Georgian Mathematical Journal Volume 9 (2002), Number 4, 607–622

CHAIN THEORIES AND SIMPLICIAL ABELIAN GROUP SPECTRA

FRIEDRICH W. BAUER

Abstract. Two different definitions of a chain theory K_{∗} lead to the same class of derived homology theories h_{∗}( ). On the category of CW pairs, these are those homology theories E_{∗} admitting a classifying simplicial abelian group spectrum E. So one has a functor from the category of chain theories into the category of simplicial abelian group spectra.

2000 Mathematics Subject Classification: 55U15, 55U10, 55P43. Key words and phrases: Chain theories, classifying spectrum, simplicial abelian group spectrum.

0. Introduction

The subject of the present paper are chain theories (Definitions 1.1 and 1.2) and their derived homology theories. Although these definitions are not equivalent, it turns out that they are associated with the same class of homology theories. If the underlying category 𝔎^{2} is the category of CW pairs, these homology theories are precisely direct sums of ordinary homology theories with coefficients in an abelian group (cf. 6.3). This is the classical result of R. O. Burdick, P. E. Conner and E. E. Floyd [4]. Here it appears as a corollary of the existence of a simplicial abelian group spectrum E = E_{K_{∗}} such that the derived homology H_{∗}(K_{∗}( )) of K_{∗} is isomorphic (as a homology theory) to the homology theory E_{∗}( ) (Theorem 6.2). In Section 1 we display the two kinds of chain theories together with a proof that they determine the same class of homology theories (as their derived homology). Since a chain theory of the second kind is a special case of a chain functor (cf. Proposition 2.1, Theorem 2.2 and [1], [2] concerning further information about chain functors), we felt obliged to include a definition of a chain functor in Section 2.

Sections 3–6 are devoted to a proof of the main Theorem 6.2 based on considerations about special chain functors (namely chain theories of the second kind). This proof is technically much simpler than the corresponding proof of a classifying spectrum (which is not any more an abelian group spectrum!) for an arbitrary chain functor (cf. [3]).

As we have already pointed out, we obtain the main result of [4] as a byproduct.

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1. Chain Theories

Let 𝔎^{2} be any category of pairs of topological spaces like the category of (based or unbased) CW pairs or any simplicial analogue. We agree to omit almost everywhere basepoints (whenever they occur) from our notation.

1.1. Definition. A functor A_{∗} : 𝔎^{2} ⟶ ch (= category of chain complexes) is a chain theory of the first kind, whenever to each (X, U) ∈ 𝔎^{2} there exists canonically a short exact sequence

0 ⟶ A_{∗}(U) ⟶^{i_{#}} A_{∗}(X) ⟶^{j_{#}} A_{∗}(X, U) ⟶ 0.  (1)

A chain theory of the first kind determines a homology theory

h_{∗}(·) = {h_{n}, ∂, n ∈ ℤ} = H_{∗}(A_{∗}(·))

by setting

h_{∗}(X, U) = H_{∗}(A_{∗}(X, U))  (2)

(the derived homology of A_{∗}) and a boundary operator

∂ : h_{n}(X, U) ⟶ h_{n−1}(U)  (3)

which is defined as follows: Suppose c ∈ Z_{n}(A_{∗}(X, U)), then there exists c̄ ∈ A_{∗}(X) such that j_{#}(c̄) = c. We define

∂[c] = [i_{#}^{−1} dc̄].  (4)

As a first consequence of (1) we notice h_{∗}(X, X) = 0.

1.2. Definition. A functor B_{∗} : 𝔎^{2} ⟶ ch is a chain theory of the second kind, whenever

B1) to each (X, U) ∈ 𝔎^{2} there are inclusions

B_{∗}(U) ⟶^{i_{#}}_{⊂} B_{∗}(X) ⟶^{j_{#}}_{⊂} B_{∗}(X, U)  (5)

such that for any cycle z ∈ B_{∗}(X, U) there exist z′ ∈ B_{∗}(X) as well as ū ∈ B_{∗}(U, U) such that

z ∼ j_{#}z′ + q_{#}ū, q : (U, U) ⊂ (X, U), dz′ ∈ im i_{#}.

Observe that the homology class of j_{#}z′ + q_{#}ū does not depend on the choice of ū.

B2) j_{#}z′ + q_{#}ū = dx in B_{∗}(X, U) ⟹ ∃u ∈ B_{∗}(U): du = i_{#}^{−1}dz′ in B_{∗−1}(U). B3) z ∈ Z_{n}(B_{∗}(X)), j_{#}(z) ∼ 0 in B_{∗}(X, U) ⟹ ∃x ∈ B_{∗}(X), u ∈ B_{∗}(U),

such that dx = z + i_{#}u.

B4) B_{∗}(X, X) is acyclic; all inclusions k : (X, U) ⊂ (Y, V) induce monomorphisms.

A morphism λ : K_{∗} ⟶ L_{∗} between chain theories of the first or the second kind is simply a natural transformation of functors.

Again we obtain a homology theory h_{∗} = {h_{n}, ˜∂, n ∈ ℤ} with homology groups

h_{n}(X, U) = H_{n}(B_{∗}(X, U))

and the boundary operator

˜∂: h_{n}(X, U) ⟶ h_{n−1}(U)

which is defined as follows: Let z ∈ B_{∗}(X, U) be a cycle, then z ∼ z′ + ū (inclusions omitted from the notation). We deduce dz′ = −dū, hence dz′ ∈ im i_{#} and we set

˜∂[z] = [i_{#}^{−1} dz′].  (6)

1.3. Lemma. 1) ∂[z] resp. ˜∂[z] do not depend on the choices involved and are natural homomorphisms.

2) The homology sequence of a pair (X, U) ∈ 𝔎^{2} is exact.

Proof. Ad 1): We treat only ˜∂, the case of ∂ is standard. z′_{1} +ū_{1} ∼ z′_{2} +ū_{2} ⟹

z′_{1}−z′_{2}+(ū_{1}−ū_{2}) = dx ⟹^{B2)} dz′_{1}−dz′_{2} = du, u ∈ B_{∗}(U) ⟹ [i_{#}^{−1}dz′_{1}] = [i_{#}^{−1}dz′_{2}].

