as a linear combination of h-torsion generators τ_{h}(B) where B = (m′_{1}, …, m′_{n}) is a string all of whose elements have prime power order, say o(m′_{j}) = P_{j}^{e_{j}} for P_{j} ∈ R prime, e_{j} ∈ ℤ^{+}. If P_{j} ≠ (unit) P_{ℓ} for some j ≠ ℓ, then o(m′_{j}) is relatively prime to o(m′_{ℓ}), which by linearity implies o(m′_{j})τ_{h}(B) = 0 = o(m′_{ℓ})τ_{h}(B), hence τ_{h}(B) = 0. Thus the only non-zero τ_{h}(B) occurring are those in which all the P_{j} are equal (up to a unit).

If σ is the permutation on n letters rearranging the order of m′_{j} so that e_{j} ≤ e_{ℓ} for j ≤ ℓ, then τ_{h}(B) = ±τ_{h}(σB) by 1.3.2(v). In τ_{h}(σB), we have o(m′_{1})|o(m′_{j}) as required. Consequently τ_{h}(A) can be expressed as a linear combination of h-torsion generators of the requisite form.

For 𝒞, a subcategory of_{R}ℳ closed under colimits, define the functor Σ: ℱ(𝒞) → 𝒞 by Σ(M_{I}) = ⨁_{i∈I} M_{i}, then the multilinear torsion algebra is actually the torsion algebra of a single module.

Proposition 1.4.5. For M_{I} ∈ ℱ(𝒞), there is an R-algebra isomorphism g: Torl(M_{I}) → Torl(ΣM_{I}) which is natural for morphisms in ℱ(𝒞).

Proof. Let q_{i}: M_{i} → ΣM_{I} be the inclusion and define

g(τ_{h}(m_{1}, …, m_{n})) = τ_{h}(q_{i_{1}}(m_{1}), …, q_{i_{n}}(m_{n}))

where m_{j} ∈ M_{i_{j}}. This map is well defined and clearly induces an R-algebra homomorphism. Define an inverse f to g by

f(τ_{h}(m_{1}, …, m_{n})) = ∑_{i_{1},...,i_{n}∈I} τ_{h}(P_{i_{1}}(m_{1}), …, P_{i_{n}}(m_{n}))

where P_{i}: ΣM_{I} → M_{i} is the projection. The sum is finite and well defined since Σq_{i}P_{i} = identity is finite and is well defined. It is trivial to verify that f preserves the defining relations and so induces a homomorphism. A direct calculation shows that f and g are inverses, and that g is natural for morphisms in ℱ(𝒞).

2. Homology is a Tor quotient

2.1. Capping Operations. In what follows, we restrict our consideration to the category _{R}𝒟ℱ𝒜. To establish notation, let _{h}M = {m | hm = 0} where M is an R-module, h ∈ R − {0}, and denote by R_{h} the quotient R/hR.

If A ∈ _{R}𝒟ℱ𝒜 and h, k ∈ R − {0}, let H_{∗}(A) = H_{∗}(A; R) be the “integral” homology algebra; H_{∗}(A; R_{h}) the mod-h homology algebra; h_{∗}: H_{∗}(A) → H_{∗}(A) the map induced by multiplication by h in A; α_{h}: H_{∗}(A) → H_{∗}(A; R_{h}) the map induced by the natural projection R → R_{h}. Let μ_{h}^{hk}: H_{∗}(A; R_{h}) → H_{∗}(A; R_{hk}) be induced by R_{h} → R_{hk}; λ_{hk}^{h}: H_{∗}(A; R_{hk}) → H_{∗}(A; R_{h}) be induced by R_{hk} → R_{h}; β_{h}: H_{∗}(A; R_{h}) → H_{∗−1}(A) be induced by the coefficient sequence

0 → R ⟶^{h} R ⟶ R_{h} ⟶ 0.

All of these maps are natural with respect to morphisms in _{R}𝒟ℱ𝒜.

The following proposition is a consequence of the long exact sequence arising from the coefficient sequence above.

Proposition 2.1.1. If A ∈ _{R}𝒟ℱ𝒜, the following sequence is exact

0 ⟶ _{h}(H_{∗}A) ⟶^{incl} H_{∗}A ⟶^{h_{∗}} H_{∗}A ⟶^{α_{h}} H_{∗}(A; R_{h}) ⟶^{β_{h}} _{h}(H_{∗−1}A) ⟶ 0 (2.1.2)

where h ∈ R − {0}.

Definition 2.1.3. A capping operation ψ on A ∈ _{R}𝒟ℱ𝒜 is an indexed family {ψ_{h} | h ∈ R − 0} of set maps ψ_{h}: _{h}(H_{∗}A) → H_{∗+1}(A; R_{h}) raising degree by 1 and satisfying the conditions:

(i) β_{h}ψ_{h} = identity on _{h}(H_{∗}A) (ii) ψ_{h}(0) = 0 (iii) the following diagram commutes for all h, k ∈ R − {0}

\xymatrix{_{hk}(H_{∗}A) \ar[r]^{k_{∗}} & _{h}(H_{∗}A) \ar[r]^{incl} & _{hk}(H_{∗}A) \\ H_{∗+1}(A; R_{hk}) \ar[r]_{λ_{hk}^{h}} \ar[u]_{β_{hk}} & H_{∗+1}(A; R_{h}) \ar[r]_{μ_{h}^{hk}} \ar[u]_{β_{h}} & H_{∗+1}(A; R_{hk}) \ar[u]_{β_{hk}} \\ _{hk}(H_{∗}A) \ar[r]_{k∗} \ar[u]_{ψ_{hk}} & _{h}(H_{∗}A) \ar[r]_{incl} \ar[u]_{ψ_{h}} & _{hk}(H_{∗}A) \ar[u]_{ψ_{hk}}}

Note: The commutativity of the upper two squares is a consequence of the definitions of β_{h}, λ_{hk}^{h}, μ_{h}^{hk} having nothing to do with capping operations.

The following examples of algebras with capping operations will be important later.

Example. Let R = ℤ and take A = E(x), the exterior algebra over ℤ on a generator of degree 1. Give A a differential by ∂1 = 0, ∂x = 0, then by Proposition 2.1.1, β_{h} = 0, so define ψ_{h} by ψ_{h}(0) = 0 and it is trivial to verify that ψ = {ψ_{h}} is the unique capping operation for A.

Example. Let R = ℤ and take A = P (y), the divided power algebra over ℤ on a generator of degree 2 (i.e., A is generated by symbols γ_{i}(y), i ≥ 0, of degree 2i such that γ_{0}(y) = 1, γ_{1}(y) = y, γ_{i}(y)γ_{j}(y) = (i, j)γ_{i+j}(y) where (i, j) = (i + j)!/i!j!). Give A a differential by ∂γ_{i}(y) = 0, i ≥ 0, then ψ_{h}(0) = 0 for all h ∈ R − {0} and ψ = {ψ_{h}} is the unique capping operation on A.

Example. Let R = ℤ and take A_{m}^{−} = E(x) ⊗ P (y), m ∈ ℤ^{+}, where x has degree 1 and y has degree 2. Give A_{m}^{−} a differential by ∂1 = 0, ∂x = 0, ∂γ_{k}(y) = mxγ_{k−1}(y), if k > 0. Then the homogeneous cycles of A_{m}^{−} are 1 and xγ_{k}(y) for k ≥ 0, of odd degree. From this and Proposition 2.1.1, it follows that A_{m}^{−} has a unique capping operation with ψ_{h} = 0 in even positive degrees, and with ψ_{h} = β_{h}^{−1} in odd degrees.

If ω_{i}^{m}(x), i > 0, is the integral homology class of degree 2i − 1 represented by the cycle xγ_{i−1}(y), an explicit formula for the non-trivial cases of ψ is

ψ_{h}(nω_{i}^{m}(x)) = cls \{\frac{hn}{m}γ_{i}(y) ⊗ 1_{h}\} ∈ H_{2i}(A_{m}^{−}; R_{h})  (2.1.4)

where hnω_{i}^{m}(x) = 0 in H_{∗}(A_{m}^{−}) and 1_{h} is the unit in R_{h}.

Remark: The differential graded R-algebras A_{m}^{−} is the same as the algebra M(m) or Eilenberg and Mac Lane [4]. It is essential in Hamsher’s [5] calculation of H_{∗}(π, 1; ℤ), and will arise in our determination of H_{∗}(π, n; ℤ) in Chapter 3, following the work of Cartan [3]. There the algebra A_{m}^{−} is called an elementary complex.

Example. Let R = ℤ and take A_{m}^{+} = P (x) ⊗ E(y), where x has degree 2 and y has degree 3. Give A_{m}^{+} a differential by ∂γ_{i}(x) = 0, i ≥ 0, and ∂y = mx. Then the homogeneous cycles of positive degree are γ_{i}(x), i > 0, and ∂(γ_{i−1}(x)y) = γ_{i−1}(x) · mx = miγ_{i}(x). As in the previous example, A_{m}^{+} has a unique capping operation with ψ_{h} = 0 in odd degrees, and ψ_{h} = β_{h}^{−1} in positive even degrees.

If γ_{i}^{m}(x), i > 0, is the integral homology class of degree 2i represented by the cycle γ_{i}(x), an explicit formula for the nontrivial cases of ψ is

ψ_{h}(nγ_{i}^{m}(x)) = cls \{\frac{hn}{mi}γ_{i−1}(x)y ⊗ 1_{h}\} ∈ H_{2i+1}(A_{m}^{+}; R_{h})  (2.1.5)

where hnγ_{i}^{m}(x) = 0 in H_{∗}(A_{m}^{+}).

Example. If R is a field, any A ∈ _{R}𝒟ℱ𝒜 has a unique trivial capping operation since β_{h} = 0 for all h ∈ R − {0}.

Definition 2.1.6. The category with capping operations, 𝒞^{ψ}, is the category whose objects are pairs (A, ψ^{A}) where A ∈ _{R}𝒟ℱ𝒜 and ψ^{A} = {ψ_{h} | h ∈ R−0} is a capping operation for A. A morphism in 𝒞^{ψ} is a morphism in _{R}𝒟ℱ𝒜 which commutes with capping operations, that is g: A → A′ such that

\xymatrix{H_{∗}(A; R_{h}) \ar[r]^{g_{∗}} & H_{∗}(A′; R_{h}) \\ _{h}(H_{∗}A) \ar[r]^{g_{∗}} \ar[u]_{ψ^{A}} & _{h}(H_{∗}A′) \ar[u]_{ψ^{A′}}}  (2.1.7)

commutes, where h ∈ R − {0} and g_{∗} is induced by g.

The identity and composition are those of _{R}𝒟ℱ𝒜, and it is immediate that {A | (A, ψ^{A}) ∈ 𝒞^{ψ}} together with the morphisms of 𝒞^{ψ} is a subcategory (not necessarily full) of _{R}𝒟ℱ𝒜.

2.2. The Map Ψ̃. Our objective is to define a morphism in _{R}𝒜 from the torsion algebra of a family of homology algebras of differential algebras with capping operations to the homology algebra of their tensor product.

Let 𝒞^{ψ} be the category with capping operations and A_{I} ∈ ℱ(𝒞^{ψ}) a family, then TA_{I} = ⨂_{ℓ∈I} A_{ℓ} and ℱ(H_{∗})(A_{I}) = HA_{I} ∈ ℱ(_{R}𝒜). Denote by κ_{ℓ}: A_{ℓ} → T A_{I}, ℓ ∈ I, the natural inclusion (each A_{ℓ} is connected); by {ψ_{h}^{ℓ} | h ∈ R − 0} the capping op-

erations in A; and let ρ_{h}^{ℓ}: _{h}(H_{∗}A_{ℓ}) ⟶^{ψ_{h}^{ℓ}} H_{∗+1}(A_{ℓ}; R_{h}) ⟶^{κ_{ℓ∗}} H_{∗+1}(T A_{I}; R_{h}) denote

the composition.

If β_{h}: H_{∗}(T A_{I}; R_{h}) → H_{∗−1}(T A_{I}) is the mod-h Bockstein, we define a morphism Ψ̃(A_{I}): Tor(HA_{I}) → H_{∗}(T A_{I}) in _{R}𝒜 (which will be abbreviated Ψ̃, A_{I} being fixed for the moment) as follows:

(i) Ψ̃(1) = 1 ∈ H_{0}(T A_{I}), Ψ̃(τ_{h}( )) = 0, h ∈ R (ii) Ψ̃(τ̃_{0}(a)) = κ_{ℓ∗}(a), where a ∈ (H_{∗}A_{ℓ}), κ_{ℓ∗} induced by κ_{ℓ} (2.2.1) (iii) Ψ̃(τ_{h}(a_{1}, …, a_{n})) = β_{h}(ρ_{h}^{ℓ_{1}}(a_{1}) ∗ ⋯ ∗ ρ_{h}^{ℓ_{n}}(a_{n})), where h ∈ R − {0}, a_{j} ∈

h(H_{∗}A_{ℓ_{j}}) for j = 1, …, n, and ∗ is the multiplication in H_{∗}(T A_{I}; R_{h}) induced by the multiplication in T A_{I} and the natural pairing R_{h} ⊗ R_{h} ≃ R_{h}.

Ψ̃ is clearly well defined, preserves the unit and augmentation, and is degree preserving since κ_{ℓ∗} is, with

|τ_{h}(a_{1}, …, a_{n})| = ∑_{j=1}^{n} |a_{j}| + n − 1 = ∑_{j=1}^{n}(|a_{j}| + 1) − 1

which is the degree of the right-hand side of 2.2.1(iii). For Ψ̃ to induce an Ralgebra homomorphism on Tor(HA_{I}) to H_{∗}(T A_{I}), it only remains to establish that relations 1.3.2 hold. This reduces to establishing certain properties of the homology classes β_{h}(ρ_{h}^{ℓ_{1}}(a_{ℓ_{1}}) ∗ ⋯ ∗ ρ_{h}^{ℓ_{n}}(a_{ℓ_{n}})) which we do in the following lemmas.

We first record several basic properties that are immediate consequences of the definitions. Let A_{1} ⟶^{κ_{1}} Ā, A_{2} ⟶^{κ_{2}} Ā be inclusions of R-algebras, A_{1}, A_{2}, Ā ∈ R𝒟ℱ𝒜. If κ_{1∗}, κ_{2∗} denote the maps induced on integral or mod-h homology and ∗ is multiplication in Ā,

μ_{h}^{hk}λ_{hk}^{h} = k_{∗}, λ_{hk}^{h}μ_{h}^{hk} = k_{∗}, for h, k ∈ R − {0}  (2.2.2)
μ_{h}^{hk}(κ_{1∗}λ_{hk}^{h}(a_{1}) ∗ κ_{2∗}a_{2}) = κ_{1∗}a_{1} ∗ κ_{2∗}μ_{h}^{hk}(a_{2})  (2.2.3)
if a_{1} ∈ H_{∗}(A_{1}; R_{hk}), a_{2} ∈ H_{∗}(A_{2}; R_{h})
λ_{hk}^{h}(κ_{1∗}a_{1} ∗ κ_{2∗}a_{2}) = κ_{1∗}λ_{hk}^{h}(a_{1}) ∗ κ_{2∗}λ_{hk}^{h}(a_{2})  (2.2.4)

if a_{1} ∈ H_{∗}(A_{1}; R_{hk}), a_{2} ∈ H_{∗}(A_{2}; R_{hk})

β_{h}(κ_{1∗}α_{h}(a_{1}) ∗ κ_{2∗}a_{2}) = (−1)^{|a_{1}|}κ_{1∗}a_{1} ∗ β_{h}κ_{2∗}(a_{2})  (2.2.5)
if a_{1} ∈ H_{∗}(A_{1}), a_{2} ∈ H_{∗}(A_{2}; R_{h})
β_{h}(κ_{1∗}a_{1} ∗ κ_{2∗}α_{h}(a_{2})) = β_{h}κ_{1∗}(a_{1}) ∗ κ_{2∗}a_{2}  (2.2.5′)
if a_{1} ∈ H_{∗}(A_{1}; R_{h}), a_{2} ∈ H_{∗}(A_{2})

All maps involved are natural for morphisms in _{R}𝒟ℱ𝒜.

Lemma 2.2.6. With notation as above, let {A_{1}, …, A_{n}} (n > 0) be a finite subfamily of A_{I}. Then, in H_{∗}(T A_{I}),

∗_{ℓ=1}^{n}λ_{hk}^{h}ρ_{hk}^{ℓ}(a_{ℓ}) = λ_{hk}^{h} (∗_{ℓ=1}^{n}ρ_{hk}^{ℓ}(a_{ℓ})) if a_{ℓ} ∈ _{hk}(H_{∗}A_{ℓ})  (2.2.7)
β_{h} [κ_{1∗}α_{h}(a_{1}) ∗ (∗_{ℓ=2}^{n}ρ_{n}^{ℓ}(a_{ℓ}))] = (−1)^{|a_{1}|}κ_{1∗}(a_{1}) ∗ β_{h} [∗_{ℓ=2}^{n}ρ_{h}^{ℓ}(a_{ℓ})]  (2.2.8)
if a_{1} ∈ H_{∗}A_{1}, a_{ℓ} ∈ _{h}(H_{∗}A_{ℓ}), for ℓ = 2, …, n
k_{∗}^{n−1}μ_{h}^{hk} (∗_{ℓ=1}^{n}ρ_{h}^{ℓ}(a_{ℓ})) = ∗_{ℓ=1}^{n}ρ_{hk}^{ℓ}(a_{ℓ}) if a_{ℓ} ∈ _{h}(H_{∗}A_{ℓ}) ⊆ _{hk}(H_{∗}A_{ℓ})  (2.2.9)

β_{h}[(∗_{ℓ=1}^{m}ρ_{h}^{ℓ}k_{∗}(a_{ℓ})) ∗ (∗_{ℓ=m+1}^{n}ρ_{h}^{ℓ}(a_{1})))] = β_{hk}[(∗_{ℓ=1}^{m}ρ_{hk}^{ℓ}(a_{ℓ})) ∗ μ_{h}^{hk}(∗_{ℓ=m+1}^{n}ρ_{h}^{ℓ}(a_{ℓ}))]

if a_{ℓ} ∈ _{hk}(H_{∗}A_{ℓ}) for ℓ = 1, …, m, and a_{ℓ} ∈ _{h}(H_{∗}A_{ℓ}) for ℓ = m + 1, …, n. (2.2.10)

Proof. 2.2.7 follows from 2.2.4 by induction and naturality; 2.2.8 follows from 2.2.5 by naturality. For 2.2.9, if n = 1, the result follows from 2.1.3(iii) and naturality. By induction for n > 1, replacing ∗_{ℓ=2}^{n}ρ_{h}^{ℓ}(a_{ℓ}) by κ_{2∗}a_{2} gives

k_{∗}^{n−1}μ_{h}^{hk}(ρ_{h}^{1}(a_{1}) ∗ κ_{2∗}a_{2}) = k_{∗}^{n−2}μ_{h}^{hk}(k_{∗}ρ_{h}^{1}(a_{1}) ∗ κ_{2∗}a_{2})  by naturality
= k_{∗}^{n−2}μ_{h}^{hk}(λ_{hk}^{h}μ_{h}^{hk}ρ_{h}^{1}(a_{1}) ∗ κ_{2∗}a_{2})  by 2.2.2
= k_{∗}^{n−2}μ_{h}^{hk}(κ_{1∗}λ_{hk}^{h}μ_{h}^{hk}ψ_{h}^{1}(a_{1}) ∗ κ_{2∗}a_{2}) by natuarality
= k_{∗}^{n−2}(κ_{1∗}μ_{h}^{hk}ψ_{h}^{1}(a_{1}) ∗ κ_{2∗}μ_{h}^{hk}a_{2})  by 2.2.3
= k_{∗}^{n−2}(κ_{1∗}ψ_{hk}^{1}(a_{1}) ∗ μ_{h}^{hk}κ_{2∗}a_{2})  by 2.1.3(iii)
= κ_{1∗}ψ_{hk}^{1}(a_{1}) ∗ k_{∗}^{n−2}μ_{h}^{hk}κ_{2∗}a_{2}  by naturality
= ρ_{hk}^{1}(a_{1}) ∗ (∗_{ℓ=2}^{n}ρ_{hk}^{ℓ}(a_{ℓ}))  by induction.

