CLASSIFICATION OF STABLE HOMOTOPY TYPES

WITH TORSION FREE HOMOLOGY

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. There are only a few explicit results on this problem in the literature; see [W1], [CH], [BH], [H], [B1], [B2], [BD1], [BD2]. A best possible solution is the classification by a complete list of indecomposable homotopy types as in the following surprising result.

Theorem A. Let n ≥ 6. There is a minimal list X(L_{5}) of 451 polyhedra such that each (n − 1)-connected (n + 5)-dimensional polyhedron X with finitely generated torsion free homology admits a homotopy equivalence

X ≃ X_{1} ∨ X_{2} ∨ … ∨ X_{t}

with X_{i} ∈ X(L_{5}) for 1 ≤ i ≤ t. Here X_{1} ∨ … ∨ X_{t} denotes the one point union of the spaces X_{i}.

We describe the elementary polyhedra in the list X(L_{5}) explicitly in § 1. They turn out to be CW-complexes with at most 6 non-trivial cells.

Let F_{n}^{k} be the homotopy category of (n − 1)-connected (n + k)-dimensional polyhedra or CW-complexes with finitely generated torsion free homology. In the stable range n ≥ k + 2 the category F^{k} = F_{n}^{k} is an additive category which does not depend on n. Recall that the isomorphism class group K_{0}(F^{k}) is generated by the homotopy types {X} in F^{k} with the relations

{X} + {Y } = {X ∨ Y }

It is a remarkable result of Freyd [F] that K_{0}(F^{k}) is actually a countably generated free abelian group for all k ≥ 0. It is well known that F^{0} is equivalent to the category of finitely generated free abelian groups so that K_{0}(F^{0}) = ℤ. We compute K_{0}(F^{k}) for all k ≥ 0 and we compute the number I(F^{k}) ≤ ∞ of indecomposable homotopy types in F^{k} for all k ≥ 0 as follows:

Theorem B.

k  0 1 2 3 4  5 k ≥ 6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
K_{0}(F^{k})  ℤ ℤ^{2} ℤ^{4} ℤ^{7} ℤ^{29} ℤ^{87} ℤ^{∞}
I(F^{k})  1 2 4 7 67 451 ∞

Typeset by AMS-TEX

Hence k = 5 is the maximal dimension k for which K_{0}(F^{k}) is finitely generated. We describe a list of generators of K_{0}(F^{5}) in (2.10) below. Theorem A, B and [BD2] imply the following result concerning the representation type of the additive category F^{k}.

Theorem C. The category F^{k} has finite representation type if and only if k ≤ 5. Moreover for k ≥ 10 the category F^{k}has wild representation type.

Hence only the representation type of F^{6}, F^{7}, F^{8}, F^{9} are unknown. The computation of the representation type of F^{6} involves a matrix problem given by a matrix with 328 rows and columns.

§ 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.

(1.1) Definition. We define a list L_{5} of 451 elements as follows. The spherical elements S^{0}, S^{1}, S^{2}, S^{3}, S^{4}, S^{5} belong to L_{5} and the Hopf elements η_{0}, η_{1}, η_{2}, η_{3} and (ηη)_{0}, (ηη)_{1}, (ηη)_{2} belong to L_{5}. Moreover the following words consisting of letters η, v_{0}, w_{1} belong to L_{5}. Here v_{0} and w_{1} are numbers with an index in {0, 1}; for example for v = 3 the letter v_{0} is 3_{0} and for w = 6 the letter w_{1} is 6_{1}. The words are

(1)  η v_{0} η, η w_{1} η with v, w ∈ {1, 2, 3}, and
(2)  v_{0}, w_{1} with v, w ∈ {1, …, 12},

and for v, w ∈ {1, …, 6} all subwords of

(3)  ηη v_{0} ηη w_{1} ηη

which are not of the form (1) or (2) or η or ηη. Here a subword of a_{0}a_{1} … a_{t} is a connected subsequence a_{n}a_{n+1} … a_{n+k} with 0 ≤ n ≤ n + k ≤ t. We call a subword of (3) containing both letters v_{0} and w_{1} a 5-dimensional word.

We can visualize the elements of L_{5} by graphs as follows. First we describe spherical elements and Hopf elements by points and vertical edges respectively. Then we describe the words in L_{5} by graphs consisting of such vertical edges and diagonal edges which represent the letters v_{0} and w_{1} respectively. This way we identify the elements of L_{5} with the following graphs.

Elementary polyhedra from dimension 4:

0 1 2 3 4 5

S^{0}

•
S^{1}

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S^{2}

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S^{3}

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S^{4}

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S^{5}

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η_{0} • •

η_{1}

•

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η_{2}

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η_{3}

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(ηη)_{0}

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(ηη)_{1}

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(ηη)_{2}

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•

0 1 2 3 4 5

•

•

v_{0}

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•

w_{1}
v,w∈{1,...,12}

•
•

•
•

ηv_{0}η

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•

•
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ηw_{1}η
v,w∈{1,2,3}

0 1 2 3 4 5

•

•

•
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ηηv_{0}ηη

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•

•
•

ηηw_{1}ηη

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ηηv_{0}η

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ηηw_{1}η

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ηv_{0}ηη

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•

•
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ηw_{1}ηη
v,w∈{1,...,6}

0 1 2 3 4 5

•

•

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ηηv_{0} •

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•

ηηw_{1}

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•
•
v_{0}ηη

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•
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w_{1}ηη

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•

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ηv_{0}

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•

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ηw_{1}

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•

v_{0}η

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•
•

w_{1}η
v,w∈{1,...,6}

New elementary polyhedra of dimension 5:

0 1 2 3 4 5

•

•
•

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v_{0}ηηw_{1}

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•

•
•

•
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ηv_{0}ηηw_{1}η

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•

•
•

•
•

ηηv_{0}ηηw_{1}ηη
v,w∈{1,...,6}

0 1 2 3 4 5

• •

• •

•

v_{0}ηηw_{1}η •

• •

•

•

ηv_{0}ηηw_{1}

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•
•

•
•

v_{0}ηηw_{1}ηη

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•

•
•

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ηηv_{0}ηηw_{1}

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•

•
•

•
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ηηv_{0}ηηw_{1}η

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•

•
•

•
•

ηv_{0}ηηw_{1}ηη
v,w∈{1,...,6}

Only the 5-dimensional words correspond to graphs which have vertices in level 0 and level 5. Let L_{4} be the subset of all graphs in L_{5} which have no vertex of level 5. Then L_{4} coincides with the following list of 67 elements. (The list L_{4} = L was already achieved in our previous paper [BD2] and is needed in the proof below).

Elementary polyhedra of dimension 4:

0 1 2 3 4

S^{0}

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S^{1}

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S^{2}

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S^{3}

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S^{4}

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η_{0}

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•

η_{1}

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η_{2}

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(ηη)_{0}

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(ηη)_{1}

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•

•

•

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•

ηvη_{v∈{1,2,3}}

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v_{v∈{1,...,12}}

0 1 2 3 4

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•

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ηηvηη

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•

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ηηvη

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•

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ηvηη

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ηηv

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vηη

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•

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ηv

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•
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vη
v∈{1,...,6}

Given an element g in L_{4} we obtain the suspension Σg ∈ L_{5} by shifting all levels of g by +1; for example Σ(ηv_{0}ηη) = ηv_{1}ηη. Hence the set L_{5} is the union of the subsets L_{4}, ΣL_{4} and the set of 5-dimensional words. The intersection L_{4} ∩ ΣL_{4} contains only spherical elements and Hopf elements.

(1.2) Definition. The duality operator D on L_{5} is the function

D : L_{5} → L_{5}

with DD = identity defined as follows:

D(S^{i}) = S^{5−i}  for i ∈ {0, …, 5}
D(η_{i}) = η_{3−i}  for i ∈ {0, 1, 2, 3}
D(ηη)_{i} = (ηη)_{2−i} for i ∈ {0, 1, 2}

Moreover for a word g in L let D(g) be obtained by reversing the order of g and by replacing v_{0} by v_{1} and w_{1} by w_{0}; that is, for example D(ηη v_{0} ηη w_{1}) = w_{0} ηη v_{1} ηη with w, v ∈ {1, …, 6}. Hence if we look at the graph g then the graph D(g) is obtained by turning g around.

