REPRESENTATION THEORY OF HOMOTOPY TYPES WITH

AT MOST TWO NON TRIVIAL HOMOTOPY GROUPS

Hans-Joachim Baues and Yuri Drozd

It is a classical result of Postnikov [15] that homotopy types X with at most two non trivial homotopy groups π_{m}X = A and π_{n}X = B, 2 ≤ m < n, are classified by the k-invariant

(1)  k_{X} ∈ H^{n+1}(K(A, m), B)

Here the cohomology group of the Eilenberg-Mac Lane space K(A, m) was computed by Eilenberg-Mac Lane [11] and Cartan [5]. Let p be a prime and let ℤ_{p} ⊂ ℚ be the smallest subring of ℚ containing 1/q for all primes q with q ≠ p. We consider finitely generated ℤ_{p}-modules A and B and the stable range n < 2m − 1. Hence X is a p-local space with at most two non trivial homotopy groups in a stable range. Then the homotopy type of X admits a product decomposition

(2)  X ≃ X_{1} × … × X_{j}

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 factors in (2) by the following result.

Theorem (A). Let n − m ≤ (p + 1)(2p − 2). Then the classification of indecomposable factors in (2) is

wild  for n − m = (p + 1)(2p − 2),
tame  for n − m = t(2p − 2), 1 ≤ t ≤ p,
tame  for p = 2 and n − m = 3, and
essentially finite otherwise.

For the tame and essentially finite cases we compute below a complete list of all possible indecomposable factors. Moreover we obtain a corresponding result as in theorem (A) also for all cases q = n − m > (p + 1)(2p − 2). But for this one has to use the explicit computation of the homology H_{q}K(A, m) by Cartan [5].

Let A^{k} be the stable homotopy category of (m−1)-connected (m+k)-dimensional finite CW-complexs, m ≥ k+1. Then A^{k} is an additive category and there is the old problem to classify the indecomposable objects in A^{k} or equivalently to determine the “representation type” of A^{k}. J.H.C. Whitehead [16] and Chang [6] showed that A^{2} has essentially finite representation type; compare the books of Hilton [13], [14]. The representation type of A^{2} can actually be deduced from theorem (A). Moreover Baues-Hennes [2] showed that A^{3} has tame representation type; a complete list of indecomposable objects in A^{3} is given in [2]; compare also the book [1]. Now theorem (A) implies

Typeset by AMS-TEX

Corollary (B). A^{k} has wild representation type for k ≥ 6.

Hence only the representation type of A^{4} and A^{5} remain unknown. Theorem (A) and the classification in [1] indicate the complexity of the categories A^{4} and A^{5}. It is, however, amazing that the spaces in A^{4} with torsion free homology still have finite representation type, see Baues-Drozd [3]. Moreover Henn [12] showed that the p-local version of A^{k} has a tame representation type for k ≤ 4p − 5, p odd. Theorem (A) overlaps with the result of Baues-Hennes [2] for p = 2 and n − m = 2 and with the result of Henn [12] for p = 3 and n − m = 4.

The authors would like to thank Idun Reiten and Claus M. Ringel for the organization of a wonderful conference on representation theory in Oberwolfach (1997) which led to the results in this paper.

§ 1 The decomposition problem

Let C be an additive category with zero object ∗ and biproducts A ⊕ B. An object X in C is decomposable if there exists an isomorphism X ≅ A ⊕ B where A and B are not isomorphic to ∗. A decomposition of X is an isomorphism

(1.1)  X = A_{1} ⊕ … ⊕ A_{n} with n < ∞,

where A_{i} is indecomposable for all i ∈ {1, …, n}. The decomposition of X is unique if B_{1} ⊕ … ⊕ B_{m} ≅ X ≅ A_{1} ⊕ … ⊕ A_{n} implies m = n and that there is a permutation σ with B_{σ(i)} ≅ A_{i}. A morphism f in C is indecomposable if F is indecomposable in the category of pairs in C; objects in this category of pairs are the morphisms f in C and morphisms (α, β) : f → g are pairs of morphisms in C satisfying gα = βf. The decomposition problem in C can be described by the following task: find a complete list of indecomposable isomorphism types in C and describe the possible decomposition of objects in C. This problem is considered by representation theory. We say that the decomposition problem in C is wild or equivalently that C has wild representation type if the solution of the decomposition problem would imply a solution of the following kind of problem.

(1.2) Problem. Let k be a field and consider the following additive category V^{α,β}. Objects are finite dimensional k-vector spaces V together with two endomorphism α_{V} , β_{V} : V → V. Morphisms are k-linear maps f : V → W satisfying fα_{V} = α_{W} f and fβ_{V} = β_{W} f. The decomposition problem in V^{α,β} for any field k is termed a “wild problem of representation theory”.

On the other hand we say that C has tame representation type if a complete list of indecomposable objects of C is computed. If the number of objects in this list which satisfy a given finiteness restraint is finite then we say that C has essentially finite representation type. If the list is finite then C has finite representation type.

For example consider the category of finitely generated (f_{·}g_{·}) abelian groups; this category has essentially finite representation type since the list of indecomposable objects is given by the indecomposable cyclic groups ℤ and ℤ/p^{i} where p is a prime and i ≥ 1. The finiteness restraint is given by a number N which bounds the order of the torsion subgroup. In fact, it is well known that a f_{·}g_{·} abelian group A admits a decomposition

(1.3)  A ≅ C_{1} ⊕ … ⊕ C_{t}

where C_{i} for i = 1, …, t is an indecomposable cyclic group and this decomposition is unique. We say that t is the rank of A and that the f_{·}g_{·} abelian group A is

torsion:  if C_{i} ∈ {ℤ/p^{j}, j ≥ 1, p prime}
p-torsion:  if C_{i} ∈ {ℤ/p^{j}, j ≥ 1}
elementary: if C_{i} ∈ {ℤ/p, p prime}
p-elementary: if C_{i} ∈ {ℤ/p}
free:  if C_{i} ∈ {ℤ}
p-primary:  if C_{i} ∈ {ℤ, ℤ/p^{j}, j ≥ 1}

