ON COMBINATORIAL DESCRIPTIONS OF HOMOTOPY GROUPS OF ΣK(π, 1)

JIE WU

Abstract. We give a combinatorial description of homotopy groups of ΣK(π, 1). In particular, all of the homotopy groups of the 3-sphere are combinatorially given.

1. Introduction

In this article, we study homotopy groups and some related problems by using simplicial homotopy theory. The point of view here is that combinatorial aspects of group theory provide further information about homotopy groups.

The homotopy groups of the 3-sphere, the suspension of K(π, 1) and wedges of 2-spheres are shown to be the centers of certain groups with specific generators and specific relations. We list two group theoretical descriptions of π_{∗}(S^{3}) as follows.

Definition 1.1. A bracket arrangement of weight n in a group is a set of elements defined recursively as follows:

Let G be a group and let a_{1}, a_{2}, ⋯, a_{n} be a finite sequence of elements of G. Let

β^{1}(a_{1}) = a_{1}

and if n > 1, then let

β^{n}(a_{1}, ⋯, a_{n}) = [β^{k}(a_{1}, ⋯, a_{k}), β^{l}(a_{k+1}, ⋯, a_{n})]

for some k and l > 0.

Theorem 1.2. For n ≥ 1, π_{n+2}(S^{3}) is isomorphic to the center of the group with generators y_{0}, …, y_{n} and relations

[y_{i_{1}}^{ϵ_{1}}, y_{i_{2}}^{ϵ_{2}}, …, y_{i_{t}}^{ϵ_{t}}]

with {i_{1}, …, i_{t}} = {−1, 0, …, n} as sets in which the indices i_{j} can be repeated, where ϵ_{j} = ±1, y_{−1} = (y_{0} … y_{n})^{−1} and the commutator bracket […] runs over all bracket arrangements of weight t for each t.

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

Notations 1.3. Let G be a group and let S be a subset of G. Let < S > denote the normal subgroup generated by S. Let H_{j} be a sequence of subgroups of G for 1 ≤ j ≤ k. Let [[H_{1}, …, H_{k}]] denote the subgroup of G generated by all of the commutators [h_{i_{1}}^{(1)}, …, h_{i_{t}}^{(t)}] with {i_{1}, …, i_{t}} = {1, …, k} as sets in which the indices i_{j}can be repeated, where h_{j}^{(s)} ∈ H_{j} and the commutator bracket […] runs over all of the bracket arrangements of weight t for each t.

Theorem 1.4. Let n ≥ 1. In the free group F (y_{0}, …, y_{n}) freely generated by y_{0}, …, y_{n} we have

([[<y_{0} >, <y_{1} >, …, <y_{n} >]]∩ <y_{−1} >)/[[<y_{−1} >, <y_{0} >, …, <y_{n} >]] ≅ π_{n+2}(S^{3})

where y_{−1} = (y_{0} … y_{n})^{−1}.

The method of the proofs of these theorems is to study the Moore chain complex of Milnor’s construction F (S^{1}) for the 1-sphere S^{1}. A group theoretical description of the homotopy groups π_{∗}(ΣK(π, 1)) is as follows.

Theorem 1.5. Let π be any group and let {x^{(α)}|α ∈ J} be a set of generators for π. Then, for n ≠ 1, π_{n+2}(ΣK(π, 1)) is isomorphic to the center of the quotient group of the free product

∐^{groups}_{0≤j≤n}(π)_{j}

modulo the relations

[y_{i_{1}}^{(α_{1})ϵ_{1}}, y_{i_{2}}^{(α_{2})ϵ_{2}}, ⋯, y_{i_{t}}^{(α_{t})ϵ_{t}}]

with {i_{1}, i_{2}, ⋯, i_{t}} = {−1, 0, 1, ⋯, n} as sets, where (π)_{j} is a copy of π with generators {x_{j}^{(α)}|α ∈ J}, ϵ_{j} = ±1, y_{−1}^{(α)} = x_{0}^{(α)}^{−1}, y_{j} = x_{j}^{(α)}x_{j+1}^{(α)}^{−1} for 1 ≤ j ≤ n − 1, y_{n}^{(α)} = x_{m}^{(α)} and the commutator bracket [⋯] runs over all of the commutator bracket arrangment of weight t for each t.

In particular, for π = Z/m, we have

Corollary 1.6. π_{n+2}(ΣK(Z/m, 1)) is isomorphic to the center of the group with generators x_{0}, …, x_{n} and relations: (1). x_{j}^{m} for 0 ≤ j ≤ n and (2).

[y_{i_{1}}^{ϵ_{1}}, y_{i_{2}}^{ϵ_{2}}, …, y_{i_{t}}^{ϵ_{t}}]

with {i_{1}, …, i_{t}} = {−1, 0, …, n} as sets in which the indices i_{j} can be repeated, where ϵ_{j} = ±1, y_{−1} = x_{0}^{−1}, y_{j} = x_{j−1}x_{j}^{−1} for −1 ≤ j ≤ n − 1, y_{n} = x_{n} and the commutator bracket […] runs over all bracket arrangements of weight t for each t.

One of the features of Theorems 1.2 and 1.5 is that homotopy groups embed in certain ’enveloping groups’. These ’enveloping groups’ have systematic and uniform structure. The centers of these groups are of course more complicated to analyze. Theorem 1.4 is very similar to the combinatorial decription of J. H. C. Whitehead’s conjecture [see, Bo2, pp.317]. These decriptions give a combinatorial question how to give a computable way to understand the quotient groups R ∩ S/[R, S] for certain subgroups R and S of a finite generated free group. Few informations about this question are known [see, Bo1]. The article is organized as follows.

In Section 2, we study some general properties of simplicial groups. Central extensions in the Moore-Postnikov systems will be considered. In Section 3, we study the intersection of certain subgroups in free groups. The proofs of Theorems 1.2 and 1.4 is given in Section 4, where Theorem 4.5 is Theorem 1.2 and Theorem 4.10 is Theorem 1.4. The proof of Theorem 1.5 is given in Section 5, where Theorem 5.9 is Theorem 1.5. In Section 6, we give some applications of our descriptions. One example is to compute the homotopy groups of the Cohen’s construction on the 1-sphere. A direct corollary is to give a short proof of the Milnor’s counter example for the minimal simplicial groups.

The author would like to thank F. Cohen, B. Gray, J. Harper, D. Kan, M. Mahowald, J. Moore, P. May, S. Priddy and many other mathematicians for their kindly encouragement and helpful suggestions. The author is indebted to helpful discussions with J. C. Moore and F. R. Cohen.

2. Central Extensions in simplicial group theory

In this section, we study some general properties of simplicial groups. A simplicial set K is called a Kan complex if it satisfies the extension condition,i.e, for any simplicial map f : Λ^{k}[n] → K has an extension g : Δ[n] → K, where Δ[n] is the standard n simplex, Λ^{k}[n] is the subcomplex of Δ[n] generated by all d_{i}(σ_{n}) for i ≠ k and σ_{n} is the nondegenerate n simplex in Δ[n],[K1, C2]. Recall that any simplicial group is a Kan complex [Mo1]. Given a simplicial group G, the Moore chain complex (NG, d_{0}) is defined by NG_{n} = ⋂_{j≠0} Kerd_{j} with d_{0} : NG_{n} → NG_{n−1}. The classical Moore Theorem is that π_{n}(G) ≅ H_{n}(NG) [Mo1, Theorem 4;C2, Theorem 3.7 or K2, Proposition 5.4]. Now let Z_{n} = Z_{n}(G) = ⋂_{j} Kerd_{j} denote the cycles and let B_{n} = B_{n}(G) = d_{0}(NG_{n+1}) denote the boundaries. It is easy to check that B_{n} is a normal subgroup of G_{n} for any simplicial group G.

Lemma 2.1. Let G be a simplicial group. Then the homotopy group π_{n}(G) is contained in the center of G_{n}/BG_{n} for n ≥ 1.

Proof. Notice that π_{n}(G) ≅ ZG_{n}/BG_{n}. It suffices to show that the commutator [x, y] ∈ BG_{n} for any x ∈ ZG_{n} and y ∈ G_{n}.

Notice that x is a cycle. There is a simplicial map f_{x} : S^{n} → G such that f_{x}(σ_{n}) = x, where S^{n} is the standard n-sphere with a nondegenerate n-simplex σ_{n}. Now let the simplicial map f_{y} : Δ[n] → G be the representative of y, i.e, f_{y}(τ_{n}) = y for the nondegenerate n-simplex τ_{n}. Let ϕ be the composite

S^{n} →^{j} S^{n} ∧ Δ[n] →^{[f_{x},f_{y}]} G,

where j(σ_{n}) = σ_{n} ∧ τ_{n} and [f_{x}, f_{y}](a ∧ b) = [f_{x}(a), f_{y}(b)], the commutator of f_{x}(a) and f_{y}(b). Notice that ϕ(σ_{n}) = [x, y] and S^{n} ∧ Δ[n] is contactible. Thus [x, y] ∈ B_{n}. The assertion follows.