For 2.2.10, replacing ∗_{ℓ=m+1}^{n}ρ_{h}^{ℓ}(a_{ℓ}) by κ_{2∗}a_{2}, gives

β_{h}[(∗_{ℓ=1}^{m}ρ_{h}^{ℓ}k_{∗}(a_{ℓ}) ∗ κ_{2∗}a_{2})]
= β_{hk}μ_{h}^{hk} [(∗_{ℓ=1}^{m}κ_{ℓ∗}λ_{hk}^{h}ψ_{hk}^{ℓ}(a_{ℓ})) ∗ κ_{2∗}a_{2}]  by 2.1.3(iii)
= β_{hk}μ_{h}^{hk} [λ_{hk}^{h} (∗_{ℓ=1}^{m}κ_{ℓ∗}ψ_{hk}^{ℓ}(a_{ℓ})) ∗ κ_{2∗}a_{2}]  by 2.2.4
= β_{hk} [(∗_{ℓ=1}^{m}κ_{ℓ∗}ψ_{hk}^{ℓ}(a_{ℓ})) ∗ μ_{h}^{hk} (∗_{ℓ=m+1}^{n}ρ_{h}^{ℓ}(a_{ℓ}))] by 2.2.3

This establishes the relations.

Lemma 2.2.11. Ψ̃ preserves relations 1.3.2

Proof. We first observe that the convention τ_{h}(A) = τ_{h}(A_{1}, …, A_{p}), where A = (A_{1}, …, A_{p}) is a string partition such that τ_{h}(A_{ℓ}) is defined for ℓ = 1, …, p, is honored by the classes β_{h} [∗_{ℓ=1}^{n}ρ_{h}(a_{ℓ})] by associativity of the product ∗ in H_{∗}(T A_{I}; R_{h}). Relation 1.3.2(v) follows by commutativity of the product ∗.

Relation 1.3.2(i) is trivially satisfied, and relation 1.3.2(vi) is satisfied by virtue of 2.1.3(ii) defining capping operations.

Relation 1.3.2(ii) follows from the equality κ_{ℓ∗}(a) = β_{h}κ_{ℓ∗}ψ_{h}^{ℓ}(a) for a ∈ _{h}(H_{∗}A_{ℓ}), ℓ ∈ I, resulting from naturality and 2.1.3(i). Relation 1.3.2(viii) follows since κ_{ℓ∗} is a homomorphism of R-algebras.

Relation 1.3.2(iii) follows from the equality

β_{h} [∗_{ℓ=1}^{n}ρ_{h}^{ℓ}k_{∗}(a_{ℓ})] = β_{h} [∗_{ℓ=1}^{n}κ_{ℓ∗}λ_{hk}^{h}ψ_{hk}^{ℓ}(a_{ℓ})] by naturality and 2.1.3(iii)
= β_{h} [∗_{ℓ=1}^{n}λ_{hk}^{h}ρ_{hk}^{ℓ}(a_{ℓ})]  by naturality and definitions
= β_{h}λ_{hk}^{h} [∗_{ℓ=1}^{n}ρ_{hk}^{ℓ}(a_{ℓ})]  by 2.2.7
= k_{∗}β_{hk} [∗_{ℓ=1}^{n}ρ_{hk}^{ℓ}(a_{ℓ})]  by 2.1.3(iii).

Relation 1.3.2(iv) follows from the equality

β_{hk}[(∗_{ℓ=1}^{m}ρ_{hk}^{ℓ}(a_{ℓ})) ∗ (∗_{ℓ=m+1}^{n}ρ_{hk}^{ℓ}(a_{ℓ}))]
= β_{hk} [(∗_{ℓ=1}^{m}ρ_{hk}^{ℓ}(a_{ℓ})) ∗ k_{∗}^{n−m−1}μ_{h}^{hk} (∗_{ℓ=m+1}^{n}ρ_{h}^{ℓ}(a_{ℓ}))] by 2.2.9
= k_{∗}^{n−m−1}β_{h} [(∗_{ℓ=1}^{m}ρ_{h}^{ℓ}k_{∗}(a_{ℓ})) ∗ (∗_{ℓ=m+1}^{n}ρ_{h}^{ℓ}(a_{ℓ}))]  by 2.2.10

where n = ℓ(A) + ℓ(B), m = ℓ(A). Note that the equality holds if ℓ(A) = 0 since β_{hk}μ_{h}^{hk} = β_{h} by 2.1.3(iii) applied to 2.2.10.

Finally, the Jacobi relation 1.3.2(vii) follows from the general relation

(−1)^{d}β_{hk}(a ∗ b) ∗ β_{h}c + (−1)^{e}β_{hk}(b ∗ μ_{h}^{hk}c) ∗ β_{hk}a + (−1)^{f}β_{hk}(μ_{h}^{hk}c ∗ a) ∗ β_{hk}b = 0

valid if a, b ∈ H_{∗}(Ā; R_{hk}), c ∈ H_{∗}(Ā; R_{h}) for Ā ∈ _{R}𝒟ℱ𝒜, where d = (|a| − 1)|c|, e = (|b| − 1)|a|, f = (|c| − 1)|b|. If A = (a_{1}, …, a_{m}), B = (a_{m+1}, …, a_{n}), and C = (a_{n+1}, …, a_{p}) in the Jacobi relation, take a = ∗_{ℓ=1}^{m}ρ_{hk}^{ℓ}(a_{ℓ}), b = ∗_{ℓ=m+1}^{n}ρ_{hk}^{ℓ}(a_{ℓ}), c = ∗_{ℓ=n+1}^{p}ρ_{h}^{ℓ}(a_{ℓ}), then d = |A|(|C| + 1), e = |B|(|A| + 1), f = |C|(|B| + 1). This completes the proof.

We observe that Ψ̃(A_{I}): Tor(HA_{I}) → H_{∗}(T A_{I}) for A_{I} ∈ ℱ(𝒞^{ψ}) is natural for morphisms in ℱ(𝒞^{ψ}) by naturality of the Bockstein and coefficient maps, and the definition of morphisms of algebras with capping operations. These results are summed up by the following theorem.

Theorem 2.2.12. The map Ψ̃(A_{I}): Tor ∘ ℱ(H_{∗})(A_{I}) → H_{∗} ∘ T (A_{I}) is a natural transformation of functors from ℱ(𝒞^{ψ}) to _{R}𝒜, where 𝒞^{ψ} is the category with capping operations.

2.3. The Torsion Inverse of an Algebra. The purpose of what follows is to give sufficient conditions for the map Ψ̃(A_{I}) of Section 2.2 to induce an isomorphism, giving, in effect, a means of computing H_{∗}(T A_{I}) by generators and relations.

To this end, let A ∈ _{R}𝒟ℱ𝒜 with capping operation ψ = {ψ_{h}}. For homogeneous elements of H_{∗}(A), the following elements of H_{∗}(A; R_{h}) lie in the kernel of β_{h}:

(i) ψ_{h}(a⊤a′) = ψ_{h}(a + a′) − ψ_{h}(a) − ψ_{h}(a′) for a, a′ ∈ _{h}(H_{d}A), d > 0 (ii) ψ_{h}β_{h}(ψ_{h}(a) ∗ ψ_{h}(a′)) − ψ_{h}(a) ∗ ψ_{h}(a′) for a, a′ ∈ _{h}(H_{∗}A) (2.3.1) (iii) ψ_{h}(a · a′) − (−1)^{|a|}α_{h}(a) ∗ ψ_{h}(a′) for a ∈ H_{∗}A, a′ ∈ _{h}(H_{∗}A),

· is multiplication in H_{∗}A Denote by D_{h}(A) the R-module of H_{∗}(A; R_{h}) generated by elements of the form

given in 2.3.1. Since β_{h}(D_{h}(A)) = 0, we have D_{h}(A) ⊆ image α_{h} by Proposition 2.1.1. Furthermore, it is easy to verify that the inclusion D_{h}(A) → H_{∗}(A; R_{h}) is natural for coefficient maps and for maps of R-algebras with capping operations.

Definition 2.3.2. For A, ψ^{A} as above, a torsion inverse σ^{A} on A is a family of set maps σ_{h}: D_{h}(A) → H_{∗}A for h ∈ R − {0} which preserve degree and satisfy the following conditions:

(i) α_{h}σ_{h} = identity on D_{h}(A) (ii) σ_{h}(0) = 0 for all h ∈ R − {0} (2.3.3) (iii) for all h, k ∈ R − {0} the “double diagram”

\xymatrix{D_{h}(A) \ar[r]^{σ_{h}} \ar[d]_{μ_{h}^{hk}} & H_{∗}A \ar[d]_{k_{∗}} \\ D_{hk}(A) \ar[r]^{σ_{hk}} \ar[u]_{λ_{hk}^{h}} & H_{∗}A \ar[u]_{id}}  commutes.

Example. In the differential algebras E(x), P (x) with unique capping operations ψ_{h} = 0, the modules D_{h}(A) = 0, so the unique torsion inverse in each case is trivial.

Example. In the ℤ-algebra A_{m}^{−} = E(x) ⊗ P (y) defined in Section 2.1, ψ_{h} = β_{h}^{−1} and σ_{h} are homomorphisms, so the D_{h}(A) are generated by elements of the form 2.3.1(iii). As these latter elements are of odd degree, D_{h}(A) = 0 in positive even degrees, and σ_{h} = 0. In odd degrees α_{h}: H_{∗}(A_{m}^{−}) → H_{∗}(A_{m}^{−}; R_{h}) is an epismorphism. As a ℤ-algebra, H_{∗}(A_{m}^{−}) is generated by 1 and the symbols ω_{i}^{m}(x), for i > 0. Since x^{2} = 0 in A_{m}^{−}, we have ω_{i}^{m}(x)ω_{j}^{m}(x) = 0, for i, j > 0. Given a = ω_{i}^{m}(x), a′ = nω_{j}^{m}(x), n ∈ ℤ such that m|hn,

a′′ = ψ_{h}(a · a′) − (−1)^{|a|}α_{h}(a) ∗ ψ_{h}(a′) = α_{h}(ω_{i}^{m}(x)) ∗ ψ_{h}(nω_{j}^{m}(x))

= cls \{\frac{hn}{m}(i − 1, j)xγ_{i+j−1}(y) ⊗ 1_{h}\}

= α_{h} (\frac{hn}{m}(i − 1, j)ω_{i+j}^{m}(x)) .

Define σ_{h}(a′′) = \frac{hn}{m}(i − 1, j)ω_{i+j}^{m}(x). It is then straightforward to verify that this

gives a torsion inverse for A_{m}^{−} with its capping operation.

Example. In the ℤ-algebra A_{m}^{+} = P (x) ⊗ E(y), we have ψ_{h} = β_{h}^{−1} or 0 and again D_{h}(A_{m}^{+}) is generated by elements of the form 2.3.1(iii), of odd degree. A simple calculation using the explicit formula for ψ_{h} shows that all elements of the latter form are 0 in H_{∗}(A_{m}^{+}; ℤ_{h}) hence D_{h}(A) = 0 and the unique torsion inverse for A_{m}^{+} with its capping operations is {σ_{h} = 0}.

Example. If R is a field, A ∈ _{R}𝒟ℱ𝒜, the unique capping operations are all trivial, hence D_{h}(A) = 0 and the unique torsion inversion is trivial as well.

Definition 2.3.4. The category with capping operations and torsion inverse, 𝒞^{ψσ}, is a category whose objects are triples (A, ψ^{A}, σ^{A}) where A ∈ _{R}𝒟ℱ𝒜, ψ^{A} is a capping operations on A and σ^{A} is a torsion inverse for the ψ^{A}. A morphism in 𝒞^{ψσ} is a morphism in _{R}𝒟ℱ𝒜 which preserves capping operations and their torsion inverse, i.e., g: A → A′ so that diagram 2.1.7 commutes and so that

\xymatrix{D_{h}(A) \ar[d]_{D_{h}(g)} \ar[r]^{σ_{h}^{A}} & H_{∗}A \ar[d]^{g_{∗}} \\ D_{h}(A′) \ar[r]^{σ_{h}^{A′}} & H_{∗}A′}  (2.3.5)

commutes, where D_{h}(g) and g_{∗} are induced by g.

2.4. Homology as a Tor Quotient. In this section, we prove the basic theorem relating Tor(HA_{I}) and H_{∗}(T A_{I}). We establish the result for the case of a finite family A_{I} ∈ ℱ(𝒞^{ψσ}) and extend to the general case by using the results of Chapter 1 applied to the filtered colimit of the finite subfamilies of a family.

If we consider the family A_{I} = {A_{1}, …, A_{n}} fixed for the ensuing discussion with capping operations {ψ_{h}^{ℓ}} and torsion inverse {σ_{h}^{ℓ}} associated to each R-algebra A_{ℓ} ∈ A_{I}, we defined an R-algebra homomorphism Ψ̃: Tor({HA_{1}, …, HA_{n}}) → H_{∗}(A_{1} ⊗ ⋯ ⊗ A_{n}) in Section 2.2 which is natural for morphisms in ℱ(𝒞^{ψσ}). Our claim in this case is that Ψ̃ is an epimorphism with kernel given by generators and relations.

For this purpose, we have need of the additive multiple Künneth formula treated by Mac Lane [9] for n = 3 and extended to the general (finite) case for torsion-free R-algebras (R a PID) by Bockstein [2] using the covariant functors Tor^{R}( ) and Mult_{i}^{R}( ) of graded R-modules.

Given M_{1}, …, M_{n} ∈ _{R}ℳ, n > 1, define the generalized torsion product Tor^{R}(M_{1}, …, M_{n}) to be the graded R-module with generators t_{h}(m_{1}, …, m_{n}) where h ∈ R − {0}, m_{ℓ} ∈ M_{ℓ} is a homogeneous element, and hm_{ℓ} = 0, 1 ≤ ℓ ≤ n. Assign to each generator t_{h}(m_{1}, …, m_{n}) and the degree given by ∑_{ℓ=1}^{n}(|m_{ℓ}|+1)−1. Impose the following relations (whenever both sides are defined):

(i) t_{h}(m_{1}, …, km_{ℓ}, …, m_{n}) = kt_{h}(m_{1}, …, m_{ℓ}, …, m_{n}) if 1 ≤ ℓ ≤ n, k ∈ R (ii) t_{h}(m_{1}, …, m_{ℓ}⊤m′_{ℓ}, …, m_{n}) = 0 if m_{ℓ}, m′_{ℓ} ∈ M_{ℓ}, |m_{ℓ}| = |m′_{ℓ}| (2.4.1) (iii) t_{hk}(m_{1}, …, m_{ℓ}, …, m_{n}) = t_{h}(km_{1}, …, km_{ℓ−1}, m_{ℓ}, km_{ℓ+1}, …, km_{n})

These relations express termwise R-linearity and the “slide” formula. For n = 1, we define Tor^{R}(M) = M. If n = 2, Tor^{R}(M_{1}, M_{2}) is the standard torsion product of two R-modules as given by Mac Lane [7].

Now for i = 1, …, n, define the multivariant covariant functors Mult_{i}^{R}( ) by Mult_{i}^{R}(M_{1}, …, M_{n}) = Ẽ(i)/E(i) where

Ẽ(i) = ∑ Tor^{R}(M_{s_{1}}, …, M_{s_{i}}) ⊗ M_{r_{1}} ⊗ ⋯ M_{r_{n−i}}  (2.4.2)

the sum over all partitions {s_{1}, …, s_{i}}, {r_{1}, …, r_{n−i}} of the indices {1, …, n} such that s_{1} < ⋯ < s_{i} and r_{1} < ⋯ < r_{n−i}; and E(i) is defined by E(1) = E(n) = 0, otherwise E(i) is generated by all elements of the form

Dt_{h}(m_{s_{1}}, …, m_{s_{i+1}}) ⊗ m_{r_{1}} ⊗ … m_{r_{n−i−1}} where
Dt_{h}(m_{s_{1}}, …, m_{s_{i+1}}) = ∑_{ℓ=1}^{i+1}(−1)^{e(ℓ)}t_{h}(m_{s_{1}}, …, m̂_{s_{ℓ}}, …, m_{s_{i+1}}) ⊗ m_{s_{ℓ}} (2.4.3)
and e(ℓ) = (|m_{s_{ℓ}}| + 1)|t_{h}(m_{s_{ℓ+1}}, …, m_{s_{i+1}})|.

The carat indicates omission, and {s_{1}, …, s_{i+1}} ∪ {r_{1}, …, r_{n−i−1}} is an ordered partition of the indices. Here we allow transposition of m_{s_{ℓ}} in the tensor product by the usual sign rule for graded modules, so as to preserve the ordering of indices.

Remark: The special case i = n gives the functor Tor^{R} defined previously, and the case Mult_{2}^{R}(M_{1}, M_{2}, M_{3}) was called Trip^{R}(M_{1}, M_{2}, M_{3}) by Mac Lane [9].

Theorem 2.4.4 (Bockstein). If K_{1}, …, K_{n} are torsion free (over R) graded differential R-modules, then there is a natural filtration 0 ⊆ S_{1} ⊆ ⋯ ⊆ S_{n} = H_{∗}(K_{1} ⊗ ⋯ ⊗ K_{n}) and natural isomoprhisms

ξ_{i}: Mult_{i}^{R}(HK_{1}, …, HK_{n}) → S_{i}/S_{i−1}

of graded R-modules for 0 < i ≤ n.

