THE HOMOTOPY CLASSIFICATION OF

(n − 1)-CONNECTED (n + 4)-DIMENSIONAL POLYHEDRA

WITH TORSION FREE HOMOLOGY, n ≥ 5

Hans-Joachim Baues and Yuri Drozd

The classification of homotopy types of finite polyhedra is a classical and fundamental task of topology which in particular is an inevitable step for the classification of manifolds; see [B2]. A best possible solution is the classification by an explicit list of indecomposable homotopy types. Such final solutions were obtained for (n − 1)- connected (n + k)-dimensional polyhedra for k = 2 by Whitehead and Chang [W1],

[Ch] and for k = 3 by Baues-Hennes [BH], [B1]. In this paper we consider the case k = 4 for torsion free polyhedra and we prove the following surprising result.

Theorem. Let n ≥ 5. There is a list X(L), see (1.3), consisting of exactly 67 polyhedra such that each (n − 1)-connected (n + 4)-dimensional polyhedron X with finitely generated torsion free homology admits a homotopy equivalence

X ≃ X_{1} ∨ X_{2} ∨ … ∨ X_{k}

with X_{i} ∈ X(L) for 1 ≤ i ≤ k. Here X_{1} ∨ … ∨ X_{k} denotes the one point union of the spaces X_{i}.

We describe the elementary polyhedra in the list X(L) explicitly. They turn out to be CW-complexes with at most four non trivial cells.

Let F_{n}^{4} be the homotopy category of (n − 1)-connected (n + 4)-dimensional polyhedra with finitely generated torsion free homology. In the stable range n ≥ 6 the category F^{4} = F_{n}^{4} is an additive category which does not depend on n. Freyd [F] showed that the \underline{isomorphism class group} K_{0}(F^{4}) is a free abelian group. Using the result above we compute this group:

Corollary. K_{0}(F^{4}) = ℤ^{29}

The generators in K_{0}(F^{4}) are given by five spheres, 23 congruence classes of 2-primary elementary polyhedra in the list X(L) and one 3-primary elementary polyhedron in X(L); see (1.4).

§ 1 The list X(L) of elementary polyhedra

We need the following elements in stable homotopy groups of spheres, compare Toda [T]. Let η_{n} = η ∈ π_{n+1}(S^{n}) = ℤ/2 be the Hopf map, and let η_{n}^{2} = ηη ∈ π_{n+2}(S^{n}) = ℤ/2 be the double Hopf map, n ≥ 3. Moreover let ν = ν_{n}, α = α_{n} ∈ π_{n+3}(S^{n}) = ℤ/24 be the generator of order 8 and 3 respectively.

Typeset by AMS-TEX

(1.1) Definition. We define a list L of 67 elements as follows. The \underline{spherical elements} S^{0}, S^{1}, S^{2}, S^{3}, S^{4} belong to L and the \underline{Hopf elements} η_{0}, η_{1}, η_{2} and (ηη)_{0} and (ηη)_{1} belong to L. Moreover the following \underline{words} consisting of letters η and v belong to L where v is a number:

\{ \begin{matrix}ηηvηη, ηηvη, ηvηη \\ ηηv, vηη, ηv, vη \end{matrix}\} with v ∈ \{1, …, 6\},

ηvη with v ∈ {1, 2, 3},
v with v ∈ {1, …, 12}.

Here v is a one letter word consisting of the number v.

We point out that the words in L are subwords of ηηvηη containing v. We can visualize the elements of L by \underline{graphs} as follows. First we describe spherical elements and Hopf elements by points and vertical edges. Then we describe the words in L by graphs consisting of such vertical edges and a diagonal edge which represents the letter v.

0 1 2 3 4 S^{0} • S^{1} • S^{2} • S^{3} • S^{4} • η_{0} • • η_{1} • • η_{2} • • (ηη)_{0} • • (ηη)_{1} • •  • • • • ηvη_{v∈{1,2,3}} • • v_{v∈{1,...,12}}

0 1 2 3 4 • • • • ηηvηη  • • • • ηηvη  • • • • ηvηη  • • • ηηv  • • • vηη  • • • ηv  • • • vη_{v∈{1,...,6}}

These graphs represent in the obvious way the words in L. We define the \underline{duality operator} D on L, this is the function

(1.2)  D : L → L

with DD = 1 defined by D(S^{i}) = S^{4−i} for i ∈ {0, …, 4} and D(η_{0}) = η_{2}, D(η_{1}) = η_{1}, D(ηη)_{0} = (ηη)_{1}. Moreover for a subword w of ηηvηη let D(w) = −w be the word obtained from w by reversing the order, that is D(ηηvη) = ηvηη. Hence if we look at the graph w then the graph D(w) is obtained by turning w around.

(1.3) Definition. Let n ≥ 5. We associate with each element g ∈ L a CW-complex X(g). The vertices of the graph given by g correspond exactly to the non-trivial cells of X(g). We call X(L) = \{(X(g), g ∈ L\} the list of \underline{elementary polyhedra} associated to L. For the spherical elements we set

X(S^{i}) = S^{n+i} with i ∈ {0, 1, 2, 3, 4}

where S^{n+i} is the (n + i)-sphere. For the Hopf elements we obtain the following 2-cell CW-complexes where Σ denotes the suspension.

X(η_{0}) = S^{n} ∪_{η} e^{n+1}
X(η_{1}) = S^{n+1} ∪_{η} e^{n+2} = ΣX(η_{0})
X(η_{2}) = S^{n+2} ∪_{η} e^{n+3} = Σ^{2}X(η_{0})
X(ηη)_{0} = S^{n} ∪_{ηη} e^{n+2}
X(ηη)_{1} = S^{n+1} ∪_{ηη} e^{n+3} = ΣX(ηη).