§ 2 The classification theorem

Let p be a prime. We say that a simply connected CW-complex X is p- \underline{local finite type} if all homotopy groups or equivalently all homology groups of X are finitely generated ℤ_{p}-modules. Moreover the p-\underline{local dimension} of X satisfies dim_{p}(X) ≤ n if H_{i}X = 0 for i > n and if H_{n}X is a free ℤ_{p}-module. Let 2 ≤ m. We define homotopy categories together with a Postnikov functor

(2.1)  ϕ : K(m, n)_{p} → k(m, n)_{p}

Here the objects of k(m, n)_{p} are CW-complexes X with at most two non trivial homotopy groups π_{m}X = A and π_{n}X = B such that A and B are finitely generated ℤ_{p}-modules. Moreover the objects of K(m, n)_{p} are p-local finite type CW-complexes Y with dim_{p}(Y) ≤ n + 1 and π_{i}Y = 0 for m < i < n. The functor ϕ in (2.1) carries Y to the n − th section of the Postnikov tower of Y. The categories in (2.1) are both additive categories and ϕ is an additive functor. The biproduct of X_{1}, X_{2} ∈ k(m, n)_{p} is the product X_{1} × X_{2} of spaces and the biproduct of Y_{1}, Y_{2} ∈ K(m, n)_{p} is the one point union Y_{1} ∨ Y_{2} of spaces.

(2.2) Proposition. Objects X ∈ k(m, n)_{p} and Y ∈ K(m, n)_{p} with 2 ≤ m < n < 2m − 1 have unique decompositions

\begin{matrix}X = X_{1} × … × X_{t} \\ Y = Y_{1} ∨ … ∨ Y_{r} \end{matrix}

where the X_{i} and Y_{j} are indecomposable. Moreover the additive functor ϕ yields a bijection of the set of indecomposable homotopy types in k(m, n)_{p} and the set of indecomposable homotopy types Y in K(m, n)_{p} with Y ≠ S_{p}^{n+1} where S_{p}^{n+1} is the p-local sphere of dimension n + 1.

Proof. The unique decomposition is well known, see for example [17]. Hence the bijection is obtained by a theorem of J.H.C. Whitehead on trees of homotopy types see for examples 10.4.4 in [1]. Given an indecomposable X ∈ k(m, n)_{p} we obtain the inverse ϕ^{−1}X = Y by the unique indecomposable Y ∈ K(m, n)_{p} for which there exists a set E such that Y ∨⋁_{E}S_{p}^{n+1} is homotopy equivalent to p-local (n+1)-skeleton

of X. q.e.d.

The proposition shows that the representation types of the categories k(m, n)_{p} and K(m, n)_{p} coincide. For the classification of objects in these topological categories we use the following algebraic category; see 3.6.1 [1].

(2.3) Definition. Let Ab be the category of abelian groups and let E, F : Ab → Ab be functors. Let p be a prime. Then the category S(E, F)_{p} is defined as follows. Objects are triple of f_{·}g_{·} ℤ_{p}-modules (A, R, H) where H is free together with a chain complex

(1)  S = (H ⟶^{b} F (A) ⟶^{∂} R ⟶^{δ} E(A) → 0)

which is exact in F (A) and E(A). Moreover δ∂ = 0. Morphisms S → S′ are triple (a, r, h) of homomorphisms in Ab for which the following diagram commutes.

(2)  \begin{matrix}H & ⟶ & F (A) & ⟶ & R & ⟶^{δ} & E(A) & ⟶ & 0 \\ ↓^{h} & & ↓^{a_{∗}} & & ↓^{r} & & ↓^{a_{∗}} \\ H′ & ⟶ & F (A′) & ⟶ & R′ & ⟶_{δ′} & E(A′) & ⟶ & 0 \end{matrix}

We introduce a natural equivalence relation ∼ on S(E, F)_{p} by setting (a, r, h) ∼ (a′, r′, h′) if a = a′, h = h′ and

(3)  r_{∗} = r′_{∗} : ker(∂) → ker(∂′)

We associate with an object S the exact sequence

(4)  H ⟶^{b} F (A) ⟶^{∂} ker(δ) ⟶^{j} cok(∂) ⟶^{δ} E(A) ⟶ 0

where j is given by ker(δ) → R → cok(∂). Here ker(δ) denotes the kernel of δ and cok(∂) denotes the cokernel of ∂. If E and F are additive functors then S(E, F)_{p} and also the quotient category S(E, F)_{p}/ ∼ are additive categories. We call S an Eilenberg-Mac Lane object if A = 0 or B = ker(δ) = 0 or even A = B = 0.

We say that a functor φ : C → K between categories is a detecting functor if φ reflects isomorphisms and is full and representative. This implies that φ induces a bijection between isomorphism types in C and K respectively. For example the quotient functor

(2.4)  q : S(E, F)_{p} → S(E, F)_{p}/ ∼

is a detecting functor since q is a linear extension of categories, see 3.6.1 (6) [1]. Moreover we prove in 6.5.2 [1] the following result.

(2.5) Classification theorem. Let p be a prime and let 2 ≤ m < n < 2m − 1. Then there is an additive detecting functor

Λ : K(m, n)_{p} → S(E, F)_{p}/ ∼

where E and F are additive functors given by the homology

\{ \begin{matrix}E(A) = H_{n}K(A, m) \\ F (A) = H_{n+1}K(A, m) \end{matrix}

of an Eilenberg-Mac Lane space K(A, m).