Definitions 2.2. Let G be a simplicial group. The subsimplicial group R_{n}G is defined by setting

(R_{n}G)_{q} = {x ∈ G_{q}|f_{x}(Δ[q])^{[n]} = 1},

where f_{x} is the representative of x and X^{[n]} is the n-skeleton of the simplicial set X. The subsimplicial group \overline{R}_{n}G is defined by setting

(\overline{R}_{n}G)_{q} = \{x ∈ G_{q}|f_{x}((Δ[q])_{n}) ⊆ BG_{n}\}.

Let P_{n}G denote G/R_{n}G and let \overline{P}_{n}G denote G/\overline{R}_{n}G.

It is easy to check that both R_{n}G and \overline{R}_{n}G are normal subsimplicial group of G. Thus both P_{n}G and \overline{P}_{n}G are quotient simplicial group of G. Notice that

R_{n}G ⊆ \overline{R}_{n}G ⊆ R_{n−1}G.

The quotient simplicial group \overline{P}_{n}G is between P_{n}G and P_{n−1}G. By checking the definition of Moore-Postnikov systems of a simplicial set [Mo1;C2], we have

Lemma 2.3. The quotient simplicial group P_{n}G is the standard n-th Moore-Postnikov system of the simplicial group G.

The quotient simplicial group \overline{P}_{n}G has the same homotopy type of P_{n}G.

Proposition 2.4. The quotient simplicial homomorphism q_{n} : P_{n}G → \overline{P}_{n}G is a homotopy equivalence for each n.

Proof: The Moore chain complex of P_{n}G is as follows:

N(P_{n}G)_{q} = \{ \begin{matrix}1 & for q > n + 1, \\ NG_{n+1}/ZG_{n+1} & for q = n + 1, \\ NG_{q} & for q < n + 1. \end{matrix}

The Moore chain complex of \overline{P}_{n}G is as follows:

N(\overline{P}_{n}G)_{q} = \{ \begin{matrix}1 & for q > n, \\ NG_{n}/BG_{n} & for q = n, \\ NG_{q} & for q < n. \end{matrix}

Thus (q_{n})_{∗} : π_{∗}(P_{n}G) → π_{∗}(\overline{P}_{n}G) is an isomorphism and the assertion follows.

Let \overline{F}_{n} denote the kernel of the quotient simplicial homomorphism r_{n} : \overline{P}_{n}G → P_{n−1}G for n > 0.

Proposition 2.5. The simplicial group \overline{F}_{n} is the minimal simplicial group K(π_{n}G, n) for each n > 0.

Proof: It is directly to check that

N(\overline{F}_{n}G)_{q} = \{ \begin{matrix}1 & for q ≠ n, \\ π_{n}G & for q = n. \end{matrix}

The assertion follows.

Proposition 2.6. The short exact sequence of simplicial groups

0 → \overline{F}_{n}G → \overline{P}_{n}G → P_{n−1}G → 0

is a central extension for each n > 0.

Proof: Consider the relative commutator sussimplicial group [\overline{F}_{n}G, \overline{P}_{n}G]. By Lemma 2.1, [\overline{F}_{n}G, \overline{P}_{n}G]_{n} = 1. Notice that [\overline{F}_{n}G, \overline{P}_{n}G] is a subsimplicial group of minimal simplicial group \overline{F}_{n}G ≅ K(π_{n}G, n). Thus [\overline{F}_{n}G, \overline{P}_{n}G] = 1 and the assertion follows.

Definition 2.7. A simplicial group is said to be r-centerless if the center Z(G_{n}) = {1} for n ≥ r.

Proposition 2.8. Let G be a reduced r-centerless simplicial group. Then π_{n}(G) ≅ Z(G_{n}/B_{n}) for n ≥ r + 1

Proof. By Lemma 2.1, Z_{n}/B_{n} ⊆ Z(G_{n}/B_{n}). It suffices to show that Z(G_{n}/B_{n}) ⊆ Z_{n}/B_{n} for n ≥ r + 1. Now let x̃ ∈ Z(G_{n}/B_{n}). Choose x ∈ G_{n} with p(x) = x̃, where p : G_{n} → G_{n}/B_{n} is the quotient homomorphism. To check x̃ ∈ Z_{n}/B_{n}, it suffices to show that x ∈ Z_{n} or d_{j}x = 1 for all j. Since Z(G_{n−1}) = {1}, d_{j}x = 1 if and only if [d_{j}x, y] = 1 for all y ∈ G_{n−1}. Now [d_{j}x, y] = d_{j}[x, s_{j−1}y] for j > 0 and [d_{0}x, y] = d_{0}[x, s_{0}y]. Since x̃ ∈ Z(G_{n}/B_{n}), [x, z] ∈ B_{n} ⊆ Z_{n} for all z ∈ G_{n} and therefore [d_{j}x, y] = 1 for all y ∈ G_{n−1}. The assertion follows.

By inspecting the proof, we also have

Proposition 2.9. Let G be a reduced r-centerless simplicial group. Then

Z(G_{n}/Z_{n}) = {1}

for n ≥ r + 1.

Lemma 2.10. Let G be a reduced simplicial group so that G_{n} is cyclic or centerless for each n. Then there exists a unique integer γ_{G} > 0 so that G_{n} = {1} for n < γ_{G} and Z(G_{n}) = {1} for n > γ_{G}.

Proof. Let γ_{G} = max{γ|G_{n} = {1} for n < γ}. Then γ_{G} > 0. If γ_{G} < ∞. Then G_{γ_{G}} ≠ {1}. We show that G_{γ_{G}+q} is centerless for each q > 0. Notice that d_{0}^{q} ∘ s_{0}^{q} : G_{n} → G_{n+q} → G_{n} and d_{1}^{q} ∘ s_{1}^{q} : G_{n} → G_{n+q} → G_{n} are identities for n > 0 Thus s_{0}^{q}(G_{γ_{G}}) and s_{1}^{q}(G_{γ_{G}}) are nontrivial summands of G_{γ_{G}+q}. Now let x = s_{0}^{q}y = s_{1}^{q}z ∈ s_{0}^{q}(G_{γ_{G}}) ∩ s_{1}^{q}(G_{γ_{G}}). Then d_{q+1}x = d_{q+1}s_{0}^{q}y = s_{0}^{q}d_{1}y = 1 = d_{q+1}s_{1}^{q}z = s_{1}^{q−1}d_{2}s_{1}z = s_{1}^{q−1}z. Thus x = s_{1}^{q}z = 1. And therefore s_{0}^{q}(G_{γ_{G}}) ∩ s_{1}^{q}(G_{γ_{G}}) = {1}. The assertion follows.

Corollary 2.11. Let G be a reduced simplicial group such that G_{n} is cyclic or centerless for each n. Then π_{n}(G) ≅ Z(G_{n}/B_{n}) for n ≠ γ_{G} + 1, where γ_{G} is defined as above.

Notice that, for any free group F, rank(F) ≥ 2 ⇔ Z(F) = {1} and F ≠ {1}. We have

Lemma 2.12. Let G be a reduced simplicial group such that G_{n} is a free group for each n. Then there exits a unique integer γ_{G} > 0 so that G_{n} = {1} for n < γ_{G} and rank(G_{n}) ≥ 2 for n > γ_{G}.

Example 2.13. The 1-stem is determined in this example.

Let G = F (S^{n}), Milnor’s F -construction on the standard n-sphere for n ≥ 1. Then G_{n} ≅ F (σ) ≅ Z(σ), the free abelian group generated by σ,

G_{n+1} ≅ F (s_{0}σ, s_{1}σ, …, s_{n}σ)

and G_{n+2} ≅ F (s_{i}s_{j}σ|0 ≤ j < i ≤ n). It is easy to check that Γ^{2}G_{n+1} = Z_{n+1}, where Γ^{q}G is the q-th term in the lower central series of a group G starting with Γ^{1}G = G. By Lemma 2.1,

Γ^{3}G_{n+1} = [Z_{n+1}, G_{n+1}] ⊆ B_{n+1}.

If n = 1, then it will be shown that Γ^{3}G_{n+1} = B_{n+1} in Section 4 and therefore π_{3}(S^{2}) ≅ π_{2}(F (S^{1})) ≅ Z, which is generated by [s_{0}σ, s_{1}σ].

