ON COMBINATORIAL CALCULATIONS FOR THE JAMES–HOPF MAPS

JIE WU

Abstract. We give some formulas of the James-Hopf maps by using combinatorial methods. An application is to give a product decomposition of the spaces ΩΣ^{2}(X).

1. Introduction

In this paper, we give some formulas of the James-Hopf maps by using combinatorial methods.

Let X be a pointed space. The James-Hopf map H_{k} : J(X) → J(X^{(k)}) is defined by setting

H_{k}(x_{1}x_{2} ⋯ x_{n}) = ∏_{1≤i_{1}<i_{2}<···<i_{k}≤n}(x_{i_{1}}x_{i_{2}} ⋯ x_{i_{k}})

with left lexicographical order in the product, where X^{(k)} is the k-fold self smash product of X and J(X) is the James construction of X. The first result is as follows.

Theorem 1.1. Let X = ΣX′ be a suspension of a CW-complex X′. Then the composite

J(∨_{l≥k}X^{(l)}) →^{˜ ∨_{l≥k}S_{l}} JX →^{H_{k}} JX^{(k)}

is a loop map, where S_{l} = [[E, ⋯, E] : X^{(l)} → JX is the l-fold Samelson product and ˜ ∨_{l≥k}S_{l} : J(∨_{l≥k}X^{(l)}) → JX is the homomorphism of topological monoids induced by ∨_{l≥k}S_{l}.

Corollary 1.2. Let X = ΣX′ be a suspension of a CW-complex X′. Then

JX^{(l)} →^{S̃_{l}} JX →^{H_{k}} JX^{(k)}

is a loop map for l ≥ k.

The second calculation is to give a decomposition of the compositions of James-Hopf maps.

Research at MSRI is supported in part by NSF grant DMS-9022140.

Notation 1.3. The map L_{k.l} : X^{(kl)} → JX^{(kl)} is defined by

L_{k,l}(x_{1} ∧ x_{2} ∧ ⋯ ∧ x_{k}l) =

∏_{1≤j_{1}<···<j_{l}=kl,1≤l_{1}^{js}<···<l_{k}^{js}=j_{s}}(x_{l_{1}^{j_{1}}} ∧ ⋯ ∧ x_{l_{k}^{j_{1}}} ∧ ⋯ ∧ x_{l_{1}^{js}} ∧ ⋯ ∧ x_{l_{k}^{j_{l}}})

where (l_{1}^{j_{1}}, ⋯, l_{k}^{j_{1}}, ⋯, l_{1}^{j_{l}}, ⋯, l_{k}^{j_{l}}) runs over shuffles of (1, ⋯, kl) with left lexicographical oder.

Let L̃_{k,l} : JX^{(kl)} → J^{(kl)} denote the homomorphism of topological monoids induced by L_{k,l}.

Proposition 1.4. Let X = ΣX′ be a suspension of a CW-complex X′. Then

H_{l} ∘ H_{k} ≃ L̃_{k,l} ∘ H_{kl} : JX → JX^{(kl)}

An application is to give a product decomposition of ΩΣX if X is a suspension.

Notation 1.5. Let X = ΣX′. The map β_{n} : X^{(n)} → X^{(n)} is defined by induction β_{n} = β_{n−1} ∧ 1 − (1, 2, …, n) ∘ (β_{n−1} ∧ 1) and β_{2} = id − (1, 2), where (1, 2, …, n) ∈ Σ_{n} acts on X^{(n)}by permutating coordinates. Let p be a prime. Denote L_{n}(X) = hocolim_{\frac{1}{n} β_{n}}X^{(n)} localized at p if n ≢ 0 (mod p).

Notice that the mod p homology of the spaces L_{n}(X) are represented by the Lie elements of weight n in the tensor algebra T (\overline{H}_{∗}(X, Z/pZ).

Theorem 1.6. Let X = ΣX′ be a suspension of a CW-complex X′. and let 1 < k_{1} < k_{2} < ⋯ be a sequence of integers so that

(1) k_{j} ≢ 0(p) for each j ≥ 1

(2) k_{j} is not a multiple of any k_{i} else for each j. Then

JX ≃ ∏_{j} J(L_{k_{j}}(X))×?

localized at the prime p.

The article is organized as follows. In Section 2, we introduce the groups K_{n}(k)(X). The combinatorial calculations and the proofs of Theorem 1.1 and Proposition 1.4 are given in Section 3. The proof of Theorem 1.6 is given in Section 4.

2. The groups K_{n}(k)(X)

In this section, we consider certain subgroup in the group [X^{n}, J(X^{(k)}], where X^{n} is the n-fold self Cartesion product of X and X^{(k)} is the K-fold self smash product of X. When k = 1, these groups have been studied by Fred Cohen [C3].

Notation 2.1. Let X = ΣX′ be the suspension of a space. Denote K_{n}(k)(X) the subgroup of [X^{n}, JX^{(k)}] generated by the homotopy classes of the maps

π_{i_{1}···i_{k}} : X^{n} → JX^{(k)}

with π_{i_{1}···i_{k}}(x_{1}, ⋯, x_{n}) = E(x_{i_{1}} ∧ ⋯ ∧ x_{i_{k}}), where X^{n} is the n-fold self Cartesian product of X, X^{(k)} is the k-fold self smash product of X and E : X^{(k)} → JX^{(k)} is the suspension. If there is no confusion, we simply denote K_{n}(k) for K_{n}(k)(X) and denote K_{n} or K_{n}(X) for K_{n}(1)(X). Denote {x_{i_{1}}|x_{i_{2}}| ⋯ |x_{i_{k}}} for the homotopy class of π_{i_{1}···i_{k}} and denote {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{k}}^{[n_{k}]}} for the homotopy class of the map f : X^{n} → JX^{(k)} with f(x_{1}, ⋯, x_{n}) = E([n_{1}](x_{i_{1}}) ∧ [n_{2}](x_{i_{2}}) ∧ ⋯ ∧ [n_{k}](x_{i_{k}})), where [n] : X → X is the composite

X →^{ψ} ⋁^{n} X →^{▽} X,

where ψ is the comultiplication and ▽ is the fold map.

