A HOMOTOPY OPERATION SPECTRAL SEQUENCE

FOR THE COMPUTATION OF HOMOTOPY GROUPS

Hans-Joachim Baues and Paul Goerss

Let X be an (r − 1) -connected space with r ≥ 2. J.H.C. Whitehead embedded the Hurewicz homomorphism h in the long exact sequence

… → H_{n+1}X → Γ_{n}X → π_{n}X ⟶^{h} H_{n}X → Γ_{n−1}X → …

The classical Hurewicz theorem is equivalent to the fact that Γ_{j}X = 0 for j ≤ r. Whitehead [W 1] computed the group Γ_{r+1}X and the group Γ_{r+2}X was computed by Baues in [B3]. In this paper we show that there is a ”homotopy operation spectral sequence” which converges to the groups Γ_{n}X, n > r. The E^{2} -term of the spectral sequence is given by certain homotopy operation functors Γ_{r}^{k} and their derived functors L_{i}Γ_{r}^{k}. The functors Γ_{r}^{k} are algebraically determined by properties of homotopy groups of spheres and can be described explicitly for low k. The spectral sequence is the canonical generalization of the computation of Γ_{r+1}X establishing the algebraic structure of homotopy groups Whitehead’s program asked for 50 years ago. A different but related spectral sequence converging to the image of the Hurewicz homomorphism is due to Blanc [Bl1]. We describe various examples and applications of the ”homotopy operation spectral sequence”.

§ 1 Homotopy operations and the associated homotopy operation functors

We consider the category of homotopy operations given by the homotopy category consisting of one point unions of spheres. Such homotopy operations act on the homotopy groups π_{∗}(X) of a space X. Moreover homotopy operations determine canonically a sequence of functors Γ_{r}^{k}, r ≥ 2, k ≥ 1 which we call homotopy operation functors. The derived functors L_{i}Γ_{r}^{k} yield the E_{2} -term of the spectral sequence in the next section.

Let Top^{∗}/ ≃ be the homotopy category of pointed topological spaces.

(1.1) Definition. The category Π of homotopy operations is the full subcategory of Top^{∗}/ ≃ consisting of finite one point unions of spheres of dimensions ≥ 1. We also consider the full subcategories (r ≥ 2, k ≥ 0)

Π_{r}^{k} ⊂ Π_{r} ⊂ Π

Typeset by AMS-TEX

Here Π_{r} consists of all finite one point unions of spheres of dimension ≥ r and Π_{r}^{k} consists of all finite one point unions of spheres S^{n} with r ≤ n ≤ r + k. The one point union of objects is the categorical coproduct in Π.

Recall that a theory T is a small category in which finite coproducts denoted by A ∨ B exist. The empty coproduct is the initial object ∗ of T. Let Set be the category of sets. A model of the theory T is a contravariant functor M : T → Set which carries finite coproducts to products. A morphism between such models is a natural transformation of functors. The resulting category of models is denoted by model (T). Compare Baues [B1], [B2].

For example the categories of homotopy operations Π, Π_{r}, Π_{r}^{k} above are theories. A model of Π is also termed a Π -algebra; see Stover [St], Dwyer-Kan-Smith-Stover [DKSS], Blanc [Bl1]. A model of Π_{r}^{k} is denoted by π : Π_{r}^{k} → Set with

(1.2)  π_{n} = π(S^{n}) for r ≤ n ≤ r + k

Since S^{n} is a cogroup in Π we see that π_{n} is a group which is abelian for n ≥ 2. Moreover one has the canonical functor

(1)  G_{n} : model (Π_{r}^{k}) → Ab

Here Ab is the category of abelian groups and G_{n} carries the model π to the abelian group G_{n}(π) = π_{n} in (1.2). In fact for k = 0 and r ≥ 2 the functor

(2)  G_{r} : model (Π_{r}^{0}) ≅ Ab

is an isomorphism of categories which we shall use as an identification. Since π carries finite coproducts to products we see that the values of π on objects in Π_{r}^{k} are completely determined by the sequence of groups (π_{r}, π_{r+1}, …, π_{r+k}).

(1.3) Remark. In view of Hilton’s analysis of the homotopy groups of a one point union of spheres (G.W. Whitehead [W h]XI) one can consider a Π -algebra π as a graded group (π_{1}, π_{2}, …) with π_{n} abelian for n ≥ 2, together with Whitehead product homomorphisms

[−, −] : π_{n} ⊗ π_{m} → π_{n+m−1}

(n, m ≥ 2) and composition functions α^{∗} : π_{m} → π_{n} for α ∈ π_{n}(S^{m}) and a left action of π_{1} on the π_{n}(n ≥ 2) which commutes with these operations. Moreover these operations satisfy ”all the identities that hold for the homotopy groups of pointed topological spaces”. This statement (as used in [DKSS], [Bl2]) is rather vague. A complete and deep algebraic analysis of the relations in terms of James-Hopf invariants was achieved in the thesis of Dreckmann [D1], compare [D2].

One has the canonical functor

(1.4)  Top^{∗}/ ≃→ model (Π_{r}^{k})

which carries a topological space X to the model

π_{∗}(X) = [−, X] : Π_{r}^{k} → Set

defined on S ∈ Π_{r}^{k} by the set of homotopy classes [S, X]. If S is the finite one point union S = S^{n_{1}} ∨ … ∨ S^{n_{t}} of spheres S^{n_{i}} with r ≤ n_{i} ≤ r + k then clearly

[S, K] = π_{n_{1}}(X) × … × π_{n_{t}}(X)

is a product of homotopy groups of X. A map S → S′ in Π_{r}^{k} induces a ”homotopy operation” between such products of homotopy groups.

(1.5) Definition. The homotopy operation functor Γ_{r}^{k+1} with r ≥ 2, k ≥ 0 is defined by the composite

Γ_{r}^{k+1} : model (Π_{r}^{k}) ⟶^{Δ} model (Π_{r}^{k+1}) ⟶^{G_{r+k+1}} Ab

where Δ is the left adjoint of the functor

λ^{∗} : model (Π_{r}^{k+1}) → model (Π_{r}^{k})

induced by the inclusion λ : Π_{r}^{k} ⊂ Π_{r}^{k+1}. In 7.4 [B2] an explicit description of the left adjoint Δ is given.

In the next result we use the following well known Grothendieck-construction ΓA of a functor Γ : C → A. The objects of the category ΓA are triple (X, A, η) where X is an object in C and η : Γ(X) → A is a homomorphism in A. Morphisms (X, A, η) → (Y, B, λ) in ΓA are pairs (f, g) where f : X → Y is a morphism in C such that the diagram

Γ(X) ⟶^{Γ(f)} Γ(Y)
↓^{η}  ↓^{λ}
A ⟶^{g} B

commutes in A. An object (X, A, η) is also denoted by η.

(1.6) Proposition. Let Γ_{r}^{k+1}Ab be the Grothendieck construction of the homotopy operation functor Γ_{r}^{k+1}. Then there is a canonical isomorphism of categories

model (Π_{r}^{k+1}) = Γ_{r}^{k+1}Ab

Proof. Since Π_{r}^{k+1} is a graded theory in the sense of (8.1′′) [B2] we obtain the result by (8.4) and (7.7) in [B2]. q.e.d.

Using the isomorphism (1.6) as an identification we obtain by (1.2) (2) inductively the sequence of functors Γ_{r}^{1}, Γ_{2}^{2}, … (also termed homotopy operation functors) with

(1.7)  \{ \begin{matrix}Γ_{r}^{1} : Ab → Ab \\ Γ_{r}^{k+1} : Γ_{r}^{k}Ab → Ab, k ≥ 1. \end{matrix}

This is an iterated Grothendieck construction satisfying model (Π_{r}^{k}) = Γ_{r}^{k}Ab. Hence a model π of Π_{r}^{k} can be identified with a graded abelian group π_{∗} = (π_{r}, π_{r+1}, …, π_{r+k}) together with a sequence of homomorphism

(1)  { η^{1} : Γ_{r}^{1}(π_{r}) → π_{r+1} ; η^{2} : Γ_{r}^{2}(η^{1}) → π_{r+2} ; η^{3} : Γ_{r}^{3}(η^{1}, η^{2}) → π_{r+3} ; ⋮ ; η^{k} : Γ_{r}^{k}(η^{1}, …, η^{k−1}) → π_{r+k}

Such sequences describe objects in the iterated Grothendieck construction. For any model π of Π_{r}^{k} the graded abelian group π_{∗} has a graded subgroup P (π) ⊂ π_{∗}, generated by all elements which are in the image of a nontrivial primary homotopy operation (i.e. any homotopy operation which vanishes in homology). In fact, we have in degree t with r ≤ t ≤ r + k

(2)  P_{t}(π) = \{ \begin{matrix}0 for t = r \\ image (η^{t} : Γ_{r}^{t}(η^{1}, …, η^{t−1}) → π_{t}) for t > r \end{matrix}

The quotient group

(3)  Q_{t}(π) = π_{t}/P_{t}(π) = \{ \begin{matrix}π_{r} for t = r \\ cokernel (η^{t}) for t > r \end{matrix}

will be called the indecomposables in degree t of the model π. Hence Q_{t} yields a functor Q_{t} : model (Π_{r}^{k}) → Ab.

(1.8) Remark. A Π -algebra π is the same as a π_{1} -object in the category model(Π_{2}) and hence the Π -algebra π can be equivalently described by a group π_{1} and a sequence (π_{2}, π_{3}, …) of left π_{1} -modules together with a sequence of π_{1} -equivariant homomorphisms

\{ \begin{matrix}η^{1} : Γ_{2}^{1}(π_{2}) → π_{3} \\ η^{k+1} : Γ_{2}^{k+1}(η^{1}, …, η^{k}) → π_{2k+1}, k ≥ 1. \end{matrix}

Hence only the homotopy operation functors Γ_{2}^{k}, k ≥ 1, are needed to define a Π -algebra π.

J.H.C. Whitehead [W 1] computed the homotopy operation functor Γ_{r}^{1} by

(1.9)  Γ_{r}^{1}(π_{r}) = \{ \begin{matrix}Γ(π_{2}) for r = 2, \\ π_{r} ⊗ ℤ/2 for r ≥ 3. \end{matrix}

Here Γ is the universal quadratic functor defined as follows. A function f : A → B between abelian groups is termed quadratic if f(−a) = f(a) and if the function A × A → B with (a, b) ↦ f(a + b) − f(a) − f(b) is bilinear. For each abelian group A there is a universal quadratic function γ : A → Γ(A) which defines the functor Γ. If π = π_{∗}X is given by (1.4) then the composite

π_{2} ⟶^{γ} Γ(π_{2}) ⟶^{η^{1}} π_{3}

coincides with the map π_{2}X → π_{3}X induced by the Hopf map η_{2} ∈ π_{3}(S^{2}). Compare [B3].

As a further example the homotopy operation functor Γ_{r}^{2} is given as follows where η^{1} : Γ_{r}^{1}(π_{r}) → π_{r+1}. See (11.3.3), (9.3.3) and (8.3.5) in [B3].