Ad 2): The exactness of the homology sequences is immediate. □

Every chain theory B_{∗} of the second kind determines a chain theory of the first kind and vice-versa:

Let B_{∗} be a chain theory of the second kind, then we define B̂_{∗} by

B̂_{∗}(X) = B_{∗}(X), B̂_{∗}(X, U) = B_{∗}(X)/B_{∗}(U)

noticing that

0 ⟶ B̂_{∗}(U) ⟶ B̂_{∗}(X) ⟶ B̂_{∗}(X, U) ⟶ 0

is exact.

Suppose that A_{∗} is a chain theory of the first kind, then we define Ã_{∗} by

Ã_{∗}(X) = A_{∗}(X), Ã_{∗}(X, U) = A_{∗}(X) ⊕_{i} cone A_{∗}(U).

We have inclusions

Ã_{∗}(U) ⊂ Ã_{∗}(X) ⊂ Ã_{∗}(X, U).

Ad B1): A cycle z ∈ Ã_{∗}(X, U) is of the form z = z′ + ũ, ũ ∈ cone A_{∗}(U). Since Ã_{∗}(U, U) = cone A_{∗}(U), B1) is satisfied.

Ad B2): Suppose z′ + ũ = d(x + ṽ), ũ, ṽ ∈ cone A_{∗}(U), x ∈ A_{∗}(X), then z′ − dx = dṽ − ũ = u ∈ A_{∗}(U), hence

i_{#}^{−1}dz′ = du,

ensuring that B2) holds.

Ad B3): Let z ∈ Ã_{∗}(X) be a cycle, j_{#}z = d(x + ṽ), x ∈ A_{∗}(X), ṽ ∈ cone A_{∗}(U), then dṽ ∈ im i_{#}, hence dṽ = i_{#}u, u ∈ A_{∗}(U), so that j_{#}z = dx + i_{#}u, implying B3).

Ad B4): Ã_{∗}(X, X) = cone A_{∗}(U) is acyclic.

Suppose A_{∗} is a chain theory of the first kind, then we have

ˆÃ_{∗}(X, U) = Ã_{∗}(X)/Ã_{∗}(U) = A_{∗}(X)/A_{∗}(U) = A_{∗}(X, U),

hence

ˆÃ_{∗} = A_{∗}.

If B_{∗} is a chain theory of the second kind, then we have

˜B̂_{∗}(X) = B_{∗}(X),

but

˜B̂_{∗}(X, U) = B_{∗}(X) ⊕_{i} cone B_{∗}(U),

which does not agree with B_{∗}(X, U), however

1.4. Lemma. There exists a natural homomorphism

β : B_{∗}(X) ⊕_{i} cone B_{∗}(U) ⟶ B_{∗}(X, U)

inducing a natural isomorphism of homology groups.

Proof. We set β | B_{∗}(X) = j_{#} and notice that

B_{∗}(U) ⟶^{q_{#}} B_{∗}(U, U) ⊂ B_{∗}(X, U)

factors over B_{∗}(U) ⊂ cone B_{∗}(U), furnishing a β. Let z = z′ + ū be a cycle in B_{∗}(X, U), z′ ∈ B_{∗}(X), ū ∈ B_{∗}(U, U), then we find a ũ ∈ cone B_{∗}(U) such that −dz′ = dũ = dū ∈ B_{∗}(U). It turns out that β(z′ + ũ) ∼ z. Hence β_{∗} is epic.

Let β(z′ + ũ) = z′ + ū, ū ∈ B_{∗}(U, U), ũ ∈ cone B_{∗}(U) be bounding in B_{∗}(X, U). According to B2) we detect v ∈ B_{∗}(U) such that dz′ = dv, hence z′ − i_{#}v is a cycle. We deduce that (z′ − v) + (v + ū) = dx̄, x̄ ∈ B_{∗}(X, U), hence, because v+ū is a bounding cycle in B_{∗}(U, U), j_{#}(z′−v) ∼ 0 in B_{∗}(X, U). According to B3) we deduce z′ − v = dx + w, x ∈ B_{∗}(X), w ∈ B_{∗}(U), hence

z′ + ũ = dx + w + v + ũ = dx + dw̃, w̃ ∈ cone B_{∗}(U),

(observing that w + v + ũ is a cycle in cone B_{∗}(U), hence bounding) implying that z′ + ũ ∼ 0 in B_{∗}(X) ⊕_{i} cone B_{∗}(U). Therefore β_{∗} is monic. □

Now we compare the homology of A_{∗} and Ã_{∗} resp. the homology of B_{∗} and B̂_{∗}:

1.5. Proposition. 1) Let A_{∗} be a chain theory of the first kind, then there exists a natural transformation of functors α : Ã_{∗} ⟶ A_{∗} inducing an isomorphism α_{∗} of homology theories.

2) Let B_{∗} be a chain theory of the second kind, then there exist natural transformations

B̂_{∗} ⟵^{α} ˜ B̂_{∗}, ⟶^{β} B_{∗}

inducing isomorphisms of homology theories.

Proof. Ad 1): We set α : Ã_{∗}(X) ⟶^{=} A_{∗}(X) to be the identity and

α : Ã_{∗}(X, U)=A_{∗}(X)⊕_{i} cone A_{∗}(U) ⟶ (A_{∗}(X)⊕_{i} cone A_{∗}(U))/cone A_{∗}(U) ≈

≈ A_{∗}(X)/A_{∗}(U) = A_{∗}(X, U)

the projection, which is easily seen to become an isomorphism on the homology level.

Let [z] = [z′ + ũ] ∈ H_{n}(Ã_{∗}(X, U)) be given, then

α_{∗}∂[z] = α_{∗}[i_{#}^{−1}dz′] = [i_{#}^{−1}dz′] = ∂α_{∗}[z],

hence α_{∗} commutes with boundaries.

Ad 2): There exists a transformation of functors β : ˜ B̂_{∗}(X) → B̂_{∗}(X) and β as in Lemma 1.4, for pairs (X, U), assuring us that β_{∗} is an isomorphism of homology groups.

Let [z] = [z′ + ũ] ∈ H_{n}( ˜B̂_{∗}(X, U)) be given, then we calculate

β_{∗}∂[z′ + ũ] = [i_{#}^{−1}dz′] = ∂[j_{#}z′] = ∂β_{∗}[z′ + ũ],

ensuring that β_{∗} is an isomorphism of homology theories. □

Remarks. 1) Proposition 1.5 asserts that chain theories of the first and of the second kind reach precisely the same kind of homology theories.