Suppose that n > 1. Consider the following equations

d_{k}([s_{j−1}s_{i}σ, s_{j+1}s_{j}σ]) = \{ \begin{matrix}1 & for k ≠ j, \\ [s_{i}σ, s_{j}σ] & for k=j, \end{matrix}

for i + 1 < j ≤ n,

d_{k}[s_{i+2}s_{i+1}σ, s_{i+3}s_{i}σ] = \{ \begin{matrix}[s_{i+1}σ, s_{i+2}σ] & for k=i+1, \\ [s_{i+1}σ, s_{i}σ] & for k=i+3, \\ 1 & otherwise, \end{matrix}

and

d_{k}[s_{i+2}s_{i}σ, s_{i+3}s_{i+1}σ] = \{ \begin{matrix}[s_{i+1}σ, s_{i+2}σ] & for k=i+1, \\ [s_{i}σ, s_{i+2}σ] & for k=i+2, \\ [s_{i}σ, s_{i+1}σ] & for k=i+3, \\ 1 & otherwise. \end{matrix}

By the Homotopy Addition Theorem [C2, Theorem 2.4], [s_{i}σ, s_{j}σ] ∈ B_{n+1} for i + 1 < j, [s_{i+1}σ, s_{i+2}σ] ≡ [s_{i}σ, s_{i+1}σ] mod B_{n+1} and 0 ≡ [s_{i}σ, s_{i+2}σ] ≡ [s_{i+1}σ, s_{i+2}σ] ≡ 2[s_{i}σ, s_{i+1}σ] if i + 2 ≤ n. Notice that [s_{0}σ, s_{1}σ] ≠ 0 in π_{n+1}(Γ^{2}G/Γ^{3}G). Thus [s_{0}σ, s_{1}σ] ∉ B_{n+1} and, by the relations above,

π_{n+2}(S^{n+1}) ≅ π_{n+1}(G) ≅ Z_{n+1}/B_{n+1} ≅ Z/2

for n ≥ 2, which can be represented by [s_{i}σ, s_{i+1}σ] for 0 ≤ i ≤ n − 1.

3. Intersections of certain subgroups in free groups

In this section, we give some group theory preliminary. The intersections of certain subgroups in free groups are considered in this section. We will use these informations to determine the Moore chain complex of certain simplicial groups in the next sections.

Definition 3.1. let S be a set and let T ⊆ S a subset. The projection homomorphism

π : F (S) → F (T)

is defined by

π(x) = \{ \begin{matrix}x & x ∈ T, \\ 1 & x ∉ T. \end{matrix}

Now let π : F (S) → F (T) be a projection homomorphism and let R equal the kernel of π. Define the subsets of the free group F (S) as follows.

A_{T} (k) = {[[x, y_{1}^{ϵ_{1}}] ⋯], y_{t}^{ϵ_{t}}]|0 ≤ t ≤ k, ϵ_{j} = ±1, y = y_{1}^{ϵ_{1}} … y_{t}^{ϵ_{t}} ∈ F (T), x ∈ S − T },

where y = y_{1}^{ϵ_{1}} ⋯ y_{t}^{ϵ_{t}} ∈ F (T) runs over reduced words in F (T) with t ≤ k and y_{j} ∈ T. Furthermore define [[x, y_{1}^{ϵ_{1}}] …], y_{t}^{ϵ_{t}}] = x for t = 0. Define

B_{T} (k) = {ϕ^{−1}xϕ|ϕ ∈ F (T) a reduced word with lenth l(ϕ) ≤ k, x ∈ S − T },

A_{T} = ∪_{k≥0}A_{T} (k)

and

B_{T} = ∪_{k≥0}B_{T} (k).

By the classical Kurosch-Schreier theorem ( see [MKS, pp.243, K2, Theorem 18.1]), we have

Proposition 3.2. The subgroup R is a free group freely generated by B_{T}.

We will show that A_{T} is also a set of free generators for R. We need a lemma.

Lemma 3.3. Let ϕ : F_{1} → F_{2} be a homomorphism of free groups. Suppose that ϕ^{ab} : F_{1}^{ab} → F_{2}^{ab} is an isomorphism, where F^{ab} is the abelianlizer of the group F. Then ϕ : F_{1} → F_{2} is a monomorphism.

Proof. Notice that ϕ_{∗} : H_{∗}(F_{1}) → H_{∗}(F_{2}) is an isomorphism, where H_{∗}(G) is the homology of the group G. Thus F_{1}/Γ^{r}F_{1} → F_{2}/Γ^{r}F_{2} is an isomorphism for each r, where Γ^{r}G is the r-th term in the lower central seris of the group G starting with Γ^{1}G = G and so

lim_{r}F_{1}/Γ^{r}F_{1} → lim_{r}F_{2}/Γ^{r}F_{2}

is an isomorphism. Notice that ∩_{r}Γ^{r}F = 1 for any free group F. Thus F → lim_{r}F/Γ^{r}F is a monomorphism. The assertion follows.

Proposition 3.4. The subgroup R is a free group freely generated by A_{T}.

Proof. First we assume that both S and T are finite sets. Denote by i_{k} : A_{T} (k) → R and j_{k} : B_{T} (k) → R the natural inclusions. Notice that

R = F (B_{T}) = colim_{k}F (B_{T} (k)).

We set up the following steps.

Step1. A_{T} (k) ⊆ F (B_{T} (k)).

The proof of this statement is given by induction on k starting with A_{T} (0) = B_{T} (0) = S−T. Suppose that A_{T} (k−1) ⊆ F (B_{T} (k−1)) and let w = [[x, y_{1}^{ϵ_{1}}, ⋯, y_{t}^{ϵ_{t}}] ∈ A_{T} (k). If t < k, then w ∈ A_{T} (k − 1) ⊆ F (B_{T} (k − 1)) ⊆ F (B_{T} (k)), by induction. Now

[[x, y_{1}^{ϵ_{1}}] ⋯], y_{k}^{ϵ_{k}}] = [[x, y_{1}^{ϵ_{1}}] ⋯], y_{k−1}^{ϵ_{k−1}}]^{−1} · y_{k}^{−ϵ_{k}}[[x, y_{1}^{ϵ_{1}}] ⋯], y_{k−1}^{ϵ_{k−1}}]y_{k}^{ϵ_{k}}

Now since [[x, y_{1}^{ϵ_{1}}] ⋯], y_{k−1}^{ϵ_{k−1}}] ∈ F (B_{T} (k − 1)), [[x, y_{1}^{ϵ_{1}}] ⋯], y_{k−1}^{ϵ_{k−1}}] = ∏_{j=1}^{s}(ϕ_{j}^{−1}x_{j}ϕ_{j}))^{η_{j}} with ϕ_{j}^{−1}x_{j}ϕ_{j} ∈ B_{T} (k − 1) and η_{j} = ±1. Thus

w = (∏_{j=1}^{s}(ϕ_{j}^{−1}x_{j}ϕ_{j}))^{η_{j}})^{−1} · ∏_{j=1}^{s}(y_{k}^{−ϵ_{k}}ϕ_{j}^{−1}x_{j}ϕ_{j}y_{k}^{ϵ_{k}}))^{η_{j}} ∈ F (B_{T} (k))

The induction is finished.

Step 2. ĩ_{k} : F (A_{T} (k)) → F (B_{T} (k)) is an epimorphism, where the homomorphism ĩ_{k} is induced by the inclusion i_{k} : A_{T} (k) → F (B_{T} (k))

The proof of this step is given induction on k starting with F (A_{T} (0)) = F (B_{T} (0)) = F (S−T). Suppose that F (A_{T} (k−1)) → F (B_{T} (k−1)) is an epimorphism and consider ĩ_{k} : F (A_{T} (k)) → F (B_{T} (k)). Let ϕ^{−1}xϕ ∈ B_{T} (k), where ϕ = y_{1}^{ϵ_{1}} ⋯ y_{t}^{ϵ_{t}} is a reduced word with t ≤ k. If t ≤ k − 1, then ϕ^{−1}xϕ ∈ Imφ_{k} by induction. Let ϕ = y_{1}^{ϵ_{1}} ⋯ y_{k}^{ϵ_{k}} be a reduced word and let z denote the word (y_{1}^{ϵ_{1}} ⋯ y_{k−1}^{ϵ_{k−1}})^{−1}xy_{1}^{ϵ_{1}} ⋯ y_{k−1}^{ϵ_{k−1}}. Then ϕ^{−1}xϕ = z · [z, y_{k}^{ϵ_{k}}]. Notice that z ∈ Im(ĩ_{k}) by induction. It suffices to show that [w, y^{ϵ}] ∈ F (A_{T} (k)) for w ∈ A_{T} (k − 1) for all w ∈ A_{T} (k − 1), y ∈ T and ϵ = ±1 by the Witt-Hall identity that

[ab, c] = [a, c] · [[a, c], b] · [b, c].

We show this by second induction starting with

A_{T} (1) = {[x, y^{ϵ}]|y ∈ T, x ∈ S − T, ϵ = ±1}.