Lemma 2.2. Let X = ΣX′. Then, in the group [X^{n}, JX^{(k)}], the following identities hold: (1)

{x_{i_{1}}|x_{i_{2}}| ⋯ |x_{i_{k}}} = 1

if i_{s} = i_{t} for some 1 ≤ s < t ≤ k (2)

{x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{k}}^{[n_{k}]}} = {x_{i_{1}}|x_{i_{2}}| ⋯ |x_{i_{k}}}^{n_{1}n_{2}···n_{k}}

(3)

[[{x_{i_{1}}|x_{i_{2}}| ⋯ |x_{i_{k}}}, {x_{i_{k+1}}|x_{i_{k+2}}| ⋯ |x_{i_{2k}}}, ⋯, {x_{i_{(l−1)k+1}}|x_{i_{(l−1)k+2}}| ⋯ |x_{i_{lk}}}] = 1

if i_{s} = i_{t} for some 1 ≤ s < t ≤ kl, where [[a_{1}, a_{2}, ⋯, a_{l}] = [⋯ [a_{1}, a_{2}], ⋯,], a_{l}] with [x, y] = x^{−1}y^{−1}xy. (4)

[[{x_{i_{1}}|x_{i_{2}}| ⋯ |x_{i_{k}}}^{n_{1}}, {x_{i_{k+1}}|x_{i_{k+2}}| ⋯ |x_{i_{2k}}}^{n_{2}}, ⋯, {x_{i_{(l−1)k+1}}|x_{i_{(l−1)k+2}}| ⋯ |x_{i_{lk}}}^{n_{l}}] =

[[{x_{i_{1}}|x_{i_{2}}| ⋯ |x_{i_{k}}}, {x_{i_{k+1}}|x_{i_{k+2}}| ⋯ |x_{i_{2k}}}, ⋯, {x_{i_{(l−1)k+1}}|x_{i_{(l−1)k+2}}| ⋯ |x_{i_{lk}}}]^{n_{1}n_{2}···n_{l}} Proof: (1) By a shuffle map, we may assume that i_{1} = i_{2}. Notice that {x_{i_{1}}|x_{i_{2}}| ⋯ |x_{i_{k}}} is represented by the composite

X^{n} →^{p_{i_{1}···i_{k}}} X^{(k)} →^{E} JX^{(k)}

where p_{i_{1}···i_{k}}(x_{1}, ⋯, x_{n}) = x_{i_{1}} ∧ x_{i_{2}} ∧ ⋯ ∧ x_{i_{k}}. Since i_{1} = i_{2}, there is a commutative diagram

X^{n} →^{p_{i_{1}i_{2}···i_{k}}} X^{(k)} →^{E} JX^{(k)}
∥ Δ̄ ↑
X^{n} →^{p_{i_{1}i_{3}···i_{k}}} X^{(k−1)}

where Δ̄ : X → X ∧ X is the reduced diagonal map. Since X = ΣX′ is a coH-space, Δ̄ is null. Thus (1) follows.

(2) The element {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{k}}^{[n_{k}]}} is represented by the composite

X^{n} →^{p_{i_{1}···i_{k}}} X^{(k)} →^{[n_{1}]∧···∧[n_{k}]} X^{(k)} →^{E} JX^{(k)}

The assertion follows from the following homotopy commutative diagram

X^{(k)} →^{E} JX^{(k)}
ϕ_{1} ↑  ↑ ϕ_{2}
X^{(k)} →^{E} JX^{(k)},

where m : JX^{(k)} → JX^{(k)} is the power map of degree m, ϕ_{1} = [n_{1}] ∧ ⋯ ∧ [n_{k}] and ϕ_{2} = n_{1} ⋯ n_{k}.

(3) The element [[{x_{i_{1}}|x_{i_{2}}| ⋯ |x_{i_{k}}}, ⋯, {x_{i_{(l−1)k+1}}|x_{i_{(l−1)k+2}}| ⋯ |x_{i_{lk}}}] is represented by the composite

X^{n} →^{q} (X^{(k)})^{(l)} →^{ϕ} JX^{(k)}

where ϕ = [[E, E], ⋯, E] and

q(x_{1}, x_{2}, ⋯, x_{n}) = (x_{i_{1}} ∧ ⋯ ∧ x_{i_{k}}) ∧ (x_{i_{k+1}} ∧ ⋯ ∧ x_{i_{2k}}) ∧ ⋯ ∧ (x_{i_{(l−1)k+1}} ∧ ⋯ ∧ x_{i_{lk}}).

Since i_{s} = i_{t} for some 1 ≤ s < t ≤ kl, there is a commutative diagram

X^{n} →^{q} X^{(kl)}
∥  ↑ Δ̄
X^{n} →^{q′} X^{(kl−1)}

for a choice of reduced diagonal map Δ̄. Thus (3) follows.

(4) The element [[{x_{i_{1}}|x_{i_{2}}| ⋯ |x_{i_{k}}}^{[n_{1}]}, ⋯, {x_{i_{(l−1)k+1}}|x_{i_{(l−1)k+2}}| ⋯ |x_{i_{lk}}}^{[n_{l}]}] is represented by the composite

X^{n} →^{q} (X^{(k)})^{(l)} →^{E^{(l)}} (JX^{(k)})^{(l)} →^{n_{1}∧···∧n_{l}} (JX^{(k)})^{(l)} →^{[[id,··· ,id]} JX^{(k)}

where q(x_{1}, ⋯, x_{n}) = x_{i_{1}} ∧ ⋯ ∧ x_{i_{kl}}. the assertion follows from the homotopy commutative diagram

(JX^{(k)})^{(l)} →^{n_{1}∧···∧n_{l}} (JX^{(k)})^{(l)} →^{[[id,··· ,id]} JX^{(k)}
ψ ↑  ↑ ψ  ∥
(X^{(k)})^{(l)} →^{[n_{1}]∧···∧[n_{l}]} (X^{(k)})^{(l)}  →^{ϕ} JX^{(k)}
ϕ ↓  ∥
JX^{(k)}  →^{=}  JX^{(k)}  →^{n_{1}n_{3}···n_{l}} JX^{(k)},

where ϕ = [[E, E], ⋯, E] and ψ = (E)^{(l)}.