(1.10)  Γ_{r}^{2}(η^{1}) = \{ \begin{matrix}Γ_{2}^{2}(η^{1}) & for r = 2, \\ π_{4} ⊗ ℤ/2 ⊕ Λ^{2}(π_{3}) & for r = 3, \\ π_{r+1} ⊗ ℤ/2 & for r ≥ 4. \end{matrix}

Here Λ^{2}(A) = A ⊗ A/(a ⊗ a ∼ 0) is the exterior square of the abelian group A and Γ_{2}^{2}(η^{1}) is defined by the pushout diagram in Ab

(1)  \begin{matrix}π_{3} ⊗ (ℤ/2 ⊕ π_{2}) & ⟶^{q_{∗}} & Γ_{2}^{2}(η^{1}) \\ ↑^{η^{1}⊗1} & & ↑^{η_{∗}^{1}} \\ Γ(π_{2}) ⊗ (ℤ/2 ⊕ π_{2}) & ⟶^{q} & L(π_{2}, 1)_{3} ⊕ Γ(π_{2}) ⊗ ℤ/2 \end{matrix}

The group L(A, 1)_{3} is the image of the homomorphism A^{⊗3} → A^{⊗3} which carries a ⊗ b ⊗ c to

(2)  [[a, b], c] = (a ⊗ b + b ⊗ a) ⊗ c − c ⊗ (a ⊗ b + b ⊗ a)

with a, b, c ∈ A. Moreover the quotient map q is the inclusion on Γ(π_{2}) ⊗ ℤ/2 and satisfies

(3)  q(γ(a) ⊗ b) = −[[b, a], a] + (γ(a + b) − γ(a) − γ(b)) ⊗ 1

with 1 ∈ ℤ/2 and a, b ∈ π_{2}.

(1.11) Remark. For s ≥ 0 let Ab^{s} = Ab × … × Ab be the s-fold product of the category Ab. We define in § 4 [B2] a system of reduction functors

Γ_{r,ε}^{n+1} : (Γ_{r}^{n}Ab) × Ab^{|ε|} → Ab

associated to the iterated Grothendieck construction (Γ_{r}^{1}, Γ_{r}^{2}, …). Here ε = (ε^{1}, ε^{2}, …, ε^{t}) is a finite sequence of integers ε^{i} ≥ 0 with t ≥ 0 and | ε | is the number of integers ε^{i} > 0 in ε. Clearly for the empty sequence t = 0 the functor coincides with Γ_{r}^{n+1}. The reduction functors are useful to obtain information on the vanishing line for the derived functors L_{i}Γ_{r}^{k} in (1.12) below with k = n + 1 + t. As an example we consider the case k = r = 2. Using (1.10) (1) above one gets the exact sequence in the top row of the following diagram where Q(η^{1}) = cokernel (η^{1}) and where we use the reduction functors Γ_{2,(0)}^{1} and Γ_{2,(1)}^{1} of with Γ_{r}^{0}Ab = Ab.

Γ_{2,(0)}^{1}(π_{2})  Γ_{2,(1)}^{1}(π_{2}, Q(η^{1}))
∥  ∥
L(π_{2}, 1)_{3} ⊕ Γ(π_{2}) ⊗ ℤ/2 ⟶ Γ_{2}^{2}(η^{1}) ⟶ Q(η^{1}) ⊗ (ℤ/2 ⊕ π_{2}) ⟶ 0
∥  ∥  ∥
Λ(I_{1}π_{2})  ⟶ Λ(η^{1}) ⟶  Λ_{(1)}(η^{1})  ⟶ 0

By (2.6) [B3] we obtain for the functor Λ = Γ_{2}^{2} : Γ_{2}^{1}Ab → Ab the exact sequence in the bottom row of the diagram which shows that the reduction functors Γ_{2,(0)}^{1} and Γ_{2,(1)}^{1} are given by the equations in the diagram. The exact sequence is short exact if η^{1} is split injective. In section § 5 we shall use the exact sequence to study the derived functors L_{i}Γ_{2}^{2}. We point out that the exact sequence in the diagram is also studied in Appendix 11.3 A[B3] where a relation with the ”nilization functor” is described.

As usual we obtain for the category C the category sC of simplicial objects in C.

(1.12) Definition. For each object

X ∈ model (Π_{r}^{k}) = Γ_{r}^{k}Ab

there is a resolution

X_{·} ∈ s model (Π_{r}^{k}) = s Γ_{r}^{k}Ab

which either can be chosen to be a free simplicial resolution defined by Blanc (3.2.2)

[Bl1] or a resolution as in Baues 3.3 [B2]. Then the derived functors

L_{i}Γ_{r}^{k+1} : Γ_{r}^{k}Ab → Ab

are defined by the homotopy groups

L_{i}Γ_{r}^{k+1}(X) = π_{i}(Γ_{r}^{k+1}(X_{·}))

of the simplicial abelian group Γ_{r}^{k+1}(X_{·}). One has L_{0}Γ_{r}^{k+1} = Γ_{r}^{k+1}.

For example the derived functors of Γ_{r}^{1} are the derived functors in the sense of Dold-Puppe [DP] satisfying

(1.13)  L_{1}Γ_{r}^{1}(A) = \{ \begin{matrix}ΓT (A) & for r = 2 \\ A ∗ ℤ/2 & for r ≥ 3 \end{matrix}

Here ΓT (A) is the Γ -torsion in (11.12) [B3] (see also (5.5) (3) below) and A ∗ B = T or(A, B) is the torsion product of abelian group A and B. The cross effect of the Γ -torsion is ΓT (A | B) = A ∗ B and for cyclic groups we have ΓT (ℤ) = 0 and ΓT (ℤ/n) = ℤ/n ∗ ℤ/2. Moreover L_{i}Γ_{r}^{1} = 0 for i ≥ 2, see (6.14.9) [B3]. The next result is a consequence of the vanishing line of Blanc [Bl1]; an additional proof of this result is contained in [B2].

(1.14) Proposition. The derived functors L_{i}Γ_{r}^{k} are trivial, that is L_{i}Γ_{r}^{k} = 0, for r ≥ 3 and i ≥ 2k. Moreover for r = 2 one has L_{i}Γ_{2}^{k} = 0 if i > 2k.

We conjecture that also L_{i}Γ_{2}^{k} = 0 for i = 2k; for example we have L_{2}Γ_{2}^{1} = 0 by (1.13) and in (5.12) (8) we shall show L_{4}Γ_{2}^{2} = 0. For the proof of (1.14) we observe that we have by 3.6 [B2] the following connection between the derived functors L_{i}Γ_{r}^{k} and the derived functors L_{j}Q_{t} of the functor Q_{t} of indecomposables in (1.7) (3):

(1.15)  L_{i+1}Q_{r+k} = L_{i}Γ_{r}^{k} for i ≥ 1.

Moreover for a model π of Π_{r}^{k} the sequence

(1.16)  0 → (L_{1}Q_{r+k})(π) → Γ_{r}^{k}(η^{1}, …, η^{k−1}) ⟶^{η^{k}} π_{r+k} → Q_{r+k}(π) → 0

is exact with Q_{r+k} = L_{0}Q_{r+k}. The derived functors L_{j}Q_{t} were studied by Blanc [Bl1]. Therefore (1.14) is a consequence of (4.1) [Bl1]. The exact sequence (1.16) shows that L_{1}Q_{t} can be computed by use of the homotopy operation functor. In addition 5.1.1 [Bl1] shows

(1.17)  (L_{2k−1}Γ_{r}^{k})(π) = π_{r} ∗ ℤ/2 for r ≥ 1, k ≥ 1

For k = 1 this is compatible with (1.13).

For small k (k ≤ 19) the categories Π_{r}^{k} are completely known by the results in Toda’s book [T]. Therefore, in principle, it is possible to compute the corresponding homotopy operation functors Γ_{r}^{k} and the derived functors L_{i}Γ_{r}^{k}. The rich structure of homotopy groups of spheres, however, implies that such a computation is of high complexity.

§ 2 The spectral sequence based on homotopy operations

Let X be an (r − 1) -connected CW-complex with r ≥ 2. The Γ -groups Γ_{n}X in the ”certain exact sequence” of J.H.C. Whitehead [W 1]

… → H_{n+1}X ⟶^{b} Γ_{n}X ⟶^{i} π_{n}X ⟶^{h} H_{n}X ⟶^{b} Γ_{n−1}X → …

are defined by

(2.1)  Γ_{n}X = image {π_{n}X^{n−1} ⟶^{j} π_{n}X^{n}}

Here X^{n} denotes the n-skeleton of X and the map j is induced by X^{n−1} ⊂ X^{n}. Moreover the operator i in the exact sequence is induced by X^{n} ⊂ X. The operator h is the Hurewicz homomorphism and the secondary boundary b is induced by attaching maps of n-cells, compare for example [B3] where many further properties of Whitehead’s exact sequence are studied. Since X is (r − 1) -connected we may assume that X^{r−1} = ∗ is a point and therefore Γ_{j}X = 0 for j ≤ r. Hence the sequence terminates as follows where we omit X in the notation so that π_{t} = π_{t}X.

Γ_{r}^{2}(η^{1})  Γ_{r}^{1}(π_{r})
γ↓  _{↘η^{2}}  ^{γ}↓  _{↘η^{1}}
… → H_{r+3} → Γ_{r+2} ⟶ π_{r+2} →H_{r+2} → Γ_{r+1} ⟶ π_{r+1} →H_{r+1} → 0

Here η^{1}, η^{2}, … are given by the model π_{∗}(X) of Π_{r} defined as in (1.4). We claim that for k ≥ 1 there exists a natural commutative diagram

(2.2)  \begin{matrix}Γ_{r}^{k}(η^{1}, …, η^{k−1}) \\ γ & ↓ & ↘η^{k} \\ Γ_{r+k}(X) & ⟶^{i} & π_{r+k}(X) \end{matrix}

where η^{k} is defined as in (1.7) (1); that is, there is a transformation γ which is natural in X and which satisfies η^{k} = iγ. In fact using (2.1) we see that γ exists since all homotopy operations involved in the definition of Γ_{r}^{k} are defined on elements in degree < r + k and such elements live in the skeleton X^{r+k−1}. J.H.C. Whitehead [W 1] showed that

(2.3)  γ : Γ_{r}^{1}(π_{r}(X)) ⟶^{≅} Γ_{r+1}X

is an isomorphism for all (r−1) -connected spaces X. This result now is generalized by the following spectral sequence which is constructed in section § 6 below.

(2.4) Theorem. Let X be an (r − 1) -connected space with r ≥ 2. Then there is a natural spectral sequence {E_{p,q}^{r}} with differential d^{r} : E_{p,q}^{r} → E_{p−r,q+r−1}^{r} satisfying

\begin{matrix}E_{0,q}^{0} = Γ_{r}^{q}(η^{1}, …, η^{q−1}) \\ E_{p,q}^{2} = (L_{p}Γ_{r}^{q})(η^{1}, …, η^{q−1}) \end{matrix}

with p, q ≥ 1. The sequence converges to Γ_{r+∗}X such that the edge homomorphism

E_{0,q}^{2} = Γ_{r}^{q}(η^{1}, …, η^{q−1}) ↠ E_{0,q}^{∞} ⊂ Γ_{r+q}(X)

coincides with γ in (2.2).

Using the vanishing line (1.14) we see that the spectral sequence looks like

⋮
Γ_{r+4} ← • • • • • • • •
Γ_{r+3} ← • • • • • •
Γ_{r+2} ← • • • •
Γ_{r+1} ← • •

Only for p < 2q with r ≥ 3 (resp. p ≤ 2q with r = 2) the E^{2} -term is non trivial. The terms E_{p,q}^{2} with p + q = k contribute to Γ_{r+k}(X). This, in fact, proves that for k = 1 the map γ in (2.3) is an isomorphism. Moreover we derive from the spectral sequence the results in the following sections.