Moreover for the words in L we get the following CW-complexes with attaching maps defined by the edges in the graphs above. For a one point union A ∨ B let i_{1} : A → A ∨ B and i_{2} : B → A ∨ B be the inclusions and we set v = v · (ν + α) for v ∈ ℕ.

v ∈ {1, 2, 3} X(ηvη) = S^{n} ∨ S^{n+2} ∪_{i_{1}η} e^{n+2} ∪_{i_{1}v+i_{2}η} e^{n+4}

v ∈ \{1, …, 6\} \{ \begin{matrix}X(ηηvηη) & = S^{n} ∨ S^{n+1} ∪_{i_{1}ηη} e^{n+3} ∪_{i_{1}v+i_{2}ηη} e^{n+4} \\ X(ηηvη) & = S^{n} ∨ S^{n+2} ∪_{i_{1}ηη} e^{n+3} ∪_{i_{1}v+i_{2}η} e^{n+4} \\ X(ηvηη) & = S^{n} ∨ S^{n+1} ∪_{i_{1}η} e^{n+2} ∪_{i_{1}v+i_{2}ηη} e^{n+4} \\ X(ηηv) & = S^{n} ∪_{ηη} e^{n+3} ∪_{v} e^{n+4} \\ X(vηη) & = S^{n} ∨ S^{n+1} ∪_{i_{1}v+i_{2}ηη} e^{n+4} \\ X(ηv) & = S^{n} ∪_{η} e^{n+2} ∪_{v} e^{n+4} \\ X(vη) & = S^{n} ∨ S^{n+2} ∪_{i_{1}v+i_{2}η} e^{n+4} \end{matrix}

v ∈ {1, …, 12} X(v) = S^{n} ∪_{v} e^{n+4}

The cells of the CW-complexes X(w) with w ∈ L correspond exactly to the vertices of the graph w above and the edges in the graph w show precisely how these cells are attached.

(1.4) Definition. We say that X(w) with w ∈ L is 2-\underline{primary} if w is a Hopf element or if w is a subword of ηηvηη where v is divisible by 3 so that v(ν + α) is a multiple of ν. There are exactly 24 elements in L which are 2-primary with only one congruence given by (3.2) below, that is v = 3 is congruent to v = 1. The only 3-\underline{primary} polyhedron in X(L) is X(v) ≃ S^{n} ∪_{α} e^{n+4} where v = 8 so that v(ν + α) = −α. This definition is compatible with (3.5) below.

(1.5) Examples. The stabilization of the complex projective plane ℂP_{2} yields the space

X(η_{0}) = Σ^{n−2}ℂP_{2}.

Similarly the stabilization of the quaterionic projective plane ℍP_{2} gives us

X(v) = Σ^{n−4}ℍP_{2}  with  v = 1.

Moreover by (3.1) [B3] we see that the stabilization of ℂP_{3} yields

X(ηv) = Σ^{n−2}ℂP_{3}  with  v = 1

in the list above.

§ 2 The decomposition problem

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

(2.1)  X = A_{1} ⊕ … ⊕ A_{n}, n < ∞,

where A_{i} is indecomposable for all i ∈ {1, …, n}. The decomposition of X is \underline{unique} if B_{1} ⊕ … ⊕ B_{m} ≅ X ≅ A_{1} ⊕ … A_{n} implies that m = n and that there is a permutation σ with B_{σ(i)} ≅ A_{i}. The \underline{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 decompositions 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 \underline{wild representation type} if the solution of the decomposition problem would imply a solution of the following problem.

(2.2) 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 endomorphisms 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”.

If the list of all indecomposable objects of C is finite then C has \underline{finite} representation type. If the representation type of C is neither finite nor wild then C is of \underline{tame representation type}. In representation theory there are in general means to compute an explicit list of all indecomposable objects in C if C has tame representation type; in the tame cases described below such explicit lists actually are computed in the literature. If the number of objects in this list which satisfy a given torsion restraint is finite then we say that C has \underline{essentially 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. Here the torsion restraint is given by a number N which bounds the order of the torsion subgroup.

Next we describe the decomposition problem of homotopy theory. Let Top^{∗}/ ≃ be the homotopy category of pointed topological spaces. The set of morphisms X → Y in Top^{∗}/ ≃ is the set of homotopy classes [X, Y]. Isomorphisms in Top^{∗}/ ≃ are called homotopy equivalences and isomorphism types in Top^{∗}/ ≃ are homotopy types. Let A_{n}^{k} be the full subcategory of Top^{∗}/ ≃ consisting of (n − 1)-connected (n + k)-dimensional CW-complexes which are finite. The objects of A_{n}^{k} are also called A_{n}^{k}-polyhedra, see [W1]. The suspension Σ gives us sequences of functors

(2.3)  A_{1}^{k} ⟶^{Σ} A_{2}^{k} → … → A_{n}^{k} ⟶^{Σ} A_{n+1}^{k} → …

which describe the k-\underline{stem of homotopy theory}. The Freudenthal suspension theorem shows that these sequences stabelize in the sense that for k +1 < n the functor Σ : A_{n}^{k} → A_{n+1}^{k} is an equivalence of \underline{additive} categories so that

(1)  A^{k} = A_{n}^{k}  with  k + 1 < n

does not depend on n. This is the stable homotopy category of (−1)-connected finite type k-dimensional spectra. The biproduct in the additive category A^{k} is the one point union of spaces. We point out that for k + 1 = n the functor Σ is full and a 1-1 correspondence of homotopy types. The \underline{Spanier-Whitehead duality} is a contravariant additive functor

(2)  D : A^{k} → A^{k}

satisfying DD = 1 and D(S^{n+i}) = S^{n+k−i} for i ∈ {0, …, k}; compare for example [Co].

The k-stem of homotopy groups of spheres, denoted by π_{n+k}(S^{n}), n ≥ 2, is now known for fairly large k; for example one can find a complete list for k ≤ 19 in Toda’s book [T]. The k-stem of homotopy types, however, is still mysterious even for small k.

For k = 0 it is well known that each A_{n}^{0}-polyhedron X is homotopy equivalent to a one point union X ≃ S^{n} ∨ … ∨ S^{n} of n-dimensional spheres. Hence A^{0} has finite representation type. Moreover the following results are known.

• A^{1} and A^{2} have essentially finite representation type; see [W1], [Ch]

• A^{3} has tame representation type; see [BH], [B1]

• A^{k} with k ≥ 6 has wild representation type; see [BD]. Therefore only the representation types of A^{4} and A^{5} remain unknown. In [BD] we show:

• The full subcategory of A^{4}(= A_{n}^{4} with n ≥ 6) consisting of spaces X with

π_{n+1}X = π_{n+2}X = 0 has tame representation type. Clearly the main result of this paper (compare the theorem in the introduction) shows

• The full subcategory of A^{4} consisting of spaces X with torsion free homology has finite representation type.