Given Y ∈ K(m, n)_{p} we have the k-invariant

(2.6)

k_{Y} ∈ H^{n+1}(K(A, m), B)

≅^{ψ} Hom(H_{n+1}K(A, m), B) ⊕ Ext(H_{n}K(A, m), B)

= Hom(F (A), B) ⊕ Ext(E(A), B)

with A = π_{m}Y and B = π_{n}Y. Here the isomorphism ψ in (2.6) can be chosen to be natural in A and B if the assumptions in (2.5) hold, see Decker [7]. Hence the k-invariant k_{Y} yields the coordinates

(2.7)  \{ \begin{matrix}k_{Y}^{1} ∈ Hom(F (A), B) \\ k_{Y}^{2} ∈ Ext(E(A), B) \end{matrix}

Now the functor Λ in (2.5) has the property that for S = Λ(Y) there is a natural commutative diagram (2.8)

H_{n+1}K(A, m)  B  H_{n}K(A, m)
∥  ∥  ∥
H ⟶ F (A) ⟶^{∂} ker δ ⟶^{j} cok(∂) ⟶^{δ} E(A) ⟶ 0
∥  ∥  ∥  ∥  ∥
H_{n+1}Y ⟶ Γ_{n}Y ⟶ π_{n}Y ⟶ H_{n}Y ⟶ Γ_{n−1}Y ⟶ 0

Here the bottom row is part of the exact sequence of J.H.C. Whitehead, see 6.5.3 [1]. Moreover k_{Y}^{1} coincides with ∂ in (2.8) and k_{Y}^{2} coincides with the extension element {R} ∈ Ext(E(A), B) given by the chain complex S = Λ(Y). This shows that the k-invariant k_{Y} can be readily deduced from the chain complex S = Λ(Y).

We say that an additive functor E : Ab → Ab is elementary if there exist elementary f_{·}g_{·} abelian groups E_{1} and E_{2} (see (1.3)) such that

(2.9)  E(A) = A ⊗ E_{1} ⊕ A ∗ E_{2}

where we use the tensor product A ⊗ B and the torsion product A ∗ B of abelian groups A, B. Moreover E is p-elementary if E_{1} and E_{2} are p-elementary.

(2.10) Proposition. Consider the functor Ab → Ab which carries A to the homology H_{n}K(A, m) of the Eilenberg-Mac Lane space K(A, m). For m < n < 2m this functor is elementary.

In fact, Cartan showed in theorem 2 of section 11 [5] that for q < m

H_{m+q}K(A, m) = A ⊗ E_{1} ⊕ A ∗ E_{2}

with E_{1} = ⨁{E_{1}^{p}, p prime} and E_{2} = ⨁{E_{2}^{p}, p prime}. Here E_{1}^{p} (resp. E_{2}^{p}) is the ℤ/p-vector space with the basis consisting of “admissible p-words (a_{1}, a_{2}, …) of the first (resp. second) kind of stable degree q such that a_{1} ≡ 0 (mod 2p − 2)”. In particular we can deduce from Cartan’s result the following addendum which is of importance for the proof of theorem (A).

(2.11) Addendum. Let p be a prime and let m < n < 2m and m − n ≤ (p + 1)(2p − 2). Then

ℤ/p ∗ H_{n}K(A, m) = \{ \begin{matrix}A ⊗ (ℤ/p ⊕ ℤ/p) & if m − n = (p + 1)(2p − 2) \\ A ⊗ ℤ/p & if m − n = t(2p − 2), 1 ≤ t ≤ p \\ A ∗ ℤ/p & if m − n = t(2p − 2) + 1, 1 ≤ t ≤ p \\ 0 & otherwise \end{matrix}

The addendum is relevant for the application of theorem (2.5) since one readily obtains the following fact.

(2.12) Lemma. Let E and F be elementary functors Ab → Ab and let p be a prime. Then there is a canonical isomorphism of additive categories

S(E, F)_{p} = S(ℤ/p ∗ E, ℤ/p ∗ F)_{p}

where the functors ℤ/p ∗ E and ℤ/p ∗ F are p-elementary.

Using (2.4), (2.9) and (2.12) we see that the classification of indecomposable objects in K(m, n)_{p} is obtained by the classification of indecomposable objects in S(E, F)_{p} where E and F are p-elementary functors. We prove below the following result.

(2.13) Theorem. Let E and F be p-elementary functors given by (E_{1}, E_{2}) and (F_{1}, F_{2}) respectively. Then the objects in S(E, F)_{p} have a unique decomposition and the representation type of S(E, F)_{p} is

• essentially finite if at most one of the groups E_{1}, E_{3}, F_{1}, F_{2} is ℤ/p and all the others are trivial,

• tame if E(A) = A ⊗ ℤ/p and F (A) = A ∗ ℤ/p,

• tame if E(A) = A ∗ ℤ/p and F (A) = A ⊗ ℤ/p,

• wild otherwise.

For the tame cases the classification of indecomposable objects is given in § 3 below. Using (2.11) the theorem implies theorem (A) in the introduction. Theorem (2.13) is proved in section § 4.

§ 3 The indecomposable objects

We first consider the tame cases of theorem (2.13) and theorem (A) in the introduction.

For the description of indecomposable objects we use certain words. Let L be a set, the elements of which are called “letters”. A word with letters in L is an element in the free monoid generated by L. Such a word a is written a = a_{1}a_{2} … a_{n} with a_{i} ∈ L, n ≥ 0; for n = 0 this is the empty word ϕ. Let b = b_{1} … b_{k} be a word. We write w = … b if there is a word a with w = ab, similarly we write w = b … if there is a word c with w = bc. A subword of an infinite sequence … a_{−2}a_{−1}a_{0}a_{1}a_{2} … with a_{i} ∈ L, i ∈ ℤ, is a finite connected subsequence a_{n}a_{n+1} … a_{n+k}, n ∈ ℤ.

(3.1) Definition. Let p be a prime. We define a collection of finite words w = w_{1}w_{2} … w_{k}. The letters w_{i} of w are the symbols ξ, η or natural numbers s_{i}, r_{i}, i ∈ ℤ, which are powers of p. We write the letters s_{i} as upper indices, the letters r_{i} as lower indices since we have to distinguish between these numbers. For example ξ^{2}η_{8}ξ^{4}η is such a word for p = 2. The basic sequence is defined by

(1)  …^{s_{0}} η_{r_{0}}ξ^{s_{1}}η_{r_{1}}ξ^{s_{2}}η_{r_{2}} …

This is the infinite product … a(−1)a(0)a(1)a(2) … of words a(i) = ξ^{s_{i}}η_{r_{i}} with i ∈ ℤ.