Let w = [[x, y_{1}^{ϵ_{1}}], ⋯], y_{t}^{ϵ_{t}}] be in A_{T} (k − 1) with k > 1, where y_{1}^{ϵ_{1}} ⋯ y_{t}^{ϵ_{t}} is a reduced word. Let y ∈ T and let ϵ = ±1. If y_{1}^{ϵ_{1}} ⋯ y_{t}^{ϵ_{t}}y^{ϵ} is a reduced word, then [w, y^{ϵ}] ∈ F (A_{T} (k)) by definition. Suppose that y_{1}^{ϵ_{1}} ⋯ y_{t}^{ϵ_{t}}y^{ϵ} is not a reduced word. Then t > 0, y = y_{t} and ϵ = −ϵ_{t}. Let w′ denote [[x, y_{1}^{ϵ_{1}}], ⋯,], y_{t−1}^{ϵ_{t−1}}] ∈ A_{T} (k − 2). Then w = [w′, y_{t}^{ϵ_{t}}]. By the Witt-Hall identities, there is an equation

1 = [w′, y_{t}^{−ϵ_{t}}] · w · [w, y_{t}^{−ϵ_{t}}].

By induction, [w′, y_{t}^{−ϵ_{t}}] ∈ F (A_{T} (k − 1)) ⊆ F (A_{T} (k)) and so

[w, y_{t}^{−ϵ_{t}}] = w^{−1}[w′, y_{t}^{−ϵ_{t}}]^{−1} ∈ F (A_{T} (k − 1)) ⊆ F (A_{T} (k)).

The second induction is finished and so the first induction is finished. The assertion follows.

Step 3. ĩ_{k} : F (A_{T} (k)) → F (B_{T} (k)) is an isomorphism.

By Step 2, Z(A_{T} (k)) → Z(B_{T} (k)) is an epimorphism. Notice that

rank(F (B_{T} (k))) = |B_{T} (k)| = |A_{T} (k)| = rank(F (A_{T} (k))).

Thus ĩ_{k} : Z(A_{T} (k)) → Z(B_{T} (k)) is an isomorphism and so ĩ_{k} : F (A_{T} (k)) → F (B_{T} (k)) is a monomorphism. Thus ĩ_{k} is an isomorphism.

Step 4. Since F (A_{T} (k)) → F (B_{T} (k)) is an isomorphism for each k, F (A_{T}) = colim_{k}F (A_{T} (k)) → F (B_{T}) = colim_{k}F (B_{T} (k)) is an isomorphism.

Now consider the general case. By Lemma 3.6, it suffices to show that ĩ: F (A_{T}) → F (B_{T}) is an isomorphism. To check that F (A_{T}) → F (B_{T}) is an epimorphism. Let w ∈ B_{T}, there exist finite subsets S′ and T′ of S and T, respectively, so that w ∈ B_{T′}. By the special case as above, ĩ|_{F (A_{T′})} : F (A_{T′}) → F (B_{T′}) is an isomorphism and w ∈ Imĩ|_{F (A_{T′})}. Thus ĩ is an epimorphism. To check that F (A_{T}) → F (B_{T}) is a monomorphism. Let w ∈ Kerĩ, there exist finite subsets S′ and T′ of S and T, respectively, so that w ∈ F (A_{T′}). Notice that ĩ|_{F (A_{T′})} is an isomorphism. Thus w = 1 and the assertion follows.

Now let’s consider the intersection of kernels of projection homomorphisms. Let S be a set and let T_{j} be a subset of S for 1 ≤ j ≤ k. Let π_{j} : F (S) → F (T_{j}) be the projection homomorphism for 1 ≤ j ≤ k. We construct a subset A(T_{1}, ⋯, T_{k}) of F (S) by induction on k as follows.

A(T_{1}) = A_{T_{1}},

where A_{T} is defined in Definition3.1. Let

T_{2}^{(2)} = {w ∈ A(T_{1})|w = [[x, y_{1}^{ϵ_{1}}, ⋯, y_{t}^{ϵ_{t}}] with x, y_{j} ∈ T_{2} for all j}

and define

A(T_{1}, T_{2}) = A(T_{1})_{T_{2}^{(2)}}.

Suppose that A(T_{1}, T_{2}, ⋯, T_{k−1}) is well defined so that all of the elements in A(T_{1}, T_{2}, ⋯, T_{k−1}) are written down as certain commutators in F (S) in terms of elements in S. Let T_{k}^{(k)} be the subset of A(T_{1}, T_{2}, ⋯, T_{k−1}) defined by

T_{k}^{(k)} = {w ∈ A(T_{1}, T_{2}, ⋯, T_{k−1})|w = [x_{1}^{ϵ_{1}}, ⋯, x_{l}^{ϵ_{l}}] with x_{j} ∈ T_{k} for all j},

where [x_{1}^{ϵ_{1}}, ⋯, x_{l}^{ϵ_{l}}] are the elements in A(T_{1}, T_{2}, ⋯, T_{k−1}) which are written down as commutators. Then define

A(T_{1}, T_{2}, ⋯, T_{k}) = A(T_{1}, T_{2}, ⋯, T_{k−1})_{T_{k}^{(k)}}

Theorem 3.5. Let S be a set and let T_{j} be a subset of S for 1 ≤ j ≤ k. Let π_{j} : F (S) → F (T_{j}) be the projection homomorphism for 1 ≤ j ≤ k. Then the intersection ⋂_{j=1}^{k} Kerπ_{j} is a free group freely generated by A(T_{1}, T_{2}, ⋯, T_{k}).

Proof. The proof is given by induction on k. If k = 1, the assertion follows from the above lemma. Suppose that ⋂_{j=1}^{k−1} Kerπ_{j} = F (A(T_{1}, T_{2}, ⋯, T_{k−1})) and consider π_{k} : F (S) → F (T_{k}). Then

⋂_{j=1}^{k} Kerπ_{j} = Ker(π̄_{k} : F (A(T_{1}, T_{2}, ⋯, T_{k−1})) → F (T_{k}))

, where π̄_{k} is π_{k} restricted to the subgroup F (A(T_{1}, T_{2}, ⋯, T_{k−1})). Let

w = [x_{1}^{ϵ_{1}}, ⋯, x_{l}^{ϵ_{l}}] ∈ A(T_{1}, T_{2}, ⋯, T_{k−1}).

If w ∉ T_{k}^{(k)}, then x_{j} ∉ T_{k} for some j and π̄_{k}(w) = 1. Thus π̄_{k} factors through F (T_{k}^{(k)}), i.e, there is a homomorphism j : F (T_{k}^{(k)}) → F (T_{k}) so that π̄_{k} = j ∘ π, where π : F (A(T_{1}, T_{2}, ⋯, T_{k−1})) → F (T_{k}^{(k)}) is the projection homomorphism. We claim that j : F (T_{k}^{(k)}) → F (T_{k}) is a monomorphism. Consider the commutative diagram

F (A(T_{1}, T_{2}, ⋯, T_{k−1})) →^{π̄_{k}} F (T_{k}) ↪  F (S)
↑  ↑ j  ↑
F (T_{k}^{(k)})  →^{=} F (T_{k}^{(k)}) ↪ F (A(T_{1}, T_{2}, ⋯, T_{k−1})),

where F (T_{k}^{(k)}) → F (A(T_{1}, T_{2}, ⋯, T_{k−1})) and F (A(T_{1}, T_{2}, ⋯, T_{k−1})) → F (S) are inclusions of subgroups. Thus j : F (T_{k}^{(k)}) → F (T_{k}) is a monomorphism and

Kerπ̄_{k} = Ker(F (A(T_{1}, T_{2}, ⋯, T_{k−1})) → F (T_{k}^{(k)}) = F (A(T_{1}, T_{2}, ⋯, T_{k})).

The assertion follows.

Corollary 3.6. Let π_{j} be the projection homomorphisms as in Theorem 3.5. Then the intersection subgroup ⋂_{j=1}^{k} Kerπ_{j} equals the commutator subgroup [[< T_{1} >, ⋯, < T_{k} >]] which is defined in Notations 1.3.