§ 3 Decomposition and congruence of spaces with torsion free homology

In this paper we consider the full subcategory

(3.1)  F_{n}^{k} ⊂ A_{n}^{k}

consisting of (n − 1)-connected (n + k)-dimensional CW-complexes which are finite and have \underline{torsion free homology}. For example, each nontrivial element α in the (k − 1)-stem, α ∈ π_{n+k−1}(S^{n}), yields the canonical \underline{2-cell complex} S^{n} ∪_{α} e^{n+k} ∈ F_{n}^{k}, n ≥ 2, which is indecomposable. Hence elements in homotopy groups of spheres can essentially be identified with special indecomposable objects in F_{n}^{k}, k ≥ 2. The decomposition in F_{n}^{4} is not unique. For example Freyd [F] points out that for n ≥ 5 there is a homotopy equivalence

(3.2)  S^{n} ∨ (S^{n} ∪_{ν} e^{n+4}) ≃ S^{n} ∨ (S^{n} ∪_{3ν} e^{n+4})

in F_{n}^{4} where, however, the CW-complexes S^{n} ∪_{ν} e^{n+4} and S^{n} ∪_{3ν} e^{n+4} are not homotopy equivalent. Here ν ∈ π_{n+3}(S^{n}) is a generator of order 8 as in § 1.

Unsöld [U1], [U2] obtained an algebraic characterization of the homotopy types in F_{n}^{4} with n ≥ 3. Using this result for n ≥ 5 we show the following main result of this paper which implies the theorem in the introduction.

(3.3) Theorem. The list X(L) of 67 elementary polyhedra in (1.3) is a complete list of all indecomposable spaces in F^{4}. Hence for n ≥ 5 each (n − 1)-connected (n + 4)-dimensional finite polyhedron X with torsion free homology admits a decomposition

X ≃ X_{1} ∨ … ∨ X_{j}

with X_{i} ∈ X(L) for 1 ≤ i ≤ j. Moreover the Spanier-Whitehead duality functor D : F^{4} → F^{4} is completely understood on objects since D(X(w)) = X(Dw) for w ∈ L. Here we use the duality operator in (1.2).

Following Freyd [F] and Cohen [Co] 4.26 we use the following notation.

(3.4) Definition. We say that two spaces X, Y in A^{k} are \underline{congruent} and we write X ≡ Y if (a) or equivalently (b) is satisfied.

(a) There exists a space Z in A^{k} such that X ∨ Z ≃ Y ∨ Z are homotopy equivalent

(b) There exists a homotopy equivalence X ∨ B_{X} ≃ Y ∨ B_{X} where B_{X} is the unique one point union of spheres which has the same Betti numbers as X, that is H_{∗}(X)/torsion = H_{∗}(B_{X}).

(3.5) Definition. Let p be a prime. A space X in A^{k} is a p-\underline{primary space} if there exists a homotopy commutative diagram

X ⟶^{p^{N} ·1_{X}} X

↘ ↗

B

where B is a one point union of spheres. Here p^{N} is a power of the prime p and p^{N} · 1_{X} is a multiple of the identity of X in the abelian group of homotopy classes [X, X] in A^{k} and N can not be chosen to be N = 0. This implies that X is not a one point union of spheres.

Recall that for any small additive category C (for example C = A^{k} or C = F^{k}, k ≥ 0) we have the \underline{isomorphism class group} K_{0}(C). This is the abelian group with one generator [A] for each isomorphism class of objects A ∈ C with relations [A]+[B] = [A⊕B]. This is just the \underline{Grothendieck group} of C as defined by Bass [Ba]. A typical element of K_{0}(A) is a formal difference [A]−[B] with [A]−[B] = [A′]−[B′] if and only if there exists an isomorphism in C of the form A⊕B′ ⊕C ≅ A′ ⊕B ⊕C for some object C in C. The following result is due to Freyd [F]; see also Cohen [Co] 4.44.

(3.6) Theorem of Freyd. Let k ≥ 0. Then K_{0}(A^{k}), resp. K_{0}(F^{k}), is a free abelian group generated by the spheres in A^{k} and by the congruence classes of indecomposable p-primary spaces in A^{k} (resp. F^{k}) where p runs through all primes.

Such a wonderful result yields the crucial task to compute the generators of K_{0}(A^{k}), resp. K_{0}(F^{k}), explicitly. According to the remarks in (2.3) we get:

• K_{0}(A^{0}) = ℤ generated by S^{n}

• K_{0}(A^{1}) = ℤ^{∞} generated by S^{n}, S^{n+1} and all elementary Moore spaces M(C, n) where C = ℤ/p^{N} with p a prime and N ≥ 1.

• K_{0}(A^{2}) = ℤ^{∞} generated by spheres, elementary Moore spaces, and elementary Chang polyhedra, see [Ch], [B1], [Hi1], [Hi2].

• K_{0}(A^{3}) = ℤ^{∞} generated by the elementary polyhedra obtained in Baues-Hennes [BH]; see also [B1].

• The computation of generators in K_{0}(A^{k}), k ≥ 6, is a wild problem in the

sense of representation theory; this follows from [BD]. For the category F^{k} of torsion free polyhedra we get accordingly:

• K_{0}(F^{0}) = ℤ generated by S^{n}

• K_{0}(F^{1}) = ℤ^{2} generated by S^{n}, S^{n+1}

• K_{0}(F^{2}) = ℤ^{4} generated by S^{n}, S^{n+1}, S^{n+2}, X(η_{0})

• K_{0}(F^{3}) = ℤ^{7} generated by S^{n}, S^{n+1}, S^{n+2}, S^{n+3}, and X(η_{0}), X(η_{1}),

X(ηη)_{0}.

As a next step we get:

(3.7) Theorem. K_{0}(F^{4}) = ℤ^{29} is generated by the 5 spheres S^{n}, …, S^{n+4} in F^{4}, by the 23 congruence classes of 2-primary poylhedra in X(L) and by the unique 3-primary polyhedron in X(L); see (1.4).

Proof. This is a consequence of (3.6), (3.3) and the description of p-primary indecomposable polyhedra in X(L) in (1.4). The only congruence between 2-primary polyhedra in X(L) is the one described in (3.2). This shows that there are exactly 23 congruence classes of 2-primary polyhedra in X(L). q.e.d.

Moreover we see by the result (A.5) in the Appendix.

(3.8) Remark. Let p be a prime. The computation of the set of generators in K_{0}(F^{k}) given by congruence classes of p-primary polyhedra is a wild problem for k ≥ 10(p − 1).

§ 4 The algebraic classification of spaces in F^{4}

We first repeat the classification theorem of Unsöld [U1], [U2]. He defines the following algebraic category SF^{4}.