Lemma 2.3. Let X = ΣX′ and let H_{k} : JX → JX^{(k)} be the k-th James-Hopf map. Let (H_{k})_{∗} : [X^{n}, JX] → [X^{n}, JX^{(k)}] be the function induced by H_{k}. Then, for y = x_{i_{1}}^{n_{1}}x_{i_{2}}^{n_{2}} ⋯ x_{i_{l}}^{n_{l}} ∈ K_{n},

(H_{k})_{∗}(y) = ∏_{1≤j_{1}<···<j_{k}≤l}{x_{i_{j_{1}}}|x_{i_{j_{2}}}| ⋯ |x_{i_{j_{k}}} }^{n_{j_{1}}n_{j_{2}}···n_{j_{k}}}
= ∏_{1≤j_{1}<···<j_{k}≤l}{x_{i_{j_{1}}}^{[n_{j_{1}}]}|x_{i_{j_{2}}}^{[n_{j_{2}}]}| ⋯ |x_{i_{j_{k}}}^{[n_{j_{k}}]}}

in K_{n}(k) with left lexicographical order.

Proof: The element y is represented by the composite

X^{n} →^{q} X^{l} \underrightarrow{∏_{j=1}^{l}[n_{j}]} X^{l} →^{p} JX,

where p is the projection and q(x_{1}, ⋯, x_{n}) = (x_{i_{1}}, ⋯, x_{i_{l}}). The assertion follows from the definition of H_{k} and the above lemma.

3. Combinatorial Calculations

In this section, we give some combinatorial calculations and give proofs of Theorem 1.1 and Proposition 1.4, where Theorem 1.1 is Theorem 3.10 and Proposition 1.4 is Proposition 3.14.

Definition 3.1. The tensor product ⊗ : [Z, JX]×[Z, JY] → [Z, J(X ∧Y)] is defined as follows:

[f] ⊗ [g] is represented by the composite

Z →^{Δ̄} Z ∧ Z →^{f∧g} JX ∧ JY →^{c} J(X ∧ Y)

for f : Z → JX and g : Z → JY, where c((x_{1} ⋯ x_{n}) ∧ (y_{1} ⋯ y_{m})) = (x_{1} ∧ y_{1}) · (x_{2} ∧ y_{1}) ⋯ (x_{n} ∧ y_{1}) · (x_{1} ∧ y_{2}) · (x_{2} ∧ y_{2}) ⋯ (x_{n} ∧ y_{2}) ⋯ (x_{1} ∧ y_{m}) · (x_{2} ∧ y_{m}) ⋯ (x_{n} ∧ y_{m})

Remark 3.2. The tensor product was introduced by F.Cohen[C2]

Lemma 3.3. Let X = ΣX′ and let ⊗ : [X^{n}, JX^{(k)}] × [X^{n}, JX^{(l)}] → [X^{n}, JX^{(k+l)}] be the tensor product. Then

− ⊗ {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{l}}^{[n_{l}]}} : K_{n}(k) → K_{n}(k + l)

is a group homomorphism. Furthermore,

ab ⊗ {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{l}}^{[n_{l}]}} = ba ⊗ {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{l}}^{[n_{l}]}}

i.e. − ⊗ {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{l}}^{n_{l}]}} factors though the abelianization K_{n}(k)/[K_{n}(k), K_{n}(k)].

Proof: Let a = {x_{j_{1}}| ⋯ |x_{j_{k}}} be a generator in K_{n}(k). Then

a^{−1} ⊗ {x_{i_{1}}^{[n_{1}]} ⋯ x_{i_{l}}^{[n_{l}]}} = {x_{j_{1}}^{[−1]}|x_{j_{2}}| ⋯ |x_{j_{k}}} ⊗ {x_{i_{1}}^{[n_{1}]} ⋯ x_{i_{l}}^{[n_{l}]}}
= {x_{j_{1}}^{[−1]}|x_{j_{2}}| ⋯ |x_{j_{k}}|x_{i_{1}}^{[n_{1}]}| ⋯ |x_{i_{l}}^{[n_{l}]}}
= ({x_{j_{1}}|x_{j_{2}}| ⋯ |x_{j_{k}}} ⊗ {x_{i_{1}}^{[n_{1}]}| ⋯ x_{i_{l}}^{[n_{l}]}})^{−1}

Let a_{1}, a_{2}, ⋯, a_{s} be sequence of generators in K_{n}(k). Then

(a_{1} ⋯ a_{s}) ⊗ {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{l}}^{[n_{l}]}} = (a_{1} ⊗ {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{l}}^{[n_{l}]}}) ⋯
(a_{s} ⊗ {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{l}}^{[n_{l}]}}).

Thus − ⊗ {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{l}}^{[n_{l}]}} is a group homomorphism. By the Lemma 2.2,

ab ⊗ {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{l}}^{[n_{l}]}} = ba ⊗ {x_{i_{1}}^{[n_{1}]}|x_{i_{2}}^{[n_{2}]}| ⋯ |x_{i_{l}}^{[n_{l}]}}

The assertion follows.

Lemma 3.4. Let X = ΣX′ and let (H_{k})_{∗} : [X^{n}, JX] → [X^{n}, JX^{(k)}] be the function induced by H_{k} : JX → JX^{(k)}. Then, for a = x_{i_{1}}^{n_{1}} ⋯ x_{i_{p}}^{n_{p}} and y = x_{i_{p+1}}^{n_{p+1}} ⋯ x_{i_{p+q}}^{n_{p+q}} in K_{n},

(H_{k})_{∗}(a · y) = (H_{k})_{∗}(a) ∏_{j=1}^{q}( ∏_{1≤l_{1}<···<l_{s−1}<l_{s}=j,1≤s≤k}(H_{k−s})_{∗}(a) ⊗ {x_{i_{p+l_{1}}}^{[n_{p+l_{1}}]}| ⋯ |x_{i_{p+ls}}^{[n_{p+ls}]}}),

where the order of ∏_{1≤l_{1}<···<l_{s−1}<l_{s}=j,1≤s≤k} can be chosen to any order.