4. On the Homotopy Groups of the 3-sphere

In this section, we study the Moore chain complex of the Milnor’s construction F (S^{1}). The proofs Theorems 1.2 and 1.4 are given in this section, where Theorem 4.5 is Theorem 1.2 and Theorem 4.10 is Theorem 1.4. Recall that the simplicial 1-sphere S^{1} is a free simplicial set generated by a 1-simplex σ. Thus S_{0}^{1} = {∗}, S_{1}^{1} = {σ, ∗} and S_{n+1}^{1} = {∗, s_{n} ⋯ s_{i+1}s_{i−1} ⋯ s_{0}σ|0 ≤ j ≤ n}. Let x_{i} denote s_{n} ⋯ s_{i+1}s_{i−1} ⋯ s_{0}σ. Then F (S^{1})_{n+1} = F (x_{0}, x_{1}, ⋯, x_{n}) the free group freely generated by x_{0}, ⋯, x_{n}. Let y_{i} denote x_{i−1}x_{i}^{−1} for −1 ≤ i ≤ n, where we put x_{−1} = x_{n+1} = 1 in F (S^{1})_{n+1} = F (x_{0}, ⋯, x_{n}). By direct calculation, we have

Lemma 4.1. F (S^{1})_{n+1} = F (y_{0}, ⋯, y_{n}) with

d_{j}y_{k} = \{ \begin{matrix}y_{k−1} & j ≤ k, \\ 1 & j = k + 1, \\ y_{k} & j > k + 1, \end{matrix}

and

s_{j}y_{k} = \{ \begin{matrix}y_{k+1} & j ≤ k, \\ y_{k}y_{k+1} & j = k + 1,and \\ y_{k} & j > k + 1 \end{matrix}

for 0 ≤ j ≤ n + 1, where y_{−1} = (y_{0} ⋯ y_{n−1})^{−1} in F (S^{1})_{n}.

Now let C_{n+1} denote the subgroup of F (y_{0}, ⋯, y_{n}) generated by all of the commutators [y_{i_{1}}^{ϵ_{1}}, ⋯, y_{i_{t}}^{ϵ_{t}}] with {i_{1}, ⋯, i_{t}} = {0, 1, ⋯, n} as sets, i.e. each j (0 ≤ j ≤ n) appears in the index set {i_{1}, ⋯, i_{t}} at least one time, where ϵ_{j} = ±1 and the commutator [y_{i_{1}}^{ϵ_{1}}, ⋯, y_{i_{t}}^{ϵ_{t}}] runs over all of the commutator bracket arrangements of weight t for y_{i_{1}}^{ϵ_{1}}, ⋯, y_{i_{t}}^{ϵ_{t}}.

Lemma 4.2. The group C_{n+1} is a subgroup of NF (S^{1})_{n+1},i.e C_{n+1} ⊆ ∩_{j≠0}Ker(d_{j}).

Proof. Notice that d_{j}y_{j−1} = 1 for 1 ≤ j ≤ n+1. Since {y_{i_{1}}, ⋯, y_{i_{t}}} = {y_{0}, y_{1}, ⋯, y_{n}}, d_{j}[y_{i_{1}}^{ϵ_{1}}, ⋯, y_{i_{t}}^{ϵ_{t}}] = 1 for each j > 0. The assertion follows.

Theorem 4.3. NF (S^{1})_{n+1} = C_{n+1}.

Proof. For 1 ≤ j ≤ n + 1, let S = {y_{0}, y_{1}, ⋯, y_{n}} and let T_{j} = {y_{0}, ⋯, ŷ_{j} ⋯, y_{n}}. By Lemma 4.1, there is a commutative diagram

F (S)  →^{π_{j}}  F (T_{j−1})
↓ d_{j}  d̄_{j} ↓≅
F (y_{0}, ⋯, y_{n−1}) →^{=} F (y_{0}, ⋯, y_{n−1}),

where

d̄_{j}(y_{k}) = \{ \begin{matrix}y_{k} & 0 ≤ k ≤ j − 2, \\ y_{k−1} & j ≤ k ≤ n. \end{matrix}

Thus Kerd_{j} = Kerπ_{j} and NF (S^{1})_{n+1} = ⋂_{j=1}^{n+1} Kerd_{j} = ⋂_{j=1}^{n+1} Kerπ_{j}. By Theorem 3.5, NF (S^{1})_{n+1} = F (A(T_{0}, T_{1}, ⋯, T_{n})), where the notation A(T_{0}, T_{1}, ⋯, T_{n}) is given in Section 3. To check that

F (A(T_{0}, T_{1}, ⋯, T_{n})) ⊆ C_{n+1},

it suffices to show that A(T_{0}, T_{1}, ⋯, T_{n}) ⊆ C_{n+1}. This will follow from the following statement.

Statement. For each 0 ≤ j ≤ n and w = [y_{i_{1}}^{ϵ_{1}}, y_{i_{2}}^{ϵ_{2}}, ⋯, y_{i_{t}}^{ϵ_{t}}] ∈ A(T_{0}, T_{1}, ⋯, T_{j}), {y_{0}, y_{1}, ⋯, y_{j}} ⊆ {y_{i_{1}}, y_{i_{2}}, ⋯, y_{i_{t}}}.

We show this statement by induction on j. Note that F (T_{0}) = F (y_{1}, ⋯, y_{n}). For j = 0,

A(T_{0}) = {[[y_{0}, y_{i_{1}}^{ϵ_{1}}], ⋯,], y_{i_{t}}^{ϵ_{t}}]|i_{s} > 0, y_{i_{1}}^{ϵ_{1}} ⋯ y_{i_{t}}^{ϵ_{t}} a reduced word in F (T_{0})}.

Thus the assertion holds for j = 0. Suppose that the assertion holds for j − 1 with j ≤ n. Notice that T_{j} = {y_{0}, ⋯, ŷ_{j}, ⋯, y_{n}}. Thus

T_{j}^{(j)} = {w ∈ A(T_{0}, T_{1}, ⋯, T_{j−1})|w = [y_{i_{1}}^{ϵ_{1}}, ⋯, y_{i_{t}}^{ϵ_{t}}] with y_{j} ∉ {y_{i_{1}}, ⋯, y_{i_{t}}}}.

and so y_{j} ∈ {y_{i_{1}}, ⋯, y_{i_{t}}} for w = [y_{i_{1}}^{ϵ_{1}}, ⋯, y_{i_{t}}^{ϵ_{t}}] ∈ A(T_{0}, T_{1}, ⋯, T_{j−1}) − T_{j}^{(j)}. Hence, by induction,

{y_{0}, y_{1}, ⋯, y_{j}} ⊆ {y_{i_{1}}, y_{i_{2}}, ⋯, y_{i_{t}}}

for w = [y_{i_{1}}^{ϵ_{1}}, ⋯, y_{i_{t}}^{ϵ_{t}}] ∈ A(T_{0}, T_{1}, ⋯, T_{j−1}) − T_{j}^{(j)}. Notice that

A(T_{0}, T_{1}, ⋯, T_{j}) = A(T_{0}, T_{1}, ⋯, T_{j−1})_{T_{j}^{(j)}}.

Thus

{y_{0}, y_{1}, ⋯, y_{j}} ⊆ {y_{i_{1}}, y_{i_{2}}, ⋯, y_{i_{t}}}

for each w = [y_{i_{1}}^{ϵ_{1}}, ⋯, y_{i_{t}}^{ϵ_{t}}] ∈ A(T_{0}, T_{1}, ⋯, T_{j}). The induction is finished and the theorem follows.

Corollary 4.4. A(T_{0}, T_{1}, ⋯, T_{n}) is a set of free generators for NF (S^{1})_{n+1}.

Theorem 4.5 (Theorem 1.2). For n ≥ 1, π_{n+2}(S^{3}) is isomorphic to the center of the group with generators y_{0}, y_{1}, ⋯, y_{n} and relations [y_{i_{1}}^{ϵ_{1}}, y_{i_{2}}^{ϵ_{2}}, ⋯, y_{i_{t}}^{ϵ_{t}}] with

{i_{1}, i_{2}, ⋯, i_{t}} = {−1, 0, 1, ⋯, n}

as sets, where ϵ_{j} = ±1,y_{−1} = (y_{0}y_{1} ⋯ y_{n})^{−1} and the commutator bracket [⋯] runs over all bracket arrangements of weight t for each t.

Proof. Notice that π_{n+2}(S^{3}) ≅ π_{n+1}F (S^{1}) for n ≥ 1 and B_{n+1} = d_{0}(NF (S^{1})_{n+2}). By Lemma 4.1 and Theorem 4.3, B_{n+1} is generated by [y_{i_{1}}^{ϵ_{1}}, y_{i_{2}}^{ϵ_{2}}, ⋯, y_{i_{t}}^{ϵ_{t}}] so that {y_{i_{1}}, y_{i_{2}}, ⋯, y_{i_{t}}} = {y_{0}, y_{1}, ⋯, y_{n}}, where ϵ_{j} = ±1. By Proposition 2.8, it suffices to check that π_{2}(F (S^{1})) ≅ Z(F (S^{1})_{2}/B_{2}) = Z(F (y_{0}, y_{1})/B_{2}). By Example 2.13, Z_{2} = Γ^{2}F (S^{1})_{2} = Γ^{2}F (y_{0}, y_{1}) and Γ^{3}F (y_{0}, y_{1}) ⊆ B_{2}. Now , by the construction of B_{2}, B_{2} ⊆ Γ^{3}F (y_{0}, y_{1}) and therefore

π_{2}(F (S^{1})) ≅ Z_{2}/B_{2} = Γ^{2}F (y_{0}, y_{1})/Γ^{3}F (y_{0}, y_{1}) = Z(F (y_{0}, y_{1})/B_{2}).