The basic sequence (1) can be visualized by infinite graphs in two ways, either by the tensor graph or by the torsion graph:

0 1 2 3  ⋯  • • • • • • • • • • • • • • • • ⋯

Tensor graph

0 1 2 3  ⋯  • • • • • • • • • • • • • • • • ⋯

Torsion graph

The letters s_{i}, resp. r_{i} in the basic sequence correspond to vertical edges conecting levels 2 and 3 resp. the levels 0 and 1. The letters η and ξ correspond in both graphs to diagonal edges: In the tensor graph η connects level 0 and 2 and ξ connects level 1 and 3; in the torsion graph η connects level 1 and 2 and ξ connects level 0 and 3. We consider connected finite subgraphs of the tensor graph and the torsion graph. These subgraphs have an initial vertex at the left hand side and a final vertex at the right hand side.

A p-tensor word is any finite subword w of the basic sequence of the form

(2)  w = a … where a =^{i_{1}} η or a = η

This is the same as any connected finite subgraph of the tensor graph containing η with initial vertex at level 2 or 3. Moreover w ≠^{p} η and w ≠^{p} η_{r}ξ since these words correspond to indecomposable Eilenberg-Mac Lane objects.

A p-torsion word is any finite subword w of the basic sequence (1) satisfying

(3)  \{ \begin{matrix}w = a … and w = … c where \\ a = ξ or a =^{s_{i}} η or a = η and \\ c = η_{r_{j}} or c = ξ or c = ξ^{s_{j}} \end{matrix}

This is the same as any connected finite subgraph of the torsion graph containing ξ or η with the property that both the initial vertex and the final vertex are not at level 1. Moreover w ≠ ξ and w ≠^{p} η_{t}ξ since these words correspond to indecomposable Eilenberg-Mac Lane objects.

The vertices of level i of a tensor word (resp. torsion word) are defined by the vertices of level i of the corresponding graph; i ∈ {0, 1, 2, 3}. We also write | x |= i if x is a vertex of level i. We call x an ∞ -vertex if x is not a vertex of a vertical edge. Such an ∞-vertex of a tensor word w or a torsion word w (if it exists) automatically is the initial or the final vertex of the corresponding subgraph. Let β_{i}(w) be the number of ∞-vertices of level i in the word w; this is the Betti number of w satisfying 0 ≤ β_{i}(w) ≤ 1 and β_{1}(w) = 0. Let r_{α} … r_{β} and s_{μ} … s_{ν} be the words of lower indices and of upper indices respectively given by w. We define the homology of w by

(3.2)  \{ \begin{matrix}H_{0}(w) = ℤ_{p}^{β_{0}(w)} ⊕ ℤ/r_{α} ⊕ … ⊕ ℤ/r_{β} \\ H_{1}(w) = 0 \\ H_{2}(w) = ℤ_{p}^{β_{2}(w)} ⊕ ℤ/s_{μ} ⊕ … ⊕ ℤ/s_{ν} \\ H_{3}(w) = ℤ_{p}^{β_{3}(w)} \end{matrix}

It is in the definition of the Betti numbers of w where the difference between the tensor graph and the torsion graph plays an essential role.

(3.3) Definition. We associate with a p-tensor word w a chain complex

S(w) ∈ S(⊗ℤ/p, ∗ℤ/p)_{p}

Here S(w) is given by A = H_{0}(w) and the chain complex

H_{3}(w) ⟶^{b} A ∗ ℤ/p ⟶^{∂} R ⟶^{δ} A ⊗ ℤ/p ⟶ 0

with cok (∂) = H_{2}(w). The abelian group R is given as follows. Let s_{μ} … s_{ν} be the word of those upper indices of w which are exponents of ξ; that is for μ ≤ i ≤ ν the word ξ^{s_{i}} is a subword of w. Then

R^{ξ}(w) = ℤ/s_{μ} · p ⊕ … ⊕ ℤ/s_{ν} · p

is defined where the order of the cyclic summands is the product s_{i} · p of s_{i} and the prime p. Moreover we define

R_{1}^{η}(w) = \{ \begin{matrix}ℤ_{p} & if w = η … \\ ℤ/s & if w = ^{s}η … \\ 0 & otherwise \end{matrix}

R_{2}^{η}(w) = \{ \begin{matrix}ℤ/p & if w = …^{s}η_{u} \\ 0 & otherwise \end{matrix}

Using these pieces we obtain the direct sum

R = R^{ξ}(w) ⊕ R_{1}^{η}(w) ⊕ R_{2}^{η}(w)

which is part of the chain complex above. The maps b, ∂, δ in (1) are all given by the canonical maps

ℤ_{p} → ℤ/p and ℤ/p → ℤ/p^{t} → ℤ/p

where the cyclic summands correspond to indices in w which are neighbours via ξ or η respectively. For example we have the following list. We write E(r) = E(ℤ/r) and E(∞) = E(ℤ_{p}) and similarly F (r) = F (ℤ/r).

A ∗ ℤ/p  A ⊗ ℤ/p
∥  ∥
w  H_{3}(w) → F (A) →  R  → E(A)

η  0  0  ℤ_{p} → E(∞)

s ≠ p,^{s} η  0  0  ℤ/s → E(∞)

sη_{r}  0  0  ℤ/s → E(r)
0  F (r) = ℤ/p  0

s ≠ p,^{s} η_{r}ξ  0  0  ℤ/s → E(r)
ℤ_{p} → F (r)  0  0

sη_{r}ξ^{t}  0  0  ℤ/s → E(r)
0  F (r) → ℤ/t · p  0

sη_{r}ξ^{t}η  0  0  ℤ/s → E(r)
0  F (r) → ℤ/t · p → E(∞)

0  0  ℤ/s → E(r)
sη_{r}ξ^{t}η_{u}  0  F (r) → ℤ/t · p → E(u)
0  F (u) = ℤ/p  0

0  0  ℤ/s → E(r)
sη_{r}ξ^{t}η_{u}ξ  0  F (r) → ℤ/t · p → E(u)
ℤ_{p} → F (u)  0  0

0  0  ℤ/s → E(r)
sη_{r}ξ^{t}η_{u}ξ^{v}  0  F (r) → ℤ/t · p → E(u)
0  F (u) → ℤ/t · v  0

If we omit the initial upper index s in w on the left hand side of the list then we have to replace ℤ/s in the list by ℤ_{p}.