Proof: By induction on q. If q = 0, the assertion is trivial. Suppose that the assertion holds for q − 1 and y = x_{i_{p+1}}^{n_{p+1}} ⋯ x_{i_{p+q}}^{n_{p+q}}. Let b denote a · x_{i_{p+1}}^{n_{p+1}} ⋯ x_{i_{p+q−1}}^{n_{p+q−1}}. By lemma 2.3,

(H_{k})_{∗}(b · x_{i_{p+q}}^{n_{p+q}}) = (H_{k})_{∗}(b) · ∏_{1≤l_{1}<···<l_{k−1}≤p+q−1}{x_{i_{l_{1}}}^{[n_{l_{1}}]}| ⋯ |x_{i_{l_{k−1}}}^{[n_{l_{k−1}}]}|x_{i_{p+q}}^{[n_{p+q}]}}).

Let

w = ∏_{1≤l_{1}<···<l_{k−1}≤p+q−1}{x_{i_{l_{1}}}^{[n_{l_{1}}]}| ⋯ |x_{i_{l_{k−1}}}^{[n_{l_{k−1}}]}|x_{i_{p+q}}^{[n_{p+q}]}}).

By Lemma 2.2, the elements in the product w commute each other. Thus

w = (H_{k−1})_{∗}(b) ⊗ {x_{i_{p+q}}^{[n_{p+q}]}}
= (H_{k−1})_{∗}(a) ⊗ {x_{i_{p+q}}^{[n_{p+q}]}}+

∑_{j=1}^{q−1} ∑_{1≤l_{1}<···<l_{s−1}<l_{s}=j,1≤s≤k−1}(H_{k−s−1})_{∗}(a) ⊗ {x_{i_{p+l_{1}}}^{[n_{p+l_{1}}]}| ⋯ |x_{i_{p+ls}}^{[n_{p+ls}]}} ⊗ {x_{i_{p+q}}^{[n_{p+q}]}}

= ∑_{1≤l_{1}<···<l_{s}=q,1≤s≤k}(H_{k−s})_{∗}(a) ⊗ {x_{i_{p+l_{1}}}^{[n_{p+l_{1}}]}| ⋯ |x_{i_{p+ls}}^{[n_{p+ls}]}}.

The assertion follows.

Lemma 3.5. Let X = ΣX′ and let (H_{k})_{∗} : [X^{n}, JX] → [X^{n}, JX^{(k)}] be induced by H_{k} : JX → JX^{(k)}. Then

(H_{k})_{∗}([[x_{i_{1}}^{n_{1}}, ⋯, x_{i_{m}}^{n_{m}}]) ⊗ x_{j} = 1

in [X^{n}, JX^{(k+1)}] for m > k ≥ 1.

Proof: By induction on k. If k = 1, H_{1} is the identity and the assertion follows from Lemma 3.3. Suppose that the assertion holds for < k and consider (H_{k})_{∗}([[x_{i_{1}}^{n_{1}}, ⋯, x_{i_{m}}^{n_{m}}])⊗x_{j} with m > k. Let ˜ < x_{j} > denote the subgroup of K_{n}(k+1) generated by all of the elements {x_{a_{1}}| ⋯ |x_{a_{k+1}}} so that a_{k+1} = j. Then, by Lemma 2.2, ˜ < x_{j} > is abelian. Thus the product is the sum in ˜ < x_{j} >. Let z_{j} = x_{i_{j}}^{n_{j}}. Then

(H_{k})_{∗}([[z_{1}, ⋯, z_{m}]) ⊗ x_{j} = (H_{k})_{∗}([[z_{1}, ⋯, z_{m−1}]^{−1} · z_{m}^{−1} · [[z_{1}, ⋯, z_{m−1}] · z_{m}) ⊗ x_{j}

= ∑_{s+t=k}(H_{s})_{∗}([[z_{1}, ⋯, z_{m−1}]^{−1}) ⊗ (H_{t})_{∗}([[z_{1}, ⋯, z_{m−1}]) ⊗ x_{j}

+ ∑_{s+t=k−1}(H_{s})_{∗}([[z_{1}, ⋯, z_{m−1}]^{−1}) ⊗ z_{m}^{−1} ⊗ (H_{t})_{∗}([[z_{1}, ⋯, z_{m−1}]) ⊗ x_{j}

+ ∑_{s+t=k−1}(H_{s})_{∗}([[z_{1}, ⋯, z_{m−1}]^{−1}) ⊗ (H_{t})_{∗}([[z_{1}, ⋯, z_{m−1}]) ⊗ z_{m} ⊗ x_{j}

+ ∑_{s+t=k−2}(H_{s})_{∗}([[z_{1}, ⋯ z_{m−1}]^{−1}) ⊗ z_{m}^{−1} ⊗ (H_{t})_{∗}([[z_{1}, ⋯, z_{m−1}]) ⊗ z_{m} ⊗ x_{j}.

By Lemma 2.2,

(H_{s})_{∗}([[z_{1}, ⋯ z_{m−1}]^{−1}) ⊗ z_{m}^{−1} ⊗ (H_{t})_{∗}([[z_{1}, ⋯, z_{m−1}]) ⊗ z_{m} ⊗ x_{j} = 1.

By induction, for 1 ≤ t ≤ k − 1,

(H_{s})_{∗}([[z_{1}, ⋯, z_{m−1}]^{−1}) ⊗ (H_{t})_{∗}([[z_{1}, ⋯, z_{m−1}]) ⊗ x_{j} = 1,

8 JIE WU

(H_{s})_{∗}([[z_{1}, ⋯, z_{m−1}]^{−1}) ⊗ z_{m}^{−1} ⊗ (H_{t})_{∗}([[z_{1}, ⋯, z_{m−1}]) ⊗ x_{j} = 1,

and

(H_{s})_{∗}([[z_{1}, ⋯, z_{m−1}]^{−1}) ⊗ (H_{t})_{∗}([[z_{1}, ⋯, z_{m−1}]) ⊗ z_{m} ⊗ x_{j} = 1

Also, by induction, we have

(H_{k−1})_{∗}([[z_{1}, ⋯, z_{m−1}]^{−1}) ⊗ z_{m}^{−1} ⊗ x_{j} = 1

and

(H_{k−1})_{∗}([[z_{1}, ⋯, z_{m−1}]^{−1}) ⊗ z_{m} ⊗ x_{j} = 1.