(2.5) Addendum. If X is a connected pointed CW-complex with universal covering X̃ then we can apply the spectral sequence (2.4) to X̃. Hence there exists a natural spectral sequence {E_{p,q}^{r}} consisting of π_{1}(X) -modules with π_{1}(X) -equivariant differentials satisfying

\begin{matrix}E_{0,q}^{2} = Γ_{2}^{q}(η^{1}, …, η^{q−1}) \\ E_{p,q}^{2} = (L_{p}Γ_{2}^{q})(η^{1}, …, η^{q−1}) \end{matrix}

with p, q ≥ 1. This sequence converges to the π_{1}(X) -module Γ_{2+∗}(X) = Γ_{2+∗}(X̃) such that the edge homomorphism coincides with γ in (2.2). Compare (1.8).

§ 3 The group Γ_{r+2}X of an (r − 1)-connected space

The next theorem is an immediate consequence of the spectral sequence (2.4) since L_{2}Γ_{r}^{1} = 0 by (1.12). In this section we describe various specifications and applications of this result.

(3.1) Theorem. Let X be an (r − 1) -connected space with r ≥ 2. Then there is a natural short exact sequence

0 → Γ_{r}^{2}(η^{1}) → Γ_{r+2}X → (L_{1}Γ_{r}^{1})(π_{r}) → 0

where η^{1} : Γ_{r}^{1}(π_{r}) → π_{r+1} with π_{∗} = π_{∗}(X).

Using (1.10) and (1.13) we see that there are the following three cases of (3.1).

(1) Corollary. Let X be a 1-connected space and let

η^{1} : Γ(π_{2}(X)) → π_{3}(X)

be induced by the Hopf map η_{2} ∈ π_{3}(S^{2}), that is, η^{1}γ(α) = αη_{2} for α ∈ π_{2}(X); see (1.9). Then there is a natural short exact sequence

0 → Γ_{2}^{2}(η^{1}) ⟶^{γ} Γ_{4}(X) ⟶^{μ} ΓT (π_{2}(X)) → 0

where Γ_{2}^{2}(η^{1}) is defined in (1.10) (1) and where ΓT is the Γ -torsion in (1.13).

(2) Corollary. Let X be a 2-connected space and let

η^{1} : π_{3}(X) ⊗ ℤ/2 → π_{4}(X)

be induced by the Hopf map η_{3} ∈ π_{4}(S^{3}), that is η^{1}(α ⊗ 1) = αη_{3} for α ∈ π_{3}(X). Then there is a natural short exact sequence

0 → π_{4}(X) ⊗ ℤ/2 ⊕ Λ^{2}(π_{3}(X)) ⟶^{γ} Γ_{5}(X) ⟶^{μ} π_{3}(X) ∗ ℤ/2 → 0

where Λ^{2} is the exterior square in (1.10).

(3) Corollary. Let X be an (r − 1) -connected space with r ≥ 4 and let

η^{1} : π_{r}(X) ⊗ ℤ/2 → π_{r+1}(X)

be induced by the Hopf map η_{r} ∈ π_{r+1}(S^{r}), that is η^{1}(α ⊗ 1) = αη_{r} for α ∈ π_{r}(X). Then one has a natural short exact sequence

0 → π_{r+1}(X) ⊗ ℤ/2 ⟶^{γ} Γ_{r+2}(X) ⟶^{μ} π_{r}(X) ∗ ℤ/2 → 0

The exact sequence (3) was already achieved by J.H.C. Whitehead [W 2] who also determined the extension in (3). Moreover the exact sequences (1), (2) and (3) were obtained using different methods by Baues [B3] (11.3.3), (9.3.3) and (8.3.5) respectively. Also the extension problems for (1), (2) and (3) are solved in [B3] as follows.

(3.2) Remark on the extension problem. In (3.1) (3) the extension is determined by the homomorphism

(1)  E : π_{r} ∗ ℤ/2 → π_{r+1} ⊗ ℤ/2

defined by E(x) = γ^{−1}2μ^{−1}(X) for x ∈ π_{r} ∗ ℤ/2. Here E = E_{r}(η^{1}) coincides with the composite

(2) E_{r}(η^{1}) : π_{r} ∗ ℤ/2 ⟶^{i} π_{r} ⟶^{q} π_{r} ⊗ ℤ/2 ⟶^{η^{1}} π_{r+1} ⟶^{q} π_{r+1} ⊗ ℤ/2 = Γ_{r}^{2}(η^{1})

where i and q are the canonical maps, r ≥ 4. For the extension (3.1) (2) we know that Λ^{2}(π_{3}) is a direct summand of Γ_{5}X and the remaining summand is an extension given as in (2) by qη^{1}qi. Let E_{3}(η^{1}) be the composite

(3)

E_{3}(η^{1}) : π_{3}∗ℤ/2 ⟶^{i} π_{3} ⟶^{q} π_{3}⊗ℤ/2 ⟶^{η^{1}} π_{4} ⟶^{q} π_{4}⊗ℤ/2 ⟶^{j} π_{4}⊗ℤ/2⊕Λ^{2}(π_{3}) = Γ_{3}^{2}(η^{1})

where j is the inclusion.

The situation for the extension (3.2) (1) is more complicated. Let A = π_{2} and let K be a basis of the ℤ/2 -vector space A⊗ℤ/2. Then the cross effects of the functors Γ and ΓT show that the choice of such a basis yields canonically an isomorphism

ψ_{K} : ΓT (A ⊗ ℤ/2) ≅ Γ(A ⊗ ℤ/2) ⊗ ℤ/2

Hence we get an extension

0 → Γ(A) ⊗ ℤ/2 ⟶^{Δ} π′′_{4}(A) ⟶^{μ} ΓT (A) → 0

which is determined by E = Δ^{−1}(·2)μ^{−1} where E is the composite

(4) E_{K} : ΓT (A) ⟶^{q_{∗}} ΓT (A ⊗ ℤ/2) ⟶^{ψ_{K}} Γ(A ⊗ ℤ/2) ⊗ ℤ/2 = Γ(A) ⊗ ℤ/2

Now consider the following push out of abelian groups

0 ⟶ L(A, 1)_{3} ⊕ Γ(A) ⊗ ℤ/2 ⟶^{1⊕Δ} L(A, 1)_{3} ⊕ π′′_{4}(A) ⟶^{(0,μ)} ΓT (A) ⟶ 0
↓^{η_{∗}^{1}  push}  ↓  ∥
0 ⟶  Γ_{2}^{2}(η^{1})  ⟶  Γ_{4}X  ⟶ ΓT (π_{2}) ⟶ 0

Here η_{∗}^{1} is the homomorphism in (1.10) (1). Then the bottom row of (4) determines the extension in (3.1) (1). We define the homomorphism E_{2}(η^{1}) by the following composite with A = π_{2}.

(6) E_{2}(η^{1}) : ΓT (A) ⟶^{E_{K}} Γ(A) ⊗ ℤ/2 ⟶^{j} L(A, 1)_{3} ⊕ Γ(A) ⊗ ℤ/2 ⟶^{η_{∗}^{1}} Γ_{2}^{2}(η^{1})

Here j denotes the inclusion and E_{K} and η_{∗}^{1} are given by (4) and (5). The homomorphism E_{r}(η^{1}) above with r ≥ 2 are used in the following theorem. We point out that all extensions (3.1) (1), (2), (3) are completely determined by η^{1}.

Blanc [Bl3] considers the problem of realizing an abstract Π -algebra π by a space X so that π ≅ π_{∗}X in model (Π). Concerning this problem we get as a consequence of (3.1) and (3.2) the following new result which describes the first obstruction for realizability.

(3.3) Theorem. Let r ≥ 2.

(A) Let π be a model of Π_{r}^{1} given by η^{1} : Γ_{r}^{1}(π_{r}) → π_{r+1}. Then π is realizable

by an (r − 1) -connected space X. The first k-invariant of X is given by η^{1}. (B) Let π be a model of Π_{r}^{2} given by η^{1} and η^{2} : Γ_{r}^{2}(η^{1}) → π_{r+2}. Then π

is realizable by an (r − 1) -connected space if and only if the obstruction O(π) = η^{2} ∘ E_{r}(η^{1}) vanishes. Here O(π) is the composite of η^{2} and the homomorphism E_{r}(η^{1}) in (3.2).

Proof. Proposition (A) is an easy consequence of Whitehead’s classification of (r − 1) -connected (r + 2) -dimensional homotopy types; compare for example (3.5.6) [B3]. Next we prove (B). If π is realizable by a space X then the existence of the commutative diagram (2.2) with k = 2 implies that O(π) = 0; here we use (3.2). On the other hand if O(π) = 0 then there exists a commutative diagram with exact row and column as follows.

Γ_{r}^{2}(η^{1})
↓  ↘ η^{2}
H_{r+3} ⟶ Γ_{r+2} ⟶ π_{r+2} ⟶ H_{r+2} ⟶ Γ_{r}^{1}(π_{r}) ⟶^{η^{1}} π_{r+1}
↓
L_{1}Γ_{r}^{1}(π_{r})

Here the column is given by the extension in (3.2) and H_{r+3} is free abelian. Now the ”theorem on the realizability of the Hurewicz homomorphism” (3.4.7) [B3] implies that the exact row of the diagram is realizable by an (r − 1) -connected (r + 3) -dimensional space. q.e.d.

(3.4) Example. Cochran-Habegger [CH] compute the homotopy group π_{4}(M) of a closed simply connected 4-dimensional manifold M. This result is a consequence of (3.1). We have the homology groups H_{4}M = ℤ, H_{3}M = 0 and H_{2}M is free abelian. Hence we have the exact sequence

0 → Γ_{4}M → π_{4}M → H_{4}M ⟶^{b} Γ(H_{2}M) ⟶^{η^{1}} π_{3}M → 0

Here b carries the generator to the intersection form of M; see (8.10) [B4]. Hence b is injective and η^{1} is the cokernel map of b. Moreover by (3.1) we see that

π_{4}M = Γ_{4}M = Γ_{2}^{2}(η^{1})

is algebraically determined by η^{1} since ΓT (H_{2}M) = 0.

(3.5) Example. Let M be a closed simply connected 5-dimensional manifold or Poincaré complex. Then we get the natural diagram

[M] ∈ H_{5}M = ℤ
↓^{b}
0 ⟶ Γ_{2}^{2}(η^{1}) ⟶  Γ_{4}M  ⟶^{μ} ΓT (H_{2}M) ⟶ 0

by (3.1) (1). The fundamental class [M] is the generator of H_{5}M and the element

μb[M] ∈ ΓT (H_{2}M)

is a homotopy invariant of the manifold M. This is the torsion invariant in 9.8 [B5]; compare also 1.16 [BB]. We generalize this torsion invariant for simply connected 6-dimensional manifolds in (5.4) below.