The submodules S_{i} of the Theorem are generated by homology classes which contain cycles of the form z_{s_{1},...,s_{i}} ⊗z_{r_{1}} ⊗⋯⊗z_{r_{n−i}}, where z_{s_{1},...,s_{i}} is a cycle in K_{s_{1}} ⊗ ⋯⊗K_{s_{i}} and z_{r_{1}}, …, z_{r_{n−i}} are cycles from the remaining complexes K_{r_{1}}, …, K_{r_{n−i}}.

The R-homomorphisms ξ_{i} are induced by the maps ξ̃_{i}: Ẽ(i) → S_{i}/S_{i−1} given on generators by

ξ̃_{i}(t_{h}(z̄_{s_{1}}, …, z̄_{s_{i}}) ⊗ z̄_{r_{1}} ⊗ ⋯ ⊗ z̄_{r_{n−i}}) =

cls{χ_{h}(z̄_{s_{1}}, …, z̄_{s_{i}}) ⊗ z̄_{r_{1}} ⊗ ⋯ ⊗ z̄_{r_{n−i}}} mod S_{i−1}

where z̄_{ℓ} ∈ K_{ℓ} represents the homology class determined by z_{ℓ}, and χ_{h}(z̄_{s_{1}}, …, z̄_{s_{i}}) is a cycle in K_{s_{1}} ⊗⋯⊗K_{s_{i}}, an i-fold “h-cap” for the cycles z_{s_{1}}, …, z_{s_{i}}. If ψ_{h}(z̄_{s_{ℓ}}) ∈ K_{s_{ℓ}} is such that ∂ψ_{h}(z̄_{s_{ℓ}}) = hz̄_{s_{ℓ}}, then \frac{1}{h}∂[ψ_{h}(z̄_{s_{1}}) ⊗ ⋯ ⊗ ψ_{h}(z̄_{s_{i}})] is an explicit i-fold h-cap. Further, ξ̃_{i}(E(i)) = 0 mod S_{i−1} by a direct calculation using the h-cap given above, and so induces ξ_{i}.

In our application, K_{ℓ} = A_{ℓ} (ℓ = 1, …, n), where A_{ℓ} is a torsion free, graded, differential, connected R-algebra. Theorem 2.4.4 then allows, up to group extensions, the computation of the R-module H_{∗}(A_{1} ⊗ ⋯ ⊗ A_{n}).

Reverting to the notation of previous sections, let A ∈ _{R}𝒟ℱ𝒜 with capping operation ψ^{A} and torsion inverse σ^{A}. We define certain elements Δ_{h}^{j}(a, a′) ∈ H_{∗}A, for h ∈ R − {0}, j = 1, 2, 3 by

Δ_{h}^{1}(a, a′) = σ_{h}(ψ_{h}(a⊤a′)) for a, a′ ∈ _{h}(H_{d}A), d > 0
Δ_{h}^{2}(a, a′) = σ_{h}[ψ_{h}β_{h}(ψ_{h}(a) ∗ ψ_{h}(a′)) − ψ_{h}(a) ∗ ψ_{h}(a′)] for a, a′ ∈ _{h}(H_{∗}A) (2.4.5)
Δ_{h}^{3}(a, a′) = σ_{h}[ψ_{h}(a · a′) − (−1)^{|a|}α_{h}(a) ∗ ψ_{h}(a′)] a ∈ H_{∗}A, a′ ∈ _{h}(H_{∗}A)

We have α_{h}(Δ_{h}^{j}(a, a′)) ∈ D_{h}(A) and d_{1} = |Δ_{h}^{1}(a, a′)| = |a| + 1 = |a′| + 1, d_{2} = |Δ_{h}^{2}(a, a′)| = |a| + |a′| + 2, d_{3} = |Δ_{h}^{3}(a, a′)| = |a| + |a′| + 1.

Assume, now, that A ∈ A_{I} as above, and define the ideal K(A_{I}) in Tor(HA_{I}) by the relations (where all symbols are defined)

(i) τ_{0}(1) = 1 (ii) τ_{h}(ka, B) = kτ_{h}(a, b) for k ∈ R (iii) τ_{h}(a⊤a′, B) = (−1)^{d_{1}}τ_{0}(Δ_{h}^{1}(a, a′))τ_{h}(B) (2.4.6) (iv) τ_{h}(a, a′, B) = τ_{h}(a′′, B) + (−1)^{d_{2}+1}τ_{0}(Δ_{h}^{2}(a, a′))τ_{h}(B)

where a′′ = β_{h}(ψ_{h}(a) ∗ ψ_{h}(a′)) ∈ H_{∗}A

(v) τ_{0}(a)τ_{h}(a′, B) = τ_{h}(a · a′, B) + (−1)^{d_{3}+1}τ_{0}(Δ_{h}^{3}(a, a′))τ_{h}(B) where B is a (possibly empty) string of homogeneous elements of members of HA_{I}.

Lemma 2.4.7. Ψ̃(K(A_{I})) = 0.

Proof. Using the definition of Ψ̃, we verify the corresponding relations on the classes β_{h}(∗_{ℓ=1}^{n}ρ_{h}^{ℓ}(a_{ℓ})) in H_{∗}(T A_{I}).

Let b = ∗_{ℓ=2}^{n}ρ_{h}^{ℓ}(a_{ℓ}) where B = (a_{2}, …, a_{n}) and put b = 1 if B is empty. Then 2.4.6(i) is trivial by 2.2.1(ii), and 2.4.6(ii) is immediate from 2.1.3(iii).

For 2.4.6(iii),

β_{h}(ρ_{h}^{1}(a⊤a′) ∗ b] = β_{h}(κ_{1∗}α_{h}(Δ_{h}^{1}(a, a′)) ∗ b)  by 2.4.5 and 2.3.3(i)
= (−1)^{d_{1}}κ_{1∗}(Δ_{h}^{1}(a, a′)) ∗ β_{h}b  by 2.2.8

For 2.4.6(iv),

β_{h}[ρ_{h}^{1}(a) ∗ ρ_{h}^{1}(a′) ∗ b] = β_{h}[κ_{1∗}ψ_{h}^{1}β_{h}(ψ_{h}^{1}(a) ∗ ψ_{h}^{1}(a′)) ∗ b]
− β_{h}[κ_{1∗}α_{h}(Δ_{h}^{2}(a, a′)) ∗ b]  by 2.4.5
= β_{h}[κ_{1∗}ψ_{h}^{1}(a′′)] + (−1)^{d_{2}+1}κ_{1∗}(Δ_{h}^{2}(a, a′)) ∗ β_{h}b by 2.2.8

For 2.4.6(v),

β_{h}[ρ_{h}^{1}(a · a′) ∗ b] + (−1)^{d_{3}+1}κ_{1∗}(Δ_{h}^{3}(a, a′)) ∗ β_{h}b
= β_{h}[ρ_{h}^{1}(a · a′) ∗ b)] − β_{h}[κ_{1∗}α_{h}(Δ_{h}^{3}(a, a′)) ∗ b] by 2.2.8
= β_{h}[κ_{1∗}(ψ_{h}^{1}(a · a′) − α_{h}(Δ_{h}^{3}(a, a′))) ∗ b]  by linearity

and naturality

= (−1)^{|a|}β_{h}[κ_{1∗}(α_{h}(a) ∗ ψ_{h}^{1}(a′)) ∗ b]  by 2.4.5
= (−1)^{|a|+|a|}κ_{1∗}(a) ∗ β_{h}[κ_{1∗}ψ_{h}^{1}(a′) ∗ b]  by 2.2.8

This completes the verification.

In the situation above, denote by Tor_{ψσ}(HA_{I}) the connected quotient R-algebra Tor(HA_{I})/K(A_{I}), and by Ψ(A_{I}) the R-algebra homomorphism induced by Ψ̃(A_{I}). We are now in a position to prove the finite case of our central theorem.

Theorem 2.4.8. Let A_{I} = {A_{1}, …, A_{n}} ∈ ℱ(𝒞^{ψσ}) be a finite family. Then Ψ(A_{I}): Tor_{ψσ}(HA_{I}) → H_{∗}(T A_{I}) is an isomorphism of graded, augmented, connected R-algebras.

Proof. The proof consists in showing that Ψ is an R-module isomorphism by introducing a filtration on Tor(HA_{I}) whose associated graded module is isomorphic to that induced by the filtration on H_{∗}(T A_{I}) by Theorem 2.4.4, and then applying the standard 5-lemma argument.

Tor_{ψσ}( ) is generated as an R-algebra by the torsion generators τ_{h}(A) subject to the relations 1.3.2 and 2.4.6. By Proposition 1.3.9, Tor_{ψσ}(HA_{I}) is generated as an R-module by products

q = τ_{h}(a_{s_{1}}, …, a_{s_{m}})τ_{0}(a_{r_{1}}) …, τ_{0}(a_{r_{p}})  (2.4.9)

where a_{s_{ℓ}} ∈ _{h}(H_{∗}A_{s_{ℓ}}) for 1 ≤ ℓ ≤ m, and a_{r_{ℓ}} ∈ H_{∗}A_{r_{ℓ}} (where we have used relation 1.3.2(ii) identifying length one generators).

By 2.4.6(iv), any h-torsion generator of length greater than one can be reduced to a linear combination of products of torsion generators of the form 2.4.9 such that each element occurring in a torsion generator lies in a distinct one of the algebras H_{∗}A_{s_{1}}, …, H_{∗}A_{s_{m}}. By relation 2.4.6(v), any product q, in which distinct torsion generators have elements in a common algebra, can be reduced to a linear combination of products of the same form as q so that all the elements lie in distinct algebras. Finally, by the permutation relation and commutativity in Tor, the products q can be taken such that the indices respect the ordering imposed by that in the index set I. We, therefore, have that products of the form 2.4.9 such that s_{1} < ⋯ < s_{m}, and r_{1} < ⋯ < r_{p} generate Tor_{ψσ}(HA_{I}) as an R-module. Call such a product normal.

Using these products, define an increasing filtration on Tor_{ψσ}(HA_{I}) by F_{0} = 0, and for i > 0, F_{i} is the R-module generated by normal q such that m ≤ i. Clearly F_{n} = Tor_{ψσ}(HA_{I}), so the filtration is bounded. Let Ψ_{i} = Ψ|_{F_{i}}, then Ψ(F_{i}) ⊆ S_{i}, the filtration of Theorem 2.4.4, for all i, by the definition of Ψ and the remarks following the Theorem. Hence the Ψ_{i} induce maps ϕ_{i} making

\xymatrix{0 \ar[r] & F_{i−1} \ar[r] \ar[d]^{Ψ_{i−1}} & F_{i} \ar[r] \ar[d]^{Ψ_{i}} & F_{i}/F_{i−1} \ar[r] \ar[d]^{ϕ_{i}} & 0 \\ 0 \ar[r] & S_{i−1} \ar[r] & S_{i} \ar[r] & Mult_{i}( ) \ar[r] & 0}  (2.4.10)

commute for i > 0, S_{i}/S_{i−1} is identified with Mult_{i}( ) using the isomorphism ξ_{i}.

Lemma 2.4.11 below establishes that ϕ_{i} is an isomorphism of R-modules for i > 0. This and the fact that Ψ_{0} is an isomorphism imply that each Ψ_{i} is by the 5-lemma. In particular, Ψ_{n} = Ψ is an isomorphism of R-modules. This concludes the proof.

Lemma 2.4.11. ϕ_{i} is an isomorphism of R-modules for i > 0.

Proof. We construct an inverse θ_{i} to ϕ_{i}. From our previous remarks, F_{i}/F_{i−1} is generated as an R-module by the images of normal products q as 2.4.9 with m = i, p = n − i. By the discussion following Theorem 2.4.4, ϕ_{i} is defined on these generators by

ϕ_{i}(q mod F_{i−1}) = t_{h}(a_{s_{1}}, …, a_{s_{i}}) ⊗ a_{r_{1}} ⊗ ⋯ ⊗ a_{r_{n−i}} in Mult_{i}( ). (2.4.12)

Define an inverse θ_{i} by first defining an R-module homomorphism θ̃_{i}: Ẽ(i) → F_{i}/F_{i−1} as follows

θ̃_{i}(t_{h}(a_{s_{1}}, …, a_{s_{i}}) ⊗ a_{r_{1}} ⊗ ⋯ ⊗ a_{r_{n−i}}) =

τ_{h}(a_{s_{1}}, …, a_{s_{i}})τ_{0}(a_{r_{1}}) ⋯ τ_{0}(a_{r_{n−i}}) mod F_{i−1} (2.4.13)

which is well defined since relations 2.4.6 and 1.3.2(iv) imply relation 2.4.1 in Ẽ(i) for normal products reduced mod F_{i−1}. Further, θ̃_{i}(E(i)) = 0 and so θ̃_{i} induces a map θ_{i}. To see this, we calculate from 2.4.3 and 2.4.13

θ̃_{i}(Dt_{h}(a_{s_{1}}, …, a_{s_{i+1}})) = ∑_{ℓ=1}^{i+1}(−1)^{e(ℓ)}τ_{h}(a_{s_{1}}, …, â_{s_{ℓ}}, …, a_{s_{i+1}})τ_{0}(a_{s_{ℓ}}) (2.4.14)

= ∑_{ℓ=1}^{i+1}(−1)^{ε(ℓ)}τ_{h}(a_{s_{ℓ+1}}, …, a_{s_{i+1}}, a_{s_{1}}, …, a_{s_{ℓ−1}})τ_{0}(a_{s_{ℓ}})

where ε(ℓ) = e(ℓ)+(|(a_{s_{1}}, …, a_{s_{ℓ−1}})|+1)(|(a_{s_{ℓ+1}}, …, a_{s_{i+1}})|+1) by the permutation relation for h-torsion generators. Inserting the value for e(ℓ) of 2.4.3 gives

ε(ℓ) = (|a_{s_{ℓ}}| + 1)|(a_{s_{ℓ+1}}, …, a_{s_{i+1}})| + (|a_{s_{1}}, …, a_{s_{ℓ−1}})| + 1)(|(a_{s_{ℓ+1}}, …, a_{s_{i+1}})| + 1)

= (|a_{s_{ℓ}}| + |(a_{s_{1}}, …, a_{s_{ℓ−1}})| + 1)|(a_{s_{ℓ+1}}, …, a_{s_{i+1}})|

+ (|(a_{s_{1}}, …, a_{s_{ℓ−1}})| + |(a_{s_{ℓ+1}}, …, a_{s_{i+1}})| + 1)

= |(a_{s_{1}}, …, a_{s_{ℓ}})||(a_{s_{ℓ+1}}, …, a_{s_{i+1}})| + |(a_{s_{1}}, …, a_{s_{ℓ−1}}, a_{s_{ℓ+1}}, …, a_{s_{i+1}})|

By Proposition 1.3.12 with each string A_{ℓ} = (a_{s_{ℓ}}), the signed sum in 2.4.14 is zero in Tor_{ψσ}(HA_{I}), hence in F_{i}/F_{i−1}. It follows by direct calculation that θ_{i} and ϕ_{i} are inverses.

Since any family A_{I} ∈ ℱ(𝒞^{ψσ}) is realizable as the filtered colimit over its finite subfamilies directed by inclusion, and all functors involved preserve filtered colimits, we have

Theorem 2.4.15. If A_{I} ∈ ℱ(𝒞^{ψσ}), then

Ψ(A_{I}): Tor_{ψσ}(HA_{I}) → H_{∗}(T A_{I})

is a natural equivalence of graded, augmented, connected R-algebras.

Note: The augmentation ε is given by ε(τ_{h}(a_{1}, …, a_{n})) = τ_{h}(εa_{1}, …, εa_{n}).

3. H_{∗}(π, n; ℤ)

3.1. Cartan’s Homology Operations. In this chapter we determine, by generators and relations, the integral homology algebra of an Eilenberg-Mac Lane space of type K(π, n), where π is an abelian group and n ≥ 1. This is accomplished by applying the machinery of the previous chapter to the description of integral homology given by H. Cartan [3]. To this end, we summarize below the pertinent facts concerning homology operations on a K(π, n). The terminology and notation are those of Cartan.

Let R = ℤ or ℤ_{p}, p a prime, in all that follows. Denote by B̄^{(n)}(Rπ) the n-th iterate reduced bar construction on Rπ, the group ring on π. Then H_{∗}(π, n; R) = H_{∗}(B̄^{(n)}(Rπ); R) is a graded, connected, strictly commutative R-algebra, and the suspension s: B̄^{(n)}(Rπ) → B̄^{(n+1)}(Rπ) induces the homology suspension

σ: H_{q}(π, n; R) → H_{q+1}(π, n + 1; R).

Suspension is an isomorphism for q < 2n, and an epimorphism for q = 2n whose kernel consists of the decomposable elements of H_{2n}(π, n; R) [3, exposé 6, Theorem 1].

The algebras H_{∗}(π, n; R) have a natural divided power structure [3, exposé 7], namely functions γ_{k}: H_{2q}(π, n; R) → H_{2kq}(π, n; R) for k ≥ 0 such that for x, y ∈ H_{∗}(π, n; R).

(i) γ_{0}(x) = 1 and γ_{1}(x) = x (ii) γ_{k}(x)γ_{ℓ}(x) = (k, ℓ)γ_{k+ℓ}(x) (iii) γ_{k}(x + y) = ∑_{i+j=k} γ_{i}(x)γ_{j}(y) (3.1.1)

(iv) for k > 1, γ_{k}(x · y) = \begin{cases}0 & if |x|, |y| odd \\ x^{k}γ_{k}(y) & if |x|, |y| even and |y| > 0 \end{cases}

(v) γ_{ℓ}(γ_{k}(x)) = {ℓ, k}γ_{ℓk}(x), {ℓ, k} = (k, k − 1) ⋯ ((ℓ − 1)k, k − 1)

Note: The abstract algebra generated by symbols γ_{k}(x) for x ∈ π (graded in even degrees) subject to the relations above is the divided power algebra Γ(π) of [4]. We also remark that for π graded with degree 2, the homogeneous subgroup of degree 4, Γ_{4}(π), is generated by symbols γ_{2}(x) subject to the relations γ_{2}(x) = γ_{2}(−x) and γ_{2}(x⊤y) = γ_{2}(x + y) − γ_{2}(x) − γ_{2}(y) be linear in x and y, see [4].

The divided powers in H_{∗}(π, n; R) arise from the algebra B̄^{(n)}(Rπ) and the divided power structure there given by γ_{k}([x]) = [x, x, …, x] (k times), where x ∈ B̄^{(n−1)}(Rπ) is of odd degree. These elements are cycles and induce the divided powers in homology.

In characteristics p, the γ_{k} are all obtained from the single operation γ_{p} by composition and multiplication using the p-adic expansion of k and relations 3.1.1(ii) and (v). If p = 2 and n > 1, all positive degree elements have divided powers.

If R = ℤ_{p}, there is the operation

ϕ_{p}: H_{2q}(π, n; ℤ_{p}) → H_{2pq+2}(π, n + 1; ℤ_{p})

defined for q ≥ 1. This operation is linear for p odd, and annihilates decomposable elements. For p = 2, ϕ_{2}(x) = γ_{2}σ(x) [3, exposé 6, §4] and [3, exposé 8, §1].