Thus (H_{k})_{∗}([[z_{1}, ⋯, z_{m}]) ⊗ x_{j} = (H_{k})_{∗}([[z_{1}, ⋯, z_{m−1}]^{−1}) ⊗ x_{j} + (H_{k})_{∗}([[z_{1}, ⋯, z_{m−1}]) ⊗ x_{j}

= (H_{k})_{∗}([[z_{1}^{−1}, ⋯, z_{m−1}]) ⊗ x_{j} + (H_{k})_{∗}([[z_{1}, ⋯, z_{m−1}]) ⊗ x_{j}

with m − 1 ≥ k. Now (H_{k})_{∗}[[z_{1}, ⋯, z_{m−1}] is represented by

X^{n} →^{q} X^{(m−1)} →^{[n_{1}]∧···∧[n_{m−1}]} X^{(m−1)} →^{[[E,··· ,E]} JX →^{H_{k}} JX^{(k)}

and (H_{k})_{∗}[[z_{1}^{−1}, ⋯, z_{m−1}] is represented by

X^{n} →^{q} X^{(m−1)} →^{[−n_{1}]∧[n_{2}]∧···∧[n_{m−1}]} X^{(m−1)} →^{[[E,··· ,E]} JX →^{H_{k}} JX^{(k)}

where q(x_{1}, x_{2}, ⋯, x_{n}) = x_{i_{1}} ∧ ⋯ ∧ x_{i_{m−1}}. Thus

(H_{k})_{∗}[[z_{1}^{−1}, ⋯, z_{m−1}] = −(H_{k})_{∗}[[z_{1}, ⋯, z_{m−1}]

and the assertion follows. Remark 3.6. This lemma does not hold for m ≤ k, e.g. (H_{1})_{∗}(x_{1}) ⊗ x_{2} = {x_{1}|x_{2}} and (H_{2})_{∗}([x_{1}, x_{2}]) ⊗ x_{3} = {x_{1}|x_{2}|x_{3}} · {x_{2}|x_{1}|x_{3}}^{−1}. Definition 3.7. Let G be a group. The lower central series Γ^{k}G is defined by induction

Γ^{1}G = G

and

Γ^{k}G = [Γ^{k−1}G, G]

for k ≥ 2. Theorem 3.8. Let X = ΣX′ and let (H_{k})_{∗} : [X^{n}, JX] → [X^{n}, JX^{(k)}] be induced by H_{k} : JX → JX^{(k)}. Then

(H_{k})_{∗}(a · y) = (H_{k})_{∗}(a) · (H_{k})_{∗}(y)

for a ∈ Γ^{k}K_{n} and y ∈ K_{n}, i.e. (H_{k})_{∗} restricted to K_{n} is a Γ^{k}K_{n}-map.

Proof: It suffices to show that

(H_{k})_{∗}([[x_{i_{1}}^{n_{1}}, ⋯, x_{i_{m}}^{n_{m}}] · y) = (H_{k})_{∗}[[x_{i_{1}}^{n_{1}}, ⋯, x_{i_{m}}^{n_{m}}] · (H_{k})_{∗}(y)

for m ≥ k. Let y = x_{j_{1}}^{l_{1}} ⋯ x_{j_{t}}^{n_{t}}. Let z_{j} = x_{i_{j}}^{n_{j}} and y_{s} = x_{j_{s}}^{l_{s}}. By Lemma 3.4,

(H_{k})_{∗}([[z_{1}, ⋯, z_{m}] · y_{1} ⋯ y_{t}) =

(H_{k})_{∗}([[z_{1}, ⋯, z_{m}) ∏_{j=1}^{t}( ∏_{1≤i_{1}<···<i_{s−1}<i_{s}=j,1≤s≤k}(H_{k−s})_{∗}([[z_{1}, ⋯, z_{m}]) ⊗ {y_{i_{1}}| ⋯ |y_{i_{s}}}).

By Lemma 3.5,

(H_{k−s}([[z_{1}, ⋯, z_{m}]) ⊗ {y_{i_{1}}| ⋯ |y_{i_{s}}} = 1

for 1 ≤ s ≤ k − 1. Thus

(H_{k})_{∗}([[z_{1}, ⋯, z_{m}] · y_{1} ⋯ y_{t}) = (H_{k})_{∗}([[z_{1}, ⋯, z_{m}) ∏_{j=1}^{t}( ∏_{1≤i_{1}<···<i_{k−1}<i_{k}=j}{y_{i_{1}}| ⋯ |y_{i_{k}}}).

= (H_{k})_{∗}([[z_{1}, ⋯, z_{m}])(H_{k})_{∗}(y_{1} ⋯ y_{t}).

The assertion follows.

Lemma 3.9. ( see also [B1]) Let X be a path-connected CW-complex and let f, g : JX → ΩY so that f|_{J_{n}X} ≃ g|_{J_{n}X} for each n. Then f ≃ g.

Proof: There is a homotopy equivalence Φ : Σ ∨_{n=1}^{∞} X^{(n)} → ΣJX so that Φ_{n} = Φ|_{Σ∨_{j=1}^{n}X^{(j)}} : Σ∨_{j=1}^{n}X^{(j)} → ΣJ_{n}X are homotopy equivalences. Denote f′, g′ : ΣJX → Y the adjoints of f and g, respectively. Let i_{j} : ΣX^{(j)} → Σ∨_{n=1}^{∞} X^{(n)} be the canonical inclusion. Since f|_{J_{n}X} ≃ g|_{J_{n}X} for each n, f′|_{ΣJ_{n}} ∘ Φ_{n} ≃ g′|_{ΣJ_{n}} ∘ Φ_{n} for each n and therefore f′ ∘ Φ ∘ i_{j} ≃ g′ ∘ Φ ∘ i_{j} for each j. Let F_{j} : ΣX^{(j)} ∧ I^{+} → Y be a homotopy between f′ ∘ Φ ∘ i_{j} and g′ ∘ Φ ∘ i_{j}. Then F = ∨_{j=1}^{∞}F_{j} : (Σ ∨_{j=1}^{∞} X^{(j)}) ∧ I^{+} → Y is a homotopy between f′ ∘ Φ and g′ ∘ Φ. The assertion follows.

Theorem 3.10. Let X = ΣX′ be a suspension of a CW-complex X′. Then the composite

J(∨_{l≥k}X^{(l)}) →^{˜ ∨_{l≥k}S_{l}} JX →^{H_{k}} JX^{(k)}

is a loop map, where S_{l} = [[E, ⋯, E] : X^{(l)} → JX is the l-fold Samelson product and ˜ ∨_{l≥k}S_{l} : J(∨_{l≥k}X^{(l)}) → JX is the homomorphism of topological monoids induced by ∨_{l≥k}S_{l}.