(3.6) Example. Let GL(ℤ) (resp. SL(ℤ)) be the infinite general (resp. special) linear group. Let K̃(ℤ) be the simply connected cover of the algebraic K-theory space K(ℤ) = B(GL(ℤ))^{+}. Then we have

(1)  \{ \begin{matrix}K_{n}(ℤ) = π_{n}K̃(ℤ) \\ H_{n}SL(ℤ) = H_{n}K̃(ℤ) \end{matrix}

for n ≥ 2. Hence Whitehead’s exact sequence for the space K̃(ℤ) has the form (see Weibel [W e])

Γ_{5} → K_{5}ℤ →H_{5}SL(ℤ) → Γ_{4} →K_{4}ℤ →H_{4}SL(ℤ) →Γ(ℤ/2) ⟶^{η^{1}} K_{3}(ℤ)
∥  ∥  ∥  ∥  ∥  ∥
ℤ ⊕ T_{3}  (ℤ/2)^{3} 0  ℤ/2  ℤ/4  ℤ/48

Here T_{3} is 3-torsion and η^{1} is the map ℤ/4 ↠ ℤ/2 ⊂ ℤ/48. Therefore we can compute Γ_{4} by (3.1) (1) and (3.2) and this yields Γ_{4} = (ℤ/2)^{3}. Let K(η^{1}, 2) be the fiber of the map η^{1} : K(ℤ/2, 2) → K(ℤ/48, 4) given by the homomorphism η^{1}. Since K_{4}ℤ = 0 we see that Γ_{5} = H_{6}K(η^{1}, 2) is 2-torsion. In fact Γ_{5} = Γ_{5}K̃(ℤ) is computed in (5.16) below. Therefore we get the short exact sequence

(2)  0 → ℤ ⊕ T_{3} → H_{5}SL(ℤ) → (ℤ/2)^{3} → 0

This sequence is actually non split so that H_{5}SL(ℤ) = ℤ ⊕ T_{3} ⊕ (ℤ/2)^{2}. To see this

we observe that the 2-connected cover K^{≈}(ℤ) of K(ℤ) satisfies

(3)  H_{n}St(ℤ) = H_{n}K^{≈}(ℤ)

for n ≥ 3 where St(ℤ) is the infinite Steinberg group. Then the map g : K^{≈}(ℤ) → K̃(ℤ) induces the following commutative diagram for the Whitehead sequences

(4)  \begin{aligned}0 & ⟶ & K_{5}ℤ & ⟶ & H_{5}St(ℤ) & ⟶ & ℤ/2 & ⟶ & 0 \\ & & ∥ & & ↓^{g_{∗}} & & ↓ \\ 0 & ⟶ & K_{5}ℤ & ⟶ & H_{5}SL(ℤ) & ⟶ & (ℤ/2)^{3} & ⟶ & 0 \end{aligned}

in which the top row is non split so that g_{∗} is split injective with cokernel (ℤ/2)^{2}. Compare [A], [ABG]. In (5.16) we give a further argument that the bottom sequence of (4) is non split. The extensions of the short exact sequences in (4) characterize

non trivial k-invariants k_{5} of the spaces K^{≈}(ℤ) and K̃(ℤ) respectively. This follows from the ”Theorem on Postnikov invariants” (2.5.10) [B3]. The top row of (4)

shows that the k-invariant k_{5} of K^{≈}(ℤ) is the non trivial element in

(5)  0 ≠ k_{5} ∈ H^{6}(K(ℤ/48, 3), ℤ ⊕ T_{3}) = ℤ/2

One can apply the Whitehead sequence also to the space K^{≈}(ℤ) to obtain the exact sequence

(6)  K_{6}ℤ → H_{6}St(ℤ) → Γ_{5} → 0

where Γ_{5} = ℤ/2 as is easily seen by (3.1) (2). Since K_{6}ℤ is odd torsion we thus have H_{6}St(ℤ) = ℤ/2⊕ odd torsion.

§ 4 On the functors Γ_{r}^{k} for k < r − 1

In the stable range k < r−1 the suspension functor Σ : Π_{r}^{k} → Π_{r+1}^{k} is an isomorphism of additive categories. Therefore the theory Π_{r}^{k} is completely determined by the k-skeleton π_{∗≤k}^{S}(S^{0}) of the stable ring of homotopy groups of spheres π_{∗}^{S}(S^{0}). A model of Π_{r}^{k} is the same as a π_{∗≤k}^{S}(S^{0}) -module so that for k < r − 1 we have the functors Γ^{k} = Γ_{r}^{k} which do not depend on r. These functors form the iterated Grothendieck construction

(4.1)  \{ \begin{matrix}Γ^{1} : Ab → Ab \\ Γ^{k+1} : Γ^{k}Ab → Ab, k ≥ 1 \end{matrix}

Here all functors Γ^{k} and categories Γ^{k}Ab are additive. In low degrees ≤ 6 we have in π_{∗}^{S}(S^{0}) only the algebra generators η and ν (in dimension 1 and 3 respectively) π_{∗}^{S} and α in dimension 3 with the relations

2η = 0, 4ν = η^{3}, ην = νη = 0, 2ν^{2} = 0 and 3α = 0, α^{2} = 0.

This implies that the functors Γ^{k} with k ≤ 6 are given as follows. For a homomorphism f : A → B ∈ Ab we use the notation

f_{⊗} = f ⊗ ℤ/n : A ⊗ ℤ/n → B ⊗ ℤ/n

and q : B → B/fA is the quotient map for the cokernel of f. Let π_{0}, …, π_{6} be abelian groups. Then the following system of homomorphisms η^{1}, …, η^{6} defines an object in Γ^{6}Ab.

Γ^{1}(π_{0}) = π_{0} ⊗ ℤ/2 ⟶^{η^{1}} π_{1} with η^{1} = η^{∗}, (1)

Γ^{2}(η^{1}) = π_{1} ⊗ ℤ/2 ⟶^{η^{2}} π_{2} with η^{2} = η^{∗}, (2)

Γ^{3}(η^{1}, η^{2}) = (π_{0} ⊗ ℤ/3) ⊕ P ⟶^{η^{3}} π_{3} (3)

Here P is given by the push out in Ab

π_{0} ⊗ ℤ/8 ⟶  P
↑^{π_{0}⊗4}  ↑
π_{0} ⊗ ℤ/2 ⟶^{(ηη)^{∗}} π_{2} ⊗ ℤ/2

The left hand column is induced by the inclusion 4 : ℤ/2 ⊂ ℤ/8 and the bottom row is the composite

(ηη)^{∗} = η_{⊗}^{2}η_{⊗}^{1} : π_{0} ⊗ ℤ/2 → π_{1} ⊗ ℤ/2 → π_{2} ⊗ ℤ/2

Using η^{3} we obtain the action of α, η, ν ∈ π_{∗}^{S}(S^{0}) by the composites

\{ \begin{matrix}α^{∗} : π_{0} ⊗ ℤ/3 ⟶^{η^{3}} π_{3} \\ η^{∗} : π_{2} ⊗ ℤ/2 → P ⟶^{η^{3}} π_{3} \\ ν^{∗} : π_{0} ⊗ ℤ/8 → P ⟶^{η^{3}} π_{3} \end{matrix}

Next we get

(4)  Γ^{4}(η^{1}, η^{2}, η^{3}) = π_{1} ⊗ ℤ/3 ⊕ Q ⟶^{η^{4}} π_{4}

Here Q is the push out

(π_{1}/η^{∗}π_{0}) ⊗ ℤ/8 ⟶  Q
↑^{q⊗4}  ↑
π_{1} ⊗ ℤ/2  ⟶^{q(ηη)^{∗}} (π_{3}/ν^{∗}π_{0}) ⊗ ℤ/2

where (ηη)^{∗} = (η^{∗})_{⊗}η_{⊗}^{2} with (η^{∗})_{⊗} given by (3). Using η^{4} we obtain again the action of α, η, ν by the composites:

\{ \begin{matrix}α^{∗} : π_{1} ⊗ ℤ/3 ⟶^{η^{4}} π_{3} \\ η^{∗} : π_{3} ⊗ ℤ/2 → (π_{3}/ν^{∗}π_{0}) ⊗ ℤ/2 → Q ⟶^{η^{4}} π_{4} \\ ν^{∗} : π_{1} ⊗ ℤ/8 → (π_{1}/η^{∗}π_{0}) ⊗ ℤ/8 → Q ⟶^{η^{4}} π_{4} \end{matrix}

Next we describe

(5)  Γ^{5}(η^{1}, η^{2}, η^{3}, η^{4}) = π_{2} ⊗ ℤ/3 ⊕ R ⟶^{η^{5}} π_{5}

Here R is the push out

(π_{2}/η^{∗}π_{1}) ⊗ ℤ/8 ⟶  R
↑^{q⊗4}  ↑
π_{2} ⊗ ℤ/2  ⟶^{q(ηη)^{∗}} (π_{4}/ν^{∗}π_{1}) ⊗ ℤ/2

where (ηη)^{∗} = (η^{∗})_{⊗}(η^{∗})_{⊗} is given by η^{∗} in (3) and (4) respectively. Again we get by η^{5} the action

\{ \begin{matrix}α^{∗} : π_{2} ⊗ ℤ/3 ⟶^{η^{5}} π_{5} \\ η^{∗} : π_{4} ⊗ ℤ/2 → (π_{4}/ν^{∗}π_{1}) ⊗ ℤ/2 → R ⟶^{η^{5}} π_{5} \\ ν^{∗} : π_{2} ⊗ ℤ/8 → (π_{2}/η^{∗}π_{1}) ⊗ ℤ/8 → R ⟶^{η^{5}} π_{5} \end{matrix}

Finally we describe

(6)  Γ^{6}(η^{1}, η^{2}, η^{3}, η^{4}, η^{5}) = (π_{3}/α^{∗}π_{0}) ⊗ ℤ/3 ⊕ S ⟶^{η^{6}} π_{6}

where S is given by the following double push out diagram

π_{0} ⊗ ℤ/2  ⟶  S
↑  ↑
π_{0} ⊗ ℤ/8  ⟶^{qν^{∗}} (π_{3}/η^{∗}π_{2}) ⊗ ℤ/8 ⟶  S′
↑^{q⊗4}  ↑
π_{3} ⊗ ℤ/2  ⟶^{q(ηη)^{∗}} (π_{5}/ν^{∗}π_{2}) ⊗ ℤ/2

Here π_{0} ⊗ ℤ/8 → π_{0} ⊗ ℤ/2 is the quotient map and ν^{∗} is defined by (3) and (ηη)^{∗} is defined by η^{∗} in (4) and η^{∗} in (5).

In the paper [B2] one finds a general method for the computation of the derived functors L_{i}Γ_{r}^{k}. Since in the stable range all functors Γ_{r}^{k} = Γ^{k}(k < r − 1) are additive the computation of L_{i}Γ_{r}^{k} is possible only in terms of chain complexes of abelian groups. We obtain the computation of the derived functors

(4.2)  L_{i}Γ^{k+1} : Γ^{k}Ab → Ab

for small k as follows.

A weak resolution C_{∗} of an abelian group A is a chain complex C_{∗} = (… → C_{1} → C_{0}) of abelian groups satisfying

H_{i}C_{∗} = \{ \begin{matrix}A & for i = 0 \\ 0 & otherwise \end{matrix}

This is a resolution if all C_{i} are free abelian. Moreover C_{∗} has length ≤ n if C_{i} = 0 for i > n.

Given an object (π_{0}, π_{1}, …, η^{1}, η^{2}, …) in the iterated Grothendieck construction (4.1) we choose a length 1 resolution of π_{0}

(1)  C_{1}^{0} → C_{0}^{0} → π_{0}

and we choose inductively weak resolutions C_{∗}^{i} of π_{i} of length 2i + 1 together with the following commutative diagrams in which all homomorphisms η_{i}^{k}, k ≥ 1, i ≥ 0, are split injective and have a free abelian group as cokernel.