If R = ℤ, h > 1, there is the operation

ψ_{h}: _{h}π → H_{2}(π, 1; ℤ_{h})

which assigns to each x ∈ _{h}π the class of its h-cap, and has the properties [3, exposé 11, §1]

(i) β_{h}ψ_{h}(x) = σ(x) for x ∈ _{h}π (3.1.2)

(ii) ψ_{h}(x⊤x′) = \frac{h(h−1)}{2}α_{h}(σ(x) ∗ σ(x′)) for x, x′ ∈ _{h}π

Let β̄_{p}: H_{∗}(π, n; ℤ_{p}) → H_{∗−1}(π, n; ℤ_{p}) be induced by the coefficient sequence 0 → ℤ_{p} → ℤ_{p^{2}} → ℤ_{p} → 0. Then β̄_{p} = α_{p}β_{p} and the following relations hold for x ∈ H_{2q}(π, n; ℤ_{p}) [3, exposé 8, §3]

(i) β̄_{p}ϕ_{p}(x) = σγ_{p}(x) if p > 2 (3.1.3)

(ii) β̄_{2}ϕ_{2}(x) = σγ_{2}(x) + (β̄_{2}σx) ∗ (σx)

Finally, β_{h} (and β̄_{p}) commute with the homology suspension σ and are natural for maps of abelian groups, π → π′.

Definition 3.1.4 (3, exposé 9, §6). An admissible sequence α of the homology operations σ, γ_{p}, ϕ_{p} is a formal sequence (word) composed of these operations of the general form

α = \begin{cases}σ^{h_{1}}α_{1}σ^{2h_{2}}α_{2} … σ^{2h_{m}} & (m ≥ 1) \\ σ^{h_{1}}α_{1}σ^{2h_{2}}α_{2} … σ^{2h_{m}}ϕ_{p} \end{cases}

where α_{i} = γ_{p} or ϕ_{p}, h ≥ 0 for i = 1, …, m, and such that h_{1} > 0 if α_{1} = γ_{p}. The height of α, h(α), is the number of symbols σ and ϕ_{p} occurring in α; the degree, d(α), is the total degree of the corresponding composite of operations; and the stable degree, q(α) = d(α) − h(α). Such a sequence defines a mod-p homology operation (by composition, in order, of the designated basic operations)

π/pπ ⟶^{σ} H_{1}(π, 1; ℤ_{p}) ⟶^{α′} H_{d(α)}(π, h(α); ℤ_{p})

pπ ⟶^{ψ_{p}} H_{2}(π, 1; ℤ_{p}) ⟶^{α′} H_{d(α)}(π, h(α); ℤ_{p})

according as α has the form α′σ or α′ϕ_{p} (in this latter case, following Cartan, we replace the final ϕ_{p} by the function ψ_{p} defined above).

Note: The enumeration of such admissible sequences can be achieved by the following device [3, exposé 9, §6]. For α of height n, stable degree q, and i ≥ 1, write α_{i}β_{i} for the sequence (not necessarily admissible) determined by truncating from α all symbols to the left of the symbol α_{i}. Define an integer

a_{i} = stable degree of α_{i}β_{i} − stable degree of β_{i},

then a_{i} = 2k_{i}(p−1)+u_{i} where k_{i} ≥ 0 and u_{i} = 0 or 1 accordingly as α_{i} = γ_{p} or ϕ_{p}. The finite sequence of non-negative integers (a_{1}, …, a_{m}) satisfies the conditions

(i) a_{i} ≡ 0 mod 2(p − 1) [such that a_{m} = 0 or 1] (ii) a_{i} ≥ pa_{i+1} (iii) ∑ a_{i} = q (3.1.5) (iv) pa_{1} < (p − 1)(n + q) if p > 2

\{ \begin{matrix}2a_{1} ≤ n + q & for a_{1} odd \\ 2a_{1} < n + q & for a_{1} even \end{matrix} \} if p = 2

The admissible sequence α corresponding to the sequence (a_{1}, …, a_{m}) is given by

h_{1} = [n + q − \frac{pa_{1}}{p − 1}] and h_{i+1} = [\frac{a_{i} − pa_{i+1}}{2(p − 1)} ] for i > 0,

where [ ] is the greatest integer function; and α_{i} = γ_{p} or ϕ_{p} accordingly as a_{i} ≡ 0 or 1 mod 2(p − 1).

If a_{m} = 1, the sequence defines an operation of the second type (i.e., the sequence of operations terminates with ϕ_{p}), otherwise the operation is of the first type. The sequence (0) defines the operation σ^{n}, n-fold suspension.

Examples: For q = 2, n = 3, the sequence (2, 0) defines the mod-2 operation σγ_{2}σ^{2}. For q = 3, n = 4, the sequence (2, 1) defines the mod-2 operation σ^{3}γ_{2}ϕ_{2}. For q = 5, n = 2, the sequence (4, 1) defines the mod-3 operation σγ_{3}ϕ_{3}.

Define groups D_{α}(π) corresponding to operations α by

D_{α}(π) = \begin{cases}π/pπ & if α is of the first type \\ _{p}π & if α is of the second type \end{cases}

Since it can be shown [3, exposé 6, Propositions 1 and 3] that the operations σ and ϕ_{p} (or ψ_{p}) annihilate decomposable elements and ϕ_{p}(u + v) = ϕ_{p}(u) + ϕ_{p}(v) + decomposables, it follows that the function induced on homology by an admissible sequence α is a homomorphism, which we denote by f_{α}: D_{α}(π) → H_{d(α)}(π, h(α); ℤ_{p}), and further, f_{α} is an isomorphism onto a ℤ_{p} submodule unless p = 2 and α has the form ϕ_{2}α′ [see 3, exposé 9, §2, and exposé 10, §2].

There are also the homology operations

f_{σ^{n}}: π → H_{n}(π, n; ℤ) and f_{σ^{n−1}ψ_{h}}: _{h}π → H_{n+1}(π, n; ℤ_{h})

defined for n ≥ 1 by the admissible sequences σ^{n} and σ^{n−1}ψ_{h}, respectively. The former is an isomorphism of groups, and the latter is a homomorphism for n > 1 and is related to the iterated suspension by β_{h}f_{σ^{n−1}ψ_{h}} = f_{σ^{n}}|_{_{h}π}. We define D_{σ^{n}}(π) = π and D_{σ^{n−1}ψ_{h}}(π) = _{h}π (the elements of π annihilated by h).

For each prime p, we use relations 3.1.3 to reclassify the admissible sequences, α, of homology operations so that α has the form β̄_{p}σ^{ℓ}ϕ_{p}α′ or σ^{ℓ}ϕ_{p}α′ (for ℓ ≥ 0), where α′ is a sequence of operations of even degree and heights h(α) − ℓ − 1 such that σα′ is itself an admissible sequence [3, exposé 11, §4]. In this reclassification, by introduction of the operations β̄_{p}, the admissible mod-p homology operations are paired in a natural way which will be useful for describing the integral homology ring, H_{∗}(π, n; ℤ), in the next section.

By 3.1.3, σ^{ℓ+1}γ_{p}α(x) = β̄_{p}σ^{ℓ}ϕ_{p}α(x) in H_{∗}(π, n; ℤ_{p}) if p is odd or if p = 2 and ℓ > 0 (since suspension of a product is zero). By 3.1.3(ii), σγ_{2}α(x) = β̄_{2}ϕ_{2}α(x) + (β̄_{2}σα(x)) ∗ (σα(x)) in H_{∗}(π, n; ℤ_{2}) if ℓ = 0. There are three distinct cases:

(i) α = σ^{2ℓ}ϕ_{2}α′ (ℓ ≥ 0) with σα(x) = σ^{2ℓ+1}ϕ_{2}α′(x) (ii) α = σ^{2ℓ}γ_{2}α′ (ℓ ≥ 0) with σα(x) = β̄_{2}σ^{2ℓ}ϕ_{2}α′(x) (3.1.6) (iii) α = γ_{2}σ^{2ℓ}ϕ_{2}α′ (ℓ ≥ 0 and not a case of (ii)) with σα(x) = β̄_{2}ϕ_{2}σ^{2ℓ}ϕ_{2}α′(x) +

(β̄_{2}σ^{2ℓ+1}ϕ_{2}α′(x)) ∗ (σ^{2ℓ+1}ϕ_{2}α′(x))

obtained by repeated application of 3.1.3(ii). If α′ is empty in case (i) or (iii), replace ϕ_{2} by ψ_{2} and β̄_{2}σ^{2ℓ+1}ϕ_{2}α′ by σ^{2ℓ+2}.

Thus, the sequences of the form σ^{ℓ}ϕ_{p}α and β̄_{p}σ^{ℓ}ϕ_{p}α represent all admissible mod-p homology operations. These sequences correspond to finite sequences (a_{1}, …, a_{m}) of non-negative integers satisfying 3.1.5, such that a_{1} ≡ 1 or a_{1} ≡ 0 mod 2(p − 1), respectively, and such that ∑ a_{i} > 1.

If α = σ^{n−1}ψ_{h} or σ^{ℓ}ϕ_{p}α′ is an admissible sequence with height n and stable degree q, we have f_{α}: D_{α}(π) → H_{n+q}(π, n; R_{α}) where R_{α} = ℤ_{h} or ℤ_{p}, respectively. For u ∈ D_{α}(π), let α̂(u) ∈ B̄_{n+q}^{(n)}(π) be a chain representing the homology class f_{α}(u). This choice is not unique, but is so in homology, and f_{β̄_{h}α} = β̄_{h}f_{α}. Define a function

f_{β_{h}α}: D_{α}(π) → H_{n+q−1}(π, n; ℤ) by f_{β_{h}α} = β_{h}f_{α},

then α_{h}f_{β_{h}α} = α_{h}β_{h}f_{α} = β̄_{h}f_{α} = f_{β̄_{h}α}. For h odd, ℓ > 0, or α a case of 3.1.6(ii), f_{β_{h}α} is an isomorphim onto a subgroup [3, exposé 11, Appendix 1].

3.2. Elementary Complexes. For π and n fixed, define the family of elementary complexes [3, exposé 11, §2–4] associated to π, 𝒳(π, n). It is the family in ℱ(𝒞^{ψσ}) consisting of the following algebras with their associated capping operations and torsion inverses (see the relevant examples in §2.1):

(a) For each u ∈ π, associate the algebra P (x) or E(x) generated by the symbol x = σ^{n}(u) of degree n accordingly as n is even or odd;

(b) For each u ∈ _{h}π, h > 1, associate the algebra P (x) ⊗ E(y) or E(x) ⊗ P (y) generated by the pair of symbols x = σ^{n}u, y = σ^{n−1}ψ_{h}u (|x| = n, |y| = n + 1) accordingly as n is even or odd. Define the differential by ∂(σ^{n−1}ψ_{h}u) = (−1)^{n−1}hσ^{n}u;

(c) For each mod-p homology operation β̄_{p}σ^{ℓ}ϕ_{p}α of height n, stable degree q > 1, and u ∈ _{p}π or π/pπ (determined by the type of the operation), associate the algebra P (x) ⊗ E(y) or E(x) ⊗ P (y) generated by the symbols x = β̄_{p}σ^{ℓ}ϕ_{p}α(u) and

y = σ^{ℓ}ϕ_{p}α(u) (|x| = n + q, |y| = n + q + 1)

accordingly as |x| is even or odd. Define the differential by

∂(σ^{ℓ}ϕ_{p}α(u)) = (−1)^{ℓ}pβ̄_{p}σ^{ℓ}ϕ_{p}α(u).

Clearly, this defines a functor 𝒳(−, n): 𝒜b → ℱ(𝒞^{ψσ}) where 𝒜b is the category of abelian groups.

Following Cartan, we define a homomorphism g_{A}: A → B̄^{n}(π) in _{ℤ}𝒟ℱ𝒜, compatible with the respective divided powers, for each elementary complex A ∈ 𝒳(π, n) [3, exposé 11, §4].

If A is generated by x = σ^{n}u, define g_{A} by g_{A}(x) = σ̂^{n}(u), [see page 23]. If A is generated by x = σ^{n}u, y = σ^{n−1}ψ_{h}u, put g_{A}(y) = { \atop σ^{n−1}ψ_{h}}(u) [see page 23], and this completely determines the map. If A is generated by x = β̄_{p}σ^{ℓ}ϕ_{p}α(u), y = σ^{ℓ}ϕ_{p}α(u), put g_{A}(y) = { \atop σ^{ℓ}ϕ_{p}α}(u), and g_{A} is determined.

While each g_{A} is not unique, it is unique up to a homotopy of chain maps [3, exposé 11, §3], so that each g_{A∗}: H_{∗}(A) → H_{∗}(π, n; ℤ) is unique and g_{A∗}(β̄_{h}α(u)) = f_{β_{h}α}(u) for u ∈ D_{α}(π), α = σ^{n−1}ψ_{h} or σ^{ℓ}ϕ_{p}α′. Using this, we will identify the integral homology class determined by β̄_{p}α(u), a generator of a complex of type (c), by β_{p}α(u).

If X(π, n) = T 𝒳(π, n) ∈ _{ℤ}𝒟ℱ𝒜, the g_{A∗} induce a unique algebra homomorphism g_{∗}: H_{∗}(X(π, n); ℤ) → H_{∗}(π, n; ℤ) which is natural for homomorphisms of abelian groups, π.

Let 𝒳′ ⊆ 𝒳(π, n) be a subfamily of elementary complexes, then X′ = T 𝒳′ is a direct summand of X(π, n) since each elementary complex, A, is connected (i.e., A = ℤ ⊕ A^{+}, where A^{+} is the submodule of elements of positive degree).

The essential result for integral homology established by Cartan [3, exposé 11, Theorem 1] is that g_{∗} is an epimorphism with specified kernel. Furthermore, for p prime, let K_{p}(π, n) be the subcomplex of X(π, n) consisting of products of elementary complexes of type (b) (for h, a power of p) or type (c) (where α is a mod-p homology operation) and complexes of type (a) for u ∈ π not p-primary. Then, the following result holds [3, exposé 11, Theorem 5]:

Theorem 3.2.1 (Cartan). If κ_{p}: K_{p}(π, n) → X(π, n) is the inclusion, the algebra homomorphism g_{∗}κ_{p∗}: H_{∗}(K_{p}(π, n)) → H_{∗}(π, n; ℤ) has image the sum (not direct) U(π, n) + C_{p}(H_{∗}(π, n; ℤ)) and kernel the ideal N_{p}(π, n) of “elementary equivalences.” Here, U(π, n) is the subalgebra (with divided powers) of H_{∗}(π, n; ℤ) generated by π ≅ H_{n}(π, n; ℤ). In particular, U(π, n) ≅ Λ(π), the exterior algebra on π, if n is odd, and U(π, n) ≅ Γ(π), the divided power algebra on π, if n is even. C_{p}( ) denotes the p-primary component, and N_{p}(π, n) is defined by the following identifications [3, exposé 11, §8, pages 17–18] among generators of the elementary complexes in K_{p}(π, n):

(i) For u, v ∈ π, identify the generator σ^{n}(u + v) of a complex of type (a) with the element σ^{n}(u) + σ^{n}(v), and identify σ^{n}(hu) with hσ^{n}(u), for h ∈ ℤ;

(ii) For u ∈ _{hk}π (h, k powers of p), identify the pair (kx, y) = (kσ^{n}(u), σ^{n−1}ψ_{hk}(u)) with the pair (x′, y′) = (σ^{n}(ku), σ^{n−1}ψ_{h}(ku));

(iii) For u ∈ _{h}π (h, k powers of p), identify the pair of generators (x, y) = (σ^{n}(u), σ^{n−1}ψ_{hk}(u)) with the pair (x′, ky′) = (σ^{n}(u), kσ^{n−1}ψ_{h}(u));

(iv) For u ∈ _{h}π (h a power of p), identify the pair of generators (x, y) = (σ^{n}(ku), σ^{n−1}ψ_{h}(ku)) with the pair (kx′, ky′) = (kσ^{n}(u), kσ^{n−1}ψ_{h}(u)) for k ∈ ℤ. For u, v ∈ _{h}π, identify the pair of generators (x, y) = (σ^{n}(u + v), σ^{n−1}ψ_{h}(u + v)) with the pair (x′ + x′′, y′ + y′′) = (σ^{n}(u) + σ^{n}(v), σ^{n−1}ψ_{h}(u) + σ^{n−1}ψ_{h}(v)), unless p = 2 and n = 1. In this case identify ψ_{h}(u + v) with the element ψ_{h}(u) + ψ_{h}(v) + \frac{h(h−1)}{2}(σu) ∗ (σv);

(v) For a pair of mod-p homology operations β̄_{p}σ^{ℓ}ϕ_{p}α and σ^{ℓ}ϕ_{p}α with stable degree > 1, and u ∈ D_{α}(π), identify the pair of generators (x, y) = (β̄_{p}σ^{ℓ}ϕ_{p}α(ku), σ^{ℓ}ϕ_{p}α(ku)) with the pair of elements (kx′, ky′) = (kβ̄_{p}σ^{ℓ}ϕ_{p}α(u), kσ^{ℓ}ϕ_{p}α(u)), where k ∈ ℤ. For u, v ∈ D_{α}(π), identify the pair of generators (x, y) = (β̄_{p}σ^{ℓ}ϕ_{p}α(u + v), σ^{ℓ}ϕ_{p}α(u + v)) with the pair of elements (β̄_{p}σ^{ℓ}ϕ_{p}α(u) + β̄_{p}σ^{ℓ}ϕ_{p}α(v), σ^{ℓ}ϕ_{p}α(u) + σ^{ℓ}ϕ_{p}α(v)) unless p = 2 and ℓ = 0. In this case identify ϕ_{2}α(u + v) with ϕ_{2}α(u) + ϕ_{2}α(v) + (σαu) ∗ (σαv), and β̄_{2}ϕ_{2}α(u + v) with β_{2}ϕ_{2}α(u) + β_{2}ϕ_{2}α(v) + (\frac{1}{2}dσαu) ∗ (σαv) = (\frac{1}{2}dσαv) ∗ (σαu) according to the three cases of 3.1.6 by

(i) σαu = σ^{2ℓ+1}ϕ_{2}α′u and \frac{1}{2}dσαu = −β̄_{2}σ^{2ℓ+1}ϕ_{2}α′u (ii) σαu = β̄_{2}σ^{2ℓ}ϕ_{2}α′u and \frac{1}{2}dσαu = 0 (3.2.2)

(iii) σαu = β̄_{2}ϕ_{2}σ^{2ℓ}ϕ_{2}α′u + (β̄_{2}σ^{2ℓ+1}ϕ_{2}α′u) ∗ (σ^{2ℓ+1}ϕ_{2}α′u) and \frac{1}{2}dσαu = −2γ_{2}(β̄_{2}σ^{2ℓ+1}ϕ_{2}α′u)

If α′ is empty in cases (i) or (iii), we replace ϕ_{2} by ψ_{2} and β_{2}σ^{2ℓ+1}ϕ_{2}α′ by σ^{2ℓ+2}. This completes the description of the elementary equivalences of N_{p}(π, n).