2) Definition 1.1 describes the classical chain theory, while Definition 1.2 describes a special case of a chain functor, a concept we will briefly record in the next section.

2. Chain Functors and Associated Homology Theories

In contrast to the preceding section we start with a homology theory h_{∗}( ) = {h_{n}, ∂_{n}, n ∈ ℤ} and ask if there exists a chain theory of the first kind K_{∗} and an isomorphism of homology theories

h_{∗}( ) ≈ H_{∗}(K_{∗}( ))

We assume that all homology and all chain theories have compact carriers. Alternatively we can of course work with the category of finite CW pairs.

Due to a result of R. O. Burdick, P. E. Conner and E. E. Floyd ([4] or [2] for a further reference) this implies, for 𝔎^{2} = category of CW pairs, that h_{∗}( ) is a sum of ordinary homology theories i.e., of those satisfying a dimension axiom although not necessarily in dimension 0.

The non-existence of such a chain theory of the first kind (implying the nonexistence of a chain theory of the second kind with the correct homology and boundary operator) gives rise to the theory of chain functors, cf. [2] for further references.

A chain functor C_{∗} = {C_{∗}, C′_{∗}, l, i′, κ, φ} is

CH1) a pair of functors C_{∗}, C′_{∗} : 𝔎 ⟶ ch, natural inclusions i′ : C_{∗}(A) ⊂ C′_{∗}(X, A), l : C′_{∗}(X, A) ⊂ C_{∗}(X, A) non-natural chain mappings

φ : C′_{∗}(X, A) ⟶ C_{∗}(X),
κ : C_{∗}(X) ⟶ C′_{∗}(X, A),

chain homotopies φκ ≃ 1, j_{#} φ ≃ l (j : X ⊂ (X, A)), as well as an identity

κ i_{#} = i′, i : A ⊂ X.

CH2) All inclusions k : (X, A) ⊂ (Y, B) are supposed to induce monomorphisms. All C_{∗}(X, X) are acyclic.

We denote by C_{∗}(X) the chain group C_{∗}(X, X) which according to CH2) can be assumed to contain all C_{∗}(X, U) for all pairs (X, U).

Needless to say, that C′_{∗}, as well as ϕ, κ are not determined by the functor C_{∗}(⋯, ⋯) but are additional ingredients of the structure of a chain functor.

Instead of the exact sequence (1) in §1, for chain theories (of the first kind) we are now, in the case of a chain functor, dealing with the sequence

0 ⟶ C_{∗}(A) ⟶^{i′} C′_{∗}(X, A) ⟶^{p} C′_{∗}(X, A)/im i′ ⟶ 0  (1)

and there exists a homomorphism

ψ : H_{∗}(C′_{∗}(X, A)/im i′) ⟶ H_{∗}(C_{∗}(X, A))  (2)
[z′] ⟼ [l(z′) + q_{#}(ā)]

where z′ ∈ C′_{∗}(X, A), dz′ ∈ im i′, q : (A, A) ⊂ (X, A), ā ∈ C_{∗}(A, A), dā = −dz′.

CH3) It is assumed that ψ is epic.

Since C_{∗}(A, A) is acyclic, dz′ ∈ im i′, such that ā exists and [l(z′) + q_{#}(ā)] is independent of the choice of ā.

This assumption implies that each cycle z ∈ C_{∗}(X, A) is homologous to a cycle of the form l(z′) + q_{#}(ā), with z′ being a relative cycle, the analogue of a classical relative cycle z ∈ C_{∗}(X) with dz ∈ im i_{#}, whenever (1) holds, i.e., whenever we are dealing with a chain theory of the first kind.

CH4) We assume

ker ψ ⊂ ker ¯∂,  (3)

¯∂ : H_{n}(C′_{∗}(X, A)/im i′) ⟶ H_{n−1}(C_{∗}(A)) being the boundary induced by the exact sequence (2). Moreover

ker j_{∗} ⊂ ker p_{∗} κ_{∗},  (4)

with, e.g., κ_{∗} denoting the mapping induced by κ for the homology groups.

CH5) Homotopies H : (X, A) × I ⟶ (Y, B) induce chain homotopies D(H) : C_{∗}(X, A) ⟶ C_{∗+1}(Y, B), natural and compatible with i′ and l.

These are almost all ingredients of a chain functor we need. The derived (or associated) homology of a chain functor

h_{∗}(X, A) = H_{∗}(C_{∗}(X, A))

resp. for the induced mappings, is endowed with a boundary operator ∂ : H_{n}(C_{∗}(X, A)) ⟶ H_{n−1}(C_{∗}(A)) determined by ¯∂:

We seek for ζ ∈ H_{n}(C_{∗}(X, A)) a representative l(z′) + q_{#}(ā) and set

∂ ζ = ¯∂[z′] = [i′^{−1} d z′].

This turns out to be independent of the choices involved.

This h_{∗}( ) satisfies all properties of a homology theory eventually with exception for an excision axiom. Therefore it is convenient to add:

CH6) Let p : (X, A) ⟶ (X′, A′) be an excision map (of some kind, e.g. p : (X, A) ⟶ (X/A, ⋆)), then p_{∗} = H_{∗}(C_{∗}(p)) is required to be an isomorphism.

This H_{∗}(C_{∗}( )) = h_{∗}( ) turns out to be a homology theory. Moreover, under very general conditions on 𝔎^{2}, every homology theory h_{∗}( ) is isomorphic to the derived homology of some chain functor (cf. [2]).

Let λ : C_{∗} ⟶ L_{∗}, λ′ : C′_{∗} ⟶ L′_{∗} be natural transformations, where C_{∗}, L_{∗} are chain functors, compatible with i′, l and the natural homotopies of CH5), then we call λ : C_{∗} ⟶ L_{∗} a transformation of chain functors. Such a transformation induces obviously a transformation λ_{∗} : H_{∗}(C_{∗}) ⟶ H_{∗}(L_{∗}) of the derived homology. This furnishes a category ℭ𝔥 of chain functors. A weak equivalence in ℭ𝔥 is λ : C_{∗} ⟶ L_{∗} which has the homotopy inverse.

For our present purposes we do not have to require that

CH7) all chain complexes C_{∗}(X, A) be free.

However this is not a very severe restriction as it can be accomplished if it is not automatically fulfilled. This condition is usually part of the definition of a chain functor.