(2)  Γ^{1}C_{1}^{0}  →  Γ^{1}C_{0}^{0}  →  Γ^{1}π_{0}
↓ η_{1}^{1}  ↓ η_{0}^{1}  ↓ η^{1}
C_{3}^{1}  → C_{2}^{1}  →  C_{1}^{1}  →  C_{0}^{1}  →  π_{1}

(3)  Γ^{1}C_{3}^{1} → Γ^{1}C_{2}^{1} →  Γ^{2}η_{1}^{1}  →  Γ^{2}η_{0}^{1}  →  Γ^{2}η^{1}
↓ η_{3}^{2}  ↓ η_{2}^{2}  ↓ η_{1}^{2}  ↓ η_{0}^{2}  ↓ η^{2}
C_{5}^{2}  →  C_{4}^{2}  → C_{3}^{2}  → C_{2}^{2}  →  C_{1}^{2}  →  C_{0}^{2}  →  π_{2}

(4)

Γ^{1}(C_{5}^{2}) → Γ^{1}(C_{4}^{2}) → Γ^{2}(η_{3}^{2}) → Γ^{2}(η_{2}^{2}) → Γ^{3}(η_{1}^{1}, η_{1}^{2}) → Γ^{3}(η_{0}^{1}, η_{0}^{2}) → Γ^{3}(η^{1}, η^{2})
↓ η_{5}^{3}  ↓ η_{4}^{3}  ↓ η_{3}^{3}  ↓ η_{2}^{3}  ↓ η_{1}^{3}  ↓ η_{0}^{3}  ↓ η^{3}
C_{5}^{3}  →  C_{4}^{3}  → C_{3}^{3}  → C_{2}^{3}  →  C_{1}^{3}  →  C_{0}^{3}  →  π_{3}
C_{7}^{3}  →  C_{6}^{3}  →

The homology of the top row in (2) is L_{∗}Γ^{1} so that by (4.1) (1) we have

(5)  L_{1}Γ^{1}(π_{0}) = π_{0} ∗ ℤ/2

and L_{i}Γ^{1} = 0 for i > 1. Moreover the homology of the top row in (3) is L_{∗}Γ^{2} so that we get by use of (4.1) (2) and the argument below the following result.

(4.3) Proposition.

L_{1}Γ^{2}(η^{1}) = (π_{1} ∗ ℤ/2)/η^{1}(π_{0} ⊗ ℤ/2)
L_{2}Γ^{2}(η^{1}) = ker(η^{1})
L_{3}Γ^{2}(η^{1}) = π_{0} ∗ ℤ/2
L_{i}Γ^{2} = 0 for i > 3.

Next the homology of the chain complex (4) is L_{∗}Γ^{3} and inductively one gets this way the derived functors L_{i}Γ^{k} with L_{i}Γ^{k} = 0 for i > 2k − 1; compare (1.14). Now it is easy to prove (1.17) in the stable case.

Proof of (4.3). We obtain by (2) the short exact sequence of chain complexes

(6)  0 → Γ^{1}C_{∗}^{0} → C_{∗}^{1} → B_{∗}^{1} → 0

with H_{0}B_{∗}^{1} = cok (η^{1}), H_{1}B_{∗}^{1} = ker(η^{1}) and H_{2}B_{∗}^{1} = π_{0} ∗ ℤ/2. We can apply the functor ⊗ℤ/2 to (6) and we get the short exact sequence of chain complexes

(7)  0 → Γ^{1}C_{∗}^{0} → Γ^{2}(η_{∗}^{1}) → B_{∗}^{1} ⊗ ℤ/2 → 0

where the chain complex Γ^{2}(η_{∗}^{1}) coincides with (3). Now (7) yields a long exact sequence of homology groups. This shows

L_{3}Γ^{2}(η^{1}) = H_{3}Γ^{2}(η_{∗}^{1}) = H_{3}(B_{∗}^{1} ⊗ ℤ/2) = π_{0} ∗ ℤ/2

Moreover we get this way the exact sequences:

0 ⟶ L_{2}Γ^{2}(η^{1}) ⟶ H_{2}(B_{∗}^{1} ⊗ ℤ/2) ⟶ π_{0} ∗ ℤ/2 ⟶ 0
∥
ker(η^{1}) ⊕ π_{0} ∗ ℤ/2

0 ⟶ L_{1}Γ^{2}(η^{1}) ⟶  H_{1}(B_{∗}^{1} ⊗ ℤ/2)  ⟶^{∂} π_{0} ⊗ ℤ/2 ⟶^{η_{⊗}^{1}} π_{1} ⊗ ℤ/2
∥
ker(η^{1}) ⊕ cok (η^{1}) ∗ ℤ/2

Here ∂ is the canonical inclusion on ker(η^{1}) so that we get the exact sequence

0 → L_{1}Γ^{2}(η^{1}) → cok (η^{1}) ∗ ℤ/2 ⟶^{∂} im (η^{1}) → π_{1} ⊗ ℤ/2

But here ∂ is the boundary associated to the 6-term exact sequence given by the short exact sequence

0 → im (η^{1}) → π_{1} → coker(η^{1}) → 0

This yields the formulas in (4.3). q.e.d.

§ 5 On the group Γ_{r+3}X of an (r − 1)-connected space

Again the spectral sequence (2.4) yields immediately the next result which is a crucial tool for the computation of the homotopy group π_{r+3}(X). We describe various specifications and applications of this result.

(5.1) Theorem. Let X be an (r − 1) -connected space with r ≥ 2. Then there is a natural exact sequence

(L_{2}Γ_{r}^{2})(η^{1}) ⟶^{d^{2}} Γ_{r}^{3}(η^{1}, η^{2}) → Γ_{r+3}X → (L_{1}Γ_{r}^{2})(η^{1}) → 0

where η^{1} : Γ_{r}^{1}(π_{r}X) → π_{r+1}(X) and η^{2} : Γ_{r}^{2}(η^{1}) → π_{r+2}(X).

Recall that the functors Γ_{r}^{1} and Γ_{r}^{2} are completely understood by (1.9) and (1.10). In the theorem the derived functors L_{i}Γ_{r}^{2} and the functor Γ_{r}^{3} are needed.

For example we get by (4.1) (3) and (4.3) the following stable case of theorem (5.1)

(5.2) Corollary. Let X be an (r − 1) -connected space with r ≥ 5 and let

\{ \begin{matrix}η^{1} = η^{∗} : π_{r}(X) ⊗ ℤ/2 → π_{r+1}(X), \\ η^{2} = η^{∗} : π_{r+1}(X) ⊗ ℤ/2 → π_{r+2}(X) \end{matrix}

be induced by the Hopf maps η = η_{r} ∈ π_{r+1}S^{r} and η = η_{r+1} ∈ π_{r+2}S^{r+1} respectively. Then there is a natural exact sequence

ker(η^{1}) ⟶^{d^{2}} π_{r}(X) ⊗ ℤ/3 ⊕ P → Γ_{r+3}(X) → \frac{π_{r+1}(X) ∗ ℤ/2}{η^{1}(π_{r}(X) ⊗ ℤ/2)} → 0

Here P is the push out of

π_{r}(X) ⊗ ℤ/8 ⟵^{1⊗4} π_{r}(X) ⊗ ℤ/2 ⟶^{(ηη)^{∗}} π_{r+2}(X) ⊗ ℤ/2

(5.3) Remark. The exact sequence in (5.2) depends only on the (r + 2) -type of X which is classified by an algebraic invariant (termed an ”injective A^{3} -system”) in (8.1.6) [B3]. Therefore it should be possible to compute d^{2} and the extension problem in the exact sequence (5.2) only in terms of the A^{3} -system associated to X similarly as this is done in Remark (3.2).

A similar explicit form of theorem (5.1) for the unstable cases r = 2, 3, 4 is needed though the relevant computations are quite involved. For CW-complexes with torsion free homology such computation can be found in Unsöld [U1]. We need notation on quadratic functors as follows; compare 6.13 [B3].

(5.4) Definition. A quadratic ℤ-module M is a diagram

M = (M_{e} ⟶^{H} M_{ee} ⟶^{P} M_{e})

in Ab satisfying P HP = 2P and HP H = 2H. For an abelian group A we define the quadratic tensor product A ⊗ M generated by the symbols a ⊗ m, [a, b] ⊗ n with a, b ∈ A, m ∈ M_{e}, n ∈ M_{ee}. The relations are

(a + b) ⊗ m = a ⊗ m + b ⊗ m + [a, b] ⊗ H(m), [a, a] ⊗ n = a ⊗ P (n), a ⊗ m is linear in m, [a, b] ⊗ n is linear in a, b and n respectively.

Each M yields the quadratic functor ⊗M : Ab → Ab which carries A to A ⊗ M. For example Whitehead’s functor Γ and the exterior square Λ^{2} yield the equations

\begin{matrix}Γ(A) ⊗ C = A ⊗ (C ⟶^{1} C ⟶^{2} C) \\ Λ^{2}(A) ⊗ C = A ⊗ (0 → C → 0) \end{matrix}

where C is an abelian group. We identify C = (C → 0 → C) since

A ⊗ C = A ⊗ (C → 0 → C)

is the usual tensor product of abelian groups. A map α = (α_{e}, α_{ee}) : M → N between quadratic ℤ -modules is given by maps α_{e} : M_{e} → N_{e}, α_{ee} : M_{ee} → N_{ee} in Ab satisfying Hα_{e} = α_{ee}H, α_{e}P = P α_{ee}. Such a map induces a homomorphism 1 ⊗ α : A ⊗ M → A ⊗ N which is natural in A.

Homotopy groups of spheres in the metastable range yield the quadratic ℤ - modules (m < 3n − 2)

π_{m}{S^{n}} = (π_{m}S^{n} ⟶^{H} π_{m}S^{2n−1} ⟶^{P} π_{m}S^{n})

where H is the Hopf invariant and P is induced by the Whitehead product square P (α) = [i_{n}, i_{n}]α. For example we have by (6.15.4) [B3]

(5.5)  \{ \begin{matrix}π_{6}\{S^{3}\} = (ℤ/4 ⟶^{1} ℤ/2 ⟶^{0} ℤ/4) ⊕ ℤ/3 \\ π_{7}\{S^{4}\} = (ℤ ⊕ ℤ/4 ⟶^{(1,0)} ℤ ⟶^{(2,−1)} ℤ ⊕ ℤ/4) ⊕ ℤ/3 \end{matrix}

These quadratic ℤ -modules are used in the next two corollaries of (5.1).

(5.6) Corollary. Let X be a 3-connected space and let

\{ \begin{matrix}η^{1} = η^{∗} : π_{4}(X) ⊗ ℤ/2 → π_{5}(X) \\ η^{2} = η^{∗} : π_{5}(X) ⊗ ℤ/2 → π_{6}(X) \end{matrix}

be induced by the Hopf maps η = η_{4} ∈ π_{5}S^{4} and η = η_{5} ∈ π_{6}(S^{5}) respectively. Then there is a natural exact sequence

ker(η^{1}) ⟶^{d^{2}} π_{4}(X) ⊗ ℤ/3 ⊕ P_{7} → Γ_{7}(X) → \frac{π_{5}(X) ∗ ℤ/2}{η^{1}(π_{4}(X) ⊗ ℤ/2)} → 0

Here P_{7} is the push out in Ab of

π_{4}(X) ⊗ M ⟵^{id⊗α} π_{4}(X) ⊗ ℤ/2 ⟶^{(ηη)^{∗}} π_{6}(X) ⊗ ℤ/2

where (ηη)^{∗} = η_{⊗}^{2}η_{⊗}^{1} and where α is the map between quadratic ℤ -modules

α : ℤ/2 = (ℤ/2 → 0 → ℤ/2) → M = (ℤ ⊕ ℤ/4 ⟶^{(1,0)} ℤ ⟶^{(2,−1)} ℤ ⊕ ℤ/4)

given by α_{e} = (0, 2) and α_{ee} = 0.