We generalize Theorem 3.2.1 as follows to get a description of the entire homology algebra. Let N(π, n) be the ideal of H_{∗}(X(π, n)) defined by the elementary equivalences (i) and (v) for every prime, the equivalences (ii) to (iv) for h, k ∈ ℤ^{+} (not just powers of a prime), and the additional equivalence

(vi) For u ∈ _{h}π (h ∈ ℤ^{+}) identify the generator σ^{n}u of the complex of type (a) with the generator σ^{n}u of the complex of type (b), namely (σ^{n}u, σ^{n−1}ψ_{h}u).

Then g_{∗}(N(π, n)) = 0, and relation (vi) serves to reduce the subgroup of H_{∗}(X(π, n)) of degree n to π.

Clearly κ_{p∗}(N_{p}(π, n)) ⊆ N(π, n) for every p. Let κ̄_{p∗} be induced by κ_{p∗}, and ḡ_{∗} be induced by g_{∗}, then

ḡ_{∗}κ̄_{p∗}: H_{∗}(K_{p}(π, n))/N_{p}(π, n) → H_{∗}(π, n; ℤ)

is an isomorphism onto U(π, n) + C_{p}(H_{∗}(π, n; ℤ)), and it follows that

κ̄_{p∗}: H_{∗}(K_{p}(π, n))/N_{p}(π, n) → H_{∗}(X(π, n))/N(π, n)

is an isomorphism onto a subalgebra.

Theorem 3.2.3. The natural (in π) homomorphism of algebras

ḡ_{∗}: H_{∗}(X(π, n))/N(π, n) → H_{∗}(π, n; ℤ)

is an isomorphism.

Proof. It is sufficient to establish the result for π finitely generated. Let π = π_{0} ⊕ π_{t} where π_{0} is free and π_{t} is the torsion subgroup of π. If X_{0} ⊆ X(π, n) is the subcomplex consisting of products of elementary complexes of type (a) for u ∈ π_{0} and κ_{0} is the inclusion, then by [3, exposé 11, Theorem 1] g_{∗}κ_{0∗} is onto the subalgebra U(π_{0}, n) with kernel in N(π, n) given by the elementary equivalences (i) reducing the degree n component of X(π, n) to π. Again, by the Theorem, H_{∗}(π, n; ℤ) ≅ U(π_{0}, n)⊕ torsion, induced by g_{∗}. We have X(π, n) = X_{0}⊕(X_{0}⊗X_{t}^{+}), where X_{t} consists of products of elementary complexes with at least one complex of type (b) or (c), so that H_{∗}(X(π, n)) = H_{∗}(X_{0}) ⊕ H_{∗}(X_{0} ⊗ X_{t}^{+}) and the latter is a torsion group. Thus,

H_{∗}(X(π, n))/N(π, n) = H_{∗}(X_{0})/N_{0} ⊕ H_{∗}(X_{0} ⊗ X_{t})/N_{t},

where N_{0} is defined by elementary equivalences given by (i) for u ∈ π_{0}, and N_{t} = N(π, n) ∩ H_{∗}(X_{0} ⊗ X_{t}^{+}). By the previous remark, ḡ_{∗}|_{H_{∗}(X_{0})/N_{0}} is an isomorphism onto U(π_{0}, n), so the torsion-free components are isomorphic.

We establish that ḡ_{∗} induces an isomorphism of the torsion components by comparing the p-primary components for each prime. For fixed p, let K_{p} = K_{p}(π, n), then

X(π, n) = K_{p} ⊕ (K_{p} ⊗ X_{1}^{+}) ⊕ (K_{p} ⊗ X_{1} ⊗ X_{2}^{+}) = K_{p} ⊕ Y_{1} ⊕ Y_{2}

where X_{1} consists of products of elementary complexes of type (b) not occurring in K_{p}, and X_{2} consists of products of elementary complexes of type (c) for q-primary homology operations with q ≠ p. We have C_{p}(H_{∗}(Y_{2})/N ∩ H_{∗}(Y_{2})) = 0 since the quotient is generated by homology classes which are q-primary, q ≠ p; so that C_{p}[H_{∗}(K_{p} ⊕ Y_{1})/N ∩ H_{∗}] ≅ C_{p}[H_{∗}(X(π, n))/N(π, n)] induced by inclusion. From the elementary equivalences (ii) to (iv) defining N(π, n) and N_{p}(π, n), it follows that

C_{p}(H_{∗}(K_{p})/N_{p}) ≅ C_{p}[H_{∗}(K_{p} ⊕ Y_{1})/N ∩ H_{∗}]

under inclusion. By Theorem 3.2.1, ḡ_{∗}κ_{p∗} induces an isomorphism

C_{p}[H_{∗}(K_{p})/N_{p}] ≅ C_{p}[H_{∗}(π, n; ℤ)],

which establishes the isomorphism of p-primary components.

We are now able to give a calculation of H_{∗}(π, n; ℤ) by observing that Theorem 2.4.15 applies to the family 𝒳(π, n) with its natural capping operations and torsion inverses.

Theorem 3.2.4. The natural algebra homomorphism

Φ = ḡ_{∗}Ψ: Tor_{ψσ}(H𝒳(π, n)) → H_{∗}(π, n; ℤ)

is an epimorphism with kernel 𝒩(π, n) = Ψ^{−1}(N(π, n)).

Remark: Recall that for the types of algebras (elementary complexes) arising in 𝒳(π, n), the natural capping operations and torsion inverses are unique, and g_{∗} is the unique (up to homotopy) natural map to integral homology [3, exposé 11, Proposition 3 bis]. We will denote the quotient Tor_{ψσ}( ) by the symbol \underline{Tor}( ).

To give the homology algebra by generators and relations, we return to the notation of the examples in Chapter 2. Thus, the “elementary classes” [the homology classes occurring in the torsion generators and special symbols of \underline{Tor}(H𝒳(π, n))] are of the form

(i) σ^{n}u (n odd) or γ_{i}(σ^{n}u) (n even, i ≥ 0) u ∈ π (3.2.5) (ii) ω_{i}^{m}(x) (|x| odd) or γ_{i}^{m}(x) (|x| even) where x = α(u), u ∈ D_{α}(π) such that

mu = 0 for m ∈ ℤ^{+} and α = σ^{n} or β_{p}σ^{ℓ}ϕ_{p}α′ for some prime p (then m = p)

The elementary equivalences of N(π, n) result in the following relations among these classes (whenever defined)

(i) σ^{n}u = ω_{1}^{m}(σ^{n}u) γ_{i}(σ^{n}u) = γ_{i}^{m}(σ^{n}u) (i ≥ 0) (ii) ω_{i}^{hm}(x) = h^{i−1}ω_{i}^{m}(x) hω_{i}^{hm}(x) = ω_{i}^{m}(hx) (i > 0) (3.2.6) (iii) γ_{i}^{hm}(x) = γ_{i}^{m}(x) h^{i}γ_{i}^{hm}(x) = γ_{i}^{m}(hx) (i ≥ 0) where x = α(u) as above

[see the Appendix] For the ℤ-algebras of 𝒳(π, n), the special classes Δ_{m}^{j}(a, a′) were determined for

the examples of Chapter 2 [page 15]. We note that Δ_{m}^{1}(a, a′) = 0 and Δ_{m}^{2}(a, a′) = 0 in all cases, and that

Δ_{m}^{3}(a, a′) = \begin{cases}0 & for a, a′ of the form γ_{i}^{m′}(x) \\ \frac{km}{m′}(i − 1, j)ω_{i+j}^{m′}(x) & for a = ω_{i}^{m′}(x), a′ = kω_{j}^{m′}(x) \end{cases}

(Refer to page 17 for the definition of these classes).

Thus the relations 2.4.6(iii) to (v) defining the quotient torsion algebra \underline{Tor}(H𝒳(π, n)) become (iii′) τ_{m}(a⊤a, B) = 0 (3.2.7)

(iv′) \begin{cases}τ_{m}(k_{1}γ_{i}^{m′}(x), k_{2}γ_{j}^{m′}(x), B) = 0 if |x| even \\ τ_{m}(k_{1}ω_{i}^{m′}(x), k_{2}ω_{j}^{m′}(x), B) = τ_{m} (\frac{mk_{1}k_{2}}{m′} (i, j)ω_{i+j}^{m′}(x), B) if |x| odd \end{cases}

(v′) \begin{cases}τ_{0}(γ_{i}^{m′}(x))τ_{m}(kγ_{j}^{m′}(x), B) = τ_{m}(k(i, j)γ_{i+j}^{m′}(x), B) if |x| even \\ τ_{0}(ω_{i}^{m′}(x))τ_{m}(kω_{j}^{m′}(x), B) = τ_{m} (\frac{mk}{m′} (i − 1, j)ω_{i+j}^{m′}(x)) τ_{m}(B) if |x| odd \end{cases}

The relations 2.4.6(ii) and (iii′) above are just the defining relations 1.4.2 for the multilinear quotient torsion algebra, so that we may identify \underline{Tor}( ) with \underline{Torl}( ) in Theorem 3.2.4.

With these identifications, we can exhibit a more restrictive set of torsion generators for H_{∗}(π, n; ℤ) under the isomorphism of Theorem 3.2.4. By relations 3.2.6(iii) and the elementary equivalence (vi), the divided powers γ_{i}(x) and γ_{i}^{m}(x) on an element x = α(u) are identified and satisfy the relations 3.1.1 for divided powers. By Proposition 1.4.4, since H𝒳(π, n) is of finite type, every torsion generator can be reduced to a linear combination of torsion generators of the form

τ_{m}(k_{1}ω_{i_{1}}^{m_{1}}(x_{1}), …, k_{r}ω_{i_{r}}^{m_{r}}(x_{r}), k_{r+1}γ_{i_{r+1}}(x_{r+1}), …, k_{r+s}γ_{i_{r+s}}(x_{r+s})) (3.2.8)

where r, s ≥ 0 and m_{1}|m_{t} for t = 1, …, r.

Using the relations enumerated above, we may reduce further to torsion generators of the form 3.2.8 where m = m_{1} = ⋯ = m_{r} as follows: Define the integers k = gcd{k_{1}, …, k_{r+s}}, m_{t} = m_{1}u_{t}, mk_{t} = m_{t}v_{t}, k_{t} = kk̄_{t}, and mk = m_{1}k′ (since m_{1}|mk_{t} for each t). Then we calculate

τ_{m}(k_{1}ω_{i_{1}}^{m_{1}}(x_{1}), …, k_{r+s}γ_{i_{r+s}}(x_{r+s}))
= kτ_{mk}(k̄_{1}ω_{i_{1}}^{m_{1}}(x_{1}), …, k̄_{r+s}γ_{i_{r+s}}(x_{r+s})) [by 1.3.2(iii)]
= kτ_{m_{1}}(k̄_{1}ω_{i_{1}}^{m_{1}}(x_{1}), k′k̄_{2}ω_{i_{2}}^{m_{2}}(x_{2}), …, k′k̄_{r+s}γ_{i_{r+s}}(x_{r+s})) [by 1.3.2(iv)]

= k_{1}τ_{m_{1}}(ω_{i_{1}}^{m_{1}}(x_{1}), u_{2}v_{2}ω_{i_{2}}^{m_{1}u_{2}}(x_{2}), …, u_{r}v_{r}ω_{i_{r}}^{m_{1}u_{r}}(x_{r}), …, k′k̄_{r+s}γ_{i_{r+s}}(x_{r+s}))

[since k′k̄_{t} = u_{t}v_{t} (2 ≤ t ≤ r) and by 2.4.6(ii) on the first factor]

= k_{1}τ_{m_{1}}(ω_{i_{1}}^{m_{1}}(x_{1}), v_{2}ω_{i_{2}}^{m_{1}}(u_{2}x_{2}), …, v_{r}ω_{i_{r}}^{m_{1}}(u_{r}x_{r}), …, k′k̄_{r+s}γ_{i_{r+s}}(x_{r+s}))

[by 3.2.6(ii) above]

= k_{1}v_{2} ⋯ v_{r}τ_{m_{1}}(ω_{i_{1}}^{m_{1}}(x_{1}), ω_{i_{2}}^{m_{1}}(u_{2}x_{2}), …, ω_{i_{r}}^{m_{1}}(u_{r}x_{r}), …, k′k̄_{r+s}γ_{i_{r+s}}(x_{r+s}))

[by 2.4.6(ii) applied to the first r factors]

Combining these results, we have the following

Theorem 3.2.9. H_{∗}(π, n; ℤ) is generated as an algebra (over ℤ) by special symbols τ_{0}(σ^{n}u) or τ_{0}(γ_{i}(σ^{n}u)) [n even, i ≥ 0, u ∈ π] and torsion generators of the form

τ_{m}(ω_{i_{1}}^{m}(x_{1}), …, ω_{i_{r}}^{m}(x_{r}), k_{1}γ_{i_{r+1}}(x_{r+1}), …, k_{s}γ_{i_{r+s}}(x_{r+s}))

for r, s ≥ 0, x_{t} = α_{t}(u_{t}), α_{t} a homology operation of the form σ^{n} or β_{p}σ^{ℓ}ϕ_{p}α′, u_{t} ∈ D_{α_{t}}(π) such that mu_{t} = 0 in D_{α_{t}}(π) for 1 ≤ t ≤ r and mk_{t}γ_{i_{r+t}}(u_{r+t}) = 0 in Γ(D_{α_{r+t}}(π)) for 1 ≤ t ≤ s (Γ( ) is the divided power of algebra defined in [4]). These generators are subject to the relations of 1.3.2, 3.2.6, and 3.2.7 above, and the following “addition rules” whenever they are defined (the m on the ω_{i}^{m} will be understood and we identify ω_{i}^{m}(x) with τ_{m}(ω_{i}^{m}(x)) by 1.3.2(ii)):

(i) τ_{0}(σ^{n}(u + v)) = τ_{0}(σ^{n}u) + τ_{0}(σ^{n}v) (ii) τ_{m}(ω_{i}(α(u + v)), B) = τ_{m}(ω_{i}(αu), B) + τ_{m}(ω_{i}(αv), B) +

∑_{r+s=i} τ_{m}(ω_{r}(αu), ω_{s}(αv), B) + ℛ

where

ℛ = 0 if α = σ^{n} (n > 1), or β_{p}σ^{ℓ}ϕ_{p}α′ (ℓ ≥ 0), or p = 2 (ℓ > 0)
ℛ = c(α) ∑_{r+s=i+1} ω_{r}(αu)ω_{s}(αv)τ_{m}(B)
where c(α) = \frac{m(m−1)}{2} if α = σ, or c(α) = 1 if α = β_{2}ϕ_{2}σ^{2ℓ}γ_{2}α′ (ℓ > 0)
ℛ = ∑_{r+s=i−1} τ_{m}(ω_{1}(Θu), ω_{1}(Θv), ω_{r}(αu), ω_{s}(av), B)
if α = β_{2}ϕ_{2}σ^{2ℓ}ϕ_{2}α′ (ℓ > 0) with Θ = β_{2}σ^{2ℓ+1}ϕ_{2}α′
ℛ = ∑_{r+s=i−1}{ω_{1}(Θu)ω_{1}(Θv)τ_{m}(ω_{r}(αu), ω_{s}(αv), B)
− ω_{1}(Θu)τ_{m}(ω_{r}(αu), ω_{s}(αv), 2γ_{2}(Θ′v), B)
+ ω_{1}(Θv)τ_{m}(ω_{r}(αu), ω_{s}(αv), 2γ_{2}(Θ′u), B)
+ τ_{m}(ω_{r}(αu), ω_{s}(αv), 2γ_{2}(Θ′u), 2γ_{2}(Θ′v), B)}

if α = β_{2}ϕ_{2}γ_{2}σ^{2ℓ}ϕ_{2}α′ (ℓ ≥ 0) with Θ = β_{2}ϕ_{2}σ^{2ℓ}ϕ_{2}α′ and Θ′ = β_{2}σ^{2ℓ+1}ϕ_{2}α′

and the relations for divided power generators (m ≥ 0)

τ_{m}(kγ_{k}(α(u + v)), B) = ∑_{r+s=k+1}γ_{s−1}(αv)τ_{m}(rγ_{r}(αu), B) + γ_{r−1}(αu)τ_{m}(sγ_{s}(αv), B)

The formulas for the additivity relations above follow directly from the elementary equivalences and the properties of the torsion generators (see the Appendix).

3.3. Integral Homology in Special Cases. Using the results of the previous section, we determine the homology algebra for n = 1, the group structure in the “stable” and “quadratic” ranges, and several high degree groups in H_{∗}(π, 2; ℤ). A calculation for n = 1 was given independently by Hamsher [5], and several special cases of high degree groups (n > 1) have been obtained by Eilenberg and Mac Lane [4].

In H_{∗}(π, 1; ℤ), the only homology operations are σ: π → H_{1}(π, 1; ℤ) and ψ_{h}: _{h}π → H_{2}(π, 1; ℤ_{h}). Consequently, the torsion generators are of the form τ_{m}(ω_{i_{1}}(σu_{1}), …, ω_{i_{r}}(σu_{r})) where mu_{1} = ⋯ = mu_{r} = 0, or special symbols τ_{0}(σu), which we may identify with σu. Since each ω_{i}^{m}(σu) has odd degree, it follows that the torsion generators (except 1) all have odd degree. The only applicable additive relation becomes (Theorem 3.2.9, the case α = σ)

τ_{m}(ω_{i}(σu⊤σv), B) = ∑_{r+s=i} τ_{m}(ω_{r}(σu), ω_{s}(σv), B) + c ∑_{r+s=i+1} ω_{r}(σu)ω_{s}(σv)τ_{m}(B)

where c = \frac{m(m−1)}{2}. The coefficients in the Jacobi relation 1.3.2(vii) and in the symmetry relation 1.3.2(v) are all +1. Finally, the corresponding generator in [5] to the torsion generator above is denoted by the symbol C_{i_{1}...,i_{r}}^{m}(u_{1}, …, u_{r}) and is defined under the same conditions.

In H_{∗}(π, 2; ℤ) the applicable homology operations are σ^{2}, and β_{p}ϕ_{p}γ_{p^{f}} ψ_{p} of degree 2p^{f+1} + 1, for p a prime and f ≥ 0. The only operation of even degree is σ^{2}, hence the torsion generators are of the form

τ_{m}(ω_{i_{1}}(x_{1}), …, ω_{i_{r}}(x_{r}), k_{1}γ_{i_{r+1}}(σ^{2}u_{r+1}), …, k_{s}γ_{i_{r+s}}(σ^{2}u_{r+s}))

where x_{t} = α_{t}(u_{t}), u_{t} ∈ D_{α_{t}}(π) for t = 1, …, r, and α_{t} = β_{p}ϕ_{p}γ_{p^{f}} ψ_{p}. The operations β_{2}ϕ_{2}γ_{2^{f}} ψ_{2} are the only non-linear homology operations of height 2.