(3.4) Definition. We associate with a p-torsion word w a chain complex

S(w) ∈ S(∗ℤ/p, ⊗ℤ/p)_{p}

Here S(w) is again given by A = H_{0}(w) and the chain complex

H_{3}(w) ⟶^{b} A ⊗ ℤ/p ⟶^{∂} R ⟶^{δ} A ∗ ℤ/p → 0

with cok (∂) = H_{2}(w). The abelian group R is defined by

R = R^{ξ}(w) ⊕ R_{1}^{η}(w) ⊕ R̄_{2}^{η}(w)

where R^{ξ}(w) and R_{1}^{η}(w) are given as in (3.3) and where R̄_{2}^{η}(w) is the following modification of R_{2}^{η}(w) in (3.3):

R̄_{2}^{η}(w) = \{ \begin{matrix}ℤ/p & if w = … η_{u} \\ 0 & otherwise \end{matrix}

The maps b, ∂ and δ are again defined by canonical maps as in (3.3). For example we have the following list.

A ⊗ ℤ/p  A ∗ ℤ/p
∥  ∥
w  H_{3}(w) → F (A) →  R  → E(A)

ξ^{u}  0  F (∞) → ℤ/u · p  0

ξ^{u}η_{t}  0  F (∞) → ℤ/u · p → E(t)
0  F (t) = ℤ/p  0

ξ^{u}η_{t}ξ  0  F (∞) → ℤ/u · p → E(t)
ℤ_{p} → F (t)  0  0

ξ^{u}η_{t}ξ^{v}  0  F (∞) → ℤ/u · p → E(t)
0  E(t) → ℤ/v · p  0

sη_{t}  0  0  ℤ/s → E(t)
0  F (t) = ℤ/p  0

s ≠ p,^{s} η_{t}ξ  0  0  ℤ/s → E(t)
ℤ_{p} → F (t)  0  0

sη_{t}ξ^{u}  0  0  ℤ/s → E(t)
0  F (t) → ℤ/u · p  0

0  0  ℤ/s → E(t)
sη_{t}ξ^{u}η_{v}  0  F (t) → ℤ/u · p → E(v)
0  F (v) = ℤ/p  0

0  0  ℤ/s → E(t)
sη_{t}ξ^{u}η_{v}ξ  0  F (t) → ℤ/u · p → E(v)
ℤ_{p} → F (v)  0  0

0  0  ℤ/s → E(t)
sη_{t}ξ^{u}η_{v}ξ^{y}  0  F (t) → ℤ/u · p → E(v)
0  F (v) → ℤ/y · p  0

Again if we omit the initial upper index s in the word w on the left hand side of the list then we have to replace ℤ/s in the list by ℤ_{p}.

(3.5) Definition. A p-cyclic word is defined by a pair (w, φ) where w is a subword of the basic sequence of the form (q ≥ 1)

w = ξ^{s_{1}}η_{r_{1}}ξ^{s_{2}}η_{r_{2}} … ξ^{s_{q}}η_{r_{q}} = a(1) … a(q)

and where φ is an automorphism of a finite dimensional ℤ/p-vector space V = V (φ). Two cyclic words (w, φ) and (w′, φ′) are equivalent if w′ is a cyclic permutation of w, that is, w′ = a(i) … a(q)a(1) … a(i − 1), and if there is an isomorphism ψ : V (φ) ≅ V (φ′) with φ = ψ^{−1}φ′ψ. A cyclic word (w, φ) is a special p-cyclic word if φ is an indecomposable automorphism and if w is not of the form w = w′w′ … w′ where the right-hand side is a j-fold power of a word w′ with j > 1: We define the homology of a p-cyclic word by

H_{i}(w, φ) = ⨁_{v} H_{i}(w)

Here v = dim V (φ) and the right hand side is the v-fold direct sum of the homology H_{i}(w) defined in (3.2).

(3.6) Definition. We associate with a p-cyclic word (w, φ) a chain complex

\begin{matrix}S(w, φ) ∈ S(⊗ℤ/p, ∗ℤ/p)_{p} \\ S(w, φ) ∈ S(∗ℤ/p, ⊗ℤ/p)_{p} \end{matrix}

which is given as follows. Let v be the dimension of V (φ) and let U^{v} = U ⊕ … ⊕ U be the v-fold dircet sum of the abelian group U. Then S(w, φ) is given by

A = H_{0}(w)^{v}, R = R^{ξ}(w)^{v}, H = 0

and by the chain complex

0 → F (H_{0}(w))^{v} ⟶^{∂} R^{ξ}(w)^{v} ⟶^{δ^{v}} E(H_{0}(w))^{v} → 0

where H_{0}(w) is a p-torsion group. The chain complex is given by the direct sum of the following lines:

F (r_{q})^{v} ⟶^{∂^{φ}} (ℤ/s_{1} · p)^{v} ⟶^{δ^{v}} E(r_{1})^{v}
F (r_{1})^{v} ⟶^{∂^{v}} (ℤ/s_{2} · p)^{v} ⟶^{δ^{v}} E(r_{2})^{v}
⋮
F (r_{q−1})^{v} ⟶^{∂^{v}} (ℤ/s_{q} · p)^{v} ⟶^{δ^{v}} E(r_{q})^{v}

Here ∂^{v} and δ^{v} denote the v-fold direct sum of the canonical maps. The map ∂^{φ}, however, is given by the composite

F (r_{q})^{v} ≅^{φ} F (r_{q})^{v} ⟶^{∂^{v}} (ℤ/s_{1} · p)^{v}

where φ is the automorphism of V (φ) = F (r_{q})^{v} given by the cyclic word (w, φ).