For the proof of (5.6) we only observe that by (5.5) and relations in Toda [T]

(5.7)  Γ_{4}^{3}(η^{1}, η^{2}) = π_{4} ⊗ ℤ/3 ⊕ P_{7}

with P_{7} defined as in (5.6). In the next corollary we use the exterior square torsion

(5.8)  Λ^{2}T (A) = Ω(A) = (L_{1}Λ^{2})(A)

which is the first left derived functor of Λ^{2}. In the notation of Eilenberg-Mac Lane [AMNY] this is the functor Ω; compare (6.2.10) [B3]. The cross effect of Λ^{2}T is the torsion product Λ^{2}T (A | B) = A ∗ B of abelian groups and one has Λ^{2}T (ℤ) = 0 and Λ^{2}T (ℤ/n) = ℤ/n. Using (1.10) it is easy to compute the derived functors L_{i}Γ_{3}^{2}(η^{1}) by use of (4.3). This leads to the following result.

(5.9) Corollary. Let X be a 2-connected space and let

\{ \begin{matrix}η^{1} = η^{∗} : π_{3}(X) ⊗ ℤ/2 → π_{4}(X) \\ η^{2} = (η^{∗}, [−, −]) : π_{4}(X) ⊗ ℤ/2 ⊕ Λ^{2}π_{3}(X) → π_{5}(X) \end{matrix}

be induced by the Hopf maps η = η_{3} ∈ π_{4}S^{3} and η = η_{4} ∈ π_{5}S^{4} respectively and the Whitehead product [−, −] : π_{3} ⊗ π_{3} → π_{5}. Then there is a natural exact sequence

ker(η^{1}) ⟶^{d^{2}} π_{3}(X) ⊗ ℤ/3 ⊕ P_{6} → Γ_{6}(X) → \frac{π_{4}(X) ∗ ℤ/2}{η^{1}(π_{3}(X) ⊗ ℤ/2)} ⊕ Λ^{2}T (π_{3}(X)) → 0

Here P_{6} is defined by the following double push out diagram in Ab with π_{∗} = π_{∗}(X).

π_{3} ⊗ N  ⟶  P_{6}
↑^{(1⊗α,1⊗β)}  ↑
π_{3} ⊗ ℤ/2 ⊕ (Λ^{2}π_{3}) ⊗ ℤ/2 ⟶^{η_{⊗}^{1}⊕1} π_{4} ⊗ ℤ/2 ⊕ (Λ^{2}π_{3}) ⊗ ℤ/2 ⟶^{η_{⊗}^{2}} π_{5} ⊗ ℤ/2  → P′_{6}
q↑  ↑
π_{3} ⊗ π_{3} ⊗ ℤ/2  ⟶_{1⊗η^{1}} π_{3} ⊗ π_{4}

The map q is given by the quotient map π_{3} ⊗ π_{3} → Λ^{2}π_{3} and α and β are maps between quadratic ℤ -modules

\begin{matrix}α : ℤ/2 = (ℤ/2 → 0 → ℤ/2) → N = (ℤ/4 ⟶^{1} ℤ/2 ⟶^{0} ℤ/4) \\ β : (0 → ℤ/2 → 0) → N = (ℤ/4 ⟶^{1} ℤ/2 ⟶^{0} ℤ/4) \end{matrix}

with α_{e} = 2, α_{ee} = 0 and β_{e} = 0, β_{ee} = 1.

For the proof of (5.9) we observe that by (5.5) and relations in Toda [T] we have

(5.10)  Γ_{3}^{3}(η^{1}, η^{2}) = π_{3} ⊗ ℤ/3 ⊕ P_{6}

with P_{6} defined as in (5.9). The remaining case r = 2 in theorem (5.1) has the following important application.

(5.11) Example. Let M be a closed simply connected 6-dimensional manifold or Poincaré complex. Then we get the following natural diagram which is the analogue of (3.4) and (3.5) respectively.

[M] ∈ H_{6}(M) = ℤ

↓^{b}
(L_{2}Γ_{2}^{2})(η^{1}) ⟶ Γ_{2}^{3}(η^{1}, η^{2}) ⟶  Γ_{5}M  ⟶^{μ} (L_{1}Γ_{2}^{2})(η^{1}) ⟶ 0

The bottom row is exact by (5.1). The fundamental class [M] thus yields the torsion invariant

μb[M] ∈ (L_{1}Γ_{2}^{2})(η^{1})

which is a homotopy invariant of M. The classification of simply connected 6dimensional manifolds and Poincaré-complexes is a deep problem which is not completely solved because of the lack of good invariants. The new torsion invariant above should ease the problem considerably; compare Zubr [Z], Wall [W a], Jupp [J], Buth [Bu], López [A]. For example for M = ℂP_{3} we obtain the generator

μb[ℂP_{3}] ∈ Ł_{1}Γ_{2}^{2}(η^{1}) ≅ Γ_{5}CP_{3} ≅ ℤ

where η^{1} : Γ(ℤ) → 0. Here the isomorphisms are compatible with the exact sequence in (5.12) (9) below.

We now study along the lines of Baues [B2] the derived functors L_{i}Γ_{2}^{2} of Γ_{2}^{2} in (1.10) (1) which are needed for example in (5.11) above. Let N be the normalisation functor from simplicial abelian groups to chain complexes and let K be the inverse of N; compare Dold-Puppe [DP].

For η^{1} : Γ(π_{2}) → π_{3} we choose a lenght 1 resolution

(5.12)  C_{1}^{2} ⟶^{d} C_{0}^{2} → π_{2}

and we choose a weak resolution C_{∗}^{3} of π_{3} of length 3 together with the following commutative diagram in which η_{i}^{1} (i = 0, 1, 2) is split injective with torsion free cokernel B_{i}^{3}; compare (4.2).

(1) \begin{aligned}& & C_{1}^{2} ⊗ C_{1}^{2} & ⟶^{∂_{2}} & Γ(C_{1}^{2}) ⊕ C_{1}^{2} ⊗ C_{0}^{2} & ⟶^{∂_{1}} & ΓC_{0}^{2} & ⟶ & Γπ_{2} \\ & & ↓^{η_{2}^{1}} & & ↓^{η_{1}^{1}} & & ↓^{η_{0}^{1}} & & ↓^{η^{1}} \\ C_{3}^{3} & ⟶ & C_{2}^{3} & ⟶ & C_{1}^{3} & ⟶ & C_{0}^{3} & ⟶ & π_{3} \\ ∥ & & ↓ & & ↓ & & ↓ & & ↓ \\ B_{3}^{3} & ⟶ & B_{2}^{3} & ⟶ & B_{1}^{3} & ⟶ & B_{0}^{3} & ⟶ & cok(η^{1}) \end{aligned}

The columns are split short exact sequences of abelian groups. The top row in diagram (1) is given by d in (5.12) and by the Whitehead product P : A⊗A → Γ(A) for the functor Γ defined by P (a ⊗ b) = γ(a + b) − γ(a) − γ(b); that is

(2)  \{ \begin{matrix}∂_{1} = (Γ(d), P (d ⊗ 1)) \\ ∂_{2} = (P, −1 ⊗ d) \end{matrix}

We point out that ∂_{1} is injective and that C_{1}^{2} ⊗ C_{1}^{2} is free abelian. Therefore it is possible to choose a weak resolution C_{∗}^{3} of π_{3} of length 3 as in diagram (1). The homology of the top row in degree 1 is the Γ -torsion

(3)  (L_{1}Γ)(π_{2}) = ΓT (π_{2}) = ker(∂_{1})/im(∂_{2})

One can check that the top row of (1) coincides with NΓ(KC_{∗}^{2}). Hence (1) yields a short exact sequence of simplicial abelian groups

(4)  0 → Γ(K(C_{∗}^{2})) ⟶^{η_{·}^{1}} K(C_{∗}^{3}) → K(B_{∗}^{3}) → 0

We can apply the functor Γ_{2}^{2} to the simplicial object η_{·}^{1} ∈ sΓAb in (4) and we obtain the derived functors of Γ_{2}^{2} by

(5)  (L_{i}Γ_{2}^{2})(η^{1}) = π_{i}Γ_{2}^{2}(η_{·}^{1})

We now use the exact sequence (1.11) to study the derived functors L^{i}Γ_{2}^{2}. Since η_{·}^{1} in (4) is split injective in each degree we see that (1.11) induces the short exact sequences of simplicial abelian groups

(6)  0 → Γ_{2,(0)}^{1}(K(C_{∗}^{2})) → Γ_{2}^{2}(η_{·}^{1}) → Γ_{2,(1)}^{1}(K(C_{∗}^{2}), K(B_{∗}^{3})) → 0

which in turn yields a long exact sequence of homotopy groups. The functors Γ_{2,(0)}^{1} and Γ_{2,(1)}^{1} are explicitly described in (1.11). Using the Eilenberg-Zilber theorem

[DP] we see that

(7) π_{i}Γ_{2,(1)}^{1}(K(C_{∗}^{2}), K(B_{∗}^{2})) = H_{i}(B_{∗}^{3}⊗(ℤ/2⊕π_{2})) ≅ H_{i}(B_{∗}^{3})⊗(ℤ/2⊕π_{2})⊕H_{i−1}(B_{∗}^{3})∗(ℤ/2⊕π_{2})

Here the direct sum decomposition is natural in the sense of the universal coefficient theorem, that is, the inclusion of the ⊗ -summand is natural but the retraction of this summand is not natural. We deduce from the definition of B_{∗}^{3} the homology groups:

H_{i}B_{∗}^{3} = \{ \begin{matrix}cok (η^{1}) & for i = 0 \\ ker(η^{1}) & for i = 1 \\ ΓT (π_{2}) & for i = 2 \\ 0 & for i ≥ 3 \end{matrix}

By a result of Buth [Bu] we have L_{i}Γ_{2,(0)}^{1} = 0 for i ≥ 3. This shows by (6)

(8)  L_{i}Γ_{2}^{2}(η^{1}) = 0 for i ≥ 4.