(4.1) Definition. Objects in SF^{4} are tuple of abelian groups

H = (H_{0}, H_{1}, H_{2}, H_{3}, H_{4}, π_{1}, π_{2}) ∈ Ab^{7}

where H_{i} with i ∈ {0, …, 4} is finitely generated and free abelian together with the following diagrams (1), (2), (3) in Ab:

(1) An exact sequence

H_{3} → π_{1} ⊗ ℤ/2 → π_{2} → H_{2} ⟶^{b} H_{0} ⊗ ℤ/2 ⟶^{η^{1}} π_{1} → H_{1} → 0

(2) Let P = ker(H_{0} ⟶^{q} H_{0} ⊗ ℤ/2 ⟶^{η^{1}} π_{1}) where q is the quotient map. Then

0 ⟶ H_{0}  ⟶^{2}  P  ⟶ ker(η^{1}) ⟶ 0
qη^{1}q↓  ↓^{T}  ↓^{Ω}
π_{1} ⊗ ℤ/2 ⟶ π_{2} ⊗ ℤ/2 ⟶ ker(b) ⊗ ℤ/2

commutes where qη^{1}q is given by (1) and where Ω is determined by the extension

0 → ker(b) → H_{2} → ker(η^{1}) → 0

given by (1). The top row of the diagram is short exact.

(3) Moreover for the abelian group

Γ_{3} = (H_{0} ⊗ ℤ/24 ⊕ π_{2} ⊗ ℤ/2)/{(ξ ⊗ 6, T (ξ)); ξ ∈ P ⊂ H_{0}}

defined by T in (2) a homomorphism

b_{4} : H_{4} → Γ_{3}

is given. A morphism between such objects in SF^{4} is a tuple of homomorphisms H → H′ in Ab^{7} which is compatible with all arrows in the diagrams (1), (2) and (3). Clearly SF^{4} is an additive category with the direct sum of objects given by the direct sum of abelian groups and morphisms.

In [U1], [U2] one finds the proof of the following result.

(4.2) Classification theorem. There is an additive functor λ : F^{4} → SF^{4} which is full and representative and which reflects isomorphisms.

The functor carries a space X to the certain exact sequence of J.H.C. Whitehead [W2] of X together with the secondary homotopy operation T which was introduced by Unsöld.

(4.3) Corollary. There is a 1-1 correspondence of homotopy types in F^{4} and isomorphism types in SF^{4}. Moreover this is also a 1-1 correspondence between indecomposable objects.

In order to classify indecomposable objects in SF^{4} we transform Unsölds classification theorem into a “matrix problem” which can be solved by methods of representation theory. For this we observe that each space X in F^{4} as given by an attaching map

(4.4)  f_{4} : ⋁_{h} S^{3} → X^{3}

where we use \underline{stable notation}. Moreover using the homology decomposition of X we may assume that f_{4} factors through the 2-skeleton X^{2}. Then f^{4} defines the homomorphism in (4.1) (3)

(1)  b_{4} : H_{4} = ℤ^{h} → Γ_{3}X^{3} = Γ_{3}

where Γ_{3}X^{3} = image(π_{3}X^{2} → π_{3}X^{3}). We know by the classification of objects in A^{3} in Baues-Hennes [BH] that X^{3} is a one point union of spaces

(2)  \{ \begin{matrix}X_{1} = S^{0},  X_{6} = S^{1} ∪_{η} e^{6} \\ X_{2} = S^{0} ∪_{η} e^{2},  X_{7} = S^{3} \\ X_{3} = S^{0} ∪_{ηη} e^{3} \\ X_{4} = S^{1} \\ X_{5} = S^{2} \end{matrix}

For 0 ≤ d < ∞ we write

(3)  dZ = \underbrace{Z ∨ … ∨ Z}_{d}

for the d-fold one point union of d copies of the space Z. Then we can assume that f_{4} in (4.4) is given by

(4)  f_{4} : hS^{3} → X^{3} = ⋁_{i≤7} d_{i}X_{i}

where 0 ≤ d_{i}. Here, however, we have

(5)  \{ \begin{matrix}Γ_{3}(S^{1} ∪_{η} e^{6}) = Γ_{3}(X_{6}) = 0 \\ Γ_{3}(S^{3}) = Γ_{3}(X_{7}) = 0 \end{matrix}

Therefore f_{4} has a factorization f′_{4} to ⋁_{i≤5}d_{i}X_{i} and the mapping cone C(f_{4}) of f_{4}

satisfies

(6)  X = C(f_{4}) = C(f′_{4}) ∨ d_{6}X_{6} ∨ d_{7}X_{7}

Thus it suffices to consider the decompositions of C(f′_{4}). We therefore assume now that X^{3} in (4.4) satisfies

X^{3} = ⋁_{i≤5}d_{i} · X_{i}

and that f_{4} = f′_{4}. The homotopy class of f_{4} is determined by the associated homomorphism b_{4} in (1). Moreover Γ_{3}X_{i} with i ≤ 5 is computed by the following list where we write Ā = A ⊗ ℤ/2; compare (4.1) (1), (2), (3).

(7)

X  H_{3} → π̄_{1} → π_{2} → H_{2} → H̄_{0} → π_{1}  P →^{T} π̄_{2}  Γ_{3}
X_{1}  0  ℤ/2 = ℤ/2  0  ℤ/2 = ℤ/2  ℤ ↠ ℤ/2  ℤ/24
X_{2}  0  0  ℤ →^{2} ℤ → ℤ/2  0  ℤ ↠ ℤ/2  ℤ/12
X_{3}  ℤ ↠ ℤ/2  0  0  ℤ/2 = ℤ/2  ℤ → 0  ℤ/12
X_{4}  0  ℤ/2 = ℤ/2  0  0  ℤ  0 → ℤ/2  ℤ/2
X_{5}  0  0  ℤ = ℤ  0  0  0  0  ℤ/2

Now let V_{i} = ℤ^{d_{i}} be the free abelian group of rank d_{i} with d_{i} given by (4). Then the computation of Γ_{3} in this list shows that b_{4} in (1) is given by a homomorphism

(8) b_{4} : H_{4} = ℤ^{h} → Γ_{3}X^{3} = V_{1} ⊗ ℤ/24 ⊕ (V_{2} ⊕ V_{3}) ⊗ ℤ/12 ⊕ (V_{4} ⊕ V_{5}) ⊗ ℤ/2

This leads to the following matrix problem where we describe the action of automorphisms in SF^{4} on b_{4}.