Our main example of a chain functor C_{∗} in this article is a chain theory of the second kind which has a compact carrier and satisfy CH5–CH7.

We define C′_{∗}(X, U) = C_{∗}(X) and take for φ : C′_{∗}(X, U) ⟶ C_{∗}(X), κ : C_{∗}(X) ⟶ C′_{∗}(X, U) the identity. The inclusion l : C′_{∗}(X, U) ⊂ C_{∗}(X, U) is simply j_{#}, j : X ⊂ (X, U) and i′ : C_{∗}(U) ⊂ C′_{∗}(X, U) equals i_{#}, i : U ⊂ X.

We have trivially

j_{#}φ = l, φκ = 1.

We have the following correspondences between the definition of a chain theory of the second kind and that of a chain functor:

B2), B3) ⟷ CH4

B1) ⟷ CH3

B4) ⟷ CH2

Summarizing we obtain

2.1. Proposition. A chain theory C_{∗} of the second kind, satisfying CH5– CH7 with compact carriers, determines a chain functor C_{∗} = {C_{∗}, C′_{∗}, φ, κ, i′, l} with C′_{∗}(X, U) = C_{∗}(X), κ = φ = identity, l = j_{#}, i′ = i_{#}.

This correspondence is functorial, embedding the category of chain functors of the second kind as a full subcategory into the category ℭh of chain functors.

Combining this with [2] Theorem 3.3, we obtain the following assertion:

2.2. Theorem. Let h_{∗}( ) = {h_{n}( ), ∂_{n}, n ∈ ℤ} be a homology theory defined on a category of pairs of topological spaces 𝔎^{2}; then the following properties of h_{∗} are equivalent:

1) h_{∗} is the derived homology theory of a chain theory of the first kind.

2) h_{∗} is the derived homology theory of a chain theory of the second kind.

3) h_{∗} is the derived homology theory of a chain functor C_{∗}, with natural φ, κ and chain homotopies φκ ≃ 1, j_{#}φ ≃ l.

Proof. The equivalence of 1) and 2) is the subject of Proposition 1.5, while the equivalence of 1) and 3) follows from [2] Theorem 3.3. □

3. Simplicial Chains

In what follows we take for 𝔎^{2} the category of based CW pairs. The objective of the present and the following sections is to associate with any chain theory of the second kind C_{∗} canonically an abelian group spectrum E = E_{C_{∗}} such that the derived homology of C_{∗} is isomorphic to E_{∗}( ). For this purpose we assume that C_{∗} satisfies CH5, CH6 and that C_{∗} has a compact carrier.

We start with some elementary properties of C_{∗}:

3.1. Lemma. 1) Let h_{∗}( ) = H_{∗}(C_{∗})( ) be the derived homology of C_{∗}, then there exists a natural isomorphism

Σ_{∗} : h_{∗}( ) ≈ h_{∗+1}(Σ ).  (1)

2) Let CX = X ×I/{x_{0}}×I ∪X ×{1} be the (reduced) cone over a space X with top vertex ⋆ = {{x_{0}} × I ∪ X × {1}}, then there exists a natural homomorphism

τ = τ_{⋆} : C_{∗}(X) ⟶ C_{∗+1}(CX).

Proof. Follows from the excision resp. the homotopy axiom. □

Take two different cones C_{+}X, C_{−}X with top vertices ⋆_{+}, ⋆_{−}, and identifying C_{−}X ∩ C_{+}X = X, then

ΣX = C_{+}X ∪ C_{−}X

is the reduced suspension with common basepoint of C_{+}Xand C_{−}X and

Σ_{#} = τ_{⋆_{+}} − τ_{⋆_{−}} : C_{∗}(X) ⟶ C_{∗+1}(ΣX)

the suspension inducing (1).

For each c ∈ C_{n}(X) we find a finite subcomplex X′ = N ; ⋃_{1} △^{q_{j}}, and a c′ ∈

C_{n}(X′), i : X′ ⊂ X, such that i_{#}c′ = c. We assume that △^{q_{j}} are cells in some cell structure of X, of maximal dimension, i.e., that X′ does not contain △^{p} ⊃ △^{q_{j}} p > q_{j}.

This is an immediate consequence of the requirement of compact carriers. Let c ∈ C_{n}(△^{q}), now △^{q} denoting a standard q-simplex, be a chain satis-

fying dc ∈ C_{n−1}(bd △^{q}) (inclusions omitted from the notation). We have the subsimplexes ∂_{i}△^{q} ⊂ bd△^{q} ⊂ △^{q}. Suppose

dc ∼ ∑^{q} ; i=0(−1)^{i}∂_{i}c in C_{n−1}(bd △^{q})

with ∂_{i}c ∈ C_{n−1}(∂_{i}△^{q}) and observe that in this case ∂_{i}c are prescribed and not determined by c.

3.2. Definition. 1) A chain c ∈ C_{n}(△^{q}), △^{q} a standard q-simplex, is a s1-chain whenever all ∂_{i_{k}} ⋯ ∂_{i_{1}}c, 1 ≤ k ≤ q are given.

2) A chain c ∈ C_{n}(X) is an s-chain (=simplicial chain) whenever

m ; ∑_{1}σ_{i#}^{q}c_{i},

c_{i} ∈ C_{n}(△^{q_{i}}) an s1-chain σ_{i}^{q_{i}} : △^{q_{i}} ⟶ X a singular simplex.

We need

3.3. Proposition. Let z ∈ C_{n}(X) be a cycle, then there exists a simplicial cycle \overline{z} ∼ z.

This assertion follows from

3.4. Proposition. Let c ∈ C_{n}(X) be any chain such that dc is simplicial, having a carrier X̂ ⊂ X′ not containing maximal dimensional simplexes of X′. Then there exists a simplicial \overline{c} ∈ C_{n}(X), d\overline{c} = dc, c − \overline{c} ∼ 0 in C_{n}(X).

Proof. The proof is divided into a ”horizontal” and a ”vertical” part. In the first (horizontal) part, we replace c up to homology by a chain ∑ σ_{#}^{q_{j}} c_{j}, where each c_{j} ∈ C_{n}(△^{q_{j}}) is concentrated in a single simplex. In the second (vertical) part, each c = σ_{#}^{q_{j}} c_{j} is adapted so that the resulting chain becomes a s1-chain (i.e., all ∂_{i_{k}} ⋯ ∂_{i_{1}}c are specified).