(1.3) Definition. Let n ≥ 6. We associate with each element g ∈ L_{5} 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_{5}) = \{(X(g), g ∈ L_{5}\} the list of \underline{elementary polyhedra} associated to L_{5}. For the spherical elements we set

X(S^{i}) = S^{n+i} with i ∈ {0, 1, 2, 3, 4, 5}

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

X(η_{i}) = Σ^{i}(S^{n} ∪_{η} e^{n+2}) for i ∈ {0, 1, 2, 3}
X(ηη)_{i} = Σ^{i}(S^{n} ∪_{ηη} e^{n+3}) for i ∈ {0, 1, 2}

Moreover for the words g in L_{4} we get the following CW-complexes with attaching maps corresponding to the edges of the graph g. In the following definitions the attaching map v at the right hand side is obtained by identifying the number v with the v · (v + α) ∈ π_{n+3}(S^{n}) where ν and α are the generators described above. Moreover i_{k} denotes the inclusion of S^{n+k} into a wedge of spheres with k ∈ {0, 1, 2, 3}.

v ∈ {1, 2, 3}, X(ηvη) = S^{n} ∨ S^{n+2} ∪_{i_{0}η} e^{n+2} ∪_{i_{0}v+i_{2}η} e^{n+4}

v ∈ \{1, …, 6\}, \{ \begin{matrix}X(ηηvηη) = S^{n} ∨ S^{n+1} ∪_{i_{0}ηη} e^{n+3} ∪_{i_{0}v+i_{1}ηη} e^{n+4} \\ X(ηηvη) = S^{n} ∨ S^{n+2} ∪_{i_{0}ηη} e^{n+3} ∪_{i_{0}v+i_{2}η} e^{n+4} \\ X(ηvηη) = S^{n} ∨ S^{n+1} ∪_{i_{0}η} e^{n+2} ∪_{i_{0}v+i_{1}ηη} e^{n+4} \\ X(ηηv) = S^{n} ∪_{ηη} e^{n+3} ∪_{v} e^{n+4} \\ X(vηη) = S^{n} ∨ S^{n+1} ∪_{i_{0}v+i_{1}ηη} e^{n+4} \\ X(ηv) = S^{n} ∪_{η} e^{n+2} ∪_{v} e^{n+4} \\ X(vη) = S^{n} ∨ S^{n+2} ∪_{i_{0}v+i_{2}η} e^{n+4} \end{matrix}

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

Moreover for elements Σg ∈ ΣL_{4}, we define

X(Σg) = ΣX(g)

Finally we define for the 5-dimensional words g in L_{5} the following CW-complexes with v, w ∈ {0, …, 6}. Here we identify v with v(ν + α) ∈ π_{n+3}S^{n} and w with w(ν + α) ∈ π_{n+4}S^{n+1}.

\begin{aligned}X(v_{0}ηηw_{1}) & = S^{n} ∨ S^{n+1} ∪_{i_{0}v+i_{1}ηη} e^{n+4} ∪_{i_{1}w} e^{n+5} \\ X(ηv_{0}ηηw_{1}η) & = S^{n} ∨ S^{n+1} ∨ S^{n+3} ∪_{i_{0}η} e^{n+2} ∪_{i_{0}v+i_{1}ηη} e^{n+4} ∪_{i_{1}w+i_{3}η} e^{n+5} \\ X(ηηv_{0}ηηw_{1}ηη) & = S^{n} ∨ S^{n+1} ∨ S^{n+2} ∪_{i_{0}ηη} e^{n+3} ∪_{i_{0}v+i_{1}ηη} e^{n+4} ∪_{i_{1}w+i_{2}ηη} e^{n+5} \\ X(v_{0}ηηw_{1}η) & = S^{n} ∨ S^{n+1} ∨ S^{n+3} ∪_{i_{0}v+i_{1}ηη} e^{n+4} ∪_{i_{1}w+i_{3}η} e^{n+5} \\ X(ηv_{0}ηηw_{1}) & = S^{n} ∨ S^{n+1} ∪_{i_{0}η} e^{n+2} ∪_{i_{0}v+i_{1}ηη} e^{n+4} ∪_{i_{1}w} e^{n+5} \\ X(v_{0}ηηw_{1}ηη) & = S^{n} ∨ S^{n+1} ∨ S^{n+2} ∪_{i_{0}v+i_{1}ηη} e^{n+4} ∪_{i_{1}w+i_{2}ηη} e^{n+5} \\ X(ηηv_{0}ηηw_{1}) & = S^{n} ∨ S^{n+1} ∪_{i_{0}ηη} e^{n+3} ∪_{i_{0}v+i_{1}ηη} e^{n+4} ∪_{i_{1}w} e^{n+5} \\ X(ηηv_{0}ηηw_{1}η) & = S^{n} ∨ S^{n+1} ∨ S^{n+3} ∪_{i_{0}ηη} e^{n+3} ∪_{i_{0}v+i_{1}ηη} e^{n+4} ∪_{i_{1}w+i_{3}η} e^{n+5} \\ X(ηv_{0}ηηw_{1}ηη) & = S^{n} ∨ S^{n+1} ∨ S^{n+2} ∪_{i_{0}η} e^{n+2} ∪_{i_{0}v+i_{1}ηη} e^{n+4} ∪_{i_{1}w+i_{2}ηη} e^{n+5} \end{aligned}

The cells of the CW-complexes X(w) with w ∈ L_{5} correspond exactly to the vertices of the graph w above and the edges in the graph w show precisely how these cells are attached.

§ 2 Decomposition and congruence of spaces with torsion free homology

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 finite representation type. If the representation type of C is neither finite nor wild then C is of 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 finite or tame representation type.

Next we describe our 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 F_{n}^{k} be the full subcategory of Top^{∗}/ ≃ consisting of (n − 1) -connected (n + k) -dimensional CW-complexes which have finitely generated torsion free homology. The objects of F_{n}^{k} are special A_{n}^{k} -polyhedra, see [W1]. The suspension Σ gives us sequences of functors

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

with k ≥ 0. The Freudenthal suspension theorem shows that these sequences stabelize in the sense that for k+1 < n the functor Σ : F_{n}^{k} → F_{n+1}^{k} is an equivalence of \underline{additive} categories so that

(1)  F^{k} = F_{n}^{k}  with  k + 1 < n

does not depend on n. This is the stable homotopy category of (−1) -connected k-dimensional spectra with finitely generated torsion free homology. The biproduct in the additive category F^{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 functor

(2)  D : F^{k} → F^{k}

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

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}^{k} is not unique. For example Freyd [F] points out that for n ≥ 5 there is a homotopy equivalence

(2.4)  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.

In [BD2] we solved the decomposition problem in the additive category F^{4} by showing that X(L_{4}) is a complete list of indecomposable objects in F^{4}. Our main purpose in this paper is the solution of the decomposition problem in F^{5}. We show that F^{5} is again of finite representation type. On the other hand we have seen in the Appendix of [BD2] that F^{k} is of wild representation type for k ≥ 10. Hence only the representation types of F^{6}, F^{7}, F^{8}, F^{9} remain unknown.

(2.5) Theorem. The list X(L_{5}) of 451 elementary polyhedra in § 1 is a complete list of all indecomposable spaces in F^{5}. Hence for n ≥ 6 each (n − 1)-connected (n + 5)-dimensional CW-complex X with finitely generated torsion free homology admits a homotopy equivalence

X ≃ X_{1} ∨ … ∨ X_{t}

with X_{i} ∈ X(L_{5}) for 1 ≤ i ≤ t. Moreover the Spanier-Whitehead duality functor D : F^{5} → F^{5} is completely understood on objects since we have

D(X(g)) = X(D(g))

where D(g) is defined by the duality operator D in (1.2) with g ∈ L_{5}.

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

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

(a) There exists a space Z in F^{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}).

(2.7) Definition. Let p be a prime. A space X in F^{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 F^{k} and N can not be chosen to be N = 0. This implies that X is not a one point union of spheres.

(2.8) Lemma. An elementary polyhedron X(g) with g ∈ L_{5} is 2-primary if and only if g is a Hopf element or g is a word with letters in the set {η, v_{0}, w_{1}; v and w divisible by 3}. The only congruences between 2-primary polyhedra in X(L_{5}) are given by (2.4) that is X(3_{0}) ≡ X(9_{0}) and X(3_{1}) ≡ X(9_{1}). Moreover X(g) is 3primary if and only if g = 8_{0} or g = 8_{1}. For a prime p > 3 there are no p-primary spaces in X(L_{5}).

Recall that for any small additive category C (for example 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}(C) 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.

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

Such a wonderful result yields the crucial task to compute the generators of K_{0}(F^{k}) explicitly. 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}. In [BD2] we show that

• K_{0}(F^{4}) = ℤ^{29} is generated by the 5-spheres S^{n}, …, S^{n+4} in F^{4}, the 23 congruence classes of 2-primary polyhedra in X(L_{4}) and by the unique 3-primary polyhedron in X(L_{4}).

Using (2.8) and (2.5) we get accordingly:

(2.10) Theorem. The group K_{0}(F^{5}) = ℤ^{87} is generated by the 6-spheres S^{n}, …, S^{n+5}, the two 3-primary polyhedra in X(L_{5}) and the 79 congruence classes of 2-primary polyhedra in X(L_{5}).

In § 6 we show that K_{0}(F^{k}) = ℤ^{∞} for k ≥ 6.

§ 3 The algebraic classification of homotopy types in F^{5}

Let X be a CW-complex in F^{5} = F_{n}^{5} with n ≥ 7. We use stable notation so that we are allowed to omit n. Hence S^{i} corresponds to the sphere S^{n+i}, moreover e^{i} corresponds to the cell e^{n+i}, and the homotopy group π_{j} S^{i} corresponds to π_{n+j}S^{n+i}, etc. We may assume that X has a cell structure given by the homology decomposition; see [B1]. This implies (since the homology of X is free abelian) that the cells of X are in 1-1 correspondence with free generators of the homology of X. Let c_{i} be the number of i-cells in X so that H_{i}(X) = ℤ^{c_{i}}. For a space A and a natural number d ≥ 0 let

(3.1)  dA = \underbrace{A ∨ … ∨ A}_{d−times}

be the d-fold one point union of A. The attaching map of (stable) 5-cells of X is a map

(3.2)  f : c_{5}S^{4} → X^{4}

with X^{4} ∈ F^{4}. Here X^{4} is the 4-skeleton of X. Since the cell structure is given by a homology decomposition we know that f admits a factorization

(1)  f : c_{5}S^{4} → X^{3} ⊂ X^{4}

with X^{3} ∈ F^{3}. Let

Γ_{4}X = image{π_{4}X^{3} → π_{4}X^{4}}

Then the homotopy class of f in (3.2) is determined by a homomorphism

(2)  f : H_{5}(X) = ℤ^{c_{5}} → Γ_{4}(X^{4})

This is the secondary boundary in J.H.C. Whitehead’s exact sequence [W2]. In [BD2] we classified the homotopy types in F^{4} showing that X^{4} is a one point union of spaces X(g) with g ∈ L_{4}. We say that f in (2) is in normal form if X is a one point union of spaces X(g) with g ∈ L_{5} so that f is canonically given by the attaching maps of 5-cells in X(g) with g ∈ L_{5}, see (1.3).