The assertion follows.

Remark 4.6. The relations in above theorem are not minimal, i.e, many of them can be cancelled out.

Proposition 4.7. Let C′_{n+1} be the subgroup of F (S^{1})_{n+1} generated by all commutators given by [⋯ [y_{i_{1}}^{ϵ_{1}}, y_{i_{2}}^{ϵ_{2}}], ⋯,], y_{i_{t}}^{ϵ_{t}}] with {i_{1}, ⋯, i_{t}} = {0, 1, ⋯, n} and ϵ_{j} = ±1. Then

NF (S^{1})_{n+1}/Γ^{s} ∩ NF (S^{1})_{n+1} ≅ C′_{n+1}/Γ^{s} ∩ C′_{n+1}

for each s, where Γ^{s} = Γ^{s}F (S^{1})_{n+1} is the s-term in the lower central series of F (S^{1})_{n+1}.

Proof. Notice that C′_{n+1} ⊆ NF (S^{1})_{n+1}. The induced homomorphism f_{s} : C′_{n+1}/Γ^{s} ∩ C′_{n+1} → NF (S^{1})_{n+1}/Γ^{s} ∩ NF (S^{1})_{n+1} is a monomorphism. We check that f is an epimorphism. It suffices to show that, for each w ∈ NF (S^{1}), there exists a sequence of elements {x_{j}} so that x_{j} ∈ Γ^{j} ∩ C′_{n+1} and wx_{1}x_{2} ⋯ x_{s} ∈ Γ^{s+1} for each s. In fact, if this statement holds, wx_{1}x_{2} ⋯ x_{s−1} ≡ 1 mod Γ^{s} ∩ NF (S^{1})_{n+1} for each s and w = (wx_{1} ⋯ x_{s−1}) · (x_{1}x_{2} ⋯ x_{j−1})^{−1} ∈ C′_{n+1} mod Γ^{s}. Now we construct x_{j} by induction, which depends on w. Notice that, for n ≥ 1, NF (S^{1})_{n+1} ∈ Γ^{n+1} ⊆ Γ^{2}. Choose x_{j} = 1 for j ≤ n. Suppose that there are x_{1}, ⋯, x_{s−1} so that x_{j} ∈ Γ^{j} ∩ C′_{n+1} and wx_{1} ⋯ x_{s−1} ∈ Γ^{s}. Since C′_{n+1} ⊆ NF (S^{1})_{n+1}, wx_{1} ⋯ x_{s−1} ∈ Γ^{s} ∩ NF (S^{1})_{n+1} and d_{j}(wx_{1} ⋯ x_{s−1}) = 1 for j > 1. Let π : Γ^{s} → Γ^{s}/Γ^{s+1} be the quotient homomorphism. π(wx_{1} ⋯ x_{s−1}) is a linear combination of basic Lie products. We claim that π(wx_{1} ⋯ x_{s−1}) is a linear combination of basic Lie products in which each y_{j} appears in the Lie product for 0 ≤ j ≤ n. If not, there exists j so that π(wx_{1} ⋯ x_{s−1}) = b+c, where b is a nontrivial linear combination of basic Lie products in which y_{j} does not appear and c is a linear combination of basic Lie products in which y_{j} appears. Now the face homomorphism d_{j+1} : F (y_{0}, ⋯, y_{n}) → F (y_{0}, ⋯, y_{n−1}) induces a homomorphism

d̄_{j+1} : Γ^{s}F (y_{0}, ⋯, y_{n})/Γ^{s+1}F (y_{0}, ⋯, y_{n}) → Γ^{s}F (y_{0}, ⋯, y_{n−1})/Γ^{s+1}F (y_{0}, ⋯, y_{n−1})

and

1 = d̄_{j+1}π(wx_{1} ⋯ x_{s−1}) = d̄_{j+1}(b) + d̄_{j+1}(c) = d̄_{j+1}(b).

Notice that

d_{j+1}|_{F (y_{0},··· ,y_{j−1},y_{j+1},··· ,y_{n})} : F (y_{0}, ⋯, y_{j−1}, y_{j+1}, ⋯, y_{n}) → F (y_{0}, ⋯, y_{n−1})

is an isomorphism. Thus b = 1. This contradicts to that b is a nontrivial linear combination of a basis. Thus π(wx_{1} ⋯ x_{s−1}) is a linear combination of basic Lie products in which all of the y_{j} appear. By Theorem 5.12 in [MKS,pp.337], there exists x_{s} in C′_{n+1} so that π(wx_{1} ⋯ x_{s}) = 1, or wx_{1} ⋯ x_{s} ∈ Γ^{s+1}. The induction is finished now.

By Corollary 3.6, we have

Theorem 4.8. In the free group F (y_{0}, ⋯, y_{n}), NF (S^{1})_{n+1} = [[< y_{0} >, ⋯, < y_{n} >]] and BF (S^{1})_{n+1} = [[< y_{−1} >, < y_{0} >, ⋯, < y_{n} >]].

Thus Theorem 4.5 can be rewritten as follows.

Theorem 4.9. In the free group F (y_{0}, ⋯, y_{n}) for n ≥ 1, the center

Z(F (y_{0}, ⋯, y_{n})/[[< y_{−1} >, < y_{0} >, ⋯, < y_{n} >]]) ≅ π_{n+2}(S^{3})

By Lemma 4.1 and Proposition 3.5, Kerd_{0} =< x_{0} >=< y_{−1} > and therefore

Z_{n+1} = [[< y_{0} >, ⋯, < y_{n} >]]∩ < y_{−1} > .

Thus we have

Theorem 4.10 (Theorem 1.4). In the free group F (y_{0}, ⋯, y_{n}) with n ≥ 1,

[[< y_{0} >, ⋯, < y_{n} >]]∩ < y_{−1} >]]/[[< y_{−1} >, < y_{0} >, ⋯, < y_{n} >]] ≅ π_{n+2}(S^{3}).

5. On the Homotopy Groups of ΣK(π, 1)

In this section, we give group theoretical descriptions for π_{∗}(ΣK(π, 1)) for any group π. The proof of Theorem 1.5 is given in this section, where Theorem 5.9 is Theorem 1.5. We will use the notations defined in Section 3. First we extend our description for π_{∗}(S^{2}) to the case π_{∗}(∨_{α∈J}S^{2}). Recall that (∨_{α∈J}S^{1})_{0} = ∗, (∨_{α∈J}S^{1})_{1} = {σ_{α}, ∗|α ∈ J} and (∨_{α∈J}S^{1})_{n+1} = {s_{n} ⋯ ŝ_{i} ⋯ s_{0}σ_{α}, ∗|α ∈ J, 0 ≤ i ≤ n}. Let x_{i}^{(α)} denote s_{n} ⋯ ŝ_{i} ⋯ s_{0}σ_{α}. Then

F (∨_{α∈J}S^{1})_{n+1} = F (x_{0}^{(α)}, x_{1}^{(α)}, ⋯, x_{n}^{(α)}|α ∈ J).

Let y_{j}^{(α)} = x_{j}^{(α)} · x_{j+1}^{(α)} for 0 ≤ j ≤ n − 1 and y_{n}^{(α)} = x_{n}^{(α)}. By Lemma 4.1, we have

Lemma 5.1. F (∨_{α∈J}S^{1})_{n+1} = F (y_{j}^{(α)}|0 ≤ j ≤ n, α ∈ J) with

d_{j}(y_{k}^{(α)}) = \{ \begin{matrix}y_{k−1}^{(α)} & j ≤ k, \\ 1 & j = k + 1, \\ y_{k}^{(α)} & j > k + 1, \end{matrix}

and

s_{j}(y_{k}^{(α)}) = \{ \begin{matrix}y_{k+1}^{(α)} & j ≤ k, \\ y_{k}^{(α)} · y_{k+1}^{(α)} & j = k + 1, \\ y_{k}^{(α)} & j > k + 1, \end{matrix}

for 0 ≤ j ≤ n + 1, where y_{−1}^{(α)} = (y_{0}^{(α)} ⋯ y_{n−1}^{(α)})^{−1} in F (∨_{α∈J}S^{1})

Let C_{n+1}^{J} denote the subgroup of F (∨_{α∈J}S^{1})_{n+1} generated by all of the commutators [y_{i_{1}}^{(α_{1})ϵ_{1}}, ⋯, y_{i_{t}}^{(α_{t})ϵ_{t}}] with {i_{1}, ⋯, i_{t}} = {0, 1, ⋯, n} as sets, where ϵ_{j} = ±1, α_{j} ∈ J and the commutator bracket [⋯] runs over all of the commutator bracket arrangements of weight t for each t.