It is easy to compute the abelian group B = ker(δ) from the description of the chain complexes S(w) and S(w, φ) above. Recall that a chain complex S with A = 0 or B = 0 is termed on Eilenberg-Mac Lane object; see (2.3). The indecomposable Eilenberg-Mac Lane objects are completely described by elementary cyclic groups A or B and by (H = ℤ, A = B = 0) respectively.

(3.7) Theorem. Let p be a prime. A complete list of indecomposable objects in S(⊗ℤ/p, ∗ℤ/p)_{p} is given by the following objects:

• indecomposable Eilenberg-Mac Lane objects,

• chain complexes S(w) where w is a p-tensor word,

• chain complexes S(w, φ) where (w, φ) is a special p-cyclic word. A complete list of indecomposable objects in S(∗ℤ/p, ⊗ℤ/p)_{p} is given by the following objects:

• indecomposable Eilenberg-Mac Lane objects,

• chain complexes S(w) where w is a p-torsion word,

• chain complexes S(w, φ) where (w, φ) is a special p-cyclic word. For two objects S, S′ in these lists there is an isomorphism S ≅ S′ if and only if there are equivalent special p-cyclic words (w, φ) ∼ (w′, φ′) with S = S(w, φ) and S′ = S(w′, φ′).

In theorem (A) in the introduction we need the category S(⊗ℤ/p, ∗ℤ/p)_{p} for all tame cases n − m = t(2p − 2) with 1 ≤ t ≤ p. Also the category S(∗ℤ/p, ⊗ℤ/p)_{p} is needed in theorem (A) for the tame case p = 2 and n − m = 3. More precisely we get the following two corollaries of (3.7). Let M(C, n) be the Moore space of the abelian group C and let K(C, m)_{p}^{n+1} be the indecomposable part of the p-local (n+1)-skeleton of the Eilenberg-Mac Lane space K(C, m); compare (2.2). Moreover let X(w), resp. X(w, φ), be the space corresponding to S(w), resp. S(w, φ), via the detecting functor Λ in (2.5). As abelian group C is an \underline{elementary} ℤ_{p}-\underline{module} if C = ℤ_{p} or if C = ℤ/p^{i} with i ≥ 1.

(3.8) Corollary. Let p be a prime and 2 ≤ m < n with n < 2m − 1 and n − m = t(2p−2) with 1 ≤ t ≤ p. Then a complete list of indecomposable objects in K(m, n)_{p} is given by the following objects.

• S_{p}^{n+1}, M(C, n), K(C, m)_{p}^{n+1} where C is an elementary ℤ_{p}-module, • spaces X(w) where w is a p-tensor word,

• spaces X(w, φ) is a special p-cyclic word.

(3.9) Corollary. Let m ≥ 5. Then a complete list of indecomposable objects in K(m, m + 3)_{2} is given by the following objects with p = 2.

• S_{p}^{m+4}, M(C, m + 3), K(C, m)_{p}^{m+4} where C is an elementary ℤ_{p}-module,

• spaces X(w) where w is a p-torsion word,

• spaces X(w, φ) where (w, φ) is a special p-cyclic word.

Here X(w) and X(w, φ) are again defined by the detecting functor Λ in (2.5) with p = 2 and 5 ≤ m < n = m + 3.

The homology of the spaces X(w) is related to the homology of the word w by the following formula.

(3.10)  H_{i}(X(w)) = \{ \begin{matrix}H_{0}(w) & for i = m \\ H_{2}(w) & for i = n \\ H_{3}(w) & for i = n + 1 \end{matrix}

This also holds if we replace w by (w, φ). The classification of indecomposable objects by words or graphs as above thus shows by (3.10) immediately part of the homology of the corresponding spaces.

§ 4 Proof of Theorem 2.13

We are going to interpret the category S = S(E, F)_{p} in (2.3) in terms of “matrix problems”, namely, matrices over a bimodule (cf. [8, 9]). Consider the category M_{p}^{2} = M_{p} × M_{p}, where M_{p} is the category of finitely generated ℤ_{p}-modules and the bimodule U over it, i.e. the bifunctor U : (M_{p}^{2})^{op} × M_{p}^{2} → Ab, defined as follows:

U((A, R), (A′, R′)) = Hom(R, E(A′)) ⊕ Hom(F (A), R′).

We write U[A, R] for U((A, R), (A, R)). Recall that the category EL(U) of matrices over U or elements of U has the set of objects ⋃_{(A,R)} U[A, R]. A morphism from an element (δ, ∂) ∈ U[A, R] to (δ′, ∂′) ∈ U[A′, R′] is by definition a pair (α, β), where α : A → A′, β : R → R′ such that α_{∗}δ = δ′β and β∂ = ∂′α_{∗}.

A complex from S is just a triple (δ, ∂, h), where (δ, ∂) is an object from EL(U) with the property that δ is surjective and δ∂ = 0, while h is an epimorphism H → ker ∂ for some free abelian group H. Hence, essentially the classification of objects from S and from EL(U) are the same.

Denote by C_{m} the cyclic group of order p^{m} if m ∈ ℕ and C_{m} = ℤ_{p} if m = ∞. The following propositions are evident.

(4.1) Proposition.

1. C_{m} ⊗ ℤ/p = ℤ/p.

2. C_{m} ∗ ℤ/p = ℤ/p if m ∈ ℕ and C_{∞} ∗ ℤ/p = 0.

3. If m ≥ n, an epimorphism C_{m} → C_{n} induces an isomorphism C_{m} ⊗ ℤ/p → C_{n} ⊗ ℤ/p and zero map on C_{m} ∗ ℤ/p.

4. If m ≤ n < ∞, a monomorphism C_{m} → C_{n} induces an isomorphism C_{m} ∗ ℤ/p → C_{n} ∗ ℤ/p and zero map on C_{m} ⊗ ℤ/p.