Since we are in the stable range the functor Γ_{4} is additive. Therefore we can compute Γ_{4} X^{4} as in proposition (3.4) below which determines Γ_{4}X(g) for g ∈ L_{4}. We define a subset L′_{4} of L_{4} by

L′_{4} = \left\{{S^{0}, S^{4}, η_{0}, η_{2}, (ηη)_{0}, ηηvη, \atop ηvη, ηηv, ηv, vη, v}\right\}

Moreover we define the spaces Y_{i} and Y_{j}^{v} with i ∈ {1, …, 5}, j ∈ {1, 2, 3}, v ∈ {1, …, 6} by

(3.3)  \{ \begin{matrix}Y_{1} & = S^{1} = X(S^{1}) \\ Y_{2} & = S^{1} ∪_{η} e^{3} = X(η_{1}) \\ Y_{3} & = S^{1} ∪_{ηη} e^{4} = X(ηη)_{1} \\ Y_{4} & = S^{2} = X(S^{2}) \\ Y_{5} & = S^{3} = X(S^{3}) \\ Y_{1}^{v} & = X(ηηvηη) \\ Y_{2}^{v} & = X(ηvηη) \\ Y_{3}^{v} & = X(vηη) \end{matrix}

These spaces form the set X(L_{4} − L′_{4}); they are given by the following graphs

0 1 2 3 4

\begin{matrix}Y_{1} \\ d_{1} \\ V_{1} \end{matrix}

•

\begin{matrix}Y_{2} \\ d_{2} \\ V_{2} \end{matrix}

• •

\begin{matrix}Y_{3} \\ d_{3} \\ V_{3} \end{matrix}

• •

\begin{matrix}Y_{4} \\ d_{4} \\ V_{4} \end{matrix}

•

\begin{matrix}Y_{5} \\ d_{5} \\ V_{5} \end{matrix}

• •

• •

•

\begin{matrix}Y_{1}^{v} \\ d_{1}^{v} \\ U_{1}^{v} \end{matrix}

•

• •

•

\begin{matrix}Y_{2}^{v} \\ d_{2}^{v} \\ U_{2}^{v} \end{matrix}

• •

•

\begin{matrix}Y_{3}^{v} \\ d_{3}^{v} \\ U_{3}^{v} \end{matrix}

Here the numbers d_{i}, d_{j}^{v} and the corresponding free abelian groups V_{i} and U_{j}^{v} are chosen in (3.5) and (3.8) below.

(3.4) Proposition. Γ_{4}X(g) = 0 for g ∈ L′_{4} and

\{ \begin{matrix}Γ_{4}(Y_{1}) = ℤ/24 \\ Γ_{4}(Y_{4}) = Γ_{4}(Y_{5}) = ℤ/2 \\ Γ_{4}(Y_{2}) = Γ_{4}(Y_{3}) = Γ_{4}(Y_{j}^{v}) = ℤ/12 \end{matrix}

Proof. We describe three examples. Consider g = η_{0} ∈ L′_{4} so that X(η_{0}) = S^{0}∪_{η} e^{2}. Then we have the stable cofiber sequence

S^{1} ⟶^{η} S^{0} → X(η_{0}) → S^{2} ⟶^{η} S^{1}

which induces the exact sequence

π_{4}S^{0} → π_{4}X(η_{0}) → π_{4}S^{2} ⟶^{η_{∗}} π_{4}S^{1}

Here π_{4}S^{0} = 0 by [T] and η_{∗} is injective by [T]. Hence π_{4}X(η_{0}) = 0. In the same way we see π_{4}X(ηη)_{0} = 0. Moreover consider g = vη ∈ L′_{4}. Then we have

Γ_{4}X(vη) = image{π_{4}(S^{0} ∨ S^{2}) = π_{4}S^{2} → π_{4}X(vη)}

where i_{2} : S^{2} → X(vη) satisfies i_{0}v + i_{2}η = 0 by definition of X(vη) in (1.3). Hence we get i_{2}ηη = i_{0}vη = 0 since νη = 0 by [T]. Hence Γ_{4}X(vη) = 0. Moreover we use similar arguments for the computation of Γ_{4}X(g) with g ∉ L′_{4}. Here we need the relation 12(ν + α) = ηηη in π_{4}S^{1} = ℤ/24; compare [T]. For example we have the cofiber sequence

S^{2} ⟶^{η} S^{1} → X(η_{1})

inducing the exact sequence

π_{4}S^{2} → π_{4}S^{1} → π_{4}X(η_{1})

so that Γ_{4}X(η_{1}) = cokernel (η_{∗} : π_{4}S^{2} → π_{4}S^{1}) = ℤ/12 since π_{4}S^{2} = ℤ/2 is generated by ηη and π_{4}S^{1} = ℤ/24 is generated by ν + α. We point out that the generator of ℤ/12 in the proposition is the composite i_{1}(ν + α) where i_{1} is the inclusion of S^{1} into Y_{2}, Y_{3} or Y_{j}^{v}. q.e.d.

Since X^{4} is a one point union of spaces X(g) with g ∈ L_{4} we can write

(3.5)  X^{4} = L ∨ X̃^{4}.

Here L is a one point union of spaces X(g) with g ∈ L′_{4} and X̃^{4} is a one point union of spaces in (3.3), that is,

(1)  X̃^{4} = d_{1}Y_{1} ∨ d_{2}Y_{2} ∨ Z_{3} ∨ d_{4}Y_{4} ∨ d_{5}Y_{5}

where Z_{3} is a one point union of spaces Y_{3}, Y_{1}^{v}, Y_{2}^{v}, Y_{3}^{v} namely

(2)  Z_{3} = d_{3}Y_{3} ∨ ⋁_{v∈{1,... ,6}}(d_{1}^{v}Y_{1}^{v} ∨ d_{2}^{v}Y_{2}^{v} ∨ d_{3}^{v}Y_{3}^{v})

Here d_{i} and d_{j}^{v} are numbers ≥ 0. Using (3.4) we see that Γ_{4}(L) = 0 so that the attaching map (3.2) factors through the inclusion X̃^{4} ⊂ X^{4}. Hence we get the following result which simplifies the proof of the decomposition theorem (2.5) a lot.

(3.6) Proposition. For X in F^{4} there is a homotopy equivalence X ≃ L ∨ X̃. Here L is a one point union of spaces X(g) with g ∈ L′_{4} and X̃ is the cofiber of a map f̃: c_{5}S^{4} → X̃^{4} with X̃^{4} as in (3.5).

Hence in order to find all indecomposable spaces in F^{5} we only have to consider decompositions of spaces X̃ as in (3.6). We therefore assume in the following that L = ∗ so that X = X̃ satisfies X^{4} = X̃^{4} with X̃^{4} as in (3.5) and f = f̃. In this case we say that X is special.

(3.7) Proposition. The space Z = d_{1}Y_{1} ∨ d_{2}Y_{2} ∨ Z_{3} in (3.5) admits a surjection

H_{1}(Z) ⊗ ℤ/24 → Γ_{4}(Z)

which is natural in Z.

This is an easy consequence of (3.4) and the definition of Z. We associate with a special space X the following free abelian groups; see (3.5).

(3.8)  \{ \begin{matrix}V_{i} & = ℤ^{d_{i}} \\ V & = ⨁_{i=1}^{5} V_{i} \\ U_{j}^{v} & = ℤ^{d_{j}^{v}}, U_{j} = ⨁_{v=1}^{6} U_{j}^{v}, U^{v} = ⨁_{j=1}^{3} U_{j}^{v} \\ U & = ⨁_{j=1}^{3} ⨁_{v=1}^{6} U_{j}^{v} = ⨁_{j=1}^{3} U_{j} = ⨁_{v=1}^{6} U^{v} \end{matrix}

Then we obtain the homology groups H_{i} = H_{i}(X) of the special space X by the formulas:

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

We also obtain for Γ_{4} = Γ_{4}(X^{4}) the formula

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

as a consequence of (3.4).