Lemma 5.2. C_{n+1}^{J} ⊆ NF (∨_{α∈J}S^{1})_{n+1}.

Proof: For each 1 ≤ j ≤ n + 1, there exists some i_{s} = j − 1. Thus d_{j}(y_{i_{s}}^{(α_{s})ϵ_{s}}) = 1 for some i_{s} and therefore

d_{j}([y_{i_{1}}^{(α_{1})ϵ_{1}}, ⋯, y_{i_{t}}^{(α_{t})ϵ_{t}}]) = 1

for each j > 0. The assertion follows.

Lemma 5.3. For each 1 ≤ j ≤ n + 1,

Kerd_{j} ∩ F (∨_{α∈J}S^{1})_{n+1} =< y_{j−1}^{(α)}|α ∈ J >,

the normal subgroup generated by y_{j−1}^{(α)} with α ∈ J.

Proof: By the definition of d_{j}, there is a commutative diagram

F (y_{j}^{(α)}|0 ≤ j ≤ n, α ∈ J) →^{p} F (y_{0}^{(α)} ⋯ ŷ_{j−1}^{(α)} ⋯ y_{n}^{(α)}|α ∈ J)
d_{j} ↓ ≅↓ d̄_{j}
F (y_{j}^{(α)}|0 ≤ j ≤ n − 1, α ∈ J) →^{=} F (y_{j}^{(α)}|0 ≤ j ≤ n − 1, α ∈ J)

where p is the projection and

d̄_{j}y_{k}^{(α)} = \{ \begin{matrix}y_{k−1}^{(α)} & j ≤ k, \\ y_{k}^{(α)} & j > k + 1. \end{matrix}

The assertion follows.

Theorem 5.4. Let C_{n+1}^{J} be defined as above. Then

NF (∨_{α∈J}S^{1})_{n+1} = C_{n+1}^{J}

Proof: By lemma 5.1, each d_{j} with j > 0 is a projection homomorphism. Thus, by Theorem 3.5,

NF (∨_{α∈J}S^{1})_{n+1} = ⋂_{j=1}^{n+1} Kerd_{j} = F (A(T_{0}, T_{1}, ⋯, T_{n}))

where T_{j} = {y_{0}^{(α)}, ⋯, ŷ_{j}^{(α)}, ⋯, y_{n}^{(α)}|α ∈ J}. It suffices to show that

A(T_{0}, T_{1}, ⋯, T_{n}) ⊆ C_{n+1}^{J}.

This follows from the next lemma.

Lemma 5.5. For 0 ≤ j ≤ n, let W = [y_{i_{1}}^{(α_{1})ϵ_{1}}, ⋯, y_{i_{t}}^{(α_{t})ϵ_{t}}] ∈ A(T_{0}, T_{1}, ⋯, T_{j}). Then

{0, 1, ⋯, j} ⊆ {i_{1}, ⋯, i_{t}}.

Proof: The proof is given by induction on j for 0 ≤ j ≤ n. Notice that, by construction, each element in A(T_{0}, T_{1}, ⋯, T_{j}) is written as a certain commutator. If j = 0, then

A(T_{0}) = {y_{0}^{(α)}, [[y_{0}^{(α)}, y_{i_{1}}^{(α_{1})ϵ_{1}}], ⋯,], y_{i_{t}}^{(α_{t})ϵ_{t}}]}

where α, α_{j} ∈ J, ϵ_{j} = ±1 and y_{i_{1}}^{(α_{1})ϵ_{1}} ⋯ y_{i_{t}}^{(α_{t})ϵ_{t}} runs over all of the reduced words ≠ 1 in F (y_{j}^{(α)}|α ∈ J, 1 ≤ j ≤ n). Thus the assertion holds for j = 0. Suppose that the the assertion holds for j − 1 with j ≤ n. Recall that

T_{j}^{(j)} = {W ∈ A(T_{0}, T_{1}, ⋯, T_{j−1}|W = [y_{i_{1}}^{(α_{1})ϵ_{1}}, y_{i_{2}}^{(α_{2})ϵ_{2}}, ⋯, y_{i_{t}}^{(α_{t})ϵ_{t}}], with y_{i_{s}}^{(α_{s})} ∈ T_{j}}.

Notice that y_{i_{s}}^{(α_{s})} ∈ T_{j} ⟺ i_{s} ≠ j. Thus, for W = [y_{i_{1}}^{(α_{1})ϵ_{1}}, y_{i_{2}}^{(α_{2})ϵ_{2}}, ⋯, y_{i_{t}}^{(α_{t})ϵ_{t}}] ∈ A(T_{0}, ⋯, T_{j−1}) − T_{j}^{(j)}, j ∈ {i_{1}, ⋯, i_{t}}. By induction, {0, ⋯, j − 1} ⊆ {i_{1}, ⋯, i_{t}}. Hence

{0, 1, ⋯, j} ⊆ {i_{1}, i_{2}, ⋯, i_{t}}

for any W = [y_{i_{1}}^{(α_{1})ϵ_{1}}, y_{i_{2}}^{(α_{2})ϵ_{2}}, ⋯, y_{i_{t}}^{(α_{t})ϵ_{t}}] ∈ A(T_{0}, ⋯, T_{j−1} − T_{j}^{(j)}. Recall that, by construction,

A(T_{0}, ⋯, T_{j}) = A_{T_{j}^{(j)}}.
{0, 1, ⋯, j} ⊆ {i_{1}, i_{2}, ⋯, i_{t}}

for any W = [y_{i_{1}}^{(α_{1})ϵ_{1}}, y_{i_{2}}^{(α_{2})ϵ_{2}}, ⋯, y_{i_{t}}^{(α_{t})ϵ_{t}}] ∈ A(T_{0}, ⋯, T_{j}). This completes the proof.

Theorem 5.6. π_{n+2}(∨_{α∈J}S^{2}) is isomorphic to the center of the group with generators

y_{0}^{(α)}, y_{1}^{(α)}, ⋯, y_{n}^{(α)}

for α ∈ J. and relations

[y_{i_{1}}^{(α_{1})ϵ_{1}}, y_{i_{2}}^{(α_{2})ϵ_{2}}, ⋯, y_{i_{t}}^{(α_{t})ϵ_{t}}]

with {i_{1}, i_{2}, ⋯, i_{t}} = {−1, 0, 1, ⋯, n} as sets, where the indices i_{j} can be repeated, ϵ_{j} = ±1, α_{j} ∈ J and the commutator bracket runs over all of the commutator bracket arrangements of weight t for each t.

Proof: Notice that π_{n+2}(∨_{α∈J}S^{2}) ≅ π_{n+1}(F (∨_{α∈J}S^{1})). By the above theorem, B_{n+1} is generated by

[y_{i_{1}}^{(α_{1})ϵ_{1}}, y_{i_{2}}^{(α_{2})ϵ_{2}}, ⋯, y_{i_{t}}^{(α_{t})ϵ_{t}}]

By Theorem 2.8, the assertion holds for n ≥ 1. For n = 0, B_{1} = Γ^{2}(F (y_{0}^{(α)}|α ∈ J)) and

F (y_{0}^{(α)}|α ∈ J)/B_{1} ≅ ⊕_{α∈J}Zπ_{2}(∨_{α∈J}S^{2}).

The assertion holds for all of n.

For the general case, we need a simplicial group construction.

Definition 5.7. Let G be a simplicial group and let X be a pointed simplicial set with a point ∗. The simplicial group F^{G}(X) is defined by setting

F^{G}(X)_{n} = ∐_{x∈X_{n}}(G_{n})_{x},

the free product, modulo the relations (G_{n})_{∗}, where (G_{n})_{x} is a copy of G_{n}. The faces and degeneracies homomorphisms in F^{G}(X) is given in the canonical way by the universal property of the coproduct in the category of groups and group homomorphisms.

Lemma 5.8 (Ca, Theorem 9, pp.88). Let G be a simplicial group and let X be a pointed simplicial set. Then the goemetric realization |F^{G}(X)| is homotopy equivalent to Ω(B|G| ∧ |X|).

A generalization of this lemma by using fibrewise simplicial groups is given in [Wu1].