Moreover the following sequence is exact

0 → L_{3}Γ_{2}^{2}(η^{1}) → ΓT (π_{2}) ∗ (ℤ/2 ⊕ π_{2}) → ΓT (π_{2}) ∗ ℤ/2 ⊕ L_{2}L(π, 1)_{3} (9)

→ L_{2}Γ_{2}^{2}(η^{1}) → \{ \begin{matrix}ΓT (π_{2}) ⊗ (ℤ/2 ⊕ π_{2}) \\ ⊕ \\ ker(η^{1}) ∗ (ℤ/2 ⊕ π_{2}) \end{matrix}\} → \{ \begin{matrix}Γ(π_{2}) ∗ ℤ/2 ⊕ ΓT (π_{2}) ⊗ ℤ/2 \\ ⊕ \\ L_{1}L(π_{2}, 1)_{3} \end{matrix}\}

→ L_{1}Γ_{2}^{2}(η^{1}) → \{ \begin{matrix}ker(η^{1}) ⊗ (ℤ/2 ⊕ π_{2}) \\ ⊕ \\ cok(η^{1}) ∗ (ℤ/2 ⊕ π_{2}) \end{matrix}\} → Γ(π_{2}) ⊗ ℤ/2 ⊕ L(π_{2}, 1)_{3}

→ Γ_{2}^{2}(η^{1}) → coker(η^{1}) ⊗ (ℤ/2 ⊕ π_{2}) → 0

Hence this sequence is a continuation of the exact sequence in (1.11). The sequence is natural for maps α : η^{1} → η_{0}^{1} in ΓAb though the direct sum decompositions are only natural in the sense of the universal coefficient theorem for homology; see (7). The sequence (9) is an initial case of the general theory of reduction functors in Baues [B2]. We point out that Buth showed that the derived functors L_{i}L(−, 1)_{3} satisfy L_{i}L(ℤ, 1)_{3} = 0 for i ≥ 0 and for k > 0

(10)  L_{i}L(ℤ/k, 1)_{3} = \{ \begin{matrix}ℤ/gcd(3k, k^{2}) & for i = 1 \\ ℤ/gcd(3, k) & for i = 2 \\ 0 & for i ≥ 3 and i = 0 \end{matrix}

This and the cross effect formulas

L(A | B, 1)_{3} = (A ⊗ B) ⊗ B ⊕ (A ⊗ B) ⊗ A

L(A | B | C, 1)_{3} = (A ⊗ C) ⊗ B ⊕ (B ⊗ C) ⊗ A

L_{1}L(A | B, 1)_{3} = (A ⊗ B) ∗ B ⊕ (A ∗ B) ⊗ B ⊕ (A ⊗ B) ∗ A ⊕ (A ∗ B) ⊗ A L_{1}L(A | B | C, 1)_{3} = (A ⊗ C) ∗ B ⊕ (A ∗ C) ⊗ B ⊕ (B ⊗ C) ∗ A ⊕ (B ∗ C) ⊗ A L_{2}L(A | B, 1)_{3} = (A ∗ B) ∗ B ⊕ (A ∗ B) ∗ A L_{2}L(A | B | C, 1)_{3} = (A ∗ C) ∗ B ⊕ (B ∗ C) ∗ A

yield a complete computation of L_{i}L(A, 1)_{3} for all finitely generated abelian groups A.

(5.13) Definition. Let π_{2}, π_{3}, π_{4} be abelian groups and let

(1)  \{ \begin{matrix}η^{1} : Γ(π_{2}) → π_{3} \\ η^{2} : Γ_{2}^{2}(η^{1}) → π_{4} \end{matrix}

be homomorphisms in Ab. Then we define by γ : π_{2} → Γ(π_{2}) the following elements (x, y ∈ π_{2})

(2)  \{ \begin{matrix}xη = η^{1}(γ(x)) ∈ π_{3} \\ [x, y] = η^{1}(γ(x + y) − γ(x) − γ(y)) ∈ π_{3} \end{matrix}

Moreover we define by the composite

π_{3} ⊗ (ℤ/2 ⊕ π_{2}) ⟶^{q^{∗}} Γ_{2}^{2}(η^{1}) ⟶^{η^{2}} π_{4}

the elements (z ∈ π_{3})

(3)  \{ \begin{matrix}zη = η^{2}q^{∗}(z ⊗ 1) ∈ π_{4} \\ [z, y] = η^{2}q^{∗}(z ⊗ y) ∈ π_{4} \end{matrix}

Using the notation in (2) and (3) we define an equivalence relation ∼ on the direct sum

(4)  U = π_{4} ⊗ (ℤ/2 ⊕ π_{2}) ⊕ Λ^{2}(π_{3})

which is generated by the following relations (5), (6), (7), with 1 ∈ ℤ/2, z ∈ π_{3}, x, y ∈ π_{2}.

(zη) ⊗ y ∼ [z, y] ⊗ 1 (1) z ∧ (xη) ∼ [z, x] ⊗ 1 + [z, x] ⊗ x (6) [z, x] ⊗ y + [z, y] ⊗ x ∼ z ∧ [x, y] (7)

Here z ∧ z′ = q(z ⊗ z′) is defined by the quotient map q : π_{3} ⊗ π_{3} → Λ^{2}(π_{3}) with z, z′ ∈ π_{3}.

(5.14) Theorem. There is a natural isomorphism

Γ_{2}^{3}(η^{1}, η^{2}) = (π_{4} ⊗ (ℤ/2 ⊕ π_{2}) ⊕ Λ^{2}(π_{3}))/ ∼

where ∼ is the equivalence relation in (5.13).

In fact (5.13) (5), (6) are obtained by the Barcus-Barratt formula and (5.13)

(7) is deduced from the Jacobi identity for Whitehead products. The tedious proof of (5.14) is achieved along the lines of the computations of Unsöld for the proof of III.1.5 page 168 [U1]. Using (5.14) we obtain as a final case r = 2 of (5.1) the following result.

(5.15) Corollary. Let X be a simply connected space and let

\{ \begin{matrix}η^{1} : Γ(π_{2}(X)) → π_{3}(X) \\ η^{2} : Γ_{2}^{2}(η^{1}) → η_{4}(X) \end{matrix}

be induced by the Hopf maps η_{2} ∈ π_{3}(S^{2}), η_{3} ∈ π_{4}(S^{3}) and the Whitehead product [−, −] : π_{3}(X) ⊗ π_{2}(X) → π_{4}(X), that is η^{1}γ(α) = αη_{2} with α ∈ π_{2}(X) as in (3.1)

(1) and η^{2}q_{∗}(β ⊗ 1) = βη_{3} and η^{2}q_{∗}(β ⊗ α) = [β, α] with β ∈ π_{3}(X) and q_{∗} as in (1.10) (1). Then there is a natural exact sequence

L_{2}Γ_{2}^{2}(η^{1}) ⟶^{d^{2}} Γ_{2}^{3}(η^{1}, η^{2}) → Γ_{5}(X) → L_{1}Γ_{2}^{2}(η^{1}) → 0

Here the derived functors L_{i}Γ_{2}^{2} are described in (5.12) and Γ_{2}^{3}(η^{1}, η^{3}) is computed in (5.14).

(5.16) Example. We can continue the exact sequence in (3.6) and we obtain for X = K̃(ℤ) the exact sequence

(1)  K_{6}ℤ → H_{6}SL(ℤ) → Γ_{5}K̃(ℤ) → 0

where η^{1} and η^{2} for K̃(ℤ) are given by η^{1} : Γ(ℤ/2) = ℤ/4 ↠ ℤ/2 ⊂ ℤ/48 as in (3.6) and η^{2} : Γ_{2}^{2}(η^{1}) = ℤ/2 ⊕ ℤ/2 → 0. Hence we see by (5.14) that Γ_{2}^{3}(η^{1}, η^{2}) = 0 so that by (5.15)

(2)  Γ_{5}K̃(ℤ) ≅ L_{1}Γ_{2}^{2}(η^{1})

This group can be computed by (5.12) (9) as follows. Consider η_{0}^{1} : Γ(ℤ/2) → 0 and the canonical map α : η^{1} → η_{0}^{1} in ΓAb which is the identity on π_{2} = ℤ/2. Then α induces the commutative diagram:

(3) \begin{aligned}… & → & (ℤ/2)^{4} & → & (ℤ/2)^{3} & → & L_{1}Γ_{2}^{2}(η^{1}) & → & (ℤ/2)^{4} & → & ℤ/2 & → & (ℤ/2)^{2} & → & (ℤ/2)^{2} & → 0 \\ ∥ & ∥ & ↓^{α_{∗}} & ↓ & ∥ & ↓ & ↓ \\ … & → & (ℤ/2)^{4} & → & (ℤ/2)^{3} & → & L_{1}Γ_{2}^{2}(η_{0}^{1}) & → & (ℤ/2)^{2} & → & ℤ/2 & → & 0 & → & 0 & → & 0 \end{aligned}

Here the exact rows are given by (5.12) (9) and (1.12). Using (5.15) for the space K(ℤ/2, 2) we see that

(4)  L_{1}Γ_{2}^{2}(η_{0}) = Γ_{5}K(ℤ/2, 2) = H_{6}K(ℤ/2, 2) = ℤ/2

where we use the computations of H_{6}K(ℤ/2, 2) = ℤ/2 of Eilenberg-Mac Lane [EM]. Now (4) and (3) shows by (2)

(5)  Γ_{5}K̃ℤ = L_{1}Γ_{2}^{2}(η^{1}) = (ℤ/2)^{3}.

Moreover since K_{6}ℤ is odd torsion we get by (1)

(6)  H_{6}SL(ℤ) = (ℤ/2)^{3} ⊕ odd torsion

Using corollary 14 [AMNY] we see that H^{6}(SL(ℤ), ℤ/2) = (ℤ/2)^{5}. This group maps surjectively to the group Hom (H_{6}(SL(ℤ)), ℤ/2) = (ℤ/2)^{3}. Therefore Ext (H_{5}(SL(ℤ)), ℤ/2) = (ℤ/2)^{2}. This implies that the extension in (3.6) (2) is non split. According to an argument of H.W. Henn equation (6) can also be proved by use of corollary 14 [AMNY] and the Bockstein spectral sequence.

(5.17) Example. We also can continue the exact sequence in (3.6) for X = K^{≈}(ℤ). In this case we get π_{3}X = ℤ/48, π_{4}X = 0, π_{5}X = ℤ ⊕ T_{3}, so that η^{1} : ℤ/2 → 0 and η^{2} : 0 → ℤ ⊕ T_{3} = K_{5}ℤ. Now (5.9) shows that there is a commutative diagram in which the row and the column are exact:

ℤ/2

↓

ℤ/3 ⊕ ℤ/2 ⊕ ℤ/2

↓

K_{7}ℤ ⟶ H_{7}St(ℤ) ⟶  Γ_{6}K^{≈}(ℤ)  ⟶ K_{6}ℤ→ H_{6}St(ℤ)→ ℤ/2→ 0

↓

0

Since K_{6}ℤ is odd torsion this shows that the 2-torsion of cokernel (K_{7}ℤ → H_{7}St(ℤ)) is ℤ/2 or ℤ/2 ⊕ ℤ/2. The diagram is also of interest with respect to the 3-torsion

since the 3-torsion of Γ_{6}K^{≈}(ℤ) is ℤ/3.

§ 6 The construction of the homotopy operation spectral sequence

In this section we prove Theorem 2.4. The key homological ingredient is a minor modification of the “E_{2} model category” structure on simplicial spaces (as in [DKS1,2]). We begin by spelling out what we need from this work. Fix r ≥ 1.

The category of simplicial pointed spaces sTop^{∗} has a closed model category structure where f : X → Y is a weak equivalence if

(6.1)  π_{p}π_{q}f : π_{p}π_{q}X → π_{p}π_{q}Y

is an isomorphism for all p ≥ 0 and q ≥ r. The cofibrations (and, hence, cofibrant objects) can be described as follows.