(3.9) Sketch of proof. Let X be special. In order to prove the decomposition theorem (2.5) we show that there exist a homotopy equivalence α : X^{4} ≃ X^{4} and an automorphism φ of H_{5}X such that the composite

(1)  α_{∗}fφ^{−1} : H_{5}(X) ≅^{φ} H_{5}(X) ⟶^{f} Γ_{4}(X^{4}) ≅^{α_{∗}} Γ_{4}(X^{4})

is in normal form; see (3.2). This implies that a special space admits a decomposition as in (2.5). Here a map α : X^{4} → X^{4} is a homotopy equivalence if and only if α induces an isomorphism in homology. Since X^{4} is a one point union of spaces X(g) with g ∈ L_{4} − L′_{4} as in (3.5) (1) we have to consider algebraic maps

(2)  n_{g′}^{g} : H_{∗}X(g) → H_{∗}X(g′)

with g, g′ ∈ L_{4} − L′_{4} such that there exists a map

(3)  α_{g′}^{g} : X(g) → X(g′)

inducing n_{g′}^{g} in homology. Moreover we determine Γ_{4}(α_{g′}^{g}) in terms of n_{g′}^{g}. Using (2) we can describe all automorphisms n_{∗} of the homology H_{∗}(X^{4}) in (3.9) for which there exists a map α : X^{4} → X^{4} with n_{∗} = H_{∗}(α) and we can compute α_{∗} = Γ_{4}(α) since we know Γ_{4}(α_{g′}^{g}) in terms of n_{g′}^{g}. This leads to the following definitions and theorem.

(3.10) Definition. We say that an automorphism φ : Γ_{4}(X^{4}) ≅ Γ_{4}(X^{4}) is realizable if there exists a homotopy equivalences α : X^{4} ≃ X^{4} which induces φ; that is Γ_{4}(α) = φ.

The next result yields by the remarks in (3.9) a purely algebraic classification of all special homotopy types in F^{5}. Proper automorphisms are defined in (3.12) below.

(3.11) Theorem. An automorphism φ of Γ_{4}(X^{4}) is realizable if and only if there exists a proper automorphism M of the free abelian group

W = V_{1} ⊕ V_{2} ⊕ (V_{3} ⊕ U) ⊕ V_{4} ⊕ V_{5}

for which the following diagram commutes

W ⟶^{M} W
p↓  ↓^{p}
Γ_{4}(X^{4}) ⟶^{φ} Γ_{4}(X^{4})

Here p is the quotient map given by (3.8) (2).

(3.12) Definition. Let W_{3} = V_{3} ⊕ U so that

W = V_{1} ⊕ V_{2} ⊕ W_{3} ⊕ V_{4} ⊕ V_{5}

is a direct sum of 5 summands. An automorphism M of the free abelian group W is proper if and onyl if M is given by a 5 × 5 matrix 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} )

with the following properties. The coordinates of the matrix M correspond to the direct sum decomposition of W above, for example a_{15} ∈ Hom (V_{5}, V_{1}), a_{23} ∈ Hom (W_{3}, V_{2}), a_{33} ∈ Hom (W_{3}, W_{3}). The submatrix

(2)  H_{1} = ( \begin{matrix}a_{11} & 2a_{12} & 2a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & 2a_{32} & a_{33} \end{matrix} )

of M is an automorphism of V_{1} ⊕ V_{2} ⊕ W_{3}. Recall that W_{3} = V_{3} ⊕ U where U = U^{1} ⊕ … ⊕ U^{6} as in (3.8). We say that for

F, G ∈ Hom(U, U)

the homomorphism F is ℤ/12-related to G if the coordinates F_{vw}, G_{vw} ∈ Hom (U^{w}, U^{v}) of F and G satisfy for v, w ∈ {1, …, 6} the equation

(3)  (w · F_{vw}) ⊗ ℤ/12 = (v · G_{vw}) ⊗ ℤ/12

We require that there exist automorphisms

(4)  \{ \begin{matrix}H_{0} ∈ Aut (U) = Aut (U_{1} ⊕ U_{2} ⊕ U_{3}) \\ H_{4} ∈ Aut (W_{3}) = Aut (V_{3} ⊕ U) \end{matrix}

as follows. With respect to the direct sum decomposition U = U_{1} ⊕U_{2} ⊕U_{3} in (3.8) the automorphism H_{0} is given by a matrix of the form

(5)  H_{0} = ( \begin{matrix}b_{11} & 2b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ 2b_{31} & 2b_{32} & b_{33} \end{matrix} )

Moreover with respect to the direct sum decomposition W_{3} = V_{3} ⊕ U the automorphism H_{4} is given by a matrix

(6)  H_{4} = ( \begin{matrix}c_{33} & c_{3U} \\ c_{U3} & c_{UU} \end{matrix} )

We require that

(7)  H_{0} is ℤ/12-related to c_{UU} ∈ Hom (U, U)

and c_{U3} ∈ Hom (V_{3}, U) has coordinates c_{U^{v}3} ∈ Hom (V_{3}, U^{v}) with v ∈ {1, …, 6} which satisfy the equation

(8)  (v · c_{U^{v}3}) ⊗ ℤ/12 = 0.

Finally we require that H_{4} in (6) satisfies

(9)  H_{4} ⊗ ℤ/2 = a_{33} ⊗ ℤ/2

where a_{33} ∈ Hom (W_{3}, W_{3}) is the coordiante of M in (1) above.

We point out that by (3.8) (1) we have the homology groups

(3.13)  \{ \begin{matrix}H_{0} = U \\ H_{1} = V_{1} ⊕ V_{2} ⊕ W_{3} \\ H_{4} = W_{3} = V_{3} ⊕ U \end{matrix}

so that the automorphisms H_{0}, H_{1}, H_{4} in (3.12) are automorphisms of H_{0}, H_{1} and H_{4} respectively. Clearly a homotopy equivalence X^{4} ≃ X^{4} induces such automorphisms H_{0}, H_{1}, H_{4} of homology and in addition automorphisms of H_{2} and H_{3} in (3.8) (1) which we obtain as follows.

(3.14) Definition. Given M, H_{0}, H_{1}, H_{4} as in (3.12) we choose automorphism H_{2} on H_{3} as follows. Let H_{2} be an automorphism of H_{2} = V_{4} ⊕ U_{2} which is given by a matrix

H_{2} = ( \begin{matrix}d_{44} & d_{42} \\ 2d_{24} & d_{22} \end{matrix} )

satisfying the equations (1) and (2).

d_{44} ⊗ ℤ/2 = a_{44} ⊗ ℤ/2 (1) d_{22} ⊗ ℤ/2 = b_{22} ⊗ ℤ/2 (2)

Here a_{44} and b_{22} are coordinates of Γ and H_{0} respectively. The properties of M and H_{0} readily show that a_{44} ⊗ ℤ/2 and b_{22} ⊗ ℤ/2 are automorphisms. Hence we can choose H_{2} since GL_{n}(ℤ) → GL_{n}(ℤ/2) is surjective for all n; choose for example d_{42} = 0.

Moreover let H_{3} be an automorphism of H_{3} = V_{5} ⊕ V_{2} ⊕ U_{1} which is given by a matrix

H_{3} = ( \begin{matrix}e_{55} & e_{52} & e_{51} \\ 2e_{25} & e_{22} & 2e_{21} \\ 2e_{15} & e_{12} & e_{11} \end{matrix} )

satisfying the equations (3) ... (4).

e_{55} ⊗ ℤ/2 = a_{55} ⊗ ℤ/2 (3) e_{22} ⊗ ℤ/2 = a_{22} ⊗ ℤ/2 (4) e_{11} ⊗ ℤ/2 = b_{11} ⊗ ℤ/2 (5) e_{25} ⊗ ℤ/2 = a_{25} ⊗ ℤ/2 (6)

Here again a_{55}, a_{22}, a_{25} are coordinates of M in (3.12) (1) and b_{11} is a coordinate of H_{0} in (3.12) (5). Since a_{55} and a_{22} ⊗ ℤ/2 and b_{11} ⊗ ℤ/2 are automorphisms it is possible to choose H_{3}. For example take e_{52} = e_{51} = e_{21} = 0 and use the surjection GL_{n}(ℤ) → GL_{n}(ℤ/2).

Equation (6) above corresponds to the equation in 4.6 (13) of [BD2].

§ 4 Proof of theorem (3.11)

Let X^{4} = X̃^{4} be a one point union as in (3.5) (1), that is,

(4.1)  \begin{matrix}X^{4} = Z ∨ d_{4}Y_{4} ∨ d_{5}Y_{5}, \\ Z = d_{1}Y_{1} ∨ d_{2}Y_{2} ∨ Z_{3}, \\ Z_{3} = d_{3}Y_{3} ∨ ⋁_{v∈\{1,... ,6\}}(d_{1}^{v}Y_{1}^{v} ∨ d_{2}^{v}Y_{2}^{v} ∨ d_{3}^{v}Y_{3}^{v}). \end{matrix}

Here Y_{i} and Y_{j}^{v} are given by (3.3). With respect to the decomposition (4.1) a map α : X^{4} → X^{4} is given by a matrix

(4.2)  α = ( \begin{matrix}α_{ZZ} & α_{Z4} & α_{Z5} \\ α_{4Z} & α_{44} & α_{45} \\ α_{5Z} & 0 & α_{55} \end{matrix} )

with α_{Z5} : d_{5}Y_{5} → Z, etc. We clearly have α_{54} = 0 since there are no essential maps from Y_{4} = S^{2} to Y_{5} = S^{3}. Moreover α_{44} = a_{44} and α_{55} = a_{55} are determined by H_{α}(α) and H_{3}(α) respectively and α_{45} is given by a homomorphism qa_{45} : V_{5} → V_{4} → V_{4} ⊗ ℤ/2 where q is the quotient map. Clearly a_{45} is not well defined by α_{45}.