1) Let c ∈ C_{n}(X′) be a chain with dc ∈ C_{n−1}(X̂). Suppose that △^{q} is

any cell of maximal dimension of X′ = N ; ⋃_{1} △^{q_{j}}, assuming that all △^{q_{i}} are of

maximal dimension (i.e., there does not exist a cell of higher dimensions in X′, containing a given summand). We consider the subcomplex Y consisting of the closure of Y′ = bd △^{q} \(bd △^{q} ∩X̂) and X′′ = ⋃ bd △^{q_{i}}, △^{q_{i}} being a maximal dimensional cell of X′. We observe that (X′ ⊂ X′/Y)_{#}c ∼ b_{1} + b̄_{1} with b_{1} ∈ C_{n}(△^{q}/Y), b̄_{1} a chain with carrier (\overline{X′ \ △^{q}})/Y. We obtain by excision b ∈ C_{n}(△^{q}, bd △^{q}), b̄ ∈ C_{n}(\overline{X′ \ △^{q}}) and can assume without loss of generality that db − c = −db̄ and (according to B1) in Definition 1.2) that b ∈ C_{n}(△^{q}), implying that b + b̄ − c ∈ C_{n}(X′, X̂ ∩ Y), hence ((X′, X̂ ∩ Y) ⊂ (X′, X̂ ∪ Y))_{#}(b + b̄ − c) ∼ 0. We deduce from the exactness of the homology sequence of the triple (X′, X̂ ∪ Y, X̂ ∩ Y) the existence of z̄′ ∈ C_{n}(X̂ ∪ Y) with dz̄′ ∈ C_{n−1}(Y ∩ X̂) such that b + b̄ ∼ c − z̄′ in C_{n}(X′, Y ∩ X̂). This gives rise to an inductive process on the number N of maximal dimensional cells in X′. At the end of this process we reach z′ ∈ C_{n}(X′′, Ȳ ∩ X̂), Ȳ = ⋃ \overline{bd △^{q_{i}} \(bd △^{q_{i}} ∩X̂)}, c_{i} ∈ C_{n}(△^{q_{i}}), dc_{i} ∈ C_{n−1}(bd △^{q_{i}}), dz′ ∈ C_{n−1}(Ȳ ∩X̂), such that

∑^{N} ; 1c_{i} − c − z′ ∼ 0 in C_{n}(X′, Ȳ ∩ X̂).

Now we treat z′ as before c, obtaining z′′, … and finally ĉ = ∑^{\overline{N}} ; 1σ_{j#}^{\overline{q}_{j}}c_{j} such that

ĉ − c ∼ 0, c_{j} ∈ C_{n}(△^{\overline{q}^{j}}),  (2)

ĉ ∈ C_{n}(X′) and dc = dĉ, with cells △^{\overline{q}^{j}} which are not necessarily any more of maximal dimension.

2) Let c̄ = c_{j} be any of those summands, △^{\overline{q}^{j}} = △^{q}. Because of our assumption on the carrier of the boundary of the original chain c in our Assertion 3.4, we conclude that dc̄ ∈ C_{n−1}(bd △^{q}). Applying the preceding process 1) to dc̄ yields elements \overline{c}_{i} ∈ C_{n−1}(∂_{i}△^{q}) as well as a homology

dc̄ ∼ ∑^{q} ; 0(−1)^{i}\overline{c}_{i},  (3)

i.e., a chain w ∈ C_{n}(bd △^{q}) satisfying

dw = dc̄ − ∑^{q} ; 0(−1)^{i}\overline{c}_{i}.

Now we proceed further with each \overline{c}_{i} in (3) separately, to the effect that we get a s1-chain c_{j} with ∑ σ_{j#}^{q_{j}}c_{j} − c ∼ 0.

As a result, ∑ σ_{j#}^{q_{j}}c_{j} is simplicial and displays the required properties.

This completes the proof of 3.4. □

4. The Simplicial Abelian Group Spectrum E

We consider the infinite dimensional simplex △^{∞} = (a_{0}, a_{1}, …, ⋆) with final vertex ⋆. A q-simplex in △^{∞} is either of the form △^{q} = (a_{i_{0}}, …, a_{i_{q}}) or (a_{i_{0}}, … a_{i_{q−1}}, ⋆) = τ_{⋆}(a_{i_{0}}, … a_{i_{q−1}}), i_{0} ≤ ⋯ ≤ i_{q}. We define ∂_{j}△^{q} = (a_{i_{0}}, …, â_{i_{j}}, …, a_{i_{q}}) resp. for ∂_{j}τ_{⋆}△^{q} and correspondingly s_{j}△^{q} = (a_{i_{0}}, …, a_{i_{j}}, a_{i_{j}}, …, a_{i_{q}}).

4.1. Definition. A p-simplex e^{p} ∈ E_{p}, p ∈ ℤ, is an assignment, associating to each such △^{q} in △^{∞} a chain e^{p}(△^{q}) ∈ C_{p+q}(△^{q}) such that:

E1) C_{p+q−1}(bd △^{q}) ∋ de^{p}(△^{q}) ∼ ∑(−1)^{i}e^{p}(∂_{i}△^{q}), if e^{p}(△^{q}) ≠ 0.

E2) e^{p}(τ_{⋆}△^{q}) = τ_{⋆}e^{p}(△^{q}) ∈ C_{p+q+1}(τ_{⋆}△^{q}).

E3) There exists \overline{q} such that e^{p}(△^{m}) = 0 for all △^{m} ⊃ △q̄ = (a_{0}, …, aq̄).

We define ∂_{j}e^{p} (s_{j}e^{p}) in the following way: by assumption we have for e^{p}(△^{q}) ≠ 0

de^{p}(△^{q}) ∼ ∑(−1)^{i}e^{p}(∂_{i}△^{q}) ∈ C_{p+q−1}(bd △^{q}), e^{p}(∂_{i}△^{q}) ∈ C_{p+q−1}(∂_{i}△^{q}),

∂_{i} : ∂_{i}△^{q} ⊂ △^{q} and set

(∂_{i} e^{p})(△^{q}) = \begin{cases}∂_{i#}e^{p}(∂_{i}△^{q}) ∈ C_{p+q−1}(△^{q}) & i ≤ q, e^{p}(△^{q}) ≠ 0 \\ 0 & in all other cases \end{cases} , (1)

(s_{j} e^{p})(△^{q}) = s_{j#}(△^{q})(e^{p}(s_{j}△^{q})) ∈ C_{p+q+1}(△^{q}), s_{j} : s_{j}△^{q} ⟶ △^{q}.