(4.5) Matrix Problem. Let V_{1}, V_{2}, V_{3}, V_{4}, V_{5} and H_{4} be finitely generated free abelian groups. Moreover let M be an automorphism of V = V_{1} ⊕ V_{2} ⊕ V_{3} ⊕ V_{4} ⊕ V_{5} given by a matrix M_{ij} of the form

V_{1} V_{2} V_{3} V_{4} V_{5}
V_{1}  · 2\ 2\ 12\ 12\
V_{2}  ⋯ 12\ 6\
V_{3}  · 2\ · 12\ 12\
V_{4}  0 0 0 ·  ·
V_{5}  0 0 0 0  ·

Here 0 is the zero matrix and · denotes matrices. Moreover n\ with n ∈ {2, 6, 12} denotes a matrix divisible by n. The automorphisms M of the form above describe a subgroup G of Aut (V). The group G acts on the abelian group

Γ_{3} = V_{1} ⊗ ℤ/24 ⊕ (V_{2} ⊕ V_{3}) ⊗ ℤ/12 ⊕ (V_{4} ⊕ V_{5}) ⊗ ℤ/2

in such a way that the canonical quotient map V ↠ Γ_{3} is G-equivariant. We define an equivalence relation for homomorphisms

b_{4}, b′_{4} : H_{4} → Γ_{3}

by setting b_{4} ∼ b′_{4} if there exists an automorphism N of H_{4} and M ∈ G with Mb_{4}N^{−1} = b′_{4}. Each b_{4} is equivalent to a sum b_{4}^{1} ⊕ … ⊕ b_{4}^{r} where the b_{4}^{i} are indecomposable.

(4.6) Theorem. The indecomposable objects in F^{4}, resp. SF^{4}, are given by S^{3}, S^{1} ∪_{η} e^{3} and by the indecomposable homomorphisms b_{4} in the matrix problem (4.5).

Proof. Given X^{3} = ⋁_{i≤5}d_{i}X_{i} as in (4.4) we say that the homology H_{∗} of X = C(f_{4})

satisfies

(1)  \begin{matrix}H_{0} = V_{1} ⊕ V_{2} ⊕ V_{3} \\ H_{1} = V_{4} \\ H_{2} = V_{2} ⊕ V_{5} \\ H_{3} = V_{3} \\ H_{4} = ℤ^{h} \end{matrix}

Moreover according to the list (4.4) (7) we get the following homotopy groups π_{∗} of X

(2)  \begin{matrix}π_{1} = (V_{1} ⊕ V_{3}) ⊗ ℤ/2 ⊕ V_{4} \\ π_{2} = (V_{1} ⊕ V_{4}) ⊗ ℤ/2 ⊕ V_{2} ⊕ V_{5} \end{matrix}

This yields the tuple H of abelian groups in (4.1). The associated exact sequence (4.1) (1) is obtained as follows where the operators are determined by the list (4.4) (7); we again set

V̄ = V ⊗ ℤ/2.

In the following a group in the top row is given by the direct sum of groups in the corresponding column.

(3)  \begin{matrix}H_{3} & → & π̄_{1} & → & π_{2} & → & H_{2} & → & H̄_{0} & → & π_{1} & → & H_{1} \\ 0 & & V̄_{1} & →^{1} & V̄_{1} & & 0 & & V̄_{1} & →^{1} & V̄_{1} & & 0 \\ 0 & & 0 & & V_{2} & →^{2} & V_{2} & →^{1} & V̄_{2} & & 0 & & 0 \\ V_{3} & →^{1} & V̄_{3} & & 0 & & 0 & & V̄_{3} & →^{1} & V̄_{3} & & 0 \\ 0 & & V̄_{4} & →^{1} & V̄_{4} & & 0 & & 0 & & V_{4} & →^{1} & V_{4} \\ 0 & & 0 & & V_{5} & →^{1} & V_{5} & & 0 & & 0 & & 0 \end{matrix}

Moreover we obtain the following natural maps from (4.1) (2), (3) and (4.4) (7).

(4)  \begin{aligned}H_{0} & ⊃ & P & →^{T} & π̄_{2} & → & Γ_{3} & ← & H_{0} ⊗ ℤ/24 \\ V_{1} & ⊃ & 2V_{1} & →^{\frac{1}{2}} & V̄_{1} & →^{12} & V_{1} ⊗ ℤ/24 & ←^{1} & V_{1} ⊗ ℤ/24 \\ V_{2} & = & V_{2} & →^{1} & V̄_{2} & →^{6} & V_{2} ⊗ ℤ/12 & ←^{1} & V_{2} ⊗ ℤ/24 \\ V_{3} & ⊃ & 2V_{3} & & 0 & & V_{3} ⊗ ℤ/12 & ←^{1} & V_{3} ⊗ ℤ/24 \\ & & 0 & & V̄_{4} & →^{1} & V_{4} ⊗ ℤ/2 & & 0 \\ & & 0 & & V̄_{5} & →^{1} & V_{5} ⊗ ℤ/2 & & 0 \end{aligned}

The tuple H given by (1) and (2) together with (3) and (4) determine together with

(5)  b_{4} : H_{4} → Γ_{3}

an object (also denoted by H) in SF_{4}. We now consider an automorphism of this object H in SF_{4}. Such an automorphism is given by automorphisms of the abelian groups in (1) and (2) which are matrices of the following form. For this we use the compatibility of the automorphism with the operators in (3) and (4). The automorphism of H_{0} = V_{1} ⊕ V_{2} ⊕ V_{3} is given by

(6)  ( \begin{matrix}y_{0} & 2a & 2b \\ a′ & w_{0} & c \\ b′ & 2c′ & u_{0} \end{matrix} )

where for example c ∈ Hom(V_{3}, V_{2}) and y_{0} ∈ Hom(V_{1}, V_{1}). The automorphism of H_{1} = V_{4} is given by

(7)  y_{1} ∈ Aut(V_{4})

The automorphism of H_{2} = V_{5} ⊕ V_{2} is given by

(8)  ( \begin{matrix}y_{2} & d \\ 2d′ & w_{2} \end{matrix} )

Next the automorphism of H_{3} = V_{3} is given by

(9)  u_{3} ∈ Aut(V_{3})

Moreover the automorphism of π_{1} = V̄_{1} ⊕ V̄_{3} ⊕ V_{4} is given by

(10)  ( \begin{matrix}y_{0} & 0 & α \\ b′ & u_{0} & β \\ 0 & 0 & y_{1} \end{matrix} )  where u_{0} ⊗ ℤ/2 = u_{3} ⊗ ℤ/2