(4.3) Proposition. The maps α_{4Z} and α_{5Z} induce the trivial homomorphism on Γ_{4}, that is Γ_{4}(α_{4Z}) = Γ_{4}(α_{5Z}) = 0. Moreover α_{Z4} has a coordinate α_{Z_{3}4} : d_{4}Y_{4} → Z_{3} and α_{Z5} has a coordinate α_{Z_{3}5} : d_{5}Y_{5} → Z_{3} with Γ_{4}(α_{Z_{3}4}) = Γ_{4}(α_{Z_{3}5}) = 0.

Proof. We obtain Γ_{4}(α_{4Z}) = Γ_{4}(α_{5Z}) = 0 easily from (3.7) and (3.4) since composites S^{1} → Z → S^{2}, S^{3} are trivial. Moreover Γ_{4}(α_{Z_{3}4}) = Γ_{4}(α_{Z_{3}5}) = 0 is a consequence of the fact that Γ_{4}S^{2} and Γ_{4}S^{3} are generated by ηη and η respectively and that ηηη is trivial in Γ_{4}Y_{3} and Γ_{4}Y_{j}^{v} by (3.4). q.e.d.

Moreover we deduce from (3.7):

(4.4) Proposition. Γ_{4}(α_{ZZ}) = H_{1}(α_{ZZ})_{∗} is induced by H_{1}(α_{ZZ}) and with respect to the decomposition Z = d_{1}Y_{1} ∨ d_{2}Y_{2} ∨ Z_{3} the automorphism H_{1}(α_{ZZ}) is given by a matrix of the form

H_{1}(α) = ( \begin{matrix}a_{11} & 2a_{12} & 2a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & 2a_{32} & a_{33} \end{matrix} )

Proof. One readily checks that each realizable map H_{1}X(η_{1}) → H_{1}(S^{1}) is divisible by 2 so that we obtain 2a_{12}. Similarly one readily checks for Ỹ = Y_{3}, Y_{j}^{v} that realizable maps H_{1}X(η_{1}) → H_{1}(Ỹ) and H_{1}(Ỹ) → H_{1}(S^{1}) are divisible by 2. Hence we obtain 2a_{32} and 2a_{13}. These facts are easy consequences of the attaching maps q.e.d.

Using (4.4) nd (4.3) we see that with respect to the decomposition

Γ_{4}(X^{4}) = Γ_{4}(d_{1}Y_{1}) ⊕ Γ_{4}(d_{2}Y_{2}) ⊕ Γ_{4}(Z_{3}) ⊕ Γ_{4}(d_{4}Y_{4}) ⊕ Γ_{4}(d_{5}Y_{5})

the homomorphism Γ_{4}α is given by a matrix of the following form:

(4.5)  Γ_{4}α = ( \begin{matrix}· & · & · & 12a_{14} & 12a_{15} \\ · & · & · & 0 & 6a_{25} \\ · & · & · & 0 & 0 \\ 0 & 0 & 0 & a_{44} & a_{45} \\ 0 & 0 & 0 & 0 & a_{55} \end{matrix} )

Here the dotted 3×3-matrix is induced by H_{1}(α) in (4.4). The coordinates a_{14}, a_{15} and a_{25} are obtained as in the proof of [BD2].

If the map α is a homotopy equivalence then clearly H_{1}(α) is an automorphism and by the form (4.5) of the automorphism Γ_{4}α we see that also a_{44} ⊗ ℤ/2 and a_{55} ⊗ ℤ/2 are automorphisms. Using the surjection GL_{n}(ℤ) → GL_{n}(ℤ/2) we may assume that a_{44} and a_{55} are automorphisms over ℤ and hence we obtain by (4.5) a matrix M as in (3.12) for which the diagram (3.11) commutes with φ = Γ_{4}(α).

In order to prove (3.11) we have to show the following two lemmas.

(4.6) Lemma. Let α be a homotopy equivalence as above. Then H_{1} = H_{1}(α), H_{0} = H_{0}(α), and H_{4} = H_{4}(α) together with M given by Γ_{4}(α) as above have properties as described in (3.12)

Hence for each homotopy equivalence α there is a proper automorphism M such that diagram (3.11) with φ = Γ_{4}(α) commutes.

(4.7) Lemma. Let M be a proper automorphism as in (3.12) so that we can choose H_{0}, H_{1}, H_{2}, H_{3}, H_{4} as in (3.12) and (3.14). Then there exists a map α : X^{4} → X^{4} with H_{i}(α) = H_{i} such that diagram (3.11) with φ = Γ_{4}(α) commutes.

Since H_{i} are automorphisms we see that α in (4.7) is a homotopy equivalence. Therefore each proper automorphism M induces a realizable automorphism φ on Γ_{4}(α).

In order to prove (4.6) and (4.7) we repeat the classification theorem of Unsöld [U1], [U2]. He defines the following algebraic category SF^{4}.

(4.8) 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.

(4.9) 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.

Theorem (4.9) allows the computation of all realizable homology homomorphisms

(4.10)  n_{∗} : H_{∗}(Y) → H_{∗}(Y′)

with Y, Y′ ∈ X(L_{4} − L′_{4}); see (3.3). For this we use the fact that the functor λ in (4.9) is full. One readily checks that H_{i}(Y) is either 0 or ℤ so that n_{∗} is given by n_{i} ∈ ℤ with i ∈ {0, …, 4}. Below we describe the non-trivial n_{i} for which n_{∗} = (n_{0}, n_{1}, n_{2}, n_{3}, n_{4}) is realizable by a map Y → Y′. For example if Y or Y′ are spheres with Y ≠ Y′ one can check that there are only the following possibilities (1)...(5).

Y_{1} ⟶^{n_{1}} Y_{3} ⟶^{n_{1}≡0(2)} Y_{1} (1)

Y_{1} ⟶^{n_{1}} Y_{2} ⟶^{n_{1}≡0(2)} Y_{1}, Y_{5} ⟶^{n_{3}≡0(2)} Y_{2} ⟶^{n_{3}} Y_{5} (2)

Y_{1} ⟶^{n_{1}} Y_{1}^{v} ⟶^{n_{1}≡0(2)} Y_{1}, Y_{5} ⟶^{n_{3}≡0(2)} Y_{1}^{v} ⟶^{n_{3}} Y_{5} (3)

Y_{1} ⟶^{n_{1}} Y_{2}^{v} ⟶^{n_{1}≡0(2)} Y_{1}, Y_{4} ⟶^{n_{2}≡0(2)} Y_{2}^{v} ⟶^{n_{2}} Y_{4} (4)

Y_{1} ⟶^{n_{1}} Y_{3}^{v} ⟶^{n_{1}≡0(2)} Y_{1} (5)

Moreover if Y or Y′ are Hopf elements we get the following possibilities (6)...(14).

Y_{2} ⟶^{n_{1}≡n_{2}(2)} Y_{2} (6)

Y_{3} ⟶^{n_{1}≡n_{4}(2)} Y_{3} (7)

Y_{2} ⟶^{n_{1}≡0(2)} Y_{3} ⟶^{n_{1}} Y_{2} (8)

Y_{2} ⟶_{n_{3}}^{n_{1}≡0(2)} Y_{1}^{v} ⟶_{n_{3}≡0(2)}^{n_{1}} Y_{2} (9)

Y_{3} ⟶_{n_{4}v≡0(12)}^{n_{4}≡n_{1}(2)} Y_{1}^{v} ⟶^{n_{1}≡n_{4}(2)} Y_{3} (10)

Y_{2} ⟶^{n_{1}≡0(2)} Y_{2}^{v} ⟶^{n_{1}} Y_{2} (11)

Y_{3} ⟶_{n_{4}v≡0(12)}^{n_{1}≡n_{4}(2)} Y_{2}^{v} ⟶^{n_{1}≡n_{4}(2)} Y_{3} (12)

Y_{2} ⟶^{n_{1}≡0(2)} Y_{3}^{v} ⟶^{n_{1}} Y_{2} (13)

Y_{3} ⟶_{n_{4}v≡0(12)}^{n_{1}≡n_{4}(2)} Y_{3}^{v} ⟶^{n_{1}≡n_{4}(2)} Y_{3} (14)

Finally we get for Y, Y′ ∈ {Y_{1}^{v}, Y_{2}^{v}, Y_{3}^{v}} the following possibilities (15)...(23) with v, w ∈ {1, …, 6}.

Y_{1}^{w} ⟶_{n_{0}w≡n_{4}v(12)}^{n_{0}≡n_{3}(2),n_{1}≡n_{4}(2)} Y_{1}^{v} (15)

Y_{1}^{w} ⟶_{n_{0}w≡n_{4}v(12)}^{n_{4}≡n_{1}(2)} Y_{2}^{v} (16)

Y_{1}^{w} ⟶_{n_{4}v≡n_{0}w(12)}^{n_{0}≡0(2),n_{4}≡n_{1}(2)} Y_{3}^{v} (17)

Y_{2}^{w} ⟶_{n_{0}w≡n_{4}v(12)}^{n_{0}≡0(2),n_{1}≡n_{4}(2)} Y_{1}^{v} (18)

Y_{2}^{w} ⟶_{n_{0}w≡n_{4}v(12)}^{n_{2}≡n_{0}(2),n_{1}≡n_{4}(2)} Y_{2}^{v} (19)

Y_{2}^{w} ⟶_{n_{0}w≡n_{4}v(12)}^{n_{0}≡0(2),n_{4}≡n_{1}(2)} Y_{3}^{v} (20)

Y_{3}^{w} ⟶_{n_{0}w≡n_{4}v(12)}^{n_{4}≡n_{1}(2)} Y_{1}^{v} (21)

Y_{3}^{w} ⟶_{n_{0}w≡n_{4}v(12)}^{n_{1}≡n_{4}(2)} Y_{2}^{v} (22)

Y_{3}^{w} ⟶_{n_{0}w≡n_{4}v(12)}^{n_{1}≡n_{4}(2)} Y_{3}^{v} (23)

One can prove the conditions of realizability of n_{∗} = (n_{1}, …, n_{4}) in (1)...(23) either directly by use of the cell structure of Y, Y′ ∈ X(L_{4} − L′_{4}) or by use of theorem (4.9). In order to apply (4.9) we point out that the objects in SF^{4} corresponding to Y_{3}, Y_{1}^{v}, Y_{2}^{v}, Y_{3}^{v} are given by the following list.