Theorem 5.9 (Theorem 1.5). Let π be any group and let {x^{(α)}|α ∈ J} be a set of generators for π. Then, for n ≠ 1, π_{n+2}(ΣK(π, 1)) is isomorphic to the center of the quotient group of the free product

∐^{groups}_{0≤j≤n}(π)_{j}

modulo the relations

[y_{i_{1}}^{(α_{1})ϵ_{1}}, y_{i_{2}}^{(α_{2})ϵ_{2}}, ⋯, y_{i_{t}}^{(α_{t})ϵ_{t}}]

with {i_{1}, i_{2}, ⋯, i_{t}} = {−1, 0, 1, ⋯, n} as sets, where (π)_{j} is a copy of π with generators {x_{j}^{(α)}|α ∈ J}, ϵ_{j} = ±1, y_{−1}^{(α)} = x_{0}^{(α)}^{−1}, y_{j} = x_{j}^{(α)}x_{j+1}^{(α)}^{−1} for 1 ≤ j ≤ n − 1 and y_{n}^{(α)} = x_{n}^{(α)}, the commutator bracket [⋯] runs over all of the commutator bracket arrangment of weight t for each t.

Proof: Since {x^{(α)}|α ∈ J} is a set of generators for π, F (x^{(α)}|α ∈ J) → π is an epimorphism. Thus

F (∨_{α∈J}S^{1}) ≅ F^{F (x^{(α)}|α∈J)}(S^{1}) → F^{π}(S^{1})

is an epimorphism. Hence

NF (∨_{α∈J}S^{1}) → NF^{π}(S^{1})

and

BF (∨_{α∈J}S^{1}) → BF^{π}(S^{1})

are epimorphisms. The assertion follows from Theorem 5.6, Lemma 5.7 and Proposition 2.8.

6. Applications

In this section, we consider Cohen’s K-construction. Our descriptions for the homotopy groups of 3-sphere gives a calculation for the K-construction of 1-sphere.

Definition 6.1. Let X be set. The group K(X) is defined to the the quotient group of the free group F (X) modulo the normal subgroup generated by all of the commutators [[x_{1}, x_{2}], ⋯,], x_{t}] with x_{i} ∈ X and x_{i} = x_{j} for some 1 ≤ i < j ≤ t Now let S be a pointed simplicial set. The simplicial group K(S) is defined to be the quotient simplicial group of F (S) modulo the normal simplicial subgroup generated by all of the commutators [[x_{1}, x_{2}], ⋯,], x_{t}] with x_{i} ∈ S and x_{i} = x_{j} for some 1 ≤ i < j ≤ t, where F (S) is the Milnor’s F (K)-construction for the simplicial set S.

Definition 6.2. The group Lie(n) is the elments of weight n in the Lie algebra Lie(x_{1}, x_{2}, ⋯, x_{n}) which is the quotient Lie algebra of the free Lie algebra

L(x_{1}, x_{2}, ⋯, x_{n})

over Z modulo the two sided Lie ideal generated by the Lie elements [[x_{i_{1}}, x_{i_{2}}], ⋯,], x_{i_{t}}] with i_{l} = i_{k} for some 1 ≤ l < k ≤ t.

The following lemmas are due to Fred Cohen.

Lemma 6.3 (Co). Γ^{q}K(x_{1}, x_{2}, ⋯, x_{n}) = {1} for q ≥ n and

Γ^{n}K(x_{1}, x_{2}, ⋯, x_{n}) ≅ Lie(n),

where Γ^{q}G is the q-th term in the lower central series for the group G starting with Γ^{1}G = G.

Lemma 6.4 (Co). In the group K(x_{1}, x_{2}, ⋯, x_{n}), the normal subgroup grnerated by x_{j} is abelian for each 1 ≤ j ≤ n.

Lemma 6.5 (Co). The set {[x_{1}, x_{σ(2)}, ⋯, x_{σ(n)}]|σ ∈ Σ_{n−1}} is a Z-basis for Lie(n), where Σ_{n−1} acts on {2, 3, ⋯, n} by permutation.

Recall that the simplicial 1-sphere S^{1} is a free simplicial set generated by a 1simplex σ. More precisely, S_{0}^{1} = {∗}, S_{1}^{1} = {σ, ∗} and S_{n+1}^{1} = {∗, s_{n} ⋯ ŝ_{j} ⋯ s_{0}σ|0 ≤ j ≤ n}. Let x_{i} denote s_{n} ⋯ ŝ_{j} ⋯ s_{0}σ. Then

Lemma 6.6. The face functions d_{i} : S_{n+1}^{1} → S_{n}^{1} and the degenarate functions s_{i} : S_{n+1}^{1} → S_{n+2}^{1} are as follows:

d_{i}x_{j} = \{ \begin{matrix}x_{j} & j < i \\ x_{j−1} & j ≥ i \end{matrix}

and

s_{i}x_{j} = \{ \begin{matrix}x_{j} & j < i \\ x_{j+1} & j ≥ i \end{matrix} ,

where we put x_{−1} = ∗ and x_{n} = ∗ in S_{n}^{1}.

Theorem 6.7. π_{n}(K(S^{1})) is isomorphic to Lie(n)

Proof: Let π : F (S^{1}) → K(S^{1}) be the quotient homomorphism. Then NF (S^{1}) → NK(S^{1}) is an epimorphism. Recall that NF (S^{1})_{n+1} is generated by all of the commutators

[y_{i_{1}}, y_{i_{2}}, ⋯, y_{i_{t}}]

so that

{i_{1}, i_{2}, ⋯, i_{t}} = {0, 1, ⋯, n}

by Theorem 4.3. Thus NF (S^{1})_{n+1} ⊆ Γ^{n+1}F (S^{1})_{n+1} and therefore

NK(S^{1})_{n+1} ⊆ Γ^{n+1}K(S^{1})_{n+1}.

Notice that K(S^{1})_{n+1} ≅ K(x_{0}, x_{1}, ⋯, x_{n}). Thus Γ^{n+1}K(S^{1})_{n+1} ≅ Lie(n + 1) and Γ^{n+1}K(S^{1})_{n} = {1}. Thus

d_{j}|_{Γ^{n+1}K(S^{1})_{n+1}} : Γ^{n+1}K(S^{1})_{n+1} → K(S^{1})_{n}

is trivial for each j ≥ 0. And therefore

NKS^{1})_{n+1} = Γ^{n+1}K(S^{1})_{n+1} ≅ Lie(n + 1)

with d_{0} : NK(S^{1})_{n+1} → NK(S^{1})_{n} is trivial. the assertion follows.

Corollary 6.8. Let π : F (S^{1}) → K(S^{1}) be the quotient simplicial homomorphism. Then

π_{∗} : π_{n}(F (S^{1})) → π_{n}(K(S^{1}))

is an isomorphism for n = 1, 2 and zero for n > 2.

Now we consider the Samelson product in π_{∗}(K(S^{1})). Let x_{j} denote s_{n} ⋯ ŝ_{j} ⋯ s_{0}σ in S_{n+1}^{1}. The following lemma follows directly from Lemma 6.6.

Lemma 6.9. Let I = (i_{1}, i_{2}, ⋯, i_{m}) be a sequence with i_{1} < i_{2} < ⋯ < i_{m}. Then s_{I} : S_{n+1}^{1} − ∗ → S_{n+m+1}^{1} − ∗ is the composite

{x_{0}, x_{1}, ⋯, x_{n}} →^{s̄_{I}} {x_{0}, x_{1}, ⋯, x̂_{i_{1}}, ⋯, x̂_{i_{2}}, ⋯, x̂_{i_{m}}, ⋯, x_{n+m}}

↪ {x_{0}, x_{1}, ⋯, x_{n+m}}.

where s_{I} = s_{i_{m}} ⋯ s_{i_{1}} and s̄_{I} is the order preserving isomorphism.

Recall that, for σ ∈ π_{n}(G) and τ ∈ π_{m}(G), the Samelson product [C1] is defined to be

< σ, τ >= ∏_{(a,b)}[s_{b}σ, s_{a}τ]^{sign(a,b)},

where G is a simplicial group, (a, b) = (a_{1}, ⋯, a_{n}, b_{1}, ⋯, b_{m}) runs over all shuffles of (0, 1, ⋯, m + n − 1),i.e. all permutations, so that a_{1} < a_{2} < ⋯ < a_{n}, b_{1} < b_{2} < ⋯ < b_{m}, sign(a, b) is the sign of the permutation (a, b), the order of the product ∏ is right lexicographic on a and s_{a} = s_{a_{n}} ⋯ s_{a_{1}}.

Proposition 6.10. The Samelson product in π_{∗}(K(S^{1})) is as follows:

< [x_{σ(0)}, x_{σ(1)}, ⋯, x_{σ(n)}], [x_{τ(0)}, x_{τ(1)}, ⋯, x_{τ(m)}] >

= ∑_{(I,J)} sign(I, J)[[x_{i_{σ(0)}}, ⋯, x_{i_{σ(n)}}], [x_{j_{τ(0)}}, ⋯, x_{j_{τ(m)}}]]

for the commutators

[x_{σ(0)}, x_{σ(1)}, ⋯, x_{σ(n)}] ∈ π_{n+1}(KS^{1})) ≅ Lie(n + 1)

and

[x