If X ∈ sTop^{∗}, let

L_{n}X = colim_{ϕ:[W 2]→[B3]}X_{k}

where ϕ runs over the morphisms in the ordinal number category which are surjections and k < n, that is, ϕ^{∗} : X_{k} → X_{n} is a composition of degeneracies. There is an obvious map s : L_{n}X → X_{n} and f : X → Y in sTop^{∗} is a Reedy cofibration if the induced maps

X_{n} ⊔_{L_{n}X} L_{n}Y → Y_{n}

are cofibrations for all n ≥ 0. Then f : X → Y is an E_{2} cofibration if it is a Reedy cofibration and for each n ≥ 0 there is a space

Z_{n} = ⋁_{α∈I_{n}} S_{α}^{m}

where α in some index set I_{n} and m ≥ r and a map

Z_{n} → Y_{n}

so that

(6.2)  [X_{n} ⊔_{L_{n}X} L_{n}Y] ∨ Z_{n} → Y_{n}

is a homotopy equivalence. In particular, an inductive argument shows that if X is cofibrant in the E_{2} model category, then X_{n} is homotopy equivalent to a wedge of spheres S^{m}, m ≥ r.

This E_{2} model category structure is actually a simplicial model category in the standard simplicial structure on sTop^{∗} (see [HA II.§ 2]). In particular, given a simplicial set K and X ∈ sTop^{∗}, we can form an object X ⊗ K ∈ sTop^{∗} with

(X ⊗ K)_{n} = ⋁_{K_{n}} X_{n}.

If K is pointed, we can form an object X ∧ K ∈ sTop^{∗} defined by the push-out diagram

(6.3)  \begin{matrix}X ≅ X ⊗ ∗ & ⟶ & X ⊗ K \\ ↓ & & ↓ \\ ∗ & ⟶ & X ∧ K. \end{matrix}

Of particular interest is the object S^{j} ∧ Δ^{i}/∂Δ^{i}, where S^{j} ∈ sTop^{∗} is the sphere regarded as a constant simplicial object, Δ^{i} is the standard simplicial i-simplex and ∂Δ^{i} ⊆ Δ^{i} is the boundary. The simplicial set Δ^{i}/∂Δ^{i} is a simplicial model for the i-sphere. If X ∈ sTop^{∗} is E_{2}-fibrant (or Reedy fibrant), define

(6.4)  π_{i,j}X = [S^{j} ∧ Δ^{i}/∂Δ^{i}, X]

where the homotopy classes are computed in the homotopy category of the E_{2}- model category of sTop^{∗}. This can be calculated as

π_{i,j}X = π_{0}map(S^{j} ∧ Δ^{i}/∂Δ^{i}, X)
= π_{i}map(S^{i}, X)

where map(·, ·) is the simplicial mapping spaces of sTop^{∗}. The “spiral exact sequence” of [DKS2] implies that

(6.5)  π_{0}π_{j}X ≅ π_{0,j}X

and that there is a long exact sequence

… → π_{i−1,j+1}X → π_{i,j}X → π_{i}π_{j}X → π_{i−2,j+1}X

→ π_{i−1,j}X → … → π_{1,j}X → π_{1}π_{j}X → 0. (6.6)

We assume j ≥ r.

Splicing the exact triangles

π_{∗+1,j+1}X ⟶ π_{∗,j}X
↖ ↙
π_{∗}π_{j}X

together yields a spectral sequence

(6.7)  π_{p}π_{q}X ⇒ π_{p+q−r,r}X = [S^{r} ∧ Δ^{p+q−r}/∂Δ^{p+q−r}, X].

Taking geometric realizations using the fact that | Δ^{i}/∂Δ^{i} |≅ S^{i} (regarding a simplicial set as a simplicial disrete space) yields a homomorphism

[S^{n} ∧ Δ^{m}/∂Δ^{m}, X] → [S^{n+m}, X]

or, in symbols, a map

π_{p+q−r,r}X → π_{p+q} | X | .

The following is a minor variation on the main theorem of [DKS2].

(6.8) Proposition. Suppose X ∈ sTop^{∗} is Reedy cofibrant and X_{m} is (r − 1)- connected for all m ≥ 0. Then the homomorphism

π_{m−r,r}X → π_{m} | X |

is an isomorphism.

Proof. In [DKS2] it is shown that

π_{m−1,1}X → π_{m} | X |

is an isomorphism. This is the case r = 1. The spiral exact sequence for the case r = 1 shows

π_{m−j−1,j+1}X ≅ π_{m−j,j}X

for j < r. q.e.d.

Combining (6.7) and (6.8) yields that for X satisfying the hypotheses of (6.8) there is a spectral sequence

(6.9)  π_{p}π_{q}X ⇒ π_{p+q} | X |

One source of input for this spectral sequence arises from bisimplicial sets. If X = {X_{m,n}} is a bisimplicial set we call the first (or m) index to be the horizontal index and the second (or n) index to be the vertical direction. If we write | X |_{v}= {| X_{m,∗} |} to be the level-wise geometric realization, then | X |_{v} is automatically Reedy cofibrant. This is because the map

L_{m}X = colim_{ϕ:[T]→[B3]}X_{k,∗} → X_{m,∗}

is automatically an inclusion of simplicial sets, so

| L_{m}X |_{v}≅ L_{m} | X |≅ colim_{ϕ:[T]→[B3]} | X_{k,∗} |→| X_{m,∗} |

is a cofibration. If X_{m,∗} is connected as a simplcial set and X is pointed then (6.8) yields a spectral sequence

π_{p}π_{q} | X |_{v}⇒ π_{p,q} ∥ X |_{v}| .

Since ∥ X_{v} ∥=| diag X | and one can define π_{q}X_{m,∗} = π_{q} | X_{m,∗} | we recover the Bousfield-Friedlander spectral sequence

(6.10)  π_{p}π_{q}X ⇒ π_{p+q}(diag X).

See [BF § B].

With this technology in hand, we can begin to construct our spectral sequence. Fix a Reedy cofibrant simplicial space Y, which will eventually be an E_{2}-cofibrant model for some fixed pointed space regarded as a constant simplicial space. Let Sing(Y) be the bisimplicial space obtained by applying the singular functor levelwise to Y. Then the natural map

(6.11)  | Sing (Y) |_{v}→ Y

is a level-wise weak equivalence between Reedy cofibrant simplicial spaces, so

(6.12)  | diag Sing (Y) |=∥ Sing(Y) |_{v}|→| Y |

is a weak equivalence. Note that if Y is pointed and level-wise connected, so is Sing (Y).

Now let ℤSing (Y) be the free bisimplicial abelian group of Sing (Y). Assume Y is pointed and let

ℤ̄Sing(Y) = ℤSing (Y)/ℤ{∗}.

Then

π_{q}ℤ̄Sing(Y) ≅ H̃_{q}Sing (Y) ≅ H̃_{q}Y ≅ {H̃_{q}Y_{m}},

and

π_{q}Y ⟶^{≅} π_{q}Sing (Y) → π_{q}ℤ̄Sing(Y) ≅ H̃_{q}Y

is the Hurewicz homomorphism. Factor Sing (Y) → ℤ̄Sing (Y) as

Sing (Y) ⟶^{i} Z ⟶^{f} ℤ̄Sing (Y)

where i is a level-wise weak equivalence and f is a fibration in the closed model category structure of Bousfield-Friedlander [BF, Theorem B.6]. Then, among other things, f is a level-wise fibration and the realization diag(f) is a fibration. Let

(6.13)  F → Z ⟶^{f} ℤ̄Sing(Y)

be the resulting fiber sequence. The next task is to identify the homotopy type of | diag F |.

If X is a pointed, connected cofibrant space, let S_{∞}X be the infinite symmetric product on X and let ΓX be the homotopy fiber of the natural map X → S_{∞}X.

6.14 Lemma. There is a weak equivalence

| diag F |→ Γ | Y | .

Proof. We have morphisms of simplicial spaces

Y ⟵^{p}| Sing (Y) |_{v}⟶^{|i|}| Z |_{v}⟶^{|f|}| ℤ̄Sing(Y) |_{v} .

Both p and | i | are level-wise weak equivalences between Reedy cofibrant objects. Since ∥ · |_{v}|≅| diag(·) | we have

| Y |⟵^{|p|}| diag Sing (Y) |⟶^{∥i∥}| diag Z |⟶^{∥f∥}| diag ℤ̄Sing (Y) |

and | p | and ∥ i ∥ are weak equivalences. In particular | p | has a homotopy inverse, since the source and target are cofibrant. Also

diag ℤ̄ Sing (Y) = ℤ̄ diag Sing (Y),

whence

π_{∗} | diag ℤ̄ Sing(Y) |= H_{∗} | diag Sing (Y) |= H_{∗} | Y | .

Since | ℤ̄ diag Sing(Y) | is an abelian topological monoid, there is a diagram

| Y | ⟶  | diag Z |
↓  ↓
S_{∞} | Y | ⟶ | ℤ̄ diag Sing (Y) |

and the horizontal maps are weak equivalences. Now f : Z → ℤ̄ Sing (Y) was chosen so that

diag(f) : diag Z → diag ℤ̄Sing (Y)

is a fibration of simplicial sets. Furthermore the diagonal functor preserves pullbacks, so the fiber of diag(f) is diag(F). Then

| diag F |→| diag Z |→| ℤ̄ diag Sing (Y) |

is a homotopy fiber sequence. q.e.d.

We now can construct our spectral sequence. Let X be an (r − 1) connected space where r ≥ 2, regard X as a constant simplicial space. Let Y → X be a cofibrant model for X in the E_{2}-model category structure based on the spheres S^{m}, m ≥ 1. Then Y is Reedy cofibrant level-wise connected and

π_{q}π_{q}Y ≅ \{ \begin{matrix}π_{q}X & p = 0, q ≥ r; \\ 0 & p ≠ 0 or q < r. \end{matrix}

Since X is (r − 1)-connected the spectral sequence (6.9) implies that the evident map

| Y |→ X

is a weak equivalence. To finish the construction apply the process above to produce a bisimplicial set F so that

| diagF |≃ Γ | Y |≃ ΓX.

Then the spectral sequence 6.9 becomes

(6.15)  π_{p}π_{q}F ⇒ π_{p+q}ΓX = Γ_{p+q}X.

It remains to identify the E_{2}-term. For fixed p, there is a fiber sequence

F_{p,∗} → Z_{p,∗} → ℤ Sing (Y_{p})

and a weak equivalence Z_{p,∗} → Sing (Y_{p}). Since Y_{p} is a wedge of spheres, the Hurewicz homomorphism for Y_{p} is onto, so there is a short exact sequence

(6.16)  0 → π_{q}F_{p,∗} → π_{q}Z_{p,∗} → π_{q}ℤ Sing (Y_{p}) → 0.

More is true. Since π_{∗}Z_{p,∗} is a free model in Π_{r},

π_{q}ℤ Sing (Y_{p}) = H̃_{q}Y_{p} = Q_{q}(π_{∗}Z_{p,∗})

and (L_{1}Q_{p})(π_{∗}Z_{p,∗}) = 0. Hence (1.16) and (6.16) imply

π_{r+q}F_{p,∗} ≅ Γ_{r}^{q}(η^{1}, …, η^{q−1}).

Combining this with (6.15) and reindexing the q variable gives

(6.17)  L_{p}Γ_{r}^{q}(η^{1}, …, η^{q−1}) = π_{p}Γ_{r}^{q}(η^{1}, …, η^{q−1}) ⇒ Γ_{r+p+q}X.

This is the desired spectral sequence. The statement about the edge homomorphism is clear.

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