Finally the automorphism of π_{2} = V̄_{1} ⊕ V̄_{4} ⊕ V_{5} ⊕ V_{2} is given by

(11)  ( \begin{matrix}y_{0} & α & γ & δ \\ 0 & y_{1} & ρ & λ \\ 0 & 0 & y_{2} & 2d \\ 0 & 0 & d′ & w_{2} \end{matrix} )  where  \{ \begin{matrix}δ = a ⊗ ℤ/2, \\ λ = 0, \\ w_{2} ⊗ ℤ/2 = w_{0} ⊗ ℤ/2 \end{matrix}

In fact, using b_{2} : H_{2} → H̄_{0} in (3) we see that the entries 2a, 2c′ and 2d′ have to be divisible by 2 and that w_{2} ⊗ ℤ/2 = w_{0} ⊗ ℤ/2. Moreover using b_{3} : H_{3} → π̄_{1} in (3) we see that the entry 2b has to be divisible by 2 and that u_{3} ⊗ ℤ/2 = u_{0} ⊗ ℤ/2. Moreover T in (4) shows that δ = a ⊗ ℤ/2, λ = 0 and w_{2} ⊗ ℤ/2 = w_{0} ⊗ ℤ/2.

The action of the automorphism on Γ_{3} is given by π̄_{2} → Γ_{3} ← H_{0} ⊗ ℤ/24 in (4). Hence only (11) ⊗ ℤ/2 is used for the action on π̄_{2} and hence on Γ_{3}. Here y_{2} ⊗ ℤ/2 and w_{2} ⊗ ℤ/2 are automorphisms. Since GL_{n}(ℤ) → GL_{n}(ℤ/2) is surjective for all n we can find y_{2}^{♯} ∈ Aut(V_{5}) with

(12)  y_{2}^{♯} ⊗ ℤ/2 = y_{2} ⊗ ℤ/2

We now construct a new automorphism of H which has the same effect on Γ_{3} as the original automorphism chosen above. The new automorphism of H acts via (4.5) on Γ_{3} by the automorphism of V_{1} ⊕ V_{2} ⊕ V_{3} ⊕ V_{4} ⊕ V_{5} given by the matrix, compare (4.5):

(13)  ( \begin{matrix}y_{0} & 2a & 2b & α & γ \\ a′ & w_{0} & c & 0 & d^{♯} \\ b′ & 2c′ & u_{0} & 0 & 0 \\ 0 & 0 & 0 & y_{1} & δ \\ 0 & 0 & 0 & 0 & y_{2}^{♯} \end{matrix} )  with d^{♯} ⊗ ℤ/2 = d′ ⊗ Z/2

In fact, given (13) we see that y_{2}^{♯} ⊗ ℤ/2 ∈ Aut (V̄_{5}) and w_{0} ⊗ ℤ/2 ∈ Aut (V̄_{2}). Hence we can choose y_{2} ∈ Aut (V_{5}), w_{2} ∈ Aut (V_{2}) with

(14)  \{ \begin{matrix}y_{2} ⊗ ℤ/2 = y_{2}^{♯} ⊗ ℤ/2, \\ w_{2} ⊗ ℤ/2 = w_{0} ⊗ ℤ/2. \end{matrix}

Then we get

(15)  ( \begin{matrix}y_{2}, & 0 \\ 2d′, & w_{2} \end{matrix} ) ∈ Aut V_{5} ⊕ V_{2})

where we set d = 0. We are allowed to choose d = 0 since the action of d ∈ Hom (V_{2}, V_{5}) is trivial on Γ_{3} (in fact, any map S^{0} ∪_{η} e^{2} → S^{2} induces the trivial homomorphism 0 : π_{3}(S^{0} ∪_{η} e^{2}) → π_{3}S^{2}). This yields (8) and therefore (6) … (11) are defined by (13) and (15) and by choosing an automorphism u_{3} in (9) with u_{3} ⊗ ℤ/2 = u_{0} ⊗ ℤ/2 and some β in (10). This is the new automorphism of H with the property that the new automorphism induces the same automorphism of Γ_{3} as the original automorphism. q.e.d.

§ 5 Computation of the indecomposable homomorphisms b_{4} in the matrix problem (4.5)

Denote by Λ the ring of all 5 × 5 matrices of the form

(1)  M = ( \begin{matrix}a_{11} & 2a_{12} & 2a_{13} & 12a_{14} & 12a_{15} \\ a_{21} & a_{22} & a_{23} & 12a_{24} & 6a_{25} \\ a_{31} & 2a_{32} & a_{33} & 12a_{34} & 12a_{35} \\ 0 & 0 & 0 & a_{44} & a_{45} \\ 0 & 0 & 0 & 0 & a_{55} \end{matrix} )

where a_{ij} ∈ ℤ. Let U = U_{1} ⊕ U_{2} ⊕ U_{3} ⊕ U_{4} ⊕ U_{5} where U_{1} = ℤ/24, U_{2} = U_{3} = ℤ/12, U_{4} = U_{5} = ℤ/2. Consider U as Λ-ℤ-bimodule_{Λ}U_{ℤ} in an obvious way. Remind that a matrix over U is, by definition (cf. [D]), an element of P ⊗_{Λ} U ⊗ H^{∗}, where P and H are, respectively, (finitely generated right) projective modules over Λ and ℤ. It is more convenient to identify this tensor product with U(H, P) = Hom(H, P ⊗_{Λ} U). Two matrices u ∈ U(H, P) and u′ ∈ U(H′, P′) are isomorphic if there are isomorphisms α : H → H′ and β : P → P′ such that βu = u′α. Put e_{i} = e_{ii} (matrix unit) and P_{i} = e_{i}Λ. Then P can be uniquely decomposed as P = ⨁_{i=1}^{5} V_{i} ⊗ P_{i} for some free abelian groups V_{i}. In this case P ⊗_{Λ} U is just ⨁_{i=1}^{5} V_{i} ⊗U_{i}. Thus we come to the same “matrix problem” as has been formulated above in (4.5).

We shall write the elements of U(H, P) as quintuples of matrices (u_{1}, u_{2}, u_{3}, u_{4}, u_{5}), u_{i} being of size m_{i} × n if H = nℤ, V_{i} = m_{i}ℤ_{i}, with the entries from ℤ/24 for i = 1, from ℤ/12 for i = 2, 3 and from ℤ/2 for i = 4, 5. The matrices M of the form (1) define the “admissible transformations” of rows of the matrices u_{i}. For instance, as we have a_{21} in M we can add any multiple of a row of the matrix u_{1} to any row of the matrix u_{2}. On the other hand, as we have 2a_{12} in M, we can add only even multiples of rows of u_{2} to the rows of u_{1}, etc.