Proof: Let β̃ : J(∨_{l≥k}X^{(l)}) → JX^{(k)} be the homomorphism of topological monoids induced by the composite

∨_{l≥k}X^{(l)} →^{∨_{l≥k}S_{l}} JX →^{H_{k}} JX^{(k)}.

By Lemma 3.9, it suffices to show that

β̃|_{J_{n}(∨_{l≥k}X^{(l)})} ≃ H_{k} ∘ ˜ ∨_{l≥k}S_{l}|_{J_{n}(∨_{l≥k}X^{(l)})}

for each n. Let j_{n} : J_{n}(∨_{l≥k}X^{(l)}) → J(∨_{l≥k}X^{(l)}) denote the inclusion and p_{n} : (∨_{l≥k}X^{(l)})^{n} → J_{n}(∨_{l≥k}X^{(l)}) the projection. Notice that

p_{n}^{∗} : [J_{n}(∨_{l≥k}X^{(l)}), ΩY] → [(∨_{l≥k}X^{(l)})^{n}, ΩY]

is a monomorphism for any space Y. It suffice to show that

β̃ ∘ j_{n} ∘ p_{n} ≃ H_{k} ∘ ˜ ∨_{l≥k}S_{l} ∘ j_{n} ∘ p_{n}.

Now, by the splitting theorem for Σ(∨_{l≥k}X^{(l)})^{n}, it suffices to show that

β̃ ∘ j_{n} ∘ p_{n} ∘ Φ_{l_{1}···l_{n}} ≃ H_{k} ∘ ˜ ∨_{l≥k}S_{l} ∘ j_{n} ∘ p_{n} ∘ Φ_{l_{1}···l_{n}}

for all l_{1}, l_{2}, ⋯, l_{n} ≥ k, where Φ_{l_{1}···l_{n}} : X^{(l_{1})} × ⋯ × X^{(l_{n})} → (∨_{l≥k}X^{(l)})^{n} is the canonical inclusion. Let K̃_{n} be the subgroup of [X^{l_{1}+l_{2}+···+l_{n}}, J(∨_{l≥k}X^{(l)})] generated by x̃_{1}, ⋯, x̃_{n}, where x̃_{i} is represented by the composite

X^{l_{1}+···+l_{n}} →^{q} ∏_{j=1}^{n} X^{(l_{j})} →^{q} X^{(l_{i})} ↪^{i} (∨_{l≥k}X^{(l)})^{n} →^{j_{n}∘p_{n}} J(∨_{l≥k}X^{(l)}),

where q is the projection and i is the injection. Consider the monomorphism

q∗ : [X^{(l_{1})} × ⋯ × X^{(l_{n})}, J(∨_{l≥k}X^{(l)})] → [X^{l_{1}+···+l_{n}}, J(∨_{l≥k}X^{(l)})].

Then

q ∗ ([j_{n} ∘ p_{n} ∘ Φ_{l_{1}···l_{n}}]) = x̃_{1} · x̃_{2} ⋯ x̃_{n},

where [f] is the homotopy class of f. Now consider the homomorphism

˜ ∨_{l≥k}S_{l}_{∗} : [X^{l_{1}+···+l_{n}}, J(∨_{l≥k}X^{(l)})] → [X^{l_{1}+···+l_{n}}, JX].

Then

˜ ∨_{l≥k}S_{l}_{∗}(x̃_{i}) = [[x_{l_{1}+···+l_{i−1}+1}, ⋯, x_{l_{1}+l_{2}+···+l_{i}}]

By Theorem 3.8,

(H_{k})_{∗} ∘ ˜ ∨_{l≥k}S_{l}_{∗}(x̃_{1} ⋯ x̃_{n}) = ∏_{i=1}^{n}(H_{k})_{∗} ∘ ˜ ∨_{l≥k}S_{l}_{∗}(x̃_{i}) = β̃_{∗}(x̃_{1}x̃_{2} ⋯ x̃_{n}).

Thus

H_{k} ∘ ˜ ∨_{l≥k}S_{l} ∘ j_{n} ∘ p_{n} ∘ Φ_{l_{1}···l_{n}} ∘ q ≃ β̃ ∘ j_{n} ∘ p_{n} ∘ Φ_{l_{1}···l_{n}} ∘ q

and therefore

H_{k} ∘ ˜ ∨_{l≥k}S_{l} ∘ j_{n} ∘ p_{n} ∘ Φ_{l_{1}···l_{n}} ≃ β̃ ∘ j_{n} ∘ p_{n} ∘ Φ_{l_{1}···l_{n}}.

The assertion follows.

Remark 3.11. When k = 2, this is Lemma 2.5 in [C2]. The homology view of this statement was given in [CT].

Proposition 3.12. Let X = ΣX′ be a suspension of a CW-complex X′. Then the diagram

JX^{(l)} × JX →^{S̃_{l}·1_{JX}} JX
ϕ ↓  ↓ H_{k}
JX^{(k)} × JX^{(k)}  →^{m} JX^{(k)}

homotopy commutes for l ≥ k, where ϕ = H_{k} ∘ S̃_{l} × H_{k} and m is the multiplication.

Proof: Let K̄_{n,m} be the subgroup of [X^{nl+m}, JX^{(l)} × JX] generated by x̄_{1}, ⋯, x̄_{n} and ȳ_{1}, ⋯, ȳ_{m}, where x̄_{i} and ȳ_{j} are represented by q_{i}, r_{j} : X^{nl+m} = (X^{l})^{n} × X^{m} → JX^{(l)} × JX with

q_{i}(a_{1}, ⋯, a_{n}; b_{1}, ⋯, b_{m}) = p_{l}(a_{i})

and

r_{j}(a_{1}, ⋯, a_{n}; b_{1}, ⋯, b_{m}) = E(b_{j})

for a_{s} ∈ X^{l} and b_{t} ∈ X, where p_{l} is the composite X^{l} → X^{(l)} → JX^{(l)}. Let μ : JX × JX → JX be the mutiplication. Consider

μ ∘ (S̃_{l} × id)_{∗} : [X^{nl+m}, JX^{(l)} × JX] → [X^{nl+m}, JX].