H_{3} → π_{1} ⊗ ℤ/2 → π_{2} → H_{2} → H_{0} ⊗ ℤ/2 → π_{1} ↠ H_{1}, H_{0}
Y_{3} 0  ℤ/2  ℤ/2  0  0  ℤ  ℤ  0
Y_{1}^{v} ℤ ⟶^{(1,0)} ℤ/2 ⊕ ℤ/2 ⟶^{(0,1)} ℤ/2  0  ℤ/2  ℤ/2 ⊕ ℤ  ℤ ℤ
Y_{2}^{v} 0  ℤ/2  ℤ/2 ⊕ ℤ  ℤ  ℤ/2  ℤ  ℤ ℤ
Y_{3}^{v} 0  ℤ/2 ⊕ ℤ/2  ℤ/2 ⊕ ℤ/2  0  0  ℤ  ℤ  0

H_{0} ⊃ P ⟶^{T} π_{2} ⊗ ℤ/2,  H_{4} →  Γ_{3}  ↞ H_{0} ⊗ ℤ/24 ⊕ π_{2} ⊗ ℤ/2
Y_{3} 0  0  ℤ/2  ℤ →^{1}  ℤ/2  ←  0 ⊕ ℤ/2
y  y
Y_{1}^{v} ℤ  2ℤ ⟶^{0}  ℤ/2  ℤ ⟶^{(v,1)} ℤ/12 ⊕ ℤ/2 ←  ℤ/24 ⊕ ℤ/2
(x, y)  (x, y)
Y_{2}^{v} ℤ = ℤ ⟶^{(0,1)} ℤ/2 ⊕ ℤ/2  ℤ ⟶^{(v,1)} ℤ/12 ⊕ ℤ/2 ←  ℤ/24 ⊕ ℤ/2 ⊕ ℤ/2
(x + 6z, y)  (x, y, z)
Y_{3}^{v} ℤ  2ℤ ⟶^{(1/2,0)} ℤ/2 ⊕ ℤ/2  ℤ ⟶^{(v,1)} ℤ/24 ⊕ ℤ/2 ←  ℤ/24 ⊕ ℤ/2 ⊕ ℤ/2
(x + 12z, z)  (x, y, z)

Using these objects of SF^{4} corresponding to Y_{3}, Y_{1}^{v}, Y_{2}^{v}, Y_{3}^{v} it is an easy but inevitable calculation to obtain the conditions on n_{∗} in (1)...(23) above. For example if we consider maps Y_{2}^{w} → Y_{1}^{v} we obtain n_{∗} = (n_{0}, n_{1}, n_{4}) such that the following diagrams commute (these diagrams describe a morphism λ(Y_{2}^{w}) → λ(Y_{1}^{v}) in SF^{4}).

0 ⟶  ℤ/2  ⟶^{(1,0)} ℤ/2 ⊕ ℤ ⟶ ℤ ⟶ ℤ/2 ⟶^{0}  ℤ  ⟶ ℤ,  ℤ
↓  ↓^{(α,n_{1})}  ↓^{(n_{1},β)}  ↓  ↓^{n_{0}}  ↓^{(α,n_{1})}  ↓^{n_{1}} ↓^{n_{0}}
ℤ ⟶^{(1,0)} ℤ/2 ⊕ ℤ/2 ⟶^{(0,1)} ℤ/2 ⟶ 0 ⟶ ℤ/2 ⟶^{(1,0)} ℤ/2 ⊕ ℤ ⟶ ℤ,  ℤ
ℤ = ℤ ⟶ ℤ/2 ⊕ ℤ/2;  ℤ ⟶^{(w,1)} ℤ/12 ⊕ ℤ/2 ⟵ ℤ/24 ⊕ ℤ/2 ⊕ ℤ/2
n_{0}↓ ↓  ↓^{(n_{1},β)}  ^{n_{4}}↓  ↓^{n_{0}⊕n_{1}}  ^{n_{0}⊕(n_{1},0)}↓
ℤ⊃ 2ℤ ⟶^{0}  ℤ/2  ℤ ⟶^{(v,1)} ℤ/12 ⊕ ℤ/2 ⟵  ℤ/24 ⊕ ℤ/2

The first diagram shows n_{0} ≡ 0(2) and the last diagram shows n_{1} ≡ n_{4}(2) and n_{0}w ≡ n_{4}v(12). If n_{∗} = (n_{0}, n_{1}, n_{4}) satisfies these conditions then α and β = 0 can be chosen such that the diagrams commute.

Homology homomorphisms (4.10) are realizable if and only if they satisfy these conditions. This proves (4.6) and (4.7) by checking that the conditions (1)...(23) correspond exactly to the assumptions used in the definition of a proper automorphism in (3.12) and (3.14).

§ 5 Simplification of proper automorphisms

The condition describing a proper automorphism in (3.12) can be considerably simplified as follows.

(5.1) Proposition. An automorphism M of W is proper if and only if with respect to the decomposition

W = V_{1} ⊕ V_{2} ⊕ V_{3} ⊕ U^{1} ⊕ U^{2} ⊕ U^{3} ⊕ U^{4} ⊕ U^{5} ⊕ U^{6} ⊕ V_{4} ⊕ V_{5}

the matrix M is a matrix of the form

( \begin{matrix}1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 2 & 12 & 12 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 12 & 6 \\ 1 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 12 & 12 \\ 1 & 2 & 2 & ⋆ & 2 & ⋆ & 2 & ⋆ & 2 & 12 & 12 \\ 1 & 2 & 2 & 1 & ⋆ & 1 & 2 & 1 & ⋆ & 12 & 12 \\ 1 & 2 & 2 & ⋆ & 2 & ⋆ & 2 & ⋆ & 2 & 12 & 12 \\ 1 & 2 & 1 & 1 & 1 & 1 & ⋆ & 1 & 1 & 12 & 12 \\ 1 & 2 & 2 & ⋆ & 2 & ⋆ & 2 & ⋆ & 2 & 12 & 12 \\ 1 & 2 & 2 & 1 & ⋆ & 1 & 2 & 1 & ⋆ & 12 & 12 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} )

Here an integer m at some place of a matrix means that the corresponding block equals ma where a can be any integral matrix. Moreover ⋆ means that this block with respect to the decomposition

U^{v} = U_{1}^{v} ⊕ U_{2}^{v} ⊕ U_{3}^{v}

is a matrix of the form

⋆ = ( \begin{matrix}1 & 2 & 1 \\ 1 & 1 & 1 \\ 2 & 2 & 1 \end{matrix} )

Remark. If all U^{i} with i ∈ {1, …, 6} are trivial then the square in the matrix of (5.1) shrinks to a 1 × 1-matrix and in this case we get the matrix problem for F^{4} which was solved in [BD2].

Proof. The necessity of this condition follows readily from the definition of a proper automorphism. To prove sufficiency consider the submatrix N of M corresponding to the direct summand W_{3} = V_{3}⊕U. The submatrix N is indicated by the rectangle in the matrix M above. Using the notation U^{0} = V_{3} the matrix N has blocks N_{ij} : U^{j} → U^{i} given with respect to the decomposition

W_{3} = U^{0} ⊕ U^{1} ⊕ U^{2} ⊕ U^{3} ⊕ U^{4} ⊕ U^{5} ⊕ U^{6}.

Since M is invertible also N is invertible modulo 2. Hence the following submatrices of N are also invertible modulo 2.

N_{0} = ( \begin{matrix}N_{00} & N_{04} \\ N_{40} & N_{44} \end{matrix} )

N_{2} = ( \begin{matrix}N_{22} & N_{26} \\ N_{62} & N_{66} \end{matrix} )

N_{1} = ( \begin{matrix}N_{11} & N_{13} & N_{15} \\ N_{31} & N_{33} & N_{35} \\ N_{51} & N_{53} & N_{55} \end{matrix} )

Consider the ring R_{0} of integral matrices of the form

B_{0} = ( \begin{matrix}B_{00} & B_{04} \\ 3B_{40} & B_{44} \end{matrix} )

where B_{vw} has the same size as N_{vw} and B_{44} has the form ⋆. Then we get

R_{0}/2R_{0} = Mat (n, 𝔽_{2})

where n is the number of rows in N_{0}. As GL(n, 𝔽_{2}) = SL(n, 𝔽_{2}) is generated by elementary matrices the natural homomorphism

R_{0}^{×} → GL(n, 𝔽_{2})

is surjective. In particular, there exists an invertible matrix B_{0} ∈ R_{0} such that B_{0} ≡ N_{0}(mod 2). Moreover the matrix B_{44} is invertible modulo 3.