(4.2) Proposition.

1. Hom(C_{m}, ℤ/p) = ℤ/p.

2. Hom(ℤ/p, C_{m}) = ℤ/p if m ∈ ℕ and Hom(ℤ/p, C_{∞}) = 0.

3. If m > 1 then ϕψ = 0 for each ψ : ℤ/p → C_{m} and ϕ : C_{m} → ℤ/p.

4. If m ≥ n, an epimorphism C_{m} → C_{n} induces an isomorphism Hom(C_{n}, ℤ/p) → Hom(C_{m}, ℤ/p) and zero map on Hom(ℤ/p, C_{m}).

5. If m ≤ n < ∞, a monomorphism C_{m} → C_{n} induces an isomorphism Hom(ℤ/p, C_{m}) → Hom(ℤ/p, C_{n}) and zero map on Hom(C_{m}, ℤ/p).

Consider the first tame cases of Theorem 2.13. Let E = _{−} ⊗ ℤ/p and F = − ∗ ℤ/p. If A = ⨂_{m} C_{m}^{a_{m}} and R = ⨂_{m} C_{m}^{r_{m}}, an element δ ∈ Hom(R, E(A)) can be considered as a block matrix of the form:

(1)  ( \begin{matrix}δ_{11} & & & δ_{12} & & … & & δ_{1∞} \\ δ_{21} & & & δ_{22} & & … & & δ_{2∞} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ δ_{∞1} & & & δ_{∞2} & & … & & δ_{∞∞} \end{matrix} )

where δ_{ij} is a matrix with entries in ℤ/p of size a_{i} × r_{j}. Just in the same way, an element ∂ ∈ Hom(F (A), R) can be considered as a block matrix of the form:

(2)  ∂ = ( \begin{matrix}∂_{11} & & & ∂_{12} & & & … \\ ∂_{21} & & & ∂_{22} & & & … \\ . & . & . & . & . & . & . & . & . & . & . & . \end{matrix} )

where ∂_{ij} is a matrix with entries in ℤ/p of size r_{i} × a_{j}. Remark that both F (C_{∞}) and Hom(F (A), C_{∞}) are zeros.

A homomorphism α : A → A′ is also given by a matrix

(3)  α = ( \begin{matrix}α_{11} & & & α_{12} & & & … & & & α_{1∞} \\ α_{21} & & & α_{22} & & & … & & & α_{2∞} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ α_{∞1} & & & α_{∞2} & & & … & & & α_{∞∞} \end{matrix} )

where α_{ij} is a matrix with entries in Hom(C_{j}, C_{i}) of size a′_{i} × a_{j}. It is an isomorphism if and only if all diagonal blocks α_{ii} are invertible. Just in the same way a homomorphism β : R → R′ is given by an analogous block matrix (β_{ij}). For a matrix α of the form (3) put

α^{+} = ( \begin{matrix}α_{11} & & & α_{12} & & & … & & α_{1∞} \\ 0 & & & α_{22} & & & … & & α_{2∞} \\ 0 & & & 0 & & & … & & α_{3∞} \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & & & 0 & & & … & & α_{∞∞} \end{matrix} )

and

α^{−} = ( \begin{matrix}α_{11} & & & 0 & & 0 & & & … & 0 \\ α_{21} & & & α_{22} & & 0 & & & … & 0 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ α_{∞1} & & & α_{∞2} & & … & & & … & α_{∞∞} \end{matrix} )

Now the objects from EL(U) are just pairs (δ, ∂), where δ and ∂ are matrices of the forms (1) and (2) respectively, while a homomorphism from this object to another one (δ′, ∂′) is a pair of matrices (α, β) of the form (3) such that α^{+}δ = δ′β^{−} and β^{+}∂ = ∂′α^{−}. Therefore elements from EL(U) coincide with representations of a bunch of chains in the sense of [4] (cf. also [10] for the special case of chains, just what we need). Namely, this bunch consists of the chains E_{1} ≃ F_{1}ℕ ∪ {∞}, E_{2} ≃ F_{2} ≃ ℕ with the equivalence relation ∼ such that m ∈ E_{2} is equivalent to the same n ∈ E_{1}. Thus we can use the list of indecomposable representation given in [4] which implies that this classification is tame. Taking into consideration for a pair (δ, ∂) to represent an object from S and adding, when necessary, a homomorphism H → F (A), we obtain the following list of indecomposable objects from S.

Consider finite sequences w = (m_{1}, m_{2}, …, m_{n}), where n is even and m_{i} are non-negative integers or ∞ such that

(4)

m_{i} ≠ 0, except possibly m_{n},
m_{i} ≠ ∞, except possibly m_{1} and m_{n},
m_{i} ≠ 1, for odd i, except possibly m_{1} and m_{n−1} if m_{n} = 0.

For each sequence w we define the object S(w) as follows:

(5)

A = ⨁_{i odd} C_{m_{i}},

R = ⨁_{i even} C_{m_{i}},

H = \{ \begin{matrix}ℤ & if n is even and m_{n} ≠ ∞ \\ 0 & otherwise \end{matrix}

h is the surjection ℤ_{p} → C_{m_{n}} (if H ≠ 0)

while ∂ and δ are given by the following diagram:

C_{m_{1}} → E(C_{m_{2}})
F (C_{m_{2}}) → C_{m_{3}} → E(C_{m_{4}})
F (C_{m_{4}}) → C_{m_{5}} → E(C_{m_{6}})
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F (C_{m_{n−2}}) → C_{m_{n−1}} → E(C_{m_{n}})

where all “left” arrows are monomorphisms and all “right” ones are epimorphisms. In this case H and h are uniquely defined. Namely if m_{n} = 0 or m_{n} = ∞ then H = 0; otherwise H = ℤ_{p} and h is the natural epimorphism ℤ_{p} → F (C_{m}) = ℤ/p.