Certainly U is also a Λ̄-ℤ-bimodule, where ¯ always denotes the reduction modulo 24. For convenience we denote it by Ū though the elements from U and from Ū are the same. Moreover the matrices from U(P, H) coincide with the matrices from Ū(P̄, H̄) though nonisomorphic U-matrices can become isomorphic as Ū-matrices. Consider first the 2-primary part Ũ of Ū. It is a Λ̃-ℤ/8-bimodule where Λ̃ is the ring of matrices of the form:

(2)  ( \begin{matrix}a_{11} & 2a_{12} & 2a_{13} & 4a_{14} & 4a_{15} \\ a_{21} & a_{22} & a_{23} & 4a_{24} & 2a_{25} \\ a_{31} & 2a_{32} & a_{33} & 4a_{34} & 4a_{35} \\ 0 & 0 & 0 & a_{44} & a_{45} \\ 0 & 0 & 0 & 0 & a_{55} \end{matrix} )

with a_{ij} ∈ ℤ/8. Namely Ũ = ⨁_{i=1}^{5} Ũ_{i} where Ũ_{1} = ℤ/8, Ũ_{2} = Ũ_{3} = ℤ/4 and Ũ_{4} = Ũ_{5} = ℤ/2. Denote by z_{i} the image of z ∈ ℤ in Ũ_{i} and write u < v for two elements u, v ∈ Ũ if there is an element a ∈ Λ̃ such that au = v but not vice versa. Then

(3) 1_{1} < 1_{3} < 1_{2} < 2_{1} < 2_{3} < 2_{2} < 4_{1} and 1_{5} < 1_{4} < 4_{1} and 1_{5} < 2_{2}

and there are no other “<” relations between these elements. We also put a < 0 for each nonzero a.

Let u be a Ũ-matrix. Choose in the matrices u_{1}, u_{2}, u_{3} the elements divisible by the least possible power of 2. If such an element exists in u_{1} then using admissible transformations we can obtain zeroes for all other elements in the same row as well as in the same column in all three matrices u_{1}, u_{2}, u_{3}. The same is possible if there are no such elements in u_{1} but there is one in u_{3} just as in the case when such elements only exist in u_{2}. Therefore we may only consider elements u such that each row of u_{1}, u_{2}, u_{3} contains at most one nonzero element, as well as each column in all these three matrices. For each column number i denote by a_{i} the nonzero element in this column in the matrices u_{1}, u_{2}, u_{3} or 0 if there is no such element at all. For two column numbers i, j put i ≤ j if a_{i} ≤ a_{j} with respect to the order defined above.

Among the numbers of columns of the matrices u_{4}, u_{5} containing a nonzero element b find the greatest possible with respect to this order. If b is the matrix u_{5} then we can make all other elements of its row zeroes as well as all elements of its column in both matrices u_{4}, u_{5}. The same is true if there are no such elements in u_{5} at all but there is one in u_{4}. After that the element u evidently splits into a direct sum of matrices of the following forms:

(4)  \{ \begin{matrix}(1_{4}), (1_{5}), (a_{1}) with a ∈ \{1, 2, 4\}, \\ (a_{2}), (a_{3}) with a ∈ \{1, 2\}, \\ ( \begin{matrix}1_{2} \\ 1_{5} \end{matrix} ) , and \\ ( \begin{matrix}a_{1} \\ 1_{4} \end{matrix} ) , ( \begin{matrix}a_{1} \\ 1_{5} \end{matrix} ) , ( \begin{matrix}a_{2} \\ 1_{4} \end{matrix} ) , ( \begin{matrix}a_{3} \\ 1_{4} \end{matrix} ) , ( \begin{matrix}a_{3} \\ 1_{5} \end{matrix} ) , with a ∈ \{1, 2\}. \end{matrix}

Hence the lower index refers to the index in (4.4) (2) and the index of V_{i} in (4.5) respectively. With v = 3a these matrices correspond to the following 20 elements in the list L.

(5)  \{ \begin{matrix}(ηη)_{1}, η_{2}, v with v ∈ \{3, 6, 12\}, \\ ηv, ηηv with v ∈ \{3, 6\}, \\ ηvη with v = 3, and \\ vηη, vη, ηvηη, ηηvηη, ηηvη with v ∈ \{3, 6\}. \end{matrix}

These describe a complete list of all congruence classes of 2-primary polyhedra with non trivial homology H_{4}. Hence together with η_{1} in (4.5), η_{0} and (ηη)_{0} the list (5) yields a complete list of 23 congruence classes of 2-primary polyhedra.

If we consider the 3-part of U-matrices the answer is quite evident: there is only one non-trivial indecomposable matrix 1_{1} (isomorphic to 1_{2} and to 1_{3}). At last gluing together 2-part and 3-part we get the following list of indecomposable U-matrices:

(6)  \{ \begin{matrix}(1_{4}), (1_{5}), (v_{1}) with v ∈ \{1, …, 12\}, \\ (v_{2}), (v_{3}) with v ∈ \{1, …, 6\}, \\ ( \begin{matrix}v_{2} \\ 1_{5} \end{matrix} ) with v ∈ \{1, 2, 3\}, and \\ ( \begin{matrix}v_{1} \\ 1_{4} \end{matrix} ) , ( \begin{matrix}v_{1} \\ 1_{5} \end{matrix} ) , ( \begin{matrix}v_{2} \\ 1_{4} \end{matrix} ) , ( \begin{matrix}v_{3} \\ 1_{4} \end{matrix} ) , ( \begin{matrix}v_{3} \\ 1_{5} \end{matrix} ) with v ∈ \{1, …, 6\}. \end{matrix}

It is easy to check that all these U-matrices are indeed non-isomorphic and that they form a complete list of all indecomposable solutions of the matrix problem (4.5). The matrices (6) correspond to the following 59 elements in the list L.