Next let R be the ring of all matrices of the form ⋆. Then

R/3R = Mat (r, 𝔽_{3})

The group GL(r, 𝔽_{3}) is generated by elementary matrices and the matrix diag (−1, 1, …, 1). Hence the natural homomorphism

R^{×} → GL(r, 𝔽_{3})

is also surjective; in particular, there exists an invertible matrix A_{44} ∈ R such that A_{44} ≡ B_{44}(mod 3).

Similar observations show that tere exist invertible matrices

B_{2} = ( \begin{matrix}B_{22} & 3B_{26} \\ B_{62} & B_{66} \end{matrix} )

B_{1} = ( \begin{matrix}B_{11} & 3B_{13} & B_{15} \\ B_{31} & B_{33} & B_{35} \\ B_{51} & 3B_{53} & B_{55} \end{matrix} )

where all blocks B_{wv} are of the form ⋆ and for which in addition B_{2} ≡ N_{2} (mod 2) and B_{1} ≡ N_{1} (mod 2). Now put

B′_{1} = ( \begin{matrix}B_{11} & B_{13} & 5B_{15} \\ 3B_{31} & B_{33} & 3B_{35} \\ 5B_{51} & 5B_{53} & B_{55} \end{matrix} )

so that

B′_{1} ≡ ( \begin{matrix}I & 0 & 0 \\ 0 & 3I & 0 \\ 0 & 0 & 5I \end{matrix} ) B_{1} ( \begin{matrix}I & 0 & 0 \\ 0 & \frac{1}{3}I & 0 \\ 0 & 0 & 5I \end{matrix} ) (mod 12)

and det B′_{1} ≡ det B_{1} ≡ ±1(mod 12). Therefore just as above there is an invertible matrix

A_{1} = ( \begin{matrix}A_{11} & A_{13} & 5A_{15} \\ 3A_{31} & A_{33} & 3A_{35} \\ 5A_{51} & 5A_{53} & A_{55} \end{matrix} )

such that A_{1} ≡ B′_{1}(mod 12). Moreover this implies that all blocks A_{wv} have the form ⋆.

On the other hand as B_{2} is invertible its conjugate

A_{2} = ( \begin{matrix}I & 0 \\ 0 & 3I \end{matrix} ) B_{2} ( \begin{matrix}I & 0 \\ 0 & \frac{1}{3}I \end{matrix} ) = ( \begin{matrix}B_{22} & B_{26} \\ 3B_{62} & B_{66} \end{matrix} )

is also invertible. Put A_{wv} = B_{wv} for v, w ∈ {2, 6}.

We are now ready to define H_{0} and H_{4} associated to M such that the properties in (3.12) hold. This proves that M is proper and hence completes the proof of (5.1). Let H_{4} be the matrix

H_{4} = ( \begin{matrix}B_{00} & B_{01} & B_{02} & B_{03} & B_{04} & B_{05} & B_{06} \\ 0 & B_{11} & 0 & 3B_{13} & 0 & B_{15} & 0 \\ 0 & B_{21} & B_{22} & 3B_{23} & 0 & B_{25} & B_{26} \\ 0 & B_{31} & 0 & B_{33} & 0 & B_{35} & 0 \\ 3B_{40} & B_{41} & B_{42} & 3B_{43} & B_{4}^{4} & B_{45} & 3B_{4}^{6} \\ 0 & B_{51} & 0 & 3B_{53} & 0 & B_{55} & 0 \\ 0 & B_{61} & B_{62} & B_{63} & 0 & B_{65} & B_{66} \end{matrix} )

Here all blocks B_{wv} which have not yet been defined coincide with N_{wv}. Then H_{4} is invertible and H_{4} ≡ N (mod 2). Next let

H_{0} = ( \begin{matrix}A_{11} & 0 & A_{13} & 0 & 5A_{15} & 0 \\ 2A_{21} & A_{22} & 2A_{23} & 0 & 2A_{25} & A_{26} \\ 3A_{31} & 0 & A_{33} & 0 & 3A_{35} & 0 \\ 4A_{41} & 2A_{42} & 4A_{43} & A_{44} & 4A_{45} & 2A_{46} \\ 5A_{51} & 0 & 5A_{53} & 0 & A_{55} & 0 \\ 6A_{61} & 3A_{62} & 2A_{63} & 0 & 6A_{65} & A_{66} \end{matrix} )

where all blocks A_{wv} which have not yet been defined coincide with B_{wv}. Then A is also invertible and, in fact, A is ℤ/12-related to the lower rigth part of H_{4} corresponding to the direct summand U and indicated by the rectangle in H_{4} above. Hence this proves that M is proper; compare (3.12). q.e.d.

§ 6 Proof of the decomposition theorem (2.5)

Let Λ be the ring of all integral (23 × 23)-matrices of the form (5.1) whose rows and columns are numbered by the indexes i = 1, …, 5 and pairs \binom{v}{j} with v = 1, …, 6; j = 1, …, 3. Let

\{ \begin{matrix}U_{1} = ℤ/24 \\ U_{2} = U_{3} = U\binom{v}{j} = ℤ/12 \\ U_{4} = U_{5} = ℤ/2 \end{matrix}

Then the direct sum

U = ⨁_{i=1}^{5} U_{i} ⊕ ⨁_{v=1}^{6} ⨁_{j=1}^{3} U\binom{v}{j}

is in the obvious way a Λ − ℤ-bimodule denoted by U =_{Λ} U_{ℤ}. Recall that a matrix over U is by definition (cf. [D]) an element of P ⊗_{Λ} U ⊗ H^{∗} where P and H are finitely generated right projective modules over Λ and ℤ respectively. It is more convenient to identify this tensor product with

U(H, P) = Hom (H, P ⊗_{Λ} U).

Two matrices u ∈ U(H, P), u′ ∈ U(H′, P′) are isomorphic if there are isomorphisms α : H → H′ and β : P → P′ such that βu = u′α.

Let e_{i} = e_{ii} and e\binom{v}{j} = e\binom{u}{j}\binom{v}{j} be matrix units and

P_{i} = e_{i}Λ, P\binom{v}{j} = e\binom{v}{j}Λ

Then P can be uniquely decomposed as

P = (⨁_{i=1}^{5} V_{i} ⊗ P_{i}) ⊕ ⨁_{v=1}^{6} ⨁_{j=1}^{3} U_{j}^{v} ⊗ P\binom{v}{j}

for some free abelian groups V_{i}, U_{j}^{v}. Therefore

P ⊗_{Λ} U ≅ ⨁_{i=1}^{5} V_{i} ⊗ U_{i} ⊕ ⨁_{v=1}^{6} ⨁_{j=1}^{3} U_{j}^{v} ⊗ U\binom{v}{j}

This shows that isomorphism classes of matrices u above are in 1-1 correspondence with homotopy types in F^{5} (for this we use (3.11) and (5.1)).

We shall write the elements of U(H, P) as families of matrices (u_{i}, u_{j}^{v}) with u_{i} being of size d_{i} × c_{5} and u_{j}^{v} of size d_{j}^{u} × c_{5} if

\{ \begin{matrix}H = ℤ^{c_{5}} \\ V_{i} = ℤ^{d_{i}} \\ U_{j}^{v} = ℤ^{d_{j}^{v}} \end{matrix}

as in (3.5) and (3.8). The entries of u_{i}, u_{j}^{v} are in the corresponding groups U_{i} and U\binom{v}{j} respectively. The matrices M ∈ Λ define the “admissible transformations” of rows in these matrices. 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.

We point out that the ring Λ can be “Morita-reduced” since Λ is Moritaequivalent to the following ring Λ′. The ring Λ′ consists of all matrices of the same form as in Λ for which rows and columns, however, correspond only to the indexes i = 1, …, 5 and pairs \binom{v}{j} with j = 1, 2, 3 and v ∈ \{3, 4, 6\} except \binom{4}{3}. Hence the restriction of the bimodule U to Λ′ yields the Λ′ − ℤ-bimodule U′ =_{Λ′} U_{ℤ}. The categories of U-matrices and U′-matrices are equivalent.

Moreover U′ is also a Λ̄ − ℤ̄-bimodule with ℤ̄ = ℤ/24 and Λ̄ = Λ′/24 which we denote by Ū =Λ̄ Uℤ̄. The elements of U and Ū are the same and also the matrices from U(P, H) coincide with those from Ū(P̄, H̄) with P̄ = P/24 and H̄ = H/24, but non-isopmorphic U-matrices might be isomorphic as Ū-matrices.

Consider the 2-primary part Ũ of Ū and let Λ̃ be the ring of matrices with entries in ℤ/8 satisfying the same conditions as matrices in Λ′ above (i.e., in the corresponding congruences we replace (mod 24) by (mod 8), (mod 12) by (mod 4) and (mod 6) by (mod 2). Then Ũ is a Λ̃ − ℤ/8-bimodule Ũ =Λ̃ Ũ_{ℤ/8}.