We compute the groups H_{2+q}(π, 2; ℤ) for 0 ≤ q < 8. For convenience, write x = γ_{1}(x), x = ω_{1}(x), and u = σ^{2}u, as appropriate. We do this by virtue of relations 3.2.6 on the elementary classes.

For q < 3, the classical results are obtained

H_{2}(π, 2; ℤ) ≅ π  H_{3}(π, 2; ℤ) = 0
H_{4}(π, 2; ℤ) ≅ Γ_{4}(π)  (the Whitehead group, see page 20)

The first p-primary homology operation β_{2}ϕ_{2}ψ_{2} and the first torsion generator of length greater than one arise for q = 3. The generators are β_{2}ϕ_{2}ψ_{2}(u) for u ∈ _{2}π and τ_{m}(u, v) for u, v ∈ _{m}π. The latter satisfy the defining relations of generators of the abelian group Tor(π, π), as well as the “diagonal” relation 3.2.7(iv′) τ_{m}(u, u) = 0. The only other notable relation is β_{2}ϕ_{2}ψ_{2}(u⊤v) = τ_{2}(u, v) for u, v ∈ _{2}π by Theorem 3.2.9(ii). The abstract abelian group defined by such symbols was called R(π) in [4]. Since β_{2}ϕ_{2}ψ_{2}(u) = β_{2}ϕ_{2}ψ_{2}(−u) and the additive deviation is bilinear, the symbols β_{2}ϕ_{2}ψ_{2}(u) satisfy the defining relations for Γ_{4}(_{2}π) (see page 20), so that

H_{5}(π, 2) ≅ {Γ_{4}(_{2}π) ⊕ Tor(π, π)}/L_{5}(π)

where L_{5}(π) is defined by the diagonal and additive relations above.

For q = 4, there are no p-primary homology operations or torsion generators that are not special symbols. The only new generator is γ_{3}(u) for u ∈ π, satisfying the relations of 3.1.1. Hence H_{6}(π, 2) ≅ Γ_{6}(π), the degree 6 component of Γ(π) where π has degree 2.

For q = 5, the new generators are β_{3}ϕ_{3}ψ_{3}(u) for u ∈ _{3}π, and the torsion generator τ_{m}(u, kγ_{2}(v)) for u ∈ _{m}π, and mkγ_{2}(v) = 0 in Γ(π). The significant relations (other than linearity, symmetry, and the “slide” formula) are

(i) β_{3}ϕ_{3}ψ_{3}(u⊤v) = 0 u, v ∈ _{3}π (ii) τ_{m}(u, kγ_{2}(v)) behaves as a generator of Tor(π, Γ_{4}(π)) (iii) τ_{m}(u, kγ_{2}(u)) = 0 (see 3.2.7(iv′)) (3.3.1) (iv) τ_{m}(u, 2γ_{2}(v⊤w)) = τ_{m}(u, v) · w + τ_{m}(u, w) · v u, v, w ∈ _{m}π

(v) τ_{m}(u, v) · w + τ_{m}(v, w) · u + τ_{m}(w, u) · v = 0 u, v, w ∈ _{m}π The elements u · τ_{m}(v, w) satisfy the relations ((ii) and (v)) defining

Trip(π, π, π) = Mult_{2}(π, π, π)

[9] with π given degree 2, so that

H_{7}(π, 2) = {Γ_{4}(_{2}π) ⊗ π ⊕ Trip(π, π, π) ⊕ Tor(π, Γ_{4}(π)) ⊕_{3}π}/L_{7}(π)

where L_{7}(π) is generated by the relations (iii) and (iv) above as well as the relations β_{2}ϕ_{2}ψ_{2}(u⊤v) · w = τ_{2}(u, v) · w, u · τ_{m}(u, v) = τ_{m}(2γ_{2}(u), v) and τ_{m}(u, u) · w = 0 on the products of generators of degree less than 7 which arise.

For q = 6, the new generators are

τ_{m}(u, v, w), mu = mv = mw = 0, τ_{m}(u, β_{2}ϕ_{2}ψ_{2}v), u, v ∈ _{2}π, and γ_{4}(u) for u ∈ π.

The only new relations (other than linearity, symmetry, or “slide” formulas) are τ_{m}(u, u, v) = 0 and τ_{2}(u, β_{2}ϕ_{2}ψ_{2}(v⊤w)) = τ_{2}(u, v, w). The symbols τ_{m}(u, v, w) behave as generators of Tor(π, π, π) = Mult_{3}(π, π, π) and we have

H_{8}(π, 2) ≃ {Γ_{8}(π) ⊕ Tor(π, π, π) ⊕ Tor(π, Γ_{4}(_{2}π))}/L_{8}(π)

where L_{8}(π) is generated by the relations

τ_{m}(u, u, v) = 0 and τ_{2}(u, β_{2}ϕ_{2}ψ_{2}(v ⊥ w)) = τ_{2}(u, v, w).

For q = 7 the new generators are τ_{m}(u, kγ_{3}(v)), τ_{m}(k_{1}γ_{2}(u), k_{2}γ_{2}(v)), and β_{2}ϕ_{2}γ_{2}ψ_{2}(u) for u ∈ _{2}π, subject to the relations

τ_{m}(u, kγ_{3}(u)) = 0

τ_{m}(k_{1}γ_{2}(u), k_{2}γ_{2}(u)) = 0

τ_{m}(u⊤v, kγ_{3}(w)) = 0  (3.3.2)

τ_{m}(2γ_{2}(u⊤v), kγ_{2}(w)) = u · τ_{m}(v, kγ_{2}(w)) + v · τ_{m}(u, kγ_{2}(w))

τ_{m}(u, 3γ_{3}(v⊤w)) = γ_{2}(v) · τ_{m}(u, w) + γ_{2}(w) · τ_{m}(u, v)

+ v · τ_{m}(u, 2γ_{2}w) + w · τ_{m}(u, 2γ_{2}v)

β_{2}ϕ_{2}γ_{2}ψ_{2}(u⊤v) = −Θu · 2γ_{2}(Θ′v) + Θv · 2γ_{2}(Θ′u) + τ_{2}(2γ_{2}(Θ′u), 2γ_{2}(Θ′v))

where Θ = β_{2}ϕ_{2}ψ_{2}, Θ′ = β_{2}σψ_{2} = σ^{2} for u, v ∈ _{2}π. The remaining generators are products of generators of lower degree, namely, u · τ_{m}(v, kγ_{2}(w)), γ_{2}(u) · τ_{m}(v, w), u · β_{3}ϕ_{3}ψ_{3}(v), and γ_{2}(u) · β_{2}ϕ_{2}ψ_{2}(v) all subject to relations arising in lower degree, and the following additonal relations

(i) u · τ_{m}(u, kγ_{2}v) = τ_{m}(2γ_{2}u, kγ_{2}v) (ii) γ_{2}u · τ_{m}(u, v) = τ_{m}(3γ_{3}u, v) (3.3.3) (iii) τ_{m}(v, kγ_{2}w) − v · τ_{m}(u, kγ_{2}w) + kγ_{2}w · τ_{m}(u, v) = 0

Since β_{2}ϕ_{2}α(u) is defined for u ∈ _{2}π, we have that β_{2}ϕ_{2}α(u) = β_{2}ϕ_{2}α(−u). By the relation β_{2}ϕ_{2}α(u⊤v) = β_{2}[σαu ∗ σαv], we see that β_{2}ϕ_{2}α(u⊤v) is bilinear in u and v. Consequently, the symbols β_{2}ϕ_{2}ψ(u) satisfy the defining relations of the group Γ_{4}(_{2}π). In particular, we have the cases α = ψ_{2} when q = 3 and α = γ_{2}ψ_{2} when q = 7.

Putting this together, we have the following description of

H_{9}(π, 2) ≅ {A ⊕ B}/L_{9}(π),

where

A = Tor(π, Γ_{6}(π)) ⊕ Tor(Γ_{4}(π), Γ_{4}(π)) ⊕ Trip(π, π, Γ_{4}(π))
B = (π ⊗_{3}π) ⊕ (Γ_{4}(π) ⊗ Γ_{4}(_{2}π)) ⊕ Γ_{4}(_{2}π)

and L_{9}(π) is generated by the relations 3.3.2 as well as 3.3.3(i) and (ii).

Finally, we observe that if π is torsion free, the p-primary homology operations are trivial, i.e., D_{α}(π) = 0, so that the only relevant operation is σ^{2}(u), u ∈ π, which is torsion free. Hence, the generators of H_{∗}(π, 2; ℤ) are of the form of special symbols τ_{0}(γ_{i}(σ^{2}u)) which satisfy the defining relations of Γ(π), and it follows that

Proposition 3.3.4. H_{∗}(π, 2; ℤ) ≅ Γ(π) whenever π is graded with degree 2, for π a torsion free abelian group (see also [3, exposé 11, Theorem 4]).

The procedure above can be carried out for arbitrarily high dimension, and will yield a description of the integral homology groups of K(π, 2). The problem becomes one of combinatories and patience. In the following section, this program is carried out in “low” dimensions for arbitrary π and n ≥ 1.

3.4. The Stable and Quadratic Ranges. We record here the integral homology groups in the range of dimensions n ≤ q < 2n (the stable range) and 2n ≤ q < 3n (the quadratic range). The term “quadratic” refers to the fact that the functors H_{q}(−, n; ℤ) in the range 2n ≤ q < 3n are quadratic in π (i.e., the functors H_{q}(π_{1}⊤π_{2}, n; ℤ) are linear in π_{1} and π_{2} [4]). In the stable range, the functors H_{q}(−, n; ℤ) are linear.

The groups H_{n+k}(π, n; ℤ) for k < n will be denoted by A_{k}(π). The suspension σ is an isomorphism in this range, so that A_{k}(π) is independent of n. These groups were determined by Cartan [3, exposé 11, §6]. Let f_{α}: D_{α}(π) → H_{n+k}(π, n; ℤ) for α = β_{p}σ^{ℓ}ϕ_{p}α′ be given by f_{α}(u) = β_{p}σ^{ℓ}ϕ_{p}α′(u), u ∈ D_{α}(π). Then the stable groups are

A_{0}(π) ≅ π A_{1}(π) = 0 A_{k}(π) ≅ ⨁_{α} D_{α}(π)  (3.4.1)

for k > 1, where α = β_{p}σ^{ℓ}ϕ_{p}α′ and q(α) = k. These operations correspond to sequences (a_{1}, …, a_{i}, …) satisfying 3.1.5 such that a_{1} ≡ 0 mod 2(p−1) and ∑ a_{i} = k. The isomorphisms are given by f_{σ^{n}} and ⨁_{α} f_{α}, respectively. All the operations α are linear, even those corresponding to p = 2 [3, exposé 11, Appendix 1]. For convenience of notation, we write α ∈ A_{k}(π) whenever α is such an operation.

To determine the quadratic range, H_{2n+k}(π, n; ℤ), for 0 ≤ k < n, we observe that the only torsion generators of length greater than 1 arising are of the form τ_{m}(x, y), where x, y are generators in the stable range, which behave as generators in Tor(−, −). We distinguish four special cases depending on the parity of k and n + k. Define a group Q_{n,k}(π) as follows:

Q_{n,2q} = \begin{cases}Λ_{2}(A_{q}(π)) & the “exterior square” if n + q is odd \\ & (Λ_{2}(G) = G ⊗ G/ < g ⊗ g | g ∈ G >) \\ Γ_{4}(A_{q}(π)) & the Whitehead group if n + q is even \end{cases}

(3.4.2)

Q_{n,2q+1} = \begin{cases}Ω_{q}(π) & if n + q is odd \\ R_{q}(π) & if n + q is even. \end{cases}

Ω_{q}(π) is generated by the symbols ω_{2}^{m}(α(u)), α ∈ A_{q}, u ∈ D_{α}(π) and τ_{m}(α_{1}(u), α_{2}(v)) for α_{1}, α_{2} ∈ A_{q} subject to the relations (whenever defined)

(i) ω_{2}^{mn}(α(u)) = nω_{2}^{m}(α(u)) nω_{2}^{mn}(α(u)) = ω_{2}^{m}(α(nu)) (ii) ω_{2}^{m}(α(u⊤v)) = τ_{m}(αu, αv) (iii) τ_{m}(αu, αv) = 2ω_{2}^{m}(αu), and τ_{m}(α_{1}u, α_{2}v)

behaves as a generator of Tor(A_{q}, A_{q}).

R_{q}(π) is generated by the symbols α(u), u ∈ D_{α}(π), α a mod-2 homology operation of the form β_{2}ϕ_{2}σ^{2ℓ}ϕ_{2}α′ (corresponding to a stable operation of the form β_{2}σ^{2ℓ+1}ϕ_{2}α′ ∈ A_{q}) and torsion generators τ_{m}(α_{1}u, α_{2}v) for α_{1}, α_{2} ∈ A_{q} subject to the relations

(i) τ_{m}(α_{1}u, α_{2}v) behaves as a generator of Tor(A_{q}, A_{q}) (ii) τ_{m}(αu, αu) = 0 (iii) β_{2}ϕ_{2}σ^{2ℓ}ϕ_{2}α′(u⊤v) = τ_{2}(β_{2}σ^{2ℓ+1}ϕ_{2}α′u, β_{2}σ^{2ℓ+1}ϕ_{2}α′v).

For q = 0, the groups Ω_{0}(π), R_{0}(π) were defined in [4]. Note that the only non-linear homology operations arising in the quadratic range are those figuring in the description of R_{q}(π).

Let L_{n,k}(π) = ⨁_{α} D_{α}(π), where α is a homology operation such that h(α) = n and q(α) = n + k. The maps f_{α}: D_{α}(π) → H_{2n+k}(π, n; ℤ) are all monomorphisms onto a direct summand.

Putting this together, we get the following natural description of integral homology in the quadratic range.

Theorem 3.4.3. For 0 ≤ k < n

H_{2n+k}(π, n; ℤ) ≅ Q_{n,k}(π) ⊕ ∑_{\begin{matrix}r+s=k \\ r<s \end{matrix}} A_{r} ⊗ A_{s} ⊕ ∑_{\begin{matrix}r+s=k−1 \\ r<s \end{matrix}} Tor(A_{r}, A_{s}) ⊕ L_{n,k}(π)

3.5. Divided Powers. The algebra H_{∗}(π, n; ℤ) is graded, connected, strictly commutative with a divided power structure. We give this structure here in terms of the torsion generators described in §3.2.

For any such algebra, the divided powers are determined by those on a homogeneous system of algebra generators by the definiton relations 3.1.1.

In H_{∗}(π, n; ℤ) the homogeneous generators are of the form τ_{m}(X, Y) where X = (ω_{i_{1}}(x_{1}), …, ω_{i_{r}}(x_{r})) and Y = (k_{1}γ_{i_{r+1}}(x_{r+1}), …, k_{s}γ_{i_{r+s}}(x_{r+s})) with x_{j} = α_{j}(u_{j}) for j = 1, …, r + s, u_{j} ∈ D_{α_{j}}(π), α_{j} a homology operation of odd or even degree, respectively. The parity of the degree of τ_{m}(X, Y) depends only upon the parity of ℓ(Y), in particular, the generator has even degree if and only if ℓ(Y) is odd. Therefore, it suffices to give the divided powers for the generators τ_{m}(X, Y) with s = ℓ(Y) odd.

To this end, we record the following power formula of such generators.

Lemma 3.5.1. For p > 1

[τ_{m}(X, Y)]^{p} = \begin{cases}0 for ℓ(Y) > 1 \\ cτ_{m}(c_{1}ω_{pi_{1}}(x_{1}), …, c_{r}ω_{pi_{r}}(x_{r}), k^{p}c_{r+1}γ_{pi_{r+1}}(y)) \end{cases}

for r ≥ 0, where X = (ω_{i_{1}}(x_{1}), …, ω_{i_{r}}(x_{r})), Y = kγ_{i_{r+1}}(y), and c = (p!)^{r+1}, c_{j} = {p, i_{j}} for j = 1, …, r + 1.

Proof. For ℓ(Y) > 1, the Jacobi relation and the diagonal relation 3.2.7(iv′) give the first case. For ℓ(Y) = 1, the second case is actually true whenever p = 1, since {1, i} = 1. By induction, with c = (p − 1)!^{r+1} and c′_{j} = {p − 1, i_{j}},

[τ_{m}(X, Y)]^{p} = c′τ_{m}(c′_{1}ω_{(p−1)i_{1}}(x_{1}), …, k^{p−1}c′_{r+1}γ_{(p−1)i_{r+1}}(y))τ_{m}(X, Y)
= c′τ_{m}(c′_{1}ω_{(p−1)i_{1}}(x_{1}), ω_{i_{1}}(x_{1}), …, kγ_{i_{r+1}}(y))τ_{m}(k^{p−1}c′_{r+1}γ_{(p−1)i_{r+1}}(y))

(by the Jacobi relation)

= c′τ_{m}(c′′_{1}ω_{pi_{1}}(x_{1}), …, c′′_{r}ω_{pi_{r}}(x_{r}), k^{p}c′′_{r+1}γ_{pi_{r+1}}(y))

where c′′_{j} = ((p − 1)i_{j}, i_{j})c′_{j} = pc_{j} (by relations 3.2.7(iv′) and (v′))

= c′p^{r+1}τ_{m}(c_{1}, ω_{pi_{1}}(x_{1}), …, k^{p}c_{r+1}γ_{pi_{r+1}}(y)) (by relation 2.4.6(ii))

where c′p^{r+1} = (p!)^{r+1} = c. This establishes the identity in the second case.

With this lemma, we can give the divided powers in H_{∗}(π, n; ℤ) by specifying the divided powers of homogeneous generators τ_{m}(X, Y) where ℓ(Y) is odd.

γ_{p}(τ_{m}(X, Y)) = \begin{cases}1 & if p = 0 \\ τ_{m}(X, Y) & if p = 1 \\ 0 & if p > 1 and ℓ(Y) > 1 \\ cτ_{m}(c_{1}ω_{pi_{1}}(x_{1}), …, c_{r}ω_{pi_{r}}(x_{r}), k^{p}c_{r+1}γ_{pi_{r+1}}(y)) \\ if p ≥ 1, ℓ(Y) = 1, where c_{j}, X, and Y \\ are as given above, and c = (p!)^{r}. \end{cases} (3.5.2)

For ℓ(X) = 0, we retrieve the standard divided powers in the form

γ_{p}(τ_{0}(kγ_{j}(y))) = τ_{0}(k^{p}γ_{p}(γ_{j}(y))) = τ_{0}(k^{p}{p, j}γ_{pj}(y)),

as required by relation 3.1.1(v). Restricting to the elements y = σ^{n}(u) for u ∈ π, the map Γ(π) → H_{∗}(π, n; ℤ) given by γ_{p}(u) ↦ τ_{0}(γ_{p}(σ^{n}u)) is an isomorphism onto a subalgebra [3, exposé 11, Theorem 4].