(7)  \{ \begin{matrix}(ηη)_{1}, η_{2}, v with v ∈ \{1, …, 12\}, \\ ηv, ηηv with v ∈ \{1, …, 6\}, \\ ηvη with v ∈ \{1, 2, 3\}, and \\ vηη, vη, ηvηη, ηηvηη, ηηvη with v ∈ \{1, …, 6\}. \end{matrix}

The list (7) together with the sphere S^{4} describes all indecomposable polyhedra with non trivial homology H_{4}. Together with S^{0}, S^{1}, S^{2}, S^{3}, η_{1}, η_{0} and (ηη)_{0} this yields the complete list of 67 indecomposable polyhedra. q.e.d.

Appendix: The p-local decomposition problem

Hans-Joachim Baues and Hans-Werner Henn

Let p be a prime and let ℤ_{p} be the smallest subring of ℚ containing 1/q for all primes q with q ≠ p. 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 X has p-local dimension dim_{p}(X) ≤ n if H_{i}X = 0 for i > n and if H_{n}X is a free ℤ_{p}-module. Let n ≥ 2 and let A_{n}^{k}(p) be the full subcategory of the homotopy category Top^{∗}/ ≃ consisting of p-local finite type CW-complexes X which are (n−1)-connected and which satisfy dim_{p}(X) ≤ n+k. The suspension Σ gives us a sequence of functors

(A.1)  A_{2}^{k}(p) → A_{3}^{k}(p) → … → A_{n}^{k}(p) →^{Σ} A_{n+1}^{k}(p) → …

which describes the p-\underline{local k-stem} of homotopy theory. The Freudenthal suspension theorem shows that the sequence (A.1) stabilizes in the sense that for k + 1 < n the functor Σ : A_{n}^{k}(p) → A_{n+1}^{k}(p) is an equivalence of categories. Hence

(A.2)  A^{k}(p) = A_{n}^{k}(p) with k + 1 < n

does not depend on n. This is the stable homotopy category of (−1)-connected p-local finite type k-dimensional spectra. The biproduct in the additive category A^{k}(p) is the one point union of spaces. The following result is well known; compare for example Wilkerson [Wi] and Freyd [F].

(A.3) Proposition. Each space X in A^{k}(p) has a unique decomposition X = X_{1} ∨ … ∨ X_{j} where all X_{i} with i = 1, …, j are indecomposable objects in A^{k}(p). Moreover the congruence classes of indecomposable p-primary spaces in A^{k} are in 1-1 correspondence with the indecomposable spaces in A^{k}(p) which are not p-local spheres.

This leads to the problem to compute a complete list of indecomposable objects in A^{k}(p) and to determine the representation type of A^{k}(p). For example Baues-Hennes [BH] show that

• A^{3}(2) has tame representation type. Moreover Henn [H] proved that for all odd primes p

• A^{4p−5}(p) has tame representation type. Now let

(A.4)  F^{k}(p) ⊂ A^{k}(p)

be the full subcategory consisting of CW-complexes X for which all homology groups H_{i}X are free ℤ_{p}-modules. The purpose of this Appendix is the proof of the following observation.

(A.5) Proposition. For each prime p there exists k < ∞ such that the category F^{k}(p) has wild representation type.

For example we show that F^{10}(2) has wild representation type. On the other hand by the theorem in the introduction above F^{4}(2) has finite representation type. The representation types of F^{5}(2), F^{6}(2), …, F^{9}(2) are unknown. It is an interesting problem to determine for each prime p the smallest number k = k(p) for which F^{k}(p) has wild representation type. For example we have 5 ≤ k(2) ≤ 10 so that it is a nice finite task to compute k(2). The decomposition problem in F^{4}(2) is solved in (3.5) above. Using the proof below we get for all p:

(A.6) Addendum. 4(p − 1) ≤ k(p) ≤ 10(p − 1).

Proof of (A.5). For p = 2 we know the q-stem π_{q}^{s}(S^{0}) = ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2; compare Toda [T]. Hence we obtain spaces in F^{10}(2) by the stable attaching map

(1)  f : ⋁_{A} S^{q} → ⋁_{B} S^{0}

where A and B are finite index sets. Such attaching maps are in 1−1 correspondence with homomorphisms

(2)  φ : V_{A} → V_{B} ⊗ (ℤ/2 ⊕ ℤ/2 ⊕ ℤ/2)

where V_{A} and V_{B} are ℤ/2 -vector spaces with basis A and B respectively. Now φ is a quiver of the form

(3)  V_{A} ⇉^{→} V_{B}

which is easily seen to be wild since we can choose A = B and the top arrow an isomorphism. Then we get the wild problem in (2.2).

Next let p be a prime > 5. We look at p-local spectra X of finite type with p-local cells concentrated in dimension 2k(p − 1) where k ∈ {0, …, 5} such that X/X^{0} is a one point union of spheres. Then the homotopy types of such X are in 1 − 1 correspondence with the isomorphism classes of 5 linear maps

(4)  g_{k} : V_{k} → V_{0} with k ∈ {1, …, 5}

where V_{k} = H_{2k(p−1)}(X) ⊗ ℤ/p for k ≥ 0. Now (4) corresponds to the quiver with 5 arrows

(5)  \begin{aligned}& • \\ & ↑ \\ • ↖ & & ↗ • \\ & • \\ • ↙ & & ↘ • \end{aligned}

which is known to be wild; see Ringel [R]. In order to obtain (4) we only have to know that the p-torsion of stable homotopy groups of spheres satisfies for 0 < n < 2p(p − 1) − 2 the equation

(6)  π_{n}(S^{0}) = \{ \begin{matrix}ℤ/p if n = 2k(p − 1) − 1 and k ∈ \{1, …, p − 1\} \\ 0 otherwise. \end{matrix}

See Ravenel [Ra] 1.1.13 and 1.1.14.

Now X is the mapping cone of

(7)  f : A = A_{1} ∨ A_{2} ∨ A_{3} ∨ A_{4} ∨ A_{5} → A_{0} = X^{0}

where A_{k}, k ≥ 1, is a one point union of p-local spheres with

V_{k} = H_{2k(p−1)−1}(A_{k}) ⊗ ℤ/p.

Moreover the group of homotopy equivalences of A surjects to the product group Aut (V_{1}) × … × Aut (V_{5}) and for i, j ≥ 1 and i ≠ j there are only trivial maps A_{i} → A_{j} as follows from (6). This shows the statement in (4) and hence for p > 5 F_{n}(p) is wild for n = dim(X) = 10(p − 1).

For p = 3 and p = 5 we slightly modify the argument above. We use the same type of space as for p > 5 but we assume that the attaching map f in (7) is an element of order p. Then the same argument as above shows that F_{n}(p) is wild for n = 10(p − 1). q.e.d.

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