This furnishes a simplicial spectrum in the sense of [7] Definition 2.1. In particular ∂_{i}e^{p} ≠ 0 only for finitely many i ≥ 0.

In order to specify E for different chain theories C_{∗} we write sometimes E_{C_{∗}}. This construction is functorial: If λ : K_{∗} ⟶ L_{∗} is a transformation of chain theories (cf. §1), then there exists canonically an induced transformation E_{λ} : E_{K_{∗}} ⟶ E_{L_{∗}} in the category of simplicial spectra.

Moreover, E carries the structure of an abelian group spectrum ([7], Definition 4.1):

(e_{1}^{p} + e_{2}^{p})(△^{q}) = e_{1}^{p}(△^{p}) + e_{2}^{p}(△^{q}) ∈ C_{p+q}(△^{q})

so that for λ : K_{∗} ⟶ L_{∗} the induced E_{λ} is a morphism of simplicial abelian group spectra.

We summarize the above as:

4.2. Proposition. There exists a functor E from the category of chain theories of the second kind into the category of simplicial abelian group spectra.

4.3. Lemma. Let c ∈ C_{∗}(△^{q}) be any s1-chain (Definition 3.1), with prescribed ∂_{i_{k}} ⋯ ∂_{i_{1}}c, 1 ≤ k ≤ q, then there exists e^{p} ∈ E_{p} such that e^{p}(△^{q}) = c, p+q = n.

Proof. We define e^{p}(△^{q}) = c. By assumption, we have e^{p}(△^{l}) for any △^{l} ⊂ △^{q}. We set e^{p}(△^{s}) = 0, △^{q} ⊂_{≠} △^{s}. This furnishes immediately a procedure to

establish e^{p} for any △^{t} and therefore a e^{p} with the required property. □

4.4. Lemma. Let c ∈ C_{n}(X) be a chain, σ_{i}^{q_{i}} : △^{q_{i}} ⟶ X the simplicial singular simplexes, representing the cells of a carrier for c (cf. §3). Then c is simplicial whenever there exist e^{p_{i}} ∈ E_{p_{i}} such that

c = ∑ σ_{i#}^{q_{i}} e_{i}^{p_{i}}(△^{q_{i}}).  (2)

Proof. Follows immediately from 4.3 and 2) of Definition 3.1 of a simplicial chain. □

5. The Simplicial Spectrum X ∧ E

Let X be a based CW space (basepoint, as always, omitted from the notation), then an n-simplex, n ∈ ℤ of X ∧ E, is a pair σ^{q} ∧ e^{p}, p + q = n, consisting of a singular simplex σ^{q} : △^{q} ⟶ X, q ≥ 0 and a p-simplex e^{p} ∈ E_{p}, p ∈ ℤ. We define

∂_{i}(σ^{q} ∧ e^{p}) = σ^{q} ∧ ∂_{i} e^{p}

resp. for s_{j}, the degeneracies. This endows X ∧ E with the structure of a simplicial spectrum. Since X is supposed to be a CW space, we can take for singular simplexes the characteristic mappings of the cells.

We define

α(σ^{q} ∧ e^{p}) = σ_{#}^{q} e^{p}(△^{q}) ∈ C_{p+q}(X)  (1)

and deduce

α(∂_{i} σ^{q} ∧ e^{p}) = σ_{#}^{q} ∂_{i#} e^{p}(∂_{i} △^{q}) = α(σ^{q} ∧ ∂_{i} e^{p}).  (2)

We define the suspension (ΣE)_{p+1} by assignments △^{q} ⟼ Σ_{#} e^{p}(△^{q}) ∈ C_{p+q+1}(Σ △^{q}). Hence we have a bijection (ΣE)_{p+1} ⟷ E_{p}. Setting (Σ e^{p})( ) = Σ_{#}(e^{p}( )) and (Definition 4.1 E2) writing τ_{±} for τ_{⋆_{±}} )

α(σ^{q} ∧ Σe^{p}) = (Σσ^{q})_{#}e^{p}(△^{q})
α(Σσ^{q} ∧ e^{p}) = α(τ_{+} σ^{q} ∧ e^{p}) − α(τ_{−}σ^{q} ∧ e^{p}) = Σ_{#}α(σ^{q} ∧ e^{p}).

We calculate

Σ_{#} α(σ^{q} ∧ e^{p}) = α(Σσ^{q} ∧ e^{p}) = α(σ^{q} ∧ Σe^{p}).  (3)

In order to perform a stable homotopy theory, we have to convert X ∧ E into a Kan spectrum F (X ∧ E) by taking the free group spectrum generated by the simplexes of X ∧ E (cf. [7]). A stable mapping f : △^{n} ⟶ F (X ∧ E), n ∈ ℤ, is a class {f_{k}} of “geometric” mappings f_{k} : △^{n+k} ⟶ F (Σ^{k} X ∧ E). To each

such f_{k} there corresponds ω = ∏^{k} ; 1σ_{i}^{q_{i}+k} ∧ e^{p_{i}}, q_{i} + p_{i} = n in F (Σ^{k} X ∧ E). This

product is abelian up to stable homotopy, i.e. if ω̃ has the same factors as ω but eventually in a different order, then the stable f̃ is stably homotopic to f.

We define

α(f_{k}) = ∑_{i} σ_{i#}^{q_{i}+k} e^{p_{i}}(△^{q_{i}+k}) ∈ C_{n+k}(Σ^{k} X),

observing that this is a stable invariant

Σ_{#} α(f_{k}) = α(f_{k+1}).

This allows us to speak about α(f), f = {f_{k}}, as a class of chains c_{k} ∈ C_{n+k}(Σ^{k} X), which are connected by suspension. In particular α(f) ∼ 0 means that α(f_{k}) ∼ 0 for sufficiently high k. In the same manner f ∼ 0 means that f_{k} ≃ 0 for sufficiently high k. If z ∈ Z_{n}(C_{∗}(X)) is a cycle, then α(f) ∼ z, f : S^{n} ⟶ F (X ∧ E) means that there exists f_{k} : S^{n+k} ⟶ F (Σ^{k} X ∧ E) such that α(f_{k}) ∼ Σ_{#}^{k} z.