Then μ ∘ (S̃_{l} × id)_{∗}(x̄_{i}) = [[x_{il+1}, ⋯, x_{(i+1)l}] and μ ∘ (S̃_{l} × id)_{∗}(ȳ_{j}) = x_{nl+j}. Thus

μ ∘ (S̃_{l} × id)_{∗}(x̄_{1} ⋯ x̄_{n} · ȳ_{1} ⋯ ȳ_{m}) = (∏_{i=1}^{n}[[x_{il+1}, ⋯, x_{(i+1)l}]) · x_{nl+1} ⋯ x_{nl+m}.

By Theorem 3.8,

H_{k} ∘ μ ∘ (S̃_{l} × id)_{∗}(x̄_{1} ⋯ x̄_{n} · ȳ_{1} ⋯ ȳ_{m})
= (∏_{i=1}^{n}(H_{k})_{∗}[[x_{il+1}, ⋯, x_{(i+1)l}]) · (H_{k})_{∗}(x_{nl+1} ⋯ x_{nl+m})
= μ ∘ ((H_{k} ∘ S̃_{l}) × H_{k})_{∗}(x̄_{1} ⋯ x̄_{n} · ȳ_{1} ⋯ ȳ_{m}).

Thus

H_{k} ∘ μ(S̃_{l} × id) ∘ q_{n,m} ≃ μ ∘ ((H_{k} ∘ S̃_{l}) × H_{k}) ∘ q_{n,m}

where q_{n,m} is the composite X^{nl+m} = (X^{l})^{n} × X^{m} → J_{n}(X^{(l)}) × J_{m}X ↪ JX^{(l)} × JX. Since q_{n,m}^{∗} : [J_{n}(X^{(l)}) × J_{m}X, ΩY] → [(X^{l})^{n} × X^{m}, ΩY] is a monomorphism,

H_{k} ∘ μ(S̃_{l} × id)|_{J_{n}X^{(l)}×J_{m}X} ≃ μ ∘ ((H_{k} ∘ S̃_{l}) × H_{k})|_{J_{n}X^{(l)}×J_{m}X}.

The assertion follows from Lemma 3.9.

Notation 3.13. The map L_{k.l} : X^{(kl)} → JX^{(kl)} is defined by

L_{k,l}(x_{1} ∧ x_{2} ∧ ⋯ ∧ x_{k}l) =

∏_{1≤j_{1}<···<j_{l}=kl,1≤l_{1}^{js}<···<l_{k}^{js}=j_{s}}(x_{l_{1}^{j_{1}}} ∧ ⋯ ∧ x_{l_{k}^{j_{1}}} ∧ ⋯ ∧ x_{l_{1}^{js}} ∧ ⋯ ∧ x_{l_{k}^{j_{l}}}

where (l_{1}^{j_{1}}, ⋯, l_{k}^{j_{1}}, ⋯, l_{1}^{j_{l}}, ⋯, l_{k}^{j_{l}}) runs over shuffles of (1, ⋯, kl) with left lexicographical oder.

Let L̃_{k,l} : JX^{(kl)} → J^{(kl)} denote the homomorphism of topological monoids induced by L_{k,l}.

Proposition 3.14. Let X = ΣX′ be a suspension of a CW-complex X′. Then

H_{l} ∘ H_{k} ≃ L̃_{k,l} ∘ H_{kl} : JX → JX^{(kl)}

Proof: Consider K_{n} ⊆ [X^{n}, JX]. By Lemma 3.4,

(H_{k})_{∗}(x_{1} ⋯ x_{n}) = ∏_{j=1}^{n} ∑_{1≤l_{1}^{j}<···<l_{k}^{j}=j}{x_{l_{1}^{j}}| ⋯ |x_{l_{k}^{j}}}.

By Lemma 2.2,

{x_{l_{1}^{j_{1}}}| ⋯ |x_{l_{k}^{j_{1}}}| ⋯ |x_{l_{1}^{j_{l}}}| ⋯ |x_{l_{k}^{j_{l}}}} = 1

if some l_{t}^{j_{s}} repeats. Thus

(H_{l})_{∗} ∘ (H_{k})_{∗}(x_{1} ⋯ x_{n}) = ∏_{j=1}^{n} ∑_{1≤j_{1}<···<j_{l}=j,1≤l_{1}^{js}<···<l_{k}^{js}=j_{s}}{x_{l_{1}^{j_{1}}}| ⋯ |x_{l_{k}^{j_{1}}}| ⋯ |x_{l_{1}^{j_{l}}}| ⋯ |x_{l_{k}^{j_{l}}}}

so that no elements in {l_{1}^{j_{1}}, ⋯, l_{k}^{j_{1}}, ⋯, l_{1}^{j_{l}}, ⋯, l_{k}^{j_{l}}} repeat. Thus

(H_{l})_{∗} ∘ (H_{k})_{∗}(x_{1} ⋯ x_{n}) = (L̃_{k,l})_{∗} ∘ (H_{kl})_{∗}(x_{1} ⋯ x_{n})

and

H_{l} ∘ H_{k}|_{J_{n}X} ∘ p_{n} ≃ L̃_{k,l} ∘ H_{kl}|_{J_{n}X} ∘ p_{n},

where p_{n} : X^{n} → J_{n}X is the projection. Hence

H_{l} ∘ H_{k}|_{J_{n}X} ≃ L̃_{k,l} ∘ H_{kl}|_{J_{n}X}

for each n. The assertion follows.

Remark 3.15. If X = ΣX′, the order of the product in the definition of L_{k,l} can be chosen in any way.

Example 3.16. Let X = S^{n}. Then L_{2,2} : (S^{n})^{(4)} = S^{4n} → JS^{4n} is given by

L_{2,2}(x_{1} ∧ x_{2} ∧ x_{3} ∧ x_{4}) = (x_{2} ∧ x_{3} ∧ x_{1} ∧ x_{4}) · (x_{1} ∧ x_{3} ∧ x_{2} ∧ x_{4}) · (x_{1} ∧ x_{2} ∧ x_{3} ∧ x_{4})

Thus L_{2,2} is of degree 2 + (−1)^{n} and the diagram

JS^{n}  →^{H_{2}}  JS^{2n}
H_{4} ↓  ↓ H_{2}
JS^{4n} →^{J(2+(−1)^{n})} JS^{4n}

homotopy commutes. This coincides with the cohomology calculation.