Suppose now that m_{i} ∉ {0, 1, ∞} for all i and the sequence w is non-periodic, i.e. there exists no shorter sequence v such that w is just a repetition of several copies of v. Then for each polynomial f(T) ∈ ℤ/p[T], f(T) ≠ T^{d}, which is a power of an irreducible one, define the object S(w, f) as follows:

(6)  \begin{matrix}H = 0, \\ A = ⨁_{i odd} C_{m_{i}}^{d}, \\ R = ⨁_{i even} C_{m_{i}}^{d}, \end{matrix}

where d = deg f, while ∂ and δ are given by the following diagram:

F (C_{m_{n}}^{d}) →^{ϕ} C_{m_{1}}^{d} → E(C_{m_{2}}^{d})
F (C_{m_{2}}^{d}) → C_{m_{3}}^{d} → E(C_{m_{4}}^{d})
F (C_{m_{4}}^{d}) → C_{m_{5}}^{d} → E(C_{m_{6}}^{d})
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F (C_{m_{n−2}}^{d}) → C_{m_{n−1}}^{d} → E(C_{m_{n}}^{d})

In this diagram all mappings, except ϕ, are given by matrices εI, where I is the identity matrix, while ε is injection for “right” arrows and surjection for “left” ones. The mapping ϕ is given by the matrix εΦ, where Φ is the Frobenius matrix with characteristic polynomial f(T). In this case H = 0.

(4.3) Proposition. All objects S(w) and S(w, f) defined above are indecomposable and every indecomposable object in S is isomorphic either to S(w) or to S(w, f) for some choice of w and f. Moreover, the only isomorphisms between these objects are S(w, f) ≅ S(w′, f), where w′ is obtained from w by a cyclic permutation mapping m_{1} to m_{k} with odd k.

The case E = _{−} ∗ ℤ/p, F = _{−} ⊗ ℤ/p is quite analogous, except that the following values of m_{i} (forbidden above) are possible:

\begin{matrix}m_{1} = 0, m_{2} = ∞ \\ M_{n−1} = 1, m_{n} = ∞. \end{matrix}

Comparing the chain complexes S(w) in (4.3) and (3.3) we see that (3.7) is a consequence of (4.3).

If E = 0 or F = 0, while the other functor is _{−} ⊗ ℤ/p or _{−} ∗ ℤ/p, the situation is quite trivial: the indecomposable objects can only be one of the following forms:

C_{m} → E(C_{k});
F (C_{m}) → C_{k};
ℤ → F (C_{m})

(the first one if E ≠ 0, the second and the third if F ≠ 0). In particular, these cases are essentially finite.

We now prove that all other cases are wild. For instance, let E = F = _{−} ⊗ ℤ/p (other cases are similar or even easier to handle). Then an object from EL(U) is given by two matrices: δ as in (1) and ∂ as in (2). Two such objects are isomorphic if and only if there are matrices α, β of the form (3) such that all α_{ii} are invertible, α^{+}δ = δ′β^{−} and β^{+}∂ = ∂′α^{+}. If there are no direct summands C_{1} in R and the rows of δ are linear independent as well as the columns of ∂, the pair (∂, δ) defines an object from S (with H = 0).

Put now A = ⨁_{i=1}^{3} C_{i}^{n}, R = ⨁_{i=2}^{4} C_{i}^{n},

δ = ( \begin{matrix}I & 0 & 0 \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ) , ∂ = ( \begin{matrix}0 & 0 & I \\ 0 & I & I \\ I & X & Y \end{matrix} ) ,

where I is n × n identity matrix, X, Y are any n × n matrices. This gives an object S(X, Y) ∈ S. If S(X, Y) ≅ S(X′, Y′) then α^{+} = β^{−}, whence α^{+} = β^{−} = diag(α_{1}, α_{2}, α_{3}). Then β^{+}∂ = ∂′α^{+} implies that α_{1} = α_{2} = α_{3}, β_{ij} = 0 for i ≠ j and α_{3}X = X′α_{3}, α_{3}Y = Y′α_{3}, i.e. the pairs (X, Y) and (X′, Y′) are conjugate. Therefore, S is wild.

Literature

[1] Baues, H.-J., Homotopy type and homology, Oxford Math. Monographs, Oxford University Press (1976).

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

[3] Baues, H.-J. and Drozd, Y., (in preparation).

[4] Bondarenko, V.V., Representations of bundles of semichained sets and their applications, Algebra i Analiz 3, ♯5 (1991), 38–61; English translation: St. Petersburg Math. J. 3 (1992), 973–996.

[5] Cartan, H., Seminaire Cartan, 1950–1959, Paris Benjamin (1967).

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

[7] Decker, J., The integral homology algebra of an Eilenberg-Mac Lane space, Thesis University of Chicago (1974).

[8] Drozd, Yu. A., Matrix problems and categories of matrices, Zapiski Nauchn. Semin. LOMI 28 (1972), 144–153.

[9] Drozd, Yu. A., Tame and wild matrix problems, Representations and Quadratic Forms, Inst. Math., Kiev, 1979, pp. 39–74; English translation: Amer. Math. Soc. Transl. (2) 128 (1986), 31–55.

[10] Drozd, Yu. A. and Greuel, G.-M., On classification of vector bundles over projective curves (to appear).

[11] Eilenberg, S. and Mac Lane, S., On the groups H(π, n). I, Ann. of Math. 58 (1953), 55–106; II, Ann. of Math. 60 (1954), 49–139; III, Ann. of Math. 60 (1954), 513–557.

[12] Henn, H.W., Classification of p-local low dimensional spectra, J. Pure and Appl. Algebra 45 (1987), 45–71.

[13] Hilton, P.I., An introduction to homotopy theory, Cambridge University Press (1953).

[14] Hilton, P.I., Homotopy theory and duality, Nelson Gordon and Breach (1965).

[15] Postnikov, M.M., On the homotopy type of polyhedra, Dokl. Akad. Nauk SSR 76, 6 (1951), 789–791.

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

[17] Wilkerson, C., Genus and cancellation, Topology 14 (1975), 29–36.