This allows us to formulate

5.1. Lemma. 1) d α(f_{k}) = α(f_{k} | bd △^{n+k}), f_{k} : △^{n+k} ⟶ F (Σ^{k} X ∧ E). 2) f : S^{n} ⟶ F (X ∧ E), α(f) ∼ 0 ⟺ f ∼ 0 (stably).

3) z ∈ Z_{n}(C_{∗}(X)) ⟹ ∃ f : S^{n} ⟶ F (X ∧ E), α(f) ∼ z.

Proof. Ad 1) We observe

α(f_{k} | bd △^{n+k}) = ∑^{n+k} ; 0(−1)^{j} α(∏_{i}σ_{i}^{q_{i}+k} ∧ ∂_{j} e^{p_{i}}) = dα(f_{k})

because of (2).

Ad 2) ⇒: Let f_{k} : S^{n+k} ⟶ F (Σ^{k} X∧E), f = {f_{k}} be a representative of the

stable homotopy class f. To f_{k} there corresponds s ; ∏_{1}σ^{q_{i}+k} ∧ e^{p_{i}} ∈ F (Σ^{k} X ∧ E),

hence

α(f_{k}) = s ; ∑_{1}σ_{#}^{q_{i}+k}(e^{p_{i}}(△^{q_{i}+k})) ∈ C_{n+k}(Σ^{k} X),

which is simplicial due to 4.4. Since α(f_{k}) ∼ 0, there exists c ∈ C_{n+k+1}(Σ^{k} X) such that dc = α(f_{k}). Proposition 3.4 together with Lemma 4.4 yields a simpli-

cial ĉ = ∑^{m} ; 1 η_{j#}^{s_{j}+k} \overline{e}_{j}^{t_{j}}(△^{s_{j}+k}), satisfying d\overline{c} = α(f_{k}). So \overline{ω} = ∏ η_{j}^{s_{j}+k} ∧ \overline{e}_{j}^{t_{j}} is as-

sociated with a G_{k} : △^{n+k+1} ⟶ F (Σ^{k} X ∧ E), satisfying G_{k} | bd △^{n+k+1} = f_{k}. Hence f_{k} ≃ 0 and f ∼ 0.

⇐: Suppose f ∼ 0, then there exists a G_{k} : △^{n+k+1} ⟶ F (Σ^{k} X ∧ E), such that f_{k} = G_{k} | bd △^{n+k+1} = f_{k} ∈ f. Hence (1) implies

d α(G_{k}) = α(f_{k}).

Since suspension induces an isomorphism on the homology level, α(f) ∼ 0. Ad 3) Proposition 3.3 and Lemma 4.4 furnish a simplicial z ∼ \overline{z} =

s ; ∑_{1}σ_{#}^{q_{i}}(e^{p_{i}}(△^{q_{i}})), p_{i} + q_{i} = n. Hence there exists a ω = m ; ∏_{1}σ^{q_{i}+k} ∧ e^{p_{i}} ∈

F (Σ^{k} X ∧ E), which is associated with a f_{k} : S^{n+k} ⟶ F (Σ^{k} X ∧ E), such that

α(f_{k}) = α(ω) = Σ^{k} \overline{z} ∼ Σ^{k} z.

This implies that α(f) ∼ z stably, f = {f_{k}}. □

We define

E_{n}(X) = π_{n}(F (X ∧ E)), n ∈ ℤ  (4)

in the usual way, resp.

E_{n}(X, U) = E_{n}(X ∪ CU), (X, U) ∈ 𝔎^{2}.  (5)

More precisely: (X ∪ CU) = (X ∪ CU, ⋆) and the basepoint ⋆ is the common basepoint of CU, U and X, since CU denotes the reduced cone over a based space U. The group operation in (4) is well known and corresponds to the product in F (X ∧ E) stably.

We have a natural homomorphism

α_{∗} : E_{n}(X, U) ⟶ H_{n}(C_{∗}(X, U))  (6)
{f} ⟼ [α(f)].

More precisely: Let f : S^{n} ⟶ F ((X ∪ CU) ∧ E) be a stable mapping of a (stable) n-sphere S^{n}, n ∈ ℤ, represented by a f_{k} : S^{n+k} ⟶ F (Σ^{k}(X ∪ CU) ∧ E), then α_{k} = [α(f_{k})] ∈ H_{∗}(C_{∗}(Σ^{k}(X, U))) and

Σ_{∗} α_{k} = α_{k+1}.

By desuspending we obtain a α_{∗}({f}) ∈ H_{∗}(C_{∗}(X, U)). Lemma 5.1. ensures that this is a natural and well-defined homomorphism.

5.2. Lemma. α_{∗} is an isomorphism.

Proof. Suppose z ∈ ζ ∈ H_{n}((C_{∗}(X, U))), then lemma 5.1.3) yields a

f : S^{n} ⟶ F ((X ∪ CU) ∧ E)

such that α(f) ∼ z. Hence

α{f} = z

ensuring that α_{∗} is epic. □

Suppose α({f}) = 0, then α(f) ∼ 0 and lemma 5.1. 2) implies that f ∼ 0 (stably). Hence α_{∗} is monic.

6. The Classifying Spectrum of a Chain Theory

In order to realize that α_{∗} is an isomorphism of homology theories, we have to deal with the boundary operators.

The boundary operator

∂ : E_{n}(X, U) ⟶ E_{n−1}(U)

is well-known to be defined by

E_{n}(X ∪ CU) ⟶^{k_{∗}} E_{n}(C_{−}X ∪ CU) ⟶^{ϱ_{∗}}_{≈} E_{n}(ΣU) ≈^{−Σ_{∗}^{−1}} E_{n−1}(U).

Here C_{−} X denotes a second cone over X with top vertex ⋆_{−}, as in §3, k : (X ∪ CU) ⊂ (C_{−} X ∪ CU) is an inclusion, while ϱ : C_{−} X ∪ CU ⟶ ΣU is well-known and Σ_{∗}^{−1} denotes desuspension.