4. Proof of Theorem 1.6

Let X = ΣX′ be the suspension of a space X′. Recall that The map β_{n} : X^{(n)} → X^{(n)} is defined by induction

β_{n} = β_{n−1} ∧ id − (1, 2, ⋯, n) ∘ (β_{n−1} ∧ id)

and β_{2} = id − (12), where (1, 2, ⋯, n) ∈ Σ_{n} acts on X^{(n)} by permutation of coordinates. Then β_{n} ∘ β_{n} = nβ_{n} ( see [CW]). If n ≢ 0(p), we denote L_{n}(x) = hocolim_{n^{−1}β_{n}}X^{(n)} localized at p for a prime p. Proof of Theorem 1.6: Let S_{n} = [[E, ⋯, E] : X^{(n)} → JX be the n-fold Samelson product. Then S_{n} factors through L_{n}(X) if n ≢ 0(p), i.e. there a homotopy commutative diagram

X^{(n)} →^{S_{n}} JX
↓  ∥
L_{n}(X) →^{S_{n}} JX.

Denote S̃_{n} : J(L_{n}(X)) → JX the homomorphism of topological monoids induced by S_{n} : L_{n}(X) → JX. Let ϕ be the composite

∏_{j=1}^{∞} J(L_{k_{j}}(X)) \underrightarrow{∏_{j} S̃_{k_{j}}} ∏_{j=1}^{∞} JX →^{multi.} JX,

where ∏_{j=1}^{∞} is the weak infinite product. Notice that JX^{(n)} is (n − 1)-connected if X is connected. Define ψ : JX → ∏_{j=1}^{∞} J(L_{k_{j}}(X)) to be the composite

JX \underrightarrow{∏_{j} H_{k_{j}}} ∏_{j=1}^{∞} JX^{(k_{j})} → ∏_{j=1}^{∞} J(L_{k_{j}}(X)).

It suffices to show that

ψ ∘ ϕ : ∏_{j=1}^{∞} J(L_{k_{j}}(X)) → ∏_{j=1}^{∞} J(L_{k_{j}}(X))

is a homotopy equivalence. Consider

ψ ∘ ϕ_{∗} : P H_{∗} ∏_{j=1}^{∞} J(L_{k_{j}}(X)) → P H_{∗} ∏_{j=1}^{∞} J(L_{k_{j}}(X)),

where H_{∗}(−) = H_{∗}(−; F_{p}) the homology with coefficients in the field F_{p}, the prime field with p elements. Notice that P H_{∗} ∏_{j=1}^{∞} J(L_{k_{j}}(X)) ≅ ⨁_{j=1}^{∞} P H_{∗}JL_{k_{j}}(X). By Corollary 3.11, the composite

ϕ_{i,j} : JX^{(k_{j})} ↪ ∏_{j=1}^{∞} JX^{(k_{j})} → JX → ∏_{j=1}^{∞} JX^{(k_{j})} → JX^{(k_{i})}

is a loop map for k_{i} ≤ k_{j},i.e. i ≤ j. Since k_{j} is not a multiple of k_{i},

ϕ_{i,j}|_{H_{∗}X^{(k_{j})}} : H̄_{∗}X^{(k_{j})} → H̄_{∗}JX^{(k_{i})}

is zero for k_{i} < k_{j} [CT, Proposition 5.3]. Now the composite

ϕ′_{i,i} : JL_{k_{i}}(X) ↪ ∏_{j=1}^{∞} JL_{k_{j}}(X) → JX → ∏_{j=1}^{∞} JL_{k_{j}}(X) → JL_{k_{i}}(X)

is a homotopy eqivalence [CW]. Thus the composite

P H_{∗}JL_{k_{j}}(X) ↪^{l_{j}} ⨁_{j=1}^{∞} P H_{∗}JL_{k_{j}}(X) →^{ψ∘ϕ_{∗}} ⨁_{j=1}^{∞} P H_{∗}JL_{k_{j}}(X) →^{π_{i}} P H_{∗}J(L_{k_{i}}(X))

is zero for i < j and an isomorphism for i = j, where π_{i} is the projection. To check that

ψ ∘ ϕ_{∗} : ⨁_{j=1}^{∞} P H_{∗}J(L_{k_{j}}(X)) → ⨁_{j=1}^{∞} P H_{∗}J(L_{k_{j}}(X))

is a monomorphism, suppose that

(ψ ∘ ϕ)_{∗}(a_{k} + a_{k+1} + ⋯) = 0

with a_{k} ≠ 0, where a_{j} ∈ P H_{∗}JL_{k_{j}}(X). Then

π_{k} ∘ (ψ ∘ ϕ)_{∗}(a_{k} + a_{k+1} + ⋯) = π_{k} ∘ (ψ ∘ ϕ)_{∗}(a_{k}) + ∑_{j>k} π_{k} ∘ (ψ ∘ ϕ)_{∗}(a_{j})

= π_{k} ∘ (ψ ∘ ϕ)_{∗}(a_{k}) = π_{k} ∘ (ψ ∘ ϕ)_{∗} ∘ l_{k}(a_{k}) = 0.

Thus a_{k} = 0 which is a contradiction. Thus ψ ∘ ϕ_{∗} is a monomorphism and therefore an isomorphism for any finite CW-complex. Similarly, ψ∘ϕ is a rational isomorphism. Thus ψ ∘ ϕ is a homotopy equivalence for any finite CW-complex X = ΣX′. Notice that any CW-complex is a homotopy colimit of finite CW-complexes. The assertion follows.

Corollary 4.1. Let X = ΣX′ be a suspension of a CW-complex X′ and let q_{1} < q_{2} < ⋯ be all of the primes which are different from the prime p. Then

JX ≃ ∏_{j=1}^{∞} JL_{q_{j}}(X)×?

localized at p.

Corollary 4.2. There exists a space Z_{n+1} so that

ΩP^{n+1}(2) ≃ (∏_{j=1}^{∞} ΩΣL_{p_{j}}(P^{n}(2))) × Z_{n+1}

for n ≥ 3, where {p_{j}} is the set of odd primes.

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