Now denote by z_{i} (resp. z\binom{v}{j}) the image of z ∈ ℤ in Ũ_{i} (resp, in Ũ\binom{v}{j}). For

elements u, v ∈ Ũ we write u < v if there exists a ∈ Λ̃ with au = v. Then, in fact, all 2-primary elements from Ũ_{i} (resp. Ũ\binom{v}{j}) are linearly ordered as follows:

1_{1} < 1\binom{3}{3} < 1\binom{3}{1} < 1\binom{3}{2} < 1\binom{6}{3} < 1\binom{6}{1} < 1\binom{6}{2}
<1_{3} < 1\binom{4}{1} < 1\binom{4}{3} < 1_{2}
<2_{1} < 2\binom{3}{3} < 2\binom{3}{1} < 2\binom{3}{2} < 2\binom{6}{3} < 2\binom{6}{1} < 2\binom{6}{2}
<2_{3} < 2\binom{4}{1} < 2\binom{4}{3} < 2_{2} < 4_{1}

On the other hand 1_{5} < 1_{4} < 4_{1} and 1_{5} < 2_{2} and there are no other relations < between 2-primary elements. Therefore we can procede as in [BD2] and [D] to obtain the following list of indecomposable Ũ-matrices:

(a_{1}) with a = 1, 2, 4.
(a_{i}) with i = 2, 3 and a = 1, 2.
(1_{i}) with i = 4, 5.
(a\binom{v}{j}) with a = 1, 2 and \binom{v}{j} satisfying v ∈ \{3, 4, 6\}, j = 1, 2, 3 except \binom{4}{3}.
( \begin{matrix}a_{i} \\ 1_{k} \end{matrix} ) with a = 1, 2 and i = 1, 2, 3 and k = 4, 5 except ( \begin{matrix}2_{2} \\ 1_{5} \end{matrix} ) .
( \begin{matrix}a\binom{v}{j} \\ 1_{k} \end{matrix} ) with a = 1, 2 and k = 4, 5 and \binom{v}{j} satisfying v ∈ \{3, 4, 6\}
and j ∈ \{1, 2, 3\} except \binom{4}{3}.

If we consider the 3-part of Ū-matrices the answer is easy. There is only one nontrivial indecomposable matrix (1_{1}) (isomorphic to (1_{i}) and (1\binom{v}{j}) for all possible values of i, v, j). The Chinese remainder theorem implies that we can glue any 2-primary element from Ũ(P, H) with any 3-primary element with the same values of P and H. This gives us the following indecomposable U-matrices (taking into account those “crossed out” under Morita-reduction).

(w_{1}) with w = 1, …, 12.
(w_{i}) with i = 2, 3 and w = 1, …, 6.
(w\binom{v}{j}) with v = 1, …, 6 and j = 1, 2, 3 and w = 1, …, 6
(1_{4}) and (1_{5})
( \begin{matrix}w_{i} \\ 1_{4} \end{matrix} ) with i = 1, 3 and w = 1, …, 6.
( \begin{matrix}w\binom{v}{j} \\ 1_{4} \end{matrix} ) with v = 1, …, 6 and j = 1, 2, 3 and w = 1, …, 6.
( \begin{matrix}w_{2} \\ 1_{5} \end{matrix} ) with w = 1, 2, 3.
( \begin{matrix}w_{i} \\ 1_{5} \end{matrix} ) with i = 1, 3 and w = 1, …, 6.
( \begin{matrix}w\binom{v}{j} \\ 1_{5} \end{matrix} ) with v = 1, …, 6 and j = 1, 2, 3 and w = 1, …, 6.
( \begin{matrix}w_{2} \\ 1_{4} \end{matrix} ) with w = 1, 2, 3.

These indecomposables are in 1-1 correspondence with all elements in L_{5} − L_{4} − {S^{5}}. The correspondence is given as follows; compare the list of graphs following (1.1). The correspondence is easily deduced from the notation in (3.3) since the matrices describe the attaching map of the top cell.

(1_{5}) = η_{3},  (1_{4}) = (ηη)_{2},  (w_{1}) = w_{1},
( \begin{matrix}w_{2} \\ 1_{5} \end{matrix} ) = ηw_{1}η,  ( \begin{matrix}w_{3} \\ 1_{4} \end{matrix} ) = ηηw_{1}ηη,  ( \begin{matrix}w_{3} \\ 1_{5} \end{matrix} ) = ηηw_{1}η,
( \begin{matrix}w_{2} \\ 1_{4} \end{matrix} ) = ηw_{1}ηη,  (w_{2}) = ηηw_{1},  ( \begin{matrix}w_{1} \\ 1_{4} \end{matrix} ) = w_{1}ηη,
(w_{3}) = ηw_{1},  ( \begin{matrix}w_{1} \\ 1_{5} \end{matrix} ) = w_{1}η,  (w\binom{v}{3}) = v_{0}ηηw_{1},

( \begin{matrix}w\binom{v}{2} \\ 1_{5} \end{matrix} ) = ηv_{0}ηηw_{1}η, ( \begin{matrix}w\binom{v}{1} \\ 1_{4} \end{matrix} ) = ηηv_{0}ηηw_{1}ηη,

( \begin{matrix}w\binom{v}{3} \\ 1_{5} \end{matrix} ) = v_{0}ηηw_{1}η, (w\binom{v}{2}) = ηv_{0}ηηw_{1},

( \begin{matrix}w\binom{v}{3} \\ 1_{4} \end{matrix} ) = v_{0}ηηw_{1}ηη, (w\binom{v}{1}) = ηηv_{0}ηηw_{1},

( \begin{matrix}w\binom{v}{1} \\ 1_{5} \end{matrix} ) = ηηv_{0}ηηw_{1}η, ( \begin{matrix}w\binom{v}{2} \\ 1_{4} \end{matrix} ) = ηv_{0}ηηw_{1}ηη,

This completes the proof of the decomposition theorem (2.5). q.e.d.

§ 6 On the representation type of F^{6}

Since we classified above the indecomposable homotopy types of F^{5} we can proceed to classify the homotopy types in F^{6}. The method is similar to the computation in § 3 for F^{5}. Similarly as in (3.1) (2) we now obtain for X ∈ F^{6} a homomorphism

(6.1)  f : H_{6}X → Γ_{5}(X^{5})

where we may assume that the skeleton X^{5} again is given by a homology decomposition of X. Moreover X^{5} is a one point union of indecomposable objects in X(L_{5}) by (2.5). Hence we can compute Γ_{5}(Y) for each object Y in X(L_{5}) in order to obtain an explicit form of Γ_{5}(X^{5}). Then we have to understand the action of the group of homotopy equivalences of X^{5} on the Γ_{5}X^{5} and using this action we have to construct a “normal form” of (6.1).

(6.2) Theorem. For Y ∈ X(L_{5}) the group Γ_{5}Y is either given by (3.4) or by the following list:

Γ_{5}X(g) = 0  for g = S^{0}, v_{0}ηη, v_{0}ηηw_{1}, v_{0}ηηw_{1}η.
Γ_{5}X(g) = ℤ/24  for g = η_{0}, ηv_{0}ηη, ηv_{0}ηηw_{1}η, ηv_{0}ηηw_{1}.
Γ_{5}X(g) = ℤ/12  for g = v_{0}η, v_{0}ηηw_{1}ηη.
Γ_{5}X(g) = ℤ/2  for g = (ηη)_{0}, v_{0}, ηηv_{0}ηη, ηηv_{0}ηηw_{1}, ηηv_{0}ηηw_{1}η.
Γ_{5}X(g) = ℤ/24 ⊕ ℤ/12 for g = ηv_{0}η, ηv_{0}ηηw_{1}ηη.
Γ_{5}X(g) = ℤ/2 ⊕ ℤ/12 for g = ηηv_{0}η, ηηv_{0}ηηw_{1}ηη.
Γ_{5}X(g) = ℤ/2 ⊕ ℤ/2  for g = ηηv_{0}.
Γ_{5}X(g) = ℤ/24 ⊕ ℤ/2 for g = ηv_{0}

The theorem shows that there are exactly 328 elements g ∈ L_{5} for which Γ_{5}(g) is non zero. This shows that the 23 × 23 matrix in (5.1) has to be replaced by a 328 × 328 matrix with the additional complication that various Γ_{5}(Y) in (6.2) are given by the direct sum of two cyclic groups. We now consider the special case that X^{5} in (6.1) is a one point union of d copies of X(ηηv_{0}) with v_{0} ∈ {1, …, 6}. In this case we get the following problem.

Let H = ℤ^{h} and V = ℤ^{d} be finitely generated free abelian groups and consider homomorphisms

(6.3)  f, f′ : H → V ⊗ ℤ/2 ⊕ V ⊗ ℤ/2

Then f is equivalent to f′ if there exist automorphisms n_{0}, n_{3}, n_{4} ∈ Aut(V) and h ∈ Aut(H) with

(1)  \{ \begin{matrix}v_{0} n_{0} & ≡ v_{0} n_{4} (mod 12) \\ n_{0} & ≡ n_{3}  (mod 2) \end{matrix}

such that f′ = (n_{3} ⊕ n_{4}) ⊗ ℤ/2 ∘ f ∘ h. Equivalence classes of such homomoprhisms are in 1-1 correspondence with homotopy types of CW-complexes of the form

(2)  \underbrace{X(ηηv_{0}) ∨ … ∨ X(ηηv_{0})}_{d−times} ∪_{f} \underbrace{e^{6} ∪ … ∪ e^{6}}_{h−times} .

(6.4) Proposition. There are infinitely many indecomposable homotopy types in F^{6} and K_{0}(F^{6}) = ℤ^{∞}.

Proof. We consider the decomposition of homotopy types as in (6.3) (2) for v_{0} = 2. Then (6.3) (1) shows that the 2-primary part of this problem has the representation type of the well known Kroneker quiver

• ⇉ •

which has tame representation type and infinitely many indecomposable representations over ℤ/2. q.e.d.

Proposition (6.4) together with theorem (2.5) yields a proof of Theorem B in the introduction.

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