We need

6.1. Lemma. We have

∂α_{∗}{f} = α_{∗}∂{f}, {f} ∈ E_{n}(X, U).  (1)

Proof. To f_{k} : S^{n+k} ⟶ F (Σ^{k}X ∧E) determining the class f : S^{n} ⟶ F (X ∧E)

there corresponds ω_{1} = ∏^{m} ; 1σ^{q_{i}+k} ∧ e^{p_{i}}, which we, in a first step, change in its

stable homotopy class: We assume at first that all σ^{q_{i}+k} : △^{q_{i}+k} ⟶ Σ^{k}(X∪CU) are cellular (i.e., that they are characteristic mappings of cells). Moreover, we assume that they are of the form either η^{q_{i}+k} : △^{q_{i}+k} ⟶ Σ^{k} X or \overline{ε}^{\overline{q}_{j}+k} : △^{\overline{q}_{j}+k} ⟶ C Σ^{k} U. Without loss of generality we can assume that \overline{ε}^{\overline{q}_{j}+k} = τ_{⋆}ε^{q_{j}+k}, q_{j} + 1 = \overline{q}_{j}. Changing the order in the product ω_{1}, we assume that f_{k} is associated with ω of the form

ω = ∏^{s} ; 1η^{q_{i}+k} ∧ e^{p_{i}} ∘ ∏^{t} ; 1τ_{⋆}ε^{q_{j}+k} ∧ e^{p_{j}} = a_{1} ∘ a_{2}.  (2)

We obtain

α(ω) = v + τ_{+}b, v ∈ C_{n}(X), dv ∈ C_{n−1}(U), b ∈ C_{n−1}(U),  (3)
(X ⊂ X ∪ CU)_{#}v = α(a_{1}), (CU ⊂ X ∪ CU)_{#}τ_{+}b = α(a_{2}),  (4)

moreover, dv = −dτ_{+} b = −b + τ_{+} db. Hence τ_{+} db ∈ C_{n−1}(U), implying db = 0 and dv = −b.

We denote by τ_{+} (resp. τ_{−} ) the cone CU = C_{+} U (resp. C_{−} X) with top vertex ⋆_{+} (resp. ⋆_{−}).

Since v − τ_{−} dv is a cycle in C_{−} X and hence bounding, we have

v − τ_{−} dv = dτ_{−} v.

We calculate

α(ϱ_{#} k_{#}a_{1}) + α(ϱ_{#} k_{#}a_{2}) = v + τ_{+}b
= v − τ_{−}dv + τ_{−}dv − τ_{+}dv = dτ_{−}v − Σ_{#}dv.

Therefore

α(ϱ_{#} k_{#}a_{1}) + α(ϱ_{#} k_{#}a_{2}) ∼ Σ_{#}dv in C_{n+1}(ΣU).

Hence

α(∂f) = −Σ_{#}^{−1}α(ϱ_{#} k_{#}a_{1}) + −Σ_{#}^{−1}α(ϱ_{#} k_{#}a_{2}) ∼ (U ⊂ X)_{#}^{−1}dv in C_{n−1}(U).

On the other hand, by the definition of the boundary operator in C_{∗} (cf. §1)

∂α_{∗}([f]) = [(U ⊂ X)_{#}^{−1}dv].

This implies

∂α_{∗}{f} = α_{∗}∂{f}.
□

Remark. The sign in the definition of the boundary operator for E_{∗} disappears if one decides to define

E_{n}(X, U) = E_{n}(X ∪ C_{−} U),

exchanging C_{+} and C_{−}.

We summarize our results, recalling, that we assume that our basic category 𝔎^{2} is the category of based CW pairs:

6.2. Theorem. There exists a functor from the category of chain theories of the first kind (as well as of the second kind) assigning to each K_{∗} a simplicial abelian group spectrum E = E_{K_{∗}} and an isomorphism of homology theories

α_{∗} : E_{∗}( ) ⟶^{≈} H_{∗}(K_{∗}( )).  (5)

Proof. This is an immediate consequence of 1.5, 5.2 and 6.1. □

6.3. Corollary (O. Burdick, P. E. Conner E. E. Floyd). A homology theory h_{∗}( ) is associated with a chain theory (of the first or of the second kind) if and only if

h_{∗}( ) ≈ +∞ ; ⨁_{−∞}H_{∗+k}( ; G_{k}),  (6)

where G_{k} are abelian groups.

Proof. If (6) holds, then the summands in (6) are (shifted) singular homology, therefore the existence of a chain theory K_{∗} of the first (and therefore also of the second kind) satisfying h_{∗}( ) ≈ H_{∗}(K_{∗}( )) is well-known. The other direction is a consequence of 6.2 and the following 6.4. Lemma. Every simplicial abelian group spectrum E is homotopy equivalent (in the category of abelian group spectra) to a direct sum of suspended Eilenberg- MacLane spectra

E ≃ +∞ ; ⨁_{−∞}Σ^{k} K(G_{k}).  (7)

Proof. The proof follows the same pattern as that of [5] Proposition 2.18. for simplicial sets. For the adaptation of this proof to the case of simplicial spectra instead of simplicial sets, one has to employ the results about chain complexes in [6] instead of those in [5]. □

Remark. Every chain functor K_{∗} (§2) admits a classifying spectrum E (satisfying (5)), however this is not an abelian group spectrum (cf. [2] or [3]). The present proof employs special properties of a chain theory of the second kind, hence of a chain functor, with very special properties of κ and φ (cf. Proposition 2.1).

References

1. F. W. Bauer, Chain functors with isomorphic homology. Homology Homotopy Appl. 3(2001), No. 2, 37–53.

2. F. W. Bauer, The Boardman category of spectra, chain complexes and (co-)localizations. Homology Homotopy Appl. 1(1999), No. 3, 95–116.

3. F. W. Bauer, Classifying spectra and generalized homology theories. Ann. Mat. Pura Appl. (4) 164(1993), 365–399.

4. R. O. Burdick, P. E. Conner, and E. E. Floyd, Chain Theories and their derived homology. Proc. Amer. Math. Soc. 19(1968), 1115–1118.

5. P. G. Goerss ans J. F. Jardine, Simplicial homotopy theory. Progress in Mathematics, 174. Birkhäuser Verlag, Basel, 1999.

6. J. F. Jardine, Generalized étale cohomology theories. Progress in Mathematics, 146. Birkhäuser Verlag, Basel, 1997.

7. D. Kan and G. W. Whitehead, The reduced join of two spectra. Topology 3(1965), suppl. 2, 239–261.

(Received 17.06.2002)

Author’s address:

Fachbereich Mathematik

Johann Wolfgang Goethe Universität

60054 Frankfurt a.M., Robert-Mayer Str. 8-10

Germany

E-mail: f.w.bauer@mathematik.uni-frankfurt.de