If n = 1, there are no homogeneous non-trivial torsion generators so that divided powers on the generators are all trivial, i.e., γ_{0}(τ_{m}(X)) = 1, and γ_{1}(τ_{m}(X)) = τ_{m}(X).

4. Cohomology

4.1. The Stable Range. Using the determination of H_{n+k}(π, n; ℤ) = A_{k}(π) for k < n and the Universal Coefficient Theorem for cohomology, we give a description of the groups H^{n+k}(π, n; G), for G an abelian group. This is effected by defining groups L^{n+k}(π, n; G) consisting of finite families of functions indexed by homology operations of the appropriate height and degree. We assume that k ≠ 1.

Definition 4.1.1. For G an abelian group, and α a homology operation let

R_{α}(G) = \begin{cases}G & if α = σ^{n} or β_{p}σ^{ℓ}ϕ_{p}α′ \\ G/pG & if α = σ^{ℓ}ϕ_{p}α′ \end{cases}

Let L^{n+k}(π, n; G) be the abelian group generated by finite families

𝔤 = {g_{α} ∈ Hom(D_{α}(π), R_{α}(G)) | h(α) = n, q(α) = k}

with addition defined by pointwise addition of functions of the same index, α.

Proposition 4.1.2. The following sequence is exact and natural in π and G; for k < n and k ≠ 1 (A_{−1} = 0)

0 → ⨁_{α∈A_{k−1}} Hom(D_{α}(π), G/pG) ⟶^{λ′} L^{n+k}(π, n; G)\begin{matrix}μ′ \\ ⟶ \\ ⟵ \\ μ′′ \end{matrix} ⨁_{α∈A_{k}} Hom(D_{α}(π), G) → 0 (4.1.3)

where

λ′(g_{α})_{α′} = \begin{cases}g_{α} & if α = β_{p}α′, g_{α} ∈ Hom(D_{α}(π), G/pG) \\ 0 & otherwise \end{cases}

and for 𝔤 = {g_{α}},

μ′(𝔤) = \begin{cases}g_{α} & if α = β_{p}α′, g_{α} ∈ Hom(D_{α}(π), G) \\ 0 & otherwise \end{cases}

The sequence 4.1.3 is split by a map natural in π and G.

Proof. From the description of the stable homology groups of the previous chapter, any α ∈ A_{k−1} (k ≠ 1) has the form α = β_{p}α′ for α′ = σ^{ℓ}ϕ_{p}α′′ with h(α′) = n, q(α′) = k. With this observation, the exactness and naturality of 4.1.3 is verified immediately.

The splitting μ′′ is given by

μ′′(g_{α})_{α′} = \begin{cases}g_{α} & if α′ = α, g_{α} ∈ Hom(D_{α}(π), G), α ∈ A_{k} \\ 0 & otherwise \end{cases}

and is clearly natural in π and G.

We define a homomorphism Θ^{n+k}: H^{n+k}(π, n; G) → L^{n+k}(π, n; G). For ω ∈ H^{n+k}(π, n; G), let ω̂ ∈ Hom(B̄_{n+k}^{(n)}(π), G) represent the class ω, then ω̂ induces ω_{∗}: H_{n+k}(π, n; ℤ) → G independent of the choice of ω̂, and for α = σ^{ℓ}ϕ_{p}α′ (ℓ ≥ 0), α̂: D_{α}(π) → B_{n+k}(π) [see page 23] is a function inducing ω̂α̂ ∈ Hom(D_{α}(π), G/pG) independent of the choice of α̂ or ω̂ (indeterminacy lies in pG).

Let

(Θ^{n+k}ω)_{α} = \begin{cases}ω_{∗}f_{α} & if α = β_{p}σ^{ℓ}ϕ_{p}α′ or σ^{n} \\ ω̂α̂ & if α = σ^{ℓ}ϕ_{p}α′ \end{cases}

where f_{α}: D_{α}(π) → A_{k} is as defined in Chapter 3 [see page 23], then it is routine to verify that Θ^{n+k} is a homomorphism.

The following lemma, due to Mac Lane [8], will be useful in analyzing Θ^{n+k}.

Lemma 4.1.4. Let pπ = 0. Then there is an isomorphism natural in π and G

ρ: Ext_{ℤ}(π, G) ≅ Hom(π, G/pG)

given by ρ(E)(u) = pe(u) + pG for u ∈ π, where e(u) ∈ X is a representative for u in the extension

E : 0 → G → X → π → 0 ∈ Ext_{ℤ}(π, G).

Proof. ρ is well defined, for any other representative e′(u) of u is such that pe′(u) − pe(u) ∈ pG. Naturality is immediate, and this, together with the linear and limit properties of Hom and Ext, allows reduction to the case where π = ℤ_{p}. In this case Ext_{ℤ}(ℤ_{p}, G) ≅ G/pG ≅ Hom(ℤ_{p}, G/pG) where the isomorphims are E ↦ pe(1_{p}) ∈ G/pG and ḡ ↦ fḡ, fḡ(1_{p}) = ḡ the characteristic function for ḡ ∈ G/pG.

The following lemma, together with the 5-lemma, shows that Θ^{n+k} (k ≠ 1) is a group isomorphism.

Lemma 4.1.5. For k < n and k ≠ 1, the diagram below commutes,

\xymatrix{0 \ar[r] & Ext_{ℤ}(A_{k−1}(π), G) \ar[r]^{λ} \ar[d]^{η} & H^{n+k}(π, n; G) \ar[r]^{μ} \ar[d]^{Θ^{n+k}} & Hom(A_{k}(π), G) \ar[r] \ar[d]^{⨁_{α} f_{α}^{∗}} & 0 \\ 0 \ar[r] & ⨁_{α∈A_{k−1}} Hom(D_{α}(π), G/pG) \ar[r]^{λ′} & L^{n+k}(π, n; G) \ar[r]^{μ′} & ⨁_{α∈A_{k}} Hom(D_{α}(π), G) \ar[r] & 0}

(4.1.6) where η = ⨁_{α}(−1)^{ℓ}ρ_{α}Ext(f_{α}), the top sequence is given by the Universal Coefficient Theorem for cohomology, the bottom sequence is that of 4.1.3, and

ρ_{α}: Ext_{ℤ}(D_{α}(π), G) → Hom(D_{α}(π), G/pG)

is the isomorphism of 4.1.4 for α = β_{p}σ^{ℓ}ϕ_{p}α′ ∈ A_{k−1}.

Proof. Commutativity of the right-hand square follows upon observing that μ(ω) = ω_{∗} for ω ∈ H^{n+k}(π, n; G) so that μ′(Θ^{n+k}ω)_{α} = ω_{∗}f_{α} = f_{α}^{∗}ω_{∗} = f_{α}^{∗}μ(ω) for α ∈ A_{k}. For the left-hand square, consider the following diagram for E ∈ Ext_{ℤ}(A_{k−1}(π), G)

\xymatrix{ &  & B̄_{n+k}^{(n)}(π) \ar[d]^{∂} \\ E_{0} : & 0 \ar[r] & B_{n+k−1} \ar[r] \ar[d]^{ϕ_{E}} & Z_{n+k−1} \ar[r] \ar[d] & A_{k−1}(π) \ar[r] \ar[d]^{id} & 0 \\ E : & 0 \ar[r] & G \ar[r] & X \ar[r] & A_{k−1}(π) \ar[r] & 0} (4.1.7)

where B_{n+k−1} ⊆ Z_{n+k−1} ⊆ B̄_{n+k−1}^{(n)}(π) are boundaries and cycles in degree n + k − 1, respectively. For E as above, the map λ is given by λ(E) = cls{ϕ_{E}∂} ∈ H^{n+k}(π, n; G) [7, Chapter 3, Section 4] for some ϕ_{E} ∈ Hom(B_{n+k−1}, G). Let u ∈ D_{α}(π) for α = β_{p}α′ ∈ A_{k−1} so that α′ = σ^{ℓ}ϕ_{p}α′′ of height n and stable degree k, then α̂(u) ∈ Z_{n+k−1} given by ∂α̂′(u) = (−1)^{ℓ}pα̂(u) represents f_{α}(u) ∈ A_{k−1} (see page 23). From 4.1.7, let e(f_{α}u) ∈ X represent f_{α}u. We have pe(f_{α}u) = ϕ_{E}pα̂(u)+ pG. If ω = λ(E), let ω̂ = ϕ_{E}∂ represent ω. It follows that (Θ^{n+k}ω)_{α′}(u) = ω̂α̂′(u) + pG = ϕ_{E}∂α̂′(u) + pG = ϕ_{E}((−1)^{ℓ}pα̂(u)) + pG = (−1)^{ℓ}pe(f_{α}u) + pG = (−1)^{ℓ}ρ(E)(f_{α}u) = (−1)^{ℓ}ρ_{α}(Ef_{α})(u) = (−1)^{ℓ}ρ_{α}Ext(f_{α})(u). Hence the left-hand square commutes.

Since Hom and Ext_{ℤ} are additive, and each ρ_{α} is an isomorphism, the outside vertical maps in 4.1.6 are isomorphisms. By the 5-lemma, Θ^{n+k} is also an isomorphism, giving

Theorem 4.1.8. For k < n, n ≠ 1, the map

Θ^{n+k}: H^{n+k}(π, n; G) → L^{n+k}(π, n; G)

is an isomorphism natural in π and G.

The bottom sequence of 4.1.6 splits by Proposition 4.1.2, hence, by Lemma 4.1.5, the top sequence of 4.1.6 splits naturally, and we have an additive decomposition in cohomology.

Corollary 4.1.9. For k < n, k ≠ 1,

H^{n+k}(π, n; G) ≃ [ ⨁_{α∈A_{k−1}} Hom(D_{α}(π), G/pG)] ⊕ [ ⨁_{α∈A_{k}} Hom(D_{α}(π), G)]

by an isomorphism natural in π and G.

Note: This result extends several special cases of [4, Section 27].

For k = 1 < n, the Universal Coefficient Theorem gives the natural isomorphism λ: Ext_{ℤ}(π, G) → H^{n+1}(π, n; G) since A_{0}(π) = π and A_{1}(π) = 0. In general, Lemma 4.1.4 representing the group of extensions as a group of homomorphisms must be replaced by a more general result. This accounts for our omission of this case. If π is a p-primary torsion group and Ext_{f}(π, G) is the subgroup of extensions that are trivial on all finite subgroups of π, Mac Lane [8, Theorem 1] gives a description of the quotient Ext_{ℤ}(π, G)/Ext_{f}(π, G) as a group of families of homomorphisms g_{k}: _{p^{k}}π → G/p^{k}G indexed by integers k > 0 such that the “double diagrams”

\xymatrix{_{p^{k}}π \ar[r]^{incl} \ar[d]_{g_{k}} & _{p^{k+1}}π \ar[l]^{p} \ar[d]^{g_{k+1}} \\ G/p^{k}G \ar[r]^{p} & G/p^{k+1}G \ar[l]^{incl}}  commute.

For pπ = 0, this becomes Lemma 4.1.4.

4.2. The Case k = n. Using the techniques of the previous section, we define groups L^{2n}(π, n; G) for n > 1 and establish isomorphisms Θ^{2n}: H^{2n}(π, n; G) → L^{2n}(π, n; G) which are natural in π and G.

Definition 4.2.1. For π, G abelian groups and n > 1, let L^{2n}(π, n; G) be the abelian group generated by families of functions, 𝔤, indexed by homology operations of height n and total degree 2n with addition given by addition of similarity indexed functions. An element 𝔤 of L^{2n}(π, n; G) is a family of functions as follows:

g_{σ^{n},σ^{n}} ∈ Hom(π ⊗ π, G)

g_{γ_{2}σ^{n}}: π → G  (n even) a function with g_{γ_{2}σ^{n}}(0) = 0
g_{ϕ_{2}σ^{n−1}} : π → G/2G  (n odd) a function with g_{ϕ_{2}σ^{n−1}}(0) = 0  (4.2.2)
g_{α} ∈ Hom(D_{α}(π), R_{α}(G)) α a mod-p homology operation with h(α) = q(α) = n

subject to the conditions, for x, y ∈ π

(i) g_{σ^{n},σ^{n}}(x, x) = 0 for n odd (ii) g_{σ^{n},σ^{n}}(x, x) = 2g_{γ_{2}σ^{n}}(x) for n even (4.2.3) (iii) g_{γ_{2}σ^{n}}(x⊤y) = g_{σ^{n},σ^{n}}(x, y) for n even (iv) g_{ϕ_{2}σ^{n−1}}(x⊤y) = g_{σ^{n},σ^{n}}(x, y) + 2G for n odd

The following proposition embeds L^{2n}(π, n; G) in a short exact sequence similar to that of Propositon 4.1.2. Recall that

Q_{n,0}(π) = \begin{cases}Λ_{2}(π) & if n odd \\ Γ_{4}(π) & if n even \end{cases}

so that g_{σ^{n},σ^{n}} (n odd) as above induces ḡ_{σ^{n},σ^{n}} ∈ Hom(Λ_{2}(π), G) by ḡ_{σ^{n},σ^{n}}(x∧y) = g_{σ^{n},σ^{n}}(x, y); and g_{γ_{2}σ^{n}} (n even) and g_{σ^{n},σ^{n}} above induce ḡ_{γ_{2}σ^{n}} ∈ Hom(Γ_{4}(π), G) by ḡ_{γ_{2}σ^{n}}(γ_{2}(x)) = g_{γ_{2}σ^{n}}(x) (since 2g_{γ_{2}σ^{n}}(x) = −g_{σ^{n},σ^{n}}(x, x) = −g_{γ_{2}σ^{n}}(x⊤(−x)) = g_{γ_{2}σ^{n}}(x) + g_{γ_{2}σ^{n}}(−x) by (ii) and (iii) above.

Proposition 4.2.4. The sequence below is exact and natural in π and G for n > 1

0 → ⨁_{α∈A_{n−1}} Hom(D_{α}(π), G/pG)\begin{matrix}λ′ \\ ⟶ \\ ⟵ \\ λ′′ \end{matrix}L^{2n}(π, n; G) ⟶^{μ′} Hom(Q_{n,0}(π), G) ⊕ A_{n}^{∗} → 0 (4.2.5)

where A_{n}^{∗} = ⨁_{α=β_{p}α′} Hom(D_{α}(π), G), q(α) = h(α) = n with

λ′(g_{α})_{α′} = \begin{cases}g_{α} ∈ Hom(D_{α}(π), G/pG) & if α = β_{p}α′ \\ 0 & otherwise \end{cases}

and for 𝔤 = {g_{σ^{n},σ^{n}}, g_{γ_{2}σ^{n}} (or g_{ϕ_{2}σ^{n−1}}), g_{α}}

μ′(𝔤) = \begin{cases}\begin{matrix}ḡ_{σ^{n},σ^{n}} (n odd) \\ ḡ_{γ_{2}σ^{n}} (n even) \end{matrix}\} & ∈ Hom(Q_{n,0}(π), G) \\ g_{α} ∈ Hom(D_{α}(π), G) & if α = β_{p}α′ \\ 0 & otherwise. \end{cases}

For n even, the sequence is split by a map λ′′ natural in π and G.

Proof. λ′ is clearly well defined and a homomorphism. That it is a monomorphism is trivial. The map μ′ is well defined and a homomorphism in view of relations 4.2.3.

μ′λ′ = 0 is immediate from the definition, and μ′(𝔤) = 0 requires that all the functions are zero except g_{ϕ_{2}σ^{n−1}} and g_{α}, α = σ^{ℓ}ϕ_{p}α′. Since β_{2}ϕ_{2}σ^{n−1} and β_{p}α ∈ A_{n−1}, 𝔤 = λ′(𝔤′) where 𝔤′ = {g′_{β_{p}α} | β_{p}α ∈ A_{n−1}} with g′_{β_{2}ϕ_{2}σ^{n−1}} = ḡ_{ϕ_{2}σ^{n−1}} ∈ Hom(π/2π, G/2G) (induced by g_{ϕ_{2}σ^{n−1}} since g_{σ^{n},σ^{n}} = 0) and g′_{β_{p}α} = g_{α}.

It remains to establish that μ′ is an epimorphism. Any g_{α} ∈ Hom(D_{α}(π), G) with q(α) = h(α) = n, α = β_{p}α′, lies in the image of μ′. For g ∈ Hom(Q_{n,0}(π), G), we consider cases. If n is even, let g_{σ^{n},σ^{n}}(x, y) = g(γ_{1}(x) · γ_{1}(y)) and g_{γ_{2}σ^{n−1}}(x) = g(γ_{2}x), then {g_{σ^{n},σ^{n}}, g_{γ_{2}σ^{n}}} ∈ L^{2n} with image g. For n odd, g ∈ Hom(Λ_{2}(π), G). By [4, Corollary 26.6], there is a function g′ from π to G/2G such that g′(x⊤y) = g(x∧y)+2G. Let g_{σ^{n},σ^{n}}(x, y) = g(x∧y) and g_{ϕ_{2}σ^{n−1}} = g′, then {g_{σ^{n},σ^{n}}, g_{ϕ_{2}σ^{n−1}}} ∈ L^{2n}(π, n; G) with image g.

The splitting map for n even is given for 𝔤 ∈ L^{2n} by

λ′′(𝔤)|_{β_{p}α} = g_{α} ∈ Hom(D_{α}(π), G/pG)

where g_{α} ∈ 𝔤. Clearly λ′′ is well defined and a homomorphism such that λ′′λ′ = id. Naturality of the maps λ′, μ′, λ′′ is evident from their definition.

As before, we define a map Θ^{2n}: H^{2n}(π, n; G) → L^{2n}(π, n; G). For ω ∈ H^{2n}(π, n; G), let Θ^{2n}ω be given by

(Θ^{2n}ω)_{σ^{n},σ^{n}}(x ⊗ y) = ω_{∗}(f_{σ^{n}}(x) ∗ f_{σ^{n}}(y)) for x, y ∈ π
(Θ^{2n}ω)_{γ_{2}σ^{n}}(x) = ω_{∗}(γ_{2}f_{σ^{n}}(x))  for n even
(Θ^{2n}ω)_{ϕ_{2}σ^{n−1}}(x) = ω̂{ \atop ϕ_{2}σ^{n−1}}(x̄) + 2G  for n odd, (x̄ = x + 2π ∈ π/2π)

(Θ^{2n}ω)_{α} = \begin{cases}ω_{∗}f_{α} & for α = β_{p}σ^{ℓ}ϕ_{p}α′ \\ ω̂α̂ & for α = σ^{ℓ}ϕ_{p}α′ \end{cases}

where γ_{2} is the divided power on H_{∗}(π, n; ℤ). Since ϕ_{2}σ^{n−1}(x̄⊤ȳ) = σ^{n}x̄ ∗ σ^{n}ȳ for x̄, ȳ ∈ π/2π, n odd, and γ_{2}σ^{n}(x⊤y) = σ^{n}x ∗ σ^{n}y for x, y ∈ π, n even, the family Θ^{2n}ω ∈ L^{2n}(π, n; G). Recall that all other homology operations α with height and stable degree n are linear. Under the addition in L^{2n}, it follows that Θ^{2n} is a homomorphism.

