Contemporary Mathematics

Indecomposable Homotopy Types with at most two non–trivial Homology Groups

Hans–Joachim Baues and Yuri Drozd

Abstract. We classify indecomposable spaces in the stable range with at most two non–trivial finitely generated homology groups.

It is a classical result of Brown–Copeland [BC] and Eckmann–Hilton [EH] that a 1–connected homotopy type X with at most two non trivial homology groups H_{m}X = A and H_{n}X = B, 2 ≤ m < n, is a mapping cone of a map

k_{X} : M(B, n − 1) → M(A, m) (1)

Here M(A, m) denotes the Moore space with homology A in degree m.

Let p be a prime and let ℤ_{(p)} ⊂ ℚ be the smallest subring of ℚ containing 1/q for all primes q ≠ p. We consider finitely generated ℤ_{(p)}–modules A and B in the stable range n < 2m − 2. Hence X is a p–local space with at most two non– trivial homology groups in a stable range. Then the homotopy type of X admits a decomposition as a one point union

X ≃ X_{1} ∨ … ∨ X_{j} (2)

where all X_{i} with 1 ≤ i ≤ j are indecomposable and this decomposition is unique up to permutation. We classify in this paper the indecomposable summands in (2).

Theorem (A). Let A and B be finitely generated free ℤ_{(p)}–modules. Then the classification of indecomposable summands in (2) is

finite if π_{n−1}(S^{m}) ⊗ ℤ_{(p)} is cyclic,
tame if π_{n−1}(S^{m}) ⊗ ℤ_{(p)} ≅ ℤ/p ⊕ ℤ/p^{k}, k ≥ 1, and
wild otherwise.

Here π_{n−1}(S^{m}) = π_{n−m−1}^{s} is given by stable homotopy group of spheres since we assume n < 2m − 2. For example the wild case appears for n − m − 1 = 9 since π_{9}^{s} = ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2 and tame cases appear for n − m − 1 = 8, 15 since π_{8}^{s} = ℤ/2 ⊕ ℤ/2 and π_{15}^{s} = ℤ/2 ⊕ ℤ/480. The proof of theorem (A) is based on the representation theory of matrices with entries in a fixed ℤ_{(p)}–module M. For the finite and tame cases we describe the indecomposable summands explicitly.

\begin{matrix}c \\ ◯ \end{matrix}2000 American Mathematical Society

2000 Mathematics Subject Classification. 55P15.

The space X in (2) is (m − 1)–connected and we write conn(X) = m − 1 if H_{m}X ≠ 0. Moreover if H_{n}X ≠ 0 the p–local dimension of X is

dim_{(p)}(X) = \begin{cases}n & if H_{n}X is a free ℤ_{(p)}–module \\ n + 1 & otherwise \end{cases}

We say that X is trivial if X is a one point union of Moore spaces or equivalently if k_{X} in (1) is null homotopic.

Theorem (B). Let A and B be finitely generated ℤ_{(p)}–modules and let dim_{(p)}(X) − m ≤ 4p − 5. Then the classification of indecomposable summands in (2) is

tame  if n − m = 2p − 2,
essentially finite if n − m = 2p − 3, 2p − 1, and
trivial  otherwise.

For odd primes p this result can be deduced from the more general result of Henn [H]. For the prime p = 2 and dim_{(p)}(X) − m ≤ 2 the result is due to J. H. C. Whitehead [W1] and Chang [Ch]. Therefore we give a proof only for the highly sophisticated case p = 2 and dim_{(p)}(X) − m = 3. This case was also treated by Baues–Hennes [BH]; see also [B1] and [B2]. Our approach here using representation theory yields a new proof and confirms the intricate computation in [BH].

Theorem (C). Let p = 2 and let A and B be finitely generated ℤ/2–vector spaces and let n = m + 3 ≥ 9. Then the classification of indecomposable summands in (2) is wild.

This result solves an old question of homotopy theory: Let A^{k} = A_{m}^{k} be the homotopy category of (m − 1)–connected (m + k)–dimensional finite CW–spaces with m ≥ k + 1. Since the spaces of theorem (C) are objects in A^{4} we get:

Corollary (D). A^{k} has wild representation type for k ≥ 4.

It was shown in Baues–Drozd [BD1] that A^{k} has wild representation type for k ≥ 6. On the other hand J. H. C. Whitehead [W1] and Chang [Ch] computed the indecomposable objects of A^{2}; compare also the books of Hilton [H1], [H2]. Moreover Baues–Hennes computed all indecomposable objects of A^{3}. Hence since A^{k} is wild for k ≥ 4 the representation type of A^{k} is now known for all k. This answers a classification problem started by J. H. C. Whitehead 50 years ago.

In this paper a space is a CW–complex. Let Top^{∗}/ ≃ be the homotopy category of pointed spaces and pointed maps. For pointed spaces X, Y let [X, Y] be the set of homotopy classes of pointed maps X → Y; this is the set of morphism X → Y in the category Top^{∗}/ ≃.

1. The torsion free case

A space X is decomposable if there exists a homotopy equivalence X ≃ A ∨ B where A ∨ B is the one point union of non–contractible spaces A and B; otherwise X is indecomposable.

Definition 1.1. We say that X is a p–local (m, n)–atom if X is a 1–connected indecomposable space for which the homology groups H_{m}(X) and H_{n}(X) are non trivial finitely generated ℤ_{(p)}–modules and H̃_{i}(X) = 0 for i ≠ n, m. We say that X is torsion free if the homology H_{m}(X), H_{n}(X) are free ℤ_{(p)}–modules.

The indecomposable summands of the space X in the introduction are either indecomposable Moore spaces or (m, n)–atoms as in (1.1).

For a prime p let S_{(p)}^{n} be the p–local sphere. An element α ∈ π_{n−1}(S^{m}) ⊗ ℤ_{(p)} yields a map α : S_{(p)}^{n−1} → S_{(p)}^{m} and the mapping cone of this map is denoted by

C_{α} = S_{(p)}^{m} ∪_{α} e_{(p)}^{n}. (1.1)

For n ∈ ℤ with 1/n ∈ ℤ_{(p)} ⊂ ℚ there is a homotopy equivalence C_{α} ≃ C_{nα}.

Theorem 1.2. If π_{n−1}(S^{m}) ⊗ ℤ_{(p)} = ℤ/p^{k} is a cyclic group with generator ξ then the spaces C_{α} with α = ξ, pξ, …, p^{k−1}ξ form a complete list of torsion free p–local (n, m)–atoms.

This result yields the finite case of theorem (A). The next result yields the tame case of theorem (A).

Theorem 1.3. If π_{n−1}(S^{m}) ⊗ ℤ_{(p)} = ℤ/p ⊕ ℤ/p^{k} with generators η ∈ ℤ/p and ξ ∈ ℤ/p^{k} then a complete list of torsion free p–local (n, m)–atoms A(g) and A(g, φ) is given as follows.

We consider a finite connected non empty subword g of the infinite word (i ∈ ℤ)

… ξ_{i}ηξ_{i+1}ηξ_{i+2}η … (1)

where ξ_{i} ∈ {ξ, pξ, …, p^{k−1}ξ}. Hence g corresponds to a connected subgraph of the infinite graph

(2)

in which the vertical edges denote η and the diagonal edges denote elements in {ξ, pξ, …, p^{k−1}ξ}. According to the graph g we attach p–local cells e_{(p)}^{n} to a one point union of p–local spheres S_{(p)}^{n}. Here each vertex of level n in the graph g is a cell and each vertex of level m in g is a sphere and the cell is attached according to the edges of the graph g. More precisely, let B be the set of vertices of level m of g and let T be the set of vertices of level n of g and consider the one pont unions of p–local spheres

S_{T}^{n−1}= ⋁_{e∈T} S_{e}^{n−1} (T =top cells)

S_{B}^{m}= ⋁_{e∈B} S_{e}^{m}  (B=bottom spheres)

with S_{e}^{n−1} = S_{(p)}^{n}, S_{e}^{m} = S_{(p)}^{m}. Then the atom A(g) is the mapping cone of a map

α_{g} : S_{T}^{n−1} → S_{B}^{m} (3)

Here α_{g} is defined by the graph g, in particular, if e ∈ T is the top vertex of ξ_{i}η in g then the coordinate α_{g}^{e} is given by the sum i_{a}ξ_{i} + i_{b}η : S_{e}^{n−1} → S_{a}^{m} ∨ S_{b}^{m} ⊂ S_{B}^{m} where a is the bottom vertex of ξ_{i} and b is the bottom vertex of η and i_{a}, i_{b} are the inclusions. Hence the space A(g) is the p–local form of a “lightning flash space” defined in Baues–Drozd [BD3].

Next we define a cyclic word (g, φ) where g is a word as above of the form g = ξ_{1}η … ξ_{c−1}ηξ_{c} with c ≥ 1 and φ is an automorphism of a finite dimensional ℤ/p–vector space V = V (φ). Two cyclic words (g, φ), (g′, φ′) are equivalent if g′ is a cyclic permutation of g, that is (1 ≤ i ≤ c)

g′ = ξ_{i}ηξ_{i+1}η … ξ_{c}ηξ_{1}η … ξ_{i−1}η,

and there is an isomorphism ψ : V (φ) → V (φ′) with φ = ψ^{−1}φ′ψ. A cyclic word (g, φ) is a special cyclic word if g is not of the form g = g′g′ … g′ where the right hand side is a k–fold power of a word g′ with k > 1 and if φ is an indecomposable automorphism. Here we say that an automorphism of a homomorphism f between vector spaces is decomposable if f is a non–trivial direct sum of homomorphism.

For a special cyclic word (g = ξ_{1}η … ηξ_{c}, φ) we define the p–local (m, n)–atom A(g, φ) by the mapping cone of a map

α_{g,φ} : ⋁^{d} S_{T}^{n−1} → ⋁^{d} S_{B}^{m} (4)

Here d = dim_{ℤ/p}(V (φ)) and ⋁^{d} Y denotes the d–fold connected sum of the space Y with inclusion j_{δ} : S ⊂ ⋁^{d} S for δ = 1, …, d. Let b_{0} ∈ B be the bottom vertex of ξ_{1} and t_{0} ∈ T be the top vertex of ξ_{c}. Then the map α_{g,φ} on j_{δ}S_{e}^{n} with e ∈ T −{t_{0}} is defined as α_{g} above compatible with the inclusion j_{δ}. The map α_{g,φ} restricted to ⋁^{d} S_{t_{0}}^{n−1}, however, is a sum of the map ⋁_{δ=1}^{d} j_{δ}ξ_{c} and of the map

⋁^{d} S_{t_{0}}^{n−1} ⟶^{φ} ⋁^{d} S_{b_{0}}^{m}

given by η and φ. We sketch the map α_{g,φ} for d = 2 and g = ξ_{1}ηξ_{2}ηξ_{3} as follows

Now the atoms A(g, φ) and A(g′, φ′) given by special cyclic words are homotopy equivalent if and only if (g, φ) and (g′, φ′) are equivalent.

2. Proof of theorem (A)

For 1 < m < n and R ⊂ ℚ let

CW(m, n)_{R} ⊂ Top^{∗}/ ≃ (2.1)

be the full subcategory consisting of 1–connected spaces X with at most two non trivial homology groups H_{m}X = A and H_{n}X = B which are finitely generated R– modules. Hence A and B are direct sums of the cyclic R–modules R and ℤ/p^{k}, k ≥ 1, \frac{1}{p} ∉ R. We associate with X an element

k_{X} ∈ [M(B, n − 1), M(A, m)] (2.2)

obtained by the homology decomposition of X. Here M(A, m) is the Moore space of A in degree m and X is homotopy equivalent to the mapping cone of a map representing k_{X}. Now let M(m, n)_{R} be the following category. Objects are triple k_{X} = (A, B, k_{X}) with k_{X} as in (2.2) and morphisms k_{X} → k_{Y} are commutative diagrams

\xymatrix{M(B, n − 1) \ar[r]^{β} \ar[d]_{k_{X}} & M(B′, n − 1) \ar[d]^{k_{Y}} \\ M(A, m) \ar[r]^{α} & M(A′, m)} (2.3)

in Top^{∗}/ ≃. Using the homology decomposition of spaces we get the following result.

Lemma 2.1. For n ≥ m + 3 there is a functor

k : CW(m, n)_{R} → M(m, n)_{R}

which carries X to k_{X} in (2.2). This functor reflects isomorphism, is full and representative.

The lemma shows that k induces a bijection on equivalence classes of objects. For n = m + 2 there is slightly more delicate lemma which we describe in (3.1) below. For n < 2m−2 the categories in (2.1) are additive categories with the direct sum defined by one point union of spaces and maps respectively. The functor k is additive since k_{X∨Y} = k_{X} ∨ k_{Y}.

Next we define for a finitely generated R–module M the following category of matrices with entries in M denoted by Mat_{R}(M). Objects are triple (A, B, k) where A and B are finitely generated free R–modules and k ∈ Hom(B, A ⊗ M). Morphisms are pairs β : B → B′, α : A → A′ for which the diagram

\xymatrix{B \ar[r]^{β} \ar[d]_{k} & B′ \ar[d]^{k′} \\ A ⊗ M \ar[r]^{α⊗1} & A′ ⊗ M} (2.4)

commutes. Let M(m, n)_{R}^{free} be the full subcategory of M(m, n)_{R} consisting of objects (A, B, k_{X}) for which A and B are finitely generated free R–modules. Then the following result is an easy application of standard facts of homotopy theory.

Lemma 2.2. Let M = π_{n−1}(S^{m}) ⊗ R. Then for n < 2m − 2 one has an isomorphism of additive categories

M(m, n)_{R}^{free} ≅ Mat_{R}(M)

Proof. The isomorphism carries k_{X} to the induced morphism

k : B = π_{n−1}M(B, n − 1) ⟶^{(k_{X})_{∗}} π_{n−1}M(A, m) = A ⊗ M

and carries the morphism (α, β) in (2.3) to (π_{n−1}(β), π_{m}(α)). q.e.d.

Using (2.2), (2.1) we see that theorem (A) and (1.2) and (1.3) are consequences of the representation theory in the category Mat_{R}(M). Proof of (1.3). Let R = ℤ_{(p)} be given by a prime p. Let M = ℤ/p ⊕ ℤ/p^{k} be generated by elements η ∈ ℤ/p and ξ ∈ ℤ/p^{k}. Then an object k in Mat_{R}(M). is given by

k = Kη + Lξ

where K, L are a × b–matrices with entries in ℤ/p and ℤ/p^{k} respectively. We may suppose that L is of block–diagonal form

L = (\begin{matrix}I_{1} & & & 0 \\ & pI_{2} \\ & & p^{2}I_{3} \\ 0 & & & ... \end{matrix})

where the identity matrices I_{1}, I_{2}, I_{3}, … may have different sizes. Then for K one gets the known matrix problem

where the squares denote matrices which are transformed by conjugation and where the arrows indicate that we may add rows (columns) of the lower (right) stripes to those of the upper (left) ones. The answer of this problem is known by Bondarenko [B] or Drozd [D] and this yields the indecomposable objects described in (1.3). q.e.d.

We now recall some notation from representation theory. The wild quiver W consists of one vertex v and two arrows a, b : v → v. Let Vec be the category of finite dimensional vector spaces over a field k and let Wild be the category of representations of W in Vec (i. e. objects are functors A : W → Vec and morphisms A → B are natural transformations). The direct sum of vector spaces yields a direct sum A ⊕ B in Wild. The universal problem of representation theory is the computation of all objects in Wild which are indecomposable with respect to direct sum. We call a classification problem wild if it requires the solution of the universal problem of representation theory.

Proposition 2.3. Let R = ℤ_{(p)} and let M be a finitely generated ℤ_{(p)}–module which is not cyclic and not of the form ℤ/p ⊕ ℤ/p^{l}. Then the classification of indecomposable objects in Mat_{R}(M) is a wild problem.

This result proves by (2.2) and (2.1) the wild case of theorem (A). Proof of (2.3). The proposition is well known if M has 3 cyclic summands. Hence one only has to prove wildness for M = ℤ/p^{2} ⊕ ℤ/p^{2}. Then an object in Mat(M) is a pair of a×b–matrices (K, L) over ℤ/p^{2}. We now consider special matrices K, L of the following shape. We choose r × r–matrices X, Y over ℤ/p and we choose a = b = 7r. Then let K = I_{a} be the identity matrix and let L = L(X, Y) be the matrix

L = [\begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{r} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & I_{r} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & I_{r} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & I_{r} \\ pI_{r} & 0 & 0 & pX & pY & 0 & 0 \\ 0 & 0 & 0 & pI_{r} & 0 & 0 & 0 \end{matrix}]

Then a simple though tedious calculation shows that (K = I_{a}, L) is isomorphic in Mat_{R}(M) to (K = I_{a}, L′) with L′ = L(X′, Y′) if and only if there is an invertible r × r–matrix R with RX = X′R and RY = Y′R. This shows that the classification of indecomposable objects in Mat_{R}(M) for M = ℤ/p^{2} ⊕ ℤ/p^{2} is wild. q.e.d.

3. Proof of theorem (B)

We consider the following diagram of functors (m > 4)

\xymatrix{CW(m, m + 2)_{ℤ} \ar[dr]_{k} & & M(m, m + 2)_{ℤ} \ar[dl]^{q} \\ & M(m, m + 2)_{ℤ}/ ∼} (3.1)

Here CW(m, m+2)_{ℤ} and M(m, m+2)_{ℤ} are defined as in (2.2) and (2.3) for R = ℤ. The functor k is defined as in (2.1) though this functor maps only to a quotient category of M(m, m + 2). The functor q is the quotient functor for the following natural quivalence relation ∼ in M(m, m + 2)_{ℤ}. Consider the diagram

\xymatrix{M(B, m + 1) \ar[r]^{β,β_{1}} \ar[d]_{k_{X}} & M(B′, m + 1) \ar[d]^{k_{Y}} \\ M(A, m) \ar[ur]_{ξ} \ar[r]_{α,α_{1}} & M(A′, m)}

where (α, β), (α_{1}, β_{1}) are morphisms k_{X} → k_{Y} in M(m, m + 2)_{ℤ}. Then we set (α, β) ∼ (α_{1}, β_{1}) if there is

ξ ∈ [M(A, m), M(B′, m + 1)] = Ext(A, B′)

with β_{1} = β and α_{1} = α + k_{Y} ξ. Since m > 4 we know that all categories and functors in (3.1) are additive.

Proposition 3.1. The functor k in (3.1) is well defined. Moreover k reflects isomorphisms, is full and representative.

This is a special case of the classification result 6.8.2 in Baues [B1]. Since also the quotient functor q reflects isomorphisms, is full and representative we see that the indecomposable objects in CW(m, m + 2) coincide with the mapping cones of indecomposable objects in M(m, m + 2)_{ℤ}.

In order to obtain an algebraic description of the category M(m, m + 2)_{ℤ} we recall first some general notation on bimodule problems. Let A, B be two categories (fully additive, i. e., such additive categories where every idempotent splits), U be an A–B–bimodule, i. e., a functor A^{op} × B → Ab. Define the category El(U) of elements of this bimodule as follows:

(3.2) ObEl(U) = ∐_{{A∈ObA \atop B∈ObB}} U(A, B)

A morphism u → v, where u ∈ U(A, B), v ∈ U(A′, B′), is a pair (α, β), where α : A′ → A, β : B → B′, such that βu = vα (Here we write vα for U(α, 1)v and βu for U(1, β)u).

We have the following example of a category of elements in a bimodule. Let M^{m} be the full subcategory of Top^{∗}/ ≃ consisting of Moore spaces M(A, m) of finitely generated abelian groups. Then the suspension Σ : M^{m} → M^{m+1} is an equivalence of categories for m ≥ 3 and M^{m} is an additive category for m ≥ 3. We have a bimodule

U : (M^{m+1})^{op} × M^{m} → Ab (3.3)

defined by the abelian group [M(A, m + 1), M(H, m)]. Now the category El(U) coincides with M(m, m + 2)_{ℤ}.

We describe M^{m} and the bifunctor U algebraically as follows. For an abelian group A, consider the short exact sequence

A ⊗ ℤ/2 ↣^{η^{A}} G(A) ↠ A ∗ ℤ/2 (1)

coresponding to the composition

A ∗ ℤ/2 → A → A ⊗ ℤ/2.

Applying Hom( , ℤ/4), one gets the exact sequence

(2)  Hom(A ∗ ℤ/2, ℤ/4) ; {≃ \atop Ext(A, H ⊗ ℤ/2)} ↣ Hom(G(A), ℤ/4) ↠ Hom(A ⊗ ℤ/2, ℤ/4) ; {≃ \atop Hom(A, ℤ/2)}

Now, applying ⊗ H, one gets

(3) Ext(A, ℤ/2) ⊗ H ; {≃ \atop Ext(A, H ⊗ ℤ/2)} →^{δ} Hom(G(A), ℤ/4) ⊗ H → Hom(A, ℤ/2) ⊗ H ; {≃ \atop Hom(A, H ⊗ ℤ/2)} → 0

(all groups are supposed to be finitely generated).

Now define G(A, H) by the push–down diagram:

[figure: Ext(A, H ⊗ ℤ/2); Δ; η_{∗}^{H}; Hom(G(A), ℤ/4) ⊗ H; μ; Hom(A, H ⊗ ℤ/2); 0; Ext(A, G(H)); Δ; G(A, H); μ; Hom(A, H ⊗ ℤ/2); 0] (4)

Define the category ℑ where objects are abelian groups (finitely generated) and morphisms are pairs (φ, φ̃), φ : A → B, φ̃ : G(A) → G(B) such that the following diagram commutes:

\xymatrix{A ⊗ ℤ/2 \ar[d]_{φ⊗1} \ar[r] & G(A) \ar[d]_{φ̃} \ar[r] & A ∗ ℤ/2 \ar[d]^{φ∗1} \\ B ⊗ ℤ/2 \ar[r] & G(B) \ar[r] & B ∗ ℤ/2} (5)

Then G(A, H) can be consider as ℑ–ℑ–bimodule, the (right) action of (φ, φ̃) : A → A′ induced by

(φ, φ̃)^{∗} = Ext(φ, G(H)) ⊕ Hom(φ̃, ℤ/4) ⊗ H

and the (left) action of (ψ, ψ̃) : H → H′ induced by

(ψ, ψ̃)_{∗} = Ext(A, ψ̃) ⊕ Hom(G(A), ℤ/4) ⊗ ψ.

Hence, one can consider the category El(G) of the elements of this bimodule.

Proposition 3.2. For m ≥ 3 there is an isomorphism of categories ℑ ≅ M^{m} and one has a natural isomorphism

[M(B, m + 1), M(H, m)] ≅ G(B, H)

Hence the category M(m, m + 2) is isomorphic to the category El(G).

This result is proved in (8.2.10) Baues [B1]. In order to obtain the indecomposable objects in CW(m, m + 2)_{ℤ} we thus have to classify the indecomposable objects in the category El(G).

Note that if A = ⨁_{j} A_{j} and H = ⨁_{i} H_{i}, the elements of G(A, H) can be naturally written as the matrices (g_{ij}) with g_{ij} ∈ G(A_{j}, H_{i}). Hence, we only have to calculate G(A, H) with indecomposable A, H and the action of the morphisms A → A′ and H → H′ for indecomposable A, A′; H, H′. Thus, we are interested in the cases A = ℤ/2^{a}, H = ℤ/2^{h} (putting ℤ/2^{∞} = ℤ). First note that, if A = ℤ^{a} with a > 1, the sequence (1) splits and A ⊗ ℤ/2 = A ∗ ℤ/2 = ℤ/2. Moreover, given φ : A′ → A, the homomorphisms φ̃: G(A′) → G(A) such that (φ, φ̃) is a morphism from ℑ, are given by the homomorphisms φ′ : A ∗ ℤ/2 → A ⊗ ℤ/2; namely, w. r. t. the decomposition G(A) = A ⊗ ℤ/2 ⊕ A ∗ ℤ/2, φ̃ is given by the matrix ( \begin{matrix}φ⊗1 & φ′ \\ 0 & φ∗1 \end{matrix} ). Certainly, in this case (2) and (3) are also split sequences, hence, the lower row of the push–down (4) also splits, i. e.

G(A, H) = Ext(A, G(H)) ⊕ Hom(A, H ⊗ ℤ/2).

If h > 1, one also has that Ext(A, G(H)) = Ext(A, H ∗ ℤ/2) ⊕ Ext(A, H ⊗ ℤ/2), so G(A, H) = Ext(A, H ∗ℤ/2)⊕Ext(A, H ⊗ℤ/2)⊕Hom(A, H ⊗ℤ/2) and all three direct summands here are ℤ/2. We denote them, respectively, by G(∗^{a},_{h} ∗), G(∗^{a}, ⊗_{h}) and G(^{a}⊗, ⊗_{h}). One easily sees that a homomorphism (φ, φ̃), where φ̃ = ( \begin{matrix}φ⊗1 & φ′ \\ 0 & φ∗1 \end{matrix} ) maps a triple (g_{1}, g_{2}, g_{3}) from this direct sum to (φ^{∗}g_{1}, φ^{∗}g_{2} + φ̂′g_{3}, φ^{∗}g_{3}), where φ̂′ is the composition

Hom(A, H ⊗ ℤ/2) ≃ Hom(A ⊗ ℤ/2, H ⊗ ℤ/2) ⟶^{(φ′)^{∗}} Hom(A′ ∗ ℤ/2, H ⊗ ℤ/2) ≃
≃ Ext(A′ ∗ ℤ/2, H ⊗ ℤ/2) ≃ Ext(A′, H ⊗ ℤ/2).

If (ψ, ψ′) : H′ → H, then it maps (g_{1}, g_{2}, g_{3}) to (ψ_{∗}g_{1}, ψ_{∗}g_{2} + (ψ′)_{∗}g_{1}, ψ_{∗}g_{3}) ∈ G(A, H′). If h = 1, the sequence (1) for H is indeed

ℤ/2 ↣ ℤ/4 ↠ ℤ/2,

hence, for a > 1, one still has the exact sequence

Ext(A, H ⊗ ℤ/2) ↣ Ext(A, G(H)) ↠ Ext(A, H ∗ ℤ/2);

which is indeed

ℤ/2 ↣ ℤ/4 ↠ ℤ/2.

Again a homomorphism (φ, φ̃) : A′ → A maps a pair (g_{1}, g_{3}) with g_{1} ∈ Ext(A, G(H)), g_{2} ∈ G(^{a}⊗, ⊗_{1}) = Hom(A, H ⊗ ℤ/2) to (φ^{∗}g_{1} + φ̂′g_{3}, φ^{∗}g_{3}). On the other hand, to define a homomorphism (ψ, ψ̃) : H → H′, where H = ℤ/2, one only has to define ψ̃ : ℤ/4 → H′: then ψ = ψ̃ ⊗ ℤ/2 : ℤ/2 → H′. Such a homomorphism maps a pair (g_{1}, g_{3}) as above to (ψ̃_{∗}g_{1}, ψ_{∗}g_{3}).

Suppose now that a = 1. Then (1) is indeed

ℤ/2 ↣ ℤ/4 ↠ ℤ/2.

If h > 1, the tensor multiplication by H does not change this exact sequence; η_{∗}^{H} in (4) is split monomorphism, hence, G(A, H) ≃ ℤ/4 ⊕ Ext(A, H ∗ ℤ/2), and we have an exact sequence

ℤ/2 = G(∗^{1}, ⊗_{h}) → ℤ/4 → G(^{1}⊗, ⊗_{h}) = ℤ/2.

A morphism (φ, φ̃) is given by any homomorphism φ̃ : G(A′) → ℤ/4: then φ = φ̃|_{A′⊗ℤ/2} ∈ Hom(A′ ⊗ ℤ/2, ℤ/2) ≃ Hom(A′, ℤ/2). It maps a pair (g_{1}, g_{2}) ∈ Hom(G(A), ℤ/4) ⊗ H ⊕ Ext(A, H ∗ ℤ/2) to ((φ^{∗} ⊗ 1)g_{1}, φ^{∗}g_{2}). A morphism (ψ, ψ̃) :

H → H′, where ψ̃ = ( \begin{matrix}ψ⊗1 & ψ′ \\ 0 & ψ∗1 \end{matrix} ), maps (g_{1}, g_{2}) to ((1 ⊗ ψ)_{∗}g_{1} + ψ′_{∗}g_{2}, (ψ ∗ 1)_{∗}g_{2}).

At last, if a = 1 = h, the sequence (3) is

ℤ/2 →^{0} ℤ/2 →^{1} ℤ/2 → 0 and η_{∗}^{H} = 0

Hence, G(A, H) = Ext(A, G(H)) ⊕ Hom(A, H ⊗ ℤ/2), both summand being ℤ/2 (we denote them by G(_{1}∗, ∗^{1}) and G(⊗_{1},^{1} ⊗) respectively, putting G(_{1}∗,^{1} ⊗) = 0).

Denote by G̃ the sub–bimodule of G such that, for A = ℤ/2^{a}, H = ℤ/2^{h}, G̃(A, H) = G(_{h}∗,^{a} ⊗). In the definition of the weak equivalence (6), one always has Δ(Imφ^{∗}η_{∗}^{H}, μ(β)_{∗}) ⊆ G̃(A, H). We will use this remark later to show that indeed the weak equivalence coincide with the isomorphism in El(G).

Now we are going to reformulate our bimodule problem in the matrix form. This bimodule problem can obviously be rewritten in the matrix form as follws.

We consider the “striped” matrices M with the horizontal stripes marked by the set E = {⊗_{h},_{h} ∗|h ∈ ℕ} ∪ {⊗_{∞}} and the vertical stripes marked by the set F = {^{a}⊗, ∗^{a}|a ∈ ℕ} ∪ {^{∞}⊗}. We denote by M(x, y) the block placed at the intersection of the horizontal stripe x and the vertical stripe y. These matrices should satisfy the following conditions:

(1) The number of rows in the stripes ⊗_{h} and _{h}∗ is the same, as well as the number of columns in the stripes ^{a}⊗ and ∗_{a}.

(2) M(_{h}∗,^{a} ⊗) = 0; M(⊗_{1}, ∗^{a}) = 0; M(⊗_{h}, ∗^{1}) = 0. (We consider them as “matrices over the zero ring ℤ/1”).

(3) The matrices M(⊗_{h},^{1} ⊗) with h > 1 and M(_{1}∗, ∗^{a}) with a > 1 are with the entries from ℤ/4, all other ones are with the entries from ℤ/2.

Here is the picture describing such matrices (we always indicate the ring, where the entries of the blocks are from):

1⊗ ^{2}⊗ ^{3}⊗ … ^{∞}⊗  … ∗^{3} ∗^{2} ∗^{1}
⊗_{1} ℤ/2 ℤ/2 ℤ/2 … ℤ/2 … 0 0 0
⊗_{2} ℤ/4 ℤ/2 ℤ/2 … ℤ/2  … ℤ/2 ℤ/2 0
⊗_{1} ℤ/4 ℤ/2 ℤ/2 … ℤ/2  … ℤ/2 ℤ/2 0
⋮ . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .
⊗_{∞} ℤ/4 ℤ/2 ℤ/2 … ℤ/2  … ℤ/2 ℤ/2 0
⋮ . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .
3∗ 0  0  0 … 0  … ℤ/2 ℤ/2 ℤ/2
2∗ 0  0  0 … 0  … ℤ/2 ℤ/2 ℤ/2
1∗ 0  0  0 … 0  … ℤ/4 ℤ/4 ℤ/2

The following transformations of the matrix M are called “admissible transformations” (in what follows we denote by M(x, ) and by M( , y), respectively, the horizontal stripe marked by x and the vertical stripe marked by y).

(a) Replacing the stripes M(⊗_{h}, ) and M(_{h}∗, ) by XM(⊗_{h}) and XM(_{h}∗). (a’) Replacing the stripes M( ,^{a} ⊗) and M( , ∗^{a}) by M( ,^{a} ⊗)X and M( , ∗^{a})X.

(b) Replacing M(⊗_{h}, ) by M(⊗_{h}, )+XM(⊗_{h′}, )+Y M(_{k}∗, ), where h′ > h, k any.

(b’) Replacing M( , ∗^{a}) by M( , ∗^{a}) + M( , ∗^{a′})X + M( ,^{b} ⊗), where a′ > a, b

any.

(c) Replacing M(_{h}∗, ) by M(_{h}∗, ) + XM(_{h′}∗, ) and M(⊗_{h}, )Y by M(⊗_{h}, ) + 2^{h−h′}XM(⊗_{h′}, ), where h′ < h.

(c’) Replacing M( ,^{a} ⊗) by M( ,^{a} ⊗) + M( ,^{a′} ⊗)X and M( , ∗^{a}) by M( , ∗^{a}) +

2^{a−a′}M( , ∗^{a′})X, where a′ < a.

(d) Replacing M(_{1}∗, ) by M_{1}(∗, ) + 2XM(_{h}∗, ) + 2Y M(⊗_{k}, ) for arbitrary h, k.

(d’) Replacing M( ,^{1} ⊗) by M( ,^{1} ⊗) + 2M( ,^{a} ⊗)X + 2M( , ∗^{b})Y for arbitrary

a, b.

(e) Replacing M(⊗_{h},^{1} ⊗), h > 1, by M(⊗_{h},^{1} ⊗) + 2XM(_{h′}∗, ∗^{1}), h′ any.

(e’) Replacing M(_{1}∗, ∗^{a}), a > 1, by M(_{1}∗, ∗^{a}) + 2M(⊗_{1},^{a′} ⊗)X, a′ any.

Here X, Y always denote arbitrary matrices of the appropriate size with the entries from ℤ/4; in the transformations of type (a) and (a’) the matrix X should be invertible. Certainly, if an original block was with the entries from ℤ/2, the resulting one should also be calculated modulo 2; if an original block was over zero ring, so is the resulting one.

Two matrices, M, M′ (of the same size) are called equivalent if they can be transformed to each other by a sequence of admissible transformations. Call M, M′ equivalent modulo 2 if M can be transformed by a sequence of admissible transformations to a matrix M′′ such that M′′ ≡ M′ mod 2. Of course, considering the equivalence modulo 2, we may reduce the stripes M( ,^{1} ⊗) and M(_{1}∗, ) modulo 2 (thus forgetting the transformations of types (d), (d’)), as well as always suppose X, Y being over ℤ/2.

One can easily see that for the equivalence modulo 2 we get a sort of representations of a bunch of chains in the sense of [D]. Therefore, one can write down all indecomposable matrices. Namely, we have the chain E for rows, with the order:

⊗_{1} > ⊗_{2} > … > ⊗_{∞} > … > _{3}∗ > _{2}∗ > _{1}∗,

and the chain F for columns, with the order:

1⊗ < ^{2}⊗ < … < ^{∞}⊗ < … < ∗^{3} < ∗^{2} < ∗^{1}.

The equivalence relation ∼ is given by

⊗_{h} ∼_{h} ∗(h ∈ ℕ);^{a}⊗ ∼ ∗^{a}(a ∈ ℕ).

Remind the corresponding combinatorics. Put X = E ∪ F and write x − y if x ∈ E, y ∈ F or vice versa. Then X –word is a sequence w = x_{1}r_{2}x_{2}r_{3} … r_{n}x_{n}, where x_{i} ∈ X , r_{i} ∈ {∼, −}, such that r_{i} ≠ r_{i+1}(i = 2 …, n − 1), x_{i}r_{i+1}x_{i+1} in X (i = 1, …, n − 1). Such a word is said to be full if: either r_{2} =∼ or x_{1} ≁ y for any y ≠ x_{1}; either r_{n} =∼ or x_{n} ≁ y for any y ≠ x_{n}. w is called cyclic if r_{2} = r_{n} = − and x_{n} ∼ x_{1}; it is called aperiodic if it is not of the type v ∼ v ∼ … ∼ v for a shorter word v.

Call a polynomial π(t) ∈ ℤ/2[t] primitive if it is a power of an irreducible polynomial (with the leading coefficient 1). Then the indecomposable representations of this bunch of chains are in 1–1 correspondence with the set S ∪ B, where S consists of all full words and B consists of the pairs (w, π(t)), where w is an aperiodic cyclic word and π(t) ≠ t^{d} is a primitive polynomial. More precisely, we should identify any word w with its inverse and any cycle with its cyclic shift!

We call the representations corresponding to S “strings” and those corresponding to B “bands”.

The condition (2) from the definition of the matrix M imposes some restrictions on the representations, which can occur as equivalence classes modulo 2 for such matrices. In terms of the X –words, they mean that the corresponding word contains no subword ^{a} ⊗−_{h}∗, ⊗_{1} −∗^{a}, ∗^{1} −⊗_{h} or their inverse. (Call such words admissible.) One can easily check that such a word can only contain at most one subword of the form ⊗_{h} − ∗^{a} (or its inverse). To simplify the notations, we replace the subword a ⊗ −⊗_{h} by ^{a}⊗_{h}, ⊗_{h} − ^{a}⊗ by _{h}⊗^{a}; _{h} ∗ −∗^{a} by _{h}∗^{a}, ∗^{a} − _{h}∗ by ^{a}∗_{h}; ⊗_{h} − ∗^{a} by hθ^{a}. We also omit all ∼ signs and alway replace a superscript ^{aa} by ^{a} as well as a subscript _{hh} by _{h}. At last, we omit x_{1} if r_{2} =∼ and x_{n} if r_{n} =∼. Certainly, one can always restore the original X –word having the result of all these simplifications.

Now, any admissible word (or its inverse) can be written as a subword of

(i) ^{a_{1}} ⊗_{h_{1}} ∗^{a_{2}} ⊗_{h_{2}} ∗ …^{a_{n}} ⊗_{h_{n}} (“usual word”)

or

h_{−m} ⊗ …^{a_{−2}} ∗_{h_{−2}} ⊗^{a_{−1}} ∗_{h_{−1}} θ^{a_{1}} ⊗_{h_{1}} ∗^{a_{2}} ⊗_{h_{2}} … ∗^{a_{n}} (“θ–word”) (ii)

The strings correponding to θ–words are called “θ–strings”, all other ones are called “usual string”. Any admissible cyclic word (or its shift) can be written in the form

a_{1} ⊗_{h_{1}} ∗^{a_{2}} ⊗_{h_{2}} ∗ …^{a_{n}} ⊗_{h_{n}} ∗^{a_{1}}

Moreover, the following conditions hold:

(1) a_{i} = ∞ or h_{j} = ∞ can only occur at the end of a word as ^{∞} ⊗ (⊗^{∞}) or ⊗_{∞}(_{∞}⊗);

(2) In any θ–word, h_{−1} ≠ 1 and a_{1} ∉ {1, ∞}. Note that the description of the representations of a bunch of chains [D] implies:

• Any row (column) of a string contains at most 1 non–zero element.

• There are at most 2 zero rows or columns in a string, namely, they are in

the following stripes:

– M(_{h}∗, ) if w has an end ⊗_{h} (or _{h}⊗), h ≠ ∞;

– M( , ∗^{a}) if w has an end ^{a}⊗, a ≠ ∞;

– M(⊗_{h}, ) if w has an end _{h}∗;

– M( ,^{a} ⊗) if w has an end ∗^{a} (or ^{a}∗).

• The horizontal and vertical stripes of a band can be subdivided in such a way that every new horizontal or vertical sub–stripe contains exactly 1 non–zero block, which is invertible.

Corollary 3.3. Let \overline{M} be the reduction modulo 2 of an admissible matrix M, \overline{N} be its indecomposable direct summand. If \overline{N} is a band or a θ–string, then M ≃ N ⊕ M′ such that \overline{N} ≡ N mod 2.

So, from now on, we may only consider such M that \overline{M} is a direct sum of usual strings. Denote M̃ the matrix obtained from M by replacing all invertible entries by zeroes (in particular, all its elements are of the form 2c). Let \overline{M} = ⨁_{i=1}^{r} \overline{M_{i}}, where \overline{M_{i}} is the usual string corresponding to a word w_{i}.

Corollary 3.4. M ≃ M′, where \overline{M′} = \overline{M} and the only non–zero rows and columns of M̃′ may be the following:

(1) the columns of M̃′( ,^{1} ⊗) corresponding to the ends ^{1}⊗ occurring in the word w_{i};

(2) the rows of M̃′(_{1}∗, ) corresponding to the ends _{1}∗ occurring in the words w_{i};

(3) the rows of M̃′( ,^{1} ⊗) corresponding to the ends _{h} ∗ (h > 1);

(4) the columns of M̃′(_{1}∗, ) corresponding to the ends ^{a} ⊗ (1 < a < ∞).

We call all these ends the “distinguished ends”.

In what follows, we always suppose that already M = M′.

To prove Corollaries 3.3, 3.4, one only has to use the transformations (d), (d’) and (e), (e’).

The following Lemma is now decisive.

Lemma 3.5. Suppose that two words w_{i}, w_{j} have a common distinguished end. There is a sequence of admissible transformations of M = M′ which does not change the matrix \overline{M} and the resulting transformation of M̃ adds the row (column) corresponding to the end of w_{i} to that corresponding to the end of w_{j} or vice versa.

Corollary 3.6. There is a sequence of admissible transformations of M = M′, which does not change \overline{M} and transform M̃ to a matrix having at most one non–zero element in every row and every column.

Proof of Lemma. We consider the case of the end ^{1}⊗, all other cases being quite analogous. So let w_{i} =^{1} ⊗_{h_{1}} ∗^{a_{2}} ⊗_{h_{2}} …, while w_{j} =^{1} ⊗_{k_{1}} ∗^{b_{2}} ⊗_{k_{2}} . . .. Suppose that h_{1} ≤ k_{1}. If indeed h_{1} < k_{1}, one can add the column corresponding to the end 1⊗_{h_{1}} of w_{i} to that corresponding to the end ^{1}⊗_{k_{1}} of w_{j} and then subtract the row corresponding to the latter one from that corresponding to the first one to restore \overline{M}. If h_{1} = k_{1}, compare a_{2} and b_{2}. If a_{2} < b_{2}, one can perform the same transformations as above and afterwards subtract the column corresponding to _{h_{1}}∗^{b_{2}} from that corresponding to _{h_{1}}∗^{a_{2}}, to restore \overline{M}. Continuing these considerations, one sees that we can add the column corresponding to the end of w_{i} to that corresponding to the end of w_{j} if (h_{1}, a_{2}, h_{2}, …) < (k_{1}, b_{2}, k_{2}, …) lexicographically. As the lexicographical order is linear, it proves the lemma.

Corollary 3.7. Suppose M an indecomposable matrix such that M̃′ ≠ 0 for any matrix M′ equivalent to M. Let \overline{M} = ⨁_{i=1}^{r} \overline{M_{i}}, where \overline{M_{i}} are usual strings. Then, up to equivalence, there are but the following possibilities:

(1) r = 1, \overline{M} = \overline{M_{1}} corresponds to a word w_{1} =^{1} ⊗_{h_{1}} ∗^{a_{2}} ⊗_{h_{2}} …, M̃ has one non–zero element in the matrix M(⊗_{h},^{1} ⊗). We denote this case by the word w =_{h} θ^{1} ⊗_{h_{1}} ∗^{a_{2}} ⊗_{h_{2}} …

(2) r = 1, \overline{M} = \overline{M_{1}} corresponds to a word _{1} ∗^{a_{1}} ⊗_{h_{1}} ∗^{a_{2}} …; M̃ has one non– zero element in the matrix M(_{1}∗, ∗^{a}). We denote this case by the word w = …^{a_{2}} ∗_{h_{1}} ⊗^{a_{1}} ∗_{1} θ^{a}.

(3) r = 1, \overline{M} = \overline{M_{1}} corresponds to a word ^{a_{1}} ⊗_{h_{1}} ∗^{a_{2}} …; M̃ has one non– zero element in the matrix M(_{1}∗, ∗^{a_{1}}). We denote this case by the word w =_{1} θ^{a_{1}} ⊗_{h_{1}} ∗^{a_{2}} …

(4) r = 1, \overline{M} = \overline{M_{1}} corresponds to a word _{h_{1}} ∗^{a_{1}} ⊗_{h_{2}} ∗ …; M̃ has one non– zero element in the matrix M(⊗_{h_{1}},^{1} ⊗). We denote this case by the word w = … ∗_{h_{2}} ⊗^{a_{1}} ∗_{h_{1}} θ^{1}.

(5) r = 2, \overline{M_{1}} corresponds to a word ^{1} ⊗_{h_{1}} ∗^{a_{1}} ⊗_{h_{2}} …, \overline{M_{2}} corresponds to a word _{h_{−1}} ∗^{a_{−1}} ⊗_{h_{−2}} ∗ …; the unique non–zero element of M̃ is in the matrix M(⊗_{h_{−1}},^{1} ⊗). We denote this case by the word w = …∗_{h_{−1}} ⊗^{a_{−1}} ∗_{h_{−1}} θ^{1} ⊗_{h_{1}} ∗^{a_{1}} ⊗_{h_{2}} …

(6) r = 2, \overline{M_{1}} corresponds to a word ^{a_{1}} ⊗_{h_{1}} ∗^{a_{2}} ⊗_{h_{2}} …; \overline{M_{2}} corresponds to a word _{1} ∗^{a_{−1}} ⊗_{h_{−1}} ∗^{a_{−2}} …; the unique non–zero element of M̃ is in the matrix M(_{1}∗, ∗^{a_{1}}). We denote this case by the word w = …^{a_{−2}} ⊗_{h_{−1}} ⊗^{a_{−1}} ∗_{1} θ^{a_{1}} ⊗_{h_{1}} ∗^{a_{2}} ⊗_{h_{2}} …

Obviously, the cases (1)–(4) always give indecomposable representations. On the other hand, in the case (5), if h_{1} ≥ h_{−1}, the representation M evidently decomposes as M_{1} ⊕ M_{2}; the same is with the case (6) and a_{−1} ≥ a_{1}. If h_{1} < h_{−1} − 1 in the case (5) or a_{−1} < a_{1} −1 in the case (6), the corresponding representation is evidently indecomposable. Suppose, in the case (5), h_{1} = h_{−1} −1. Then we can delete the non–zero element of M̃(⊗_{h_{−1}},^{1} ⊗) using the transformation (c), but it changes the zero entry in M(_{h_{−1}}∗, ∗^{a_{1}}). To restore it, we need that a_{−1} ≥ a_{1}, moreover, if a_{−1} = a_{1}, we also need that h_{2} ≥ h_{−2}, etc. Thus, such a representation remains indecomposable if and only if (h_{1}+1, a_{−1}, h_{2}, a_{−2}, …) < (h_{−1}, a_{1}, h_{−1}, a_{2}, …) lexicographically. Just in the same way, the representation of type (6) is indecomposable if and only if (a_{−1} + 1, h_{1}, a_{−2}, h_{2}, …) < (a_{1}, h_{−1}, a_{2}, h_{−2}, …) lexicographically.

Therefore, the complete list of indecomposable matrices looks like follows.

Theorem 3.8. Indecomposable objects in El(G) are in 1–1 correspondence with the following types of data:

(1) usual words, i. e. subword of the words ^{a_{1}} ⊗_{h_{1}} ∗^{a_{2}} ⊗_{h_{2}} ∗ …^{a_{n}} ⊗_{h_{n}}, where a_{i}, h_{j} ∈ ℕ ∪ {∞}, ∞ being only possible for a_{1} or h_{n}.

(2) θ–words, i. e. subwords containing θ of the words _{h_{−m}} ⊗…^{a_{−2}} ∗_{h_{−2}} ⊗^{a_{−1}} ∗_{h_{−1}} θ^{a_{1}} ⊗_{h_{1}} ∗^{a_{2}} ⊗_{h_{2}} … ∗^{a_{n}} ⊗_{h_{n}}, where a_{i}, h_{j} ∈ ℕ ∪ {∞}, ∞ being only possible for h_{−m} or h_{n}, (a_{1}, h_{−1}) ≠ (1, 1) and, if h_{−1} = 1, then (a_{−1}+1, h_{1}, a_{−2}, h_{2}, …) < (a_{1}, h_{−2}, a_{2}, h_{−3}, …) lexicographically, and, if a_{1} = 1, then (h_{1}+1, a_{−2}, h_{2}, a_{−3}, …) < (h_{−1}, a_{2}, h_{−2}, a_{3}, …) lexicographically. Moreover the θ–word does not coincide with _{∞}θ^{1}⊗_{∞}.

(3) Pairs (w, π(t)), where w is an aperiodic cyclic word: w =^{a_{1}} ⊗_{h_{1}}∗^{a_{2}}⊗_{h_{2}} …^{a_{n}}⊗_{h_{n}} ∗^{a_{1}} (up to shift), and π(t) ≠ t^{d} is a primitive polynomial over ℤ/2.

Corollary 3.9. Let α be an indecomposable element of the bimodule G. Denote by α̃ its component belonging to the sub–bimodule G̃ and by G̃_{α} the sub– bimodule of G̃ generated by α̃ Then G̃_{α} ∩ Imη_{∗}μ(α)_{∗} = 0.

Proof. Follows immediately from the description above.

Here are examples of indecomposable representations.

(1) Usual string corresponding to the word _{2} ∗^{3} ⊗_{2} ∗^{4} ⊗_{1} ∗^{3} ⊗_{4}

\begin{aligned}& ^{3}⊗ & ^{4}⊗ & ∗^{4} & & & ∗^{3} \\ ⊗_{1} & 0 & 0 & 1 \\ 0 & 0 & 0 \\ ⊗_{2} & 1 & 0 & 0 & 0 \\ ⊗_{4} & 0 & 1 & 0 \\ & & _{4}∗ & 0 & & 0 & & 0 \\ & & & 0 & & 1 & & 0 \\ & & _{2}∗ & 1 & & 0 & & 0 \\ & & _{1}∗ & 0 & & 0 & & 1 \end{aligned}  \begin{aligned}A & = 2 · ℤ/8 ⊕ ℤ/16 \\ H & = ℤ/2 ⊕ 2 · ℤ/4 ⊕ ℤ/16 \end{aligned}

(2) θ–string corresponding to the word _{∞} ⊗^{4} ∗_{3} ⊗^{2} ∗_{2}θ^{4} ⊗_{3} ∗^{3}

2⊗  ^{3}⊗  ^{4}⊗  ∗^{4}  ∗^{3}  ∗^{2}
⊗_{2} 0  0  0 0  0 1  0  0
1  0  0 0  0 0  0  0
⊗_{3} 0  0  0 1  0 0  0  0
⊗_{∞} 0  0  1 0  0 0  0  0
1 0  0  0
3∗  0 0  1  0
2∗  0 0  0  1

\begin{aligned}A & = ℤ/4 ⊕ ℤ/8 ⊕ 2 · ℤ/16 \\ H & = ℤ/4 ⊕ 2 · ℤ/8 ⊕ ℤ \end{aligned}

(3) θ–string corresponding to the word ^{2} ∗_{4} ⊗^{1} ∗_{3} θ^{1} ⊗_{2} ∗^{3}⊗_{1}

1⊗  ^{2}⊗  ^{3}⊗  ∗^{3}  ∗^{2}  ∗^{1}
⊗_{1} 0 0  0  1
⊗_{2} 0 1  0  0
⊗_{3} 0 2  0  0
⊗_{4} 1 0  0  0  0
1∗  0  0  0 0
2∗  1  0  0 0
3∗  0  0  1 0
4∗  0  1  0 0

\begin{aligned}A & = 2 · ℤ/2 ⊕ ℤ/4 ⊕ ℤ/8 \\ H & = ℤ/2 ⊕ ℤ/4 ⊕ ℤ/8 ⊕ ℤ/16 \end{aligned}

(4) band corresponding to the word ^{2} ⊗_{4} ∗^{3} ⊗_{1} ∗^{3} ⊗_{2} ∗^{2} and the polynomial π(t) = t^{2} + t + 1

2⊗  ^{3}⊗  ^{3}∗  ^{2}∗
1 0
⊗_{1}  0 1
1 0
⊗_{2}  0 1
0 1
⊗_{4} 1 1  0
↖  1 0
Frobenius matrix  _{4}∗ 0 1
corresponding to π(t)  1 0
2∗  0 1
1 0
1∗  0 1

\begin{aligned}A & = 2 · ℤ/4 ⊕ 4 · ℤ/8 \\ H & = 2 · ℤ/2 ⊕ 2 · ℤ/4 ⊕ 2 · ℤ/16 \end{aligned}

4. Proof of theorem (C)

Let M^{m}(ℤ/2) be the homotopy category of Moore spaces M(A, m) where A is a finitely generated ℤ/2–vector space. Then we know by Baues [B1] 1.6.7 that M^{m}(ℤ/2) for m ≥ 3 is isomorphic to the category mod(ℤ/4) of finitely generated free ℤ/4–modules. Using this result we show:

Theorem 4.1. Let m ≥ 6. Then the bimodule M^{m+3}(ℤ/2)^{op} × M^{m}(ℤ/2) → Ab given by [M(a, m + 3), M(B, m)] is natural isomorphic to the bimodule

mod(ℤ/4)^{op} × mod(ℤ/4) → Ab

which carries (\overline{A}, \overline{B}) to Hom(\overline{A} ⊗ ℤ/2, \overline{B} ⊗ (ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2)).

Since the quiver

•\overrightarrow{\overrightarrow{⟶}}•

is wild we see that theorem (4.1) implies theorem (C) in the introduction. Proof of (4.1). Since the bimodule is biadditive it suffices to compute for A = B = ℤ/2 the ℤ/4–bimodule π = [M(A, m + 3), M(B, m)]. Here the ℤ/4–module structure is already determined by the structure of π as an abelian group. We know that

π_{m+2}M(B, m) = ℤ/4
π_{m+3}M(B, m) = ℤ/2 ⊕ ℤ/2

and that the Hopf map η : S^{m+3} → S^{m+2} induces the map

η^{∗} : π_{m+2}M(B, m) → π_{m+3}M(B, m)

corresponding to the composite

η^{∗} : ℤ/4 ↠ ℤ/2 ⟶^{i_{1}} ℤ/2 ⊕ ℤ/2

of the inclusion i_{1} and the quotient map. This follows for example from Baues– Goerss [BG] 5.2. Moreover using Baues [B1] 1.6.11 we have the push out diagram of abelian groups

[figure: ℤ/2 ⊕ ℤ/2; Ext(A, π_{m+3}M(B, m)); π; ℤ/2; Ext(A, π_{m+2}(B, m) ⊗ ℤ/2); j; (η^{∗})_{∗}; G(A, π_{m+2}M(B, m))]

Here (η^{∗})_{∗} is induced by η^{∗} above so that (η^{∗})_{∗} is split injective with cokernel ℤ/2. Moreover we have by definition of the category G in Baues [B1] the isomorphism

G(A, π_{m+2}(B, m)) = G(ℤ/2, ℤ/4) = ℤ/2 ⊕ ℤ/2

with j above corresponding to the inclusion i_{1} : ℤ/2 ⊂ ℤ/2 ⊕ ℤ/2. This shows π = ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2. q.e.d.

References

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[B1] Baues, H.–J., Homotopy type and homology, Oxford Math. Monographs, Oxford University Press, 1996, 596 pages.

[B2] Baues, H.–J., Homotopy types, Handbood of algebraic topology, chapter I (1995), Edited by I. M. James, Elsevier Science, 1–71.

[BC] Brown, E. H. and Copeland, A. H., Homology analogue of Postnikov Systems, Mich. Math. Journ. 6 (1959) 313–330.

[BD1] Baues, H.–J. and Drozd, Y., Representation theory of homotopy types with at most two non trivial homotopy groups, Math. Proc. Camp. Phil. Soc. 128 (2000) 283–300.

[BD2] Baues, H.–J. and Drozd, Y., The homotopy classification of (n − 1)–connected (n + 1)– dimensional polyhedra with torsion free homology, n ≥ 5, Expositiones Mathematicae, 17 (1999) 161–180.

[BD3] Baues, H.–J. and Drozd, Y., Classification of stable homotopy types with torsion free homology, to appear in Topology.

[BG] Baues, H.–J. and Goerss, P., A homotopy operation spectral sequence for the computation of homotopy groups, Topology 39 (2000) 161–192.

[BH] Baues, H.–J. and Hennes, M., The homotopy classification of (n − 1)–connected (n + 3)– dimensional polyhedra, n ≥ 4, Topology 30 (1991) 373–408.

[Ch] Chang, S. C., Homotopy invariants and continous mappings, Proc. R. Soc. London Ser. A 202 (1950) 253–263.

[D] Y. Drozd, Finitely generated quadratic modules. Preprint, MPI, 1999.

[EH] Eckmann, B. and Hilton, P. J., On the homology and homotopy decompositions of continuous maps. Pro. Nat. Acad. Science 45 (1959) 372–375.

[H] Henn, H.–W., Classification of p–local low dimensional spectra. J. Pure Appl. Algebra 19 (1980) 159–169.

[H1] Hilton, P. J., An introduction to homotopy theory, Cambridge University Press, 1953.

[H2] Hilton, P. J., Homotopy theory and duality, Nelson Gordon and Breach, 1965.

[W1] Whitehead, J. H. C., The homotopy type of a special kind of polyhedron, Ann. Soc. Polon. Math. 21 (1948) 176–186.

Hans–Joachim Baues Max–Planck–Institut für Mathematik P.O.Box: 7280 D–53072 Bonn Germany

E-mail address: baues@mpim-bonn.mpg.de

Yuri A. Drozd Faculty of Mechanics and Mathematics Kiev Taras Shevchenko University RUS–252033 Kiev Ukraine