From Theorem 3.4.3, we have the natural isomorphism

H_{2n}(π, n; ℤ) ≅ Q_{n,0}(π) ⊕ L_{n,0}(π)

where L_{n,0}(π) = ⨁_{α} D_{α}(π) for α = β_{p}σ^{ℓ}ϕ_{p}α′ of height and stable degree n. If f_{n,0} : Q_{n,0}(π) → H_{2n}(π, n; ℤ) is the isomorphism onto a direct summand, we have f_{n,0}(γ_{2}x) = γ_{2}f_{σ^{n}}(x) for n even, and f_{n,0}(x ∧ y) = f_{σ^{n}}(x) · f_{σ^{n}}(y) for n odd.

The following lemma and the 5-lemma establish that Θ^{2n} (n > 1) is an isomorphism of abelian groups.

Lemma 4.2.6. For n > 1, the diagram below commutes,

\xymatrix{0 \ar[r] & Ext_{ℤ}(A_{n−1}(π), G) \ar[r]^{λ} \ar[d]^{η} & H^{2n}(π, n; G) \ar[r]^{μ} \ar[d]^{Θ^{2n}} & Hom(H_{2n}(π, n; ℤ), G) \ar[r] \ar[d]^{f_{n,0}^{∗}⊕[⨁_{α} f_{α}^{∗}]} & 0 \\ 0 \ar[r] & ⨁_{α∈A_{n−1}} Hom(D_{α}(π), G/pG) \ar[r]^{λ′} & L^{2n}(π, n; G) \ar[r]^{μ′} & Hom(Q_{n,0}(π), G) ⊕ A_{n}^{∗} \ar[r] & 0}

where η = ⨁_{α}(−1)^{ℓ}ρ_{α}Ext(f_{α}), α ∈ A_{n−1}, A_{n}^{∗} is defined as in 4.2.5, with the top sequence given by the Universal Coefficient Theorem for cohomology, the bottom sequence that of Propsition 4.2.4, and ρ_{α} is the isomorphism of 4.1.4 for α ∈ A_{n−1}.

Proof. Commutativity of the right-hand square follows upon observing that μ(ω) = ω_{∗} for ω ∈ H^{2n}(π, n; G), so that

μ′(Θ^{2n}ω)_{σ^{n},σ^{n}} = ω_{∗}f_{n,0} = f_{n,0}^{∗}μ(ω)  for n odd
μ′(Θ^{2n}ω)_{γ_{2}σ^{n}} = ω_{∗}f_{n,0} = f_{n,0}^{∗}μ(ω)  for n even
μ′(Θ^{2n}ω)_{α} = ω_{∗}f_{α} = f_{α}^{∗}μ(ω)  for α ∈ L_{n,0}(π).

For the left-hand square, commutativity follows by the corresponding argument in the proof of Lemma 4.1.5, where k = n. Note that β_{2}ϕ_{2}σ^{n−1} = σγ_{2}σ^{n−1} ∈ A_{n−1}, and that ϕ_{2}σ^{n−1} is the only non-additive homology operation with height and stable degree n.

The following theorem is immediate.

Theorem 4.2.7. For n > 1, Θ^{2n}: H^{2n}(π, n; G) → L^{2n}(π, n; G) is an isomorphism of abelian groups natural in π and G. Further, for n even, there is a natural decomposition of H^{2n}(π, n; G) into a direct sum

H^{2n}(π, n; G) ≅ Hom(Γ_{4}(π), G) ⊕ [ ⨁_{α∈A_{n−1}} Hom(D_{α}(π), G/pG)]

⊕ [⨁_{α} Hom(D_{α}(π), G)]

the latter sum being isomorphic to Hom(L_{n,0}(π), G) by the identification

⨁_{α} f_{α}: ⨁_{α} D_{α}(π) ≅ L_{n,0}(π)

for h(α) = q(α) = n, α = β_{p}σ^{ℓ}ϕ_{p}α′.

For the case n = 2, we have A_{1}(π) = 0 and L_{2,0}(π) = 0, so that H^{4}(π, 2; G) ≅ Hom(Γ_{4}(π), G). For n = 3, A_{2}(π) = π/2π associated with the operation β_{2}ϕ_{2}σ^{2} = σγ_{2}σ^{2} and L_{3,0}(π) = _{2}π associated with the operation β_{2}σϕ_{2}ψ_{2}, so that

L^{6}(π, 3; G) = L(π, G) ⊕ Hom(_{2}π, G)

where L(π, G) is the group generated by pairs of functions g_{σ^{3},σ^{3}} ∈ Hom(π ⊗ π, G) and g_{ϕ_{2}σ^{2}} : π → G/2G such that g_{σ^{3},σ^{3}}(x, x) = 0 and g_{ϕ_{2}σ^{2}}(x⊤y) = g_{σ^{3},σ^{3}}(x, y) + 2G. This group was first identified in [4, §27].

In general, for n odd, the same observation yields a natural decomposition L^{2n}(π, n; G) = L(π, G) ⊕ K, where

K = [ ⨁_{\begin{matrix}α∈A_{n−1} \\ α≠β_{2}ϕ_{2}σ^{n−1} \end{matrix}} Hom(D_{α}(π), G/pG)] ⊕ Hom(L_{n,0}(π), G).

This gives a decomposion for n odd as follows.

Theorem 4.2.8. For n odd, n > 1, the direct sum

H^{2n}(π, n; G) ≅ L(π, G) ⊕ [ ⨁_{α∈A′_{n−1}} Hom(D_{α}(π), G/pG)] ⊕ Hom(L_{n,0}(π), G)

is a decomposition natural in π and G, where A′_{n−1} ⊕ π/2π = A_{n−1}(π) with the summand π/2π associated with the homology operation β_{2}ϕ_{2}σ^{n−1}.

Special cases of these results for n ≤ 6 were orginally obtained by Eilenberg-Mac Lane [4] where the relevant homology operations were specified by their actual integral chain representatives, α̂, in our notation.

Appendix Elementary Equivalences in Tor(H𝒳(π, n))

In this Appendix, we show that the relations 3.2.6 and the additive relations of Theorem 3.2.9 determine the ideal 𝒩(π, n) = Ψ^{−1}(N(π, n)) of Theorem 3.2.4, where N(π, n) ⊆ H_{∗}(X(π, n)) is the ideal of elementary equivalences (page 24). Here, using relation 1.3.2(ii), we have abused notation and identified the elementary classes of 3.2.5 with their images under Φ^{−1}.

By Theorem 2.4.15, the elements of X(π, n) of the form \frac{1}{m}∂(ψ_{m}(x_{1}) ⊗ ⋯ ⊗ ψ_{m}(x_{r})), for x_{1}, …, x_{r} elementary classes as in 3.2.5(ii), and elements of the form x or γ_{i}(x), for x an elementary class as in 3.2.5(i), constitute a set of generating cycles for H_{∗}(X(π, n)). Furthermore, the image under Ψ^{−1} of the homology class of such a cycle is the torsion generator τ_{m}(x_{1}, …, x_{r}), τ_{0}(x), or τ_{0}(γ_{i}(x)), respectively. We therefore prove the assertion in the paragraph above by establishing the relations induced on these generating cycles by the elementary equivalences.

For (x, y) generators of an elementary complex E(x) ⊗ P (y) with y = mx, ω_{i}^{m}(x) = cls \{\frac{1}{m}∂(γ_{i}(y))\} (i > 0), and for P (x) ⊗ E(y) with y = mx, γ_{i}^{m}(x) = cls \{\frac{1}{mi}∂(yγ_{i−1}(x))\} (i > 0).

For x = σ^{n}u, u ∈ π and n ≥ 1 odd, such that mu = 0, we have the pair of generators (σ^{n}u, σ^{n−1}ψ_{m}u) with ∂(σ^{n−1}ψ_{m}u) = m(σ^{n}u). For h ∈ ℤ, the pair (σ^{n}u, σ^{n−1}ψ_{hm}u) is identified with the pair of elements (σ^{n}u, hσ^{n−1}ψ_{m}u) by equivalences (iii). Thus,

ω_{i}^{hm}(σ^{n}u) = cls \{\frac{1}{hm}∂(γ_{i}(σ^{n−1}ψ_{hm}u))\} = cls \{\frac{1}{hm}∂(γ_{i}(hσ^{n−1}ψ_{m}u)\}

= cls \{\frac{h^{i}}{hm}∂(γ_{i}(σ^{n−1}ψ_{m}u))\} [by 3.1.1(ii) and (iii)]

= h^{i−1}cls \{\frac{1}{m}∂(γ_{i}(σ^{n−1}ψ_{m}u))\} = h^{i−1}ω_{i}^{m}(σ^{n}u).

For u ∈ π such that hmu = 0, the pair (σ^{n}(hu), σ^{n−1}ψ_{m}(hu)) is identified to the pair (hσ^{n}u, σ^{n−1}ψ_{hm}u) by equivalence (ii). Thus,

hω_{i}^{hm}(σ^{n}u) = cls \{\frac{h}{hm}∂(γ_{i}(σ^{n−1}ψ_{hm}u))\} = cls \{\frac{1}{m}∂(γ_{i}(σ^{n−1}ψ_{m}(hu))\}
= ω_{i}^{m}(hσ^{n}u) = ω_{i}^{m}(σ^{n}(hu)).

This establishes relations 3.2.6(ii).

For x = σ^{n}u, u ∈ π (n > 1 even), such that mu = 0, we have the pair of generators (σ^{n}u, σ^{n−1}ψ_{m}u) with ∂(σ^{n−1}ψ_{m}u) = −mσ^{n}u. For h ∈ ℤ, the pair (σ^{n}u, σ^{n−1}ψ_{hm}u) is identified with the pair (σ^{n}u, hσ^{n−1}ψ_{m}u) by equivalence (iii). Thus,

γ_{i}^{hm}(σ^{n}u) = cls \{\frac{1}{hmi}∂(−σ^{n−1}ψ_{hm}u · γ_{i−1}(σ^{n}u))\}

= cls \{\frac{1}{hmi}∂(−hσ^{n−1}ψ_{m}u · γ_{i−1}(σ^{n}u))\}

= cls \{\frac{1}{mi}∂(−σ^{n−1}ψ_{m}u · γ_{i−1}(σ^{n}u))\} = γ_{i}^{m}(σ^{n}u).

For u ∈ π such that hmu = 0, the pair (σ^{n}(hu), σ^{n−1}ψ_{m}(hu)) is identified to the pair (hσ^{n}u, σ^{n−1}ψ_{hm}u) by equivalence (ii). By equivalence (iv) and (vi), hσ^{n}u = σ^{n}(hu), so that

γ_{i}^{m}(hσ^{n}u) = γ_{i}^{m}(σ^{n}(hu)) = cls \{\frac{1}{mi}∂(−σ^{n−1}ψ_{m}(hu) · γ_{i−1}(σ^{n}(hu)))\}

= cls \{\frac{1}{mi}∂(−σ^{n−1}ψ_{hm}u · γ_{i−1}(hσ^{n}u))\}

= cls \{\frac{h^{i}}{hmi}∂(−σ^{n−1}ψ_{hm}u · γ_{i−1}(σ^{n}u))\} = h^{i}γ_{i}^{hm}(σ^{n}u).

This establishes relations 3.2.6(iii).

For relations 3.2.6(i), we observe that x = σ^{n}u, generating the complex of type (a), is identified by equivalence (vi) to the generator σ^{n}u in a complex of type (b) whenever mu = 0 in π for some m > 1. Thus, for n odd, ω_{1}^{m}(σ^{n}u) = cls \{\frac{1}{m}∂(σ^{n−1}ψ_{m}u)\} = cls\{σ^{n}u\} is identified with the homology class determined by x = σ^{n}u, which we have again denoted by σ^{n}u. For n even, γ_{i}^{m}(σ^{n}u) = cls \{\frac{1}{mi}∂(−σ^{n−1}ψ_{m}u · γ_{i−1}(σ^{n}u)\} = cls\{γ_{i}(σ^{n}u)\} is identified with the homology class determined by γ_{i}(x) = γ_{i}(σ^{n}u), which we have denoted by γ_{i}(σ^{n}u).

From elementary equivalence (iv), for x = σ^{n}u, the pair of generators (σ^{n}(hu), σ^{n−1}ψ_{m}(hu)) is identified with the pair of elements (hσ^{n}u, hσ^{n−1}ψ_{m}u), so that

ω_{i}^{m}(σ^{n}(hu)) = cls \{\frac{1}{m}∂(γ_{i}(σ^{n−1}ψ_{m}(hu)))\} = cls \{\frac{1}{m}∂(γ_{i}(hσ^{n−1}ψ_{m}u))\}

= cls \{\frac{h^{i}}{m} ∂(γ_{i}(σ^{n−1}ψ_{m}u))\} = h^{i}ω_{i}^{m}(σ^{n}u), for n odd; and

γ_{i}^{m}(σ^{n}(hu)) = cls \{\frac{1}{mi}∂(−σ^{n−1}ψ_{m}(hu) · γ_{i−1}(σ^{n}(hu)))\}

= cls \{\frac{1}{mi}∂(−hσ^{n−1}ψ_{m}u · γ_{i−1}(hσ^{n}u))\} [by equivalence (vi)]

= cls \{\frac{h^{i}}{mi}∂(−σ^{n−1}ψ_{m}u · γ_{i−1}(σ^{n}u))\} = h^{i}γ_{i}^{m}(σ^{n}u) for n even.

Relations 3.2.6(ii) and (iii) result in the identifications

ω_{i}^{m}(hx) = h^{i}ω_{i}^{m}(x) and γ_{i}^{m}(hx) = h^{i}γ_{i}^{m}(x) for h ∈ ℤ  (I)

Thus the relations induced by the initial identifications of equivalence (iv) are also induced by relations 3.2.6.

For x = β_{p}σ^{ℓ}ϕ_{p}α(u), a generator of a complex of type (c), only the identifications (I) induced by relations (ii) and (iii) are applicable, and in this case, we must have m = p. The relations now follow just as above using the initial identifications of equivalence (v).

It remains to establish the consequence of the additive relations of equivalences (i), (iv) and (v). We have

ψ_{m}(ω_{i}^{m}(x)) = cls\{γ_{i}(ψ_{i}(x)) ⊗ 1_{m}\}, ψ_{m}(kγ_{i}^{m}(x)) = cls \{\frac{k}{i} ψ_{m}(x) · γ_{i−1}(x) ⊗ 1_{m}\}

for k ∈ ℤ, where

ψ_{m}(σ^{n}u) = (−1)^{n−1}σ^{n−1}ψ_{m}u and ψ_{m}(β_{p}σ^{ℓ}ϕ_{p}α(u)) = (−1)^{ℓ−1} \frac{m}{p}σ^{ℓ}ϕ_{p}α(u).

For n > 1 and odd, by equivalence (iv),

ψ_{m}(σ^{m}(u + v)) = σ^{n−1}ψ_{m}(u + v) = σ^{n−1}ψ_{m}u + σ^{n−1}ψ_{m}v = ψ_{m}(σ^{n}u) + ψ_{m}(σ^{n}v).

For n = 1, ψ_{m}(σ(u + v)) = ψ_{m}(u + v) = ψ_{m}u + ψ_{m}v + \frac{m(m−1)}{2}σu · σv.

For α = β_{p}α′, α′ = σ^{2ℓ}ϕ_{p}α′′, a mod-p homology operation, we have, by equivalence (v),

ψ_{m}(α(u + v)) = ±\frac{m}{p}α′(u + v) = ±\frac{m}{p}[α′u + α′v + L]

= ψ_{m}(αu) + ψ_{m}(αv) + (±\frac{m}{p}L)

where

L = \begin{cases}ψ_{2}(Θu) · ψ_{2}(Θv) & for α′′ = σ^{2ℓ}ϕ_{2}α′′′, Θ = β_{2}σα′′, (ℓ ≥ 0) [2.2(i)] \\ Θu · Θv & for α′′ = σ^{2ℓ}γ_{2}α′′′, Θ = β_{2}σ^{2ℓ}ϕ_{2}α′′′, (ℓ ≥ 0) [2.2 (ii)] \\ [Θu + Θ′u · ψ_{2}(Θ′u)] · [Θv + Θ′v · ψ_{2}(Θ′v)] for α′′ = γ_{2}σ^{2ℓ}ϕ_{2}α′′′, \\ Θ = β_{2}ϕ_{2}σ^{2ℓ}ϕ_{2}α′′′, Θ′ = β_{2}σ^{2ℓ+1}ϕ_{2}α′′′, \\ (ℓ ≥ 0 and not a case above) [2.2(iii)] \\ 0 & for all other cases. \end{cases}

Note: L ≠ 0 occurs only for α = β_{2}α′ of odd degree.

The additive relations of Theorem 3.2.9 now follow directly by carrying out the relevant calculation on a representative cycle \frac{1}{m}∂(ψ_{m}(x_{1}) ⊗ ⋯ ⊗ ψ_{m}(x_{r})) using the ψ_{m}(x) given above for x = α(u), and the relations 3.1.1 governing the divided powers γ_{i}(ψ_{m}(x)) and γ_{i−1}(x). To illustrate, we perform the calculation for the case x = σ(u): let B̂ = ψ_{m}(x_{2}) ⊗ ⋯ ⊗ ψ_{m}(x_{r}) [let B̂ = 1 if r = 1], then

\frac{1}{m}∂[ψ_{m}(ω_{i}^{m}(σ(u + v))) ⊗ B̂] = \frac{1}{m}∂[γ_{i}(ψ_{m}(σ(u + v))) ⊗ B̂]
= \frac{1}{m}∂[γ_{i}(ψ_{m}(σu) + ψ_{m}(σv) + c(σu) · (σv)) ⊗ B̂]
= \frac{1}{m}[(∑_{i} γ_{r}(ψ_{m}(σu)) · γ_{s}(ψ_{m}(σv))
+ c ∑_{i−1}(σu · γ_{r}(ψ_{m}u)) · (σv · γ_{s}(ψ_{m}σv)) ⊗ B̂]
= ∑_{i} \frac{1}{m}∂[γ_{r}(ψ_{m}σu) · γ_{s}(ψ_{m}σv) ⊗ B̂]
+ c ∑_{i=1} [σu · γ_{r}(ψ_{m}σu) · σv · γ_{s}(ψ_{m}σv)] ⊗ \frac{1}{m}∂[B̂]

which gives in homology the relation recorded in 3.2.9, where c = \frac{m(m−1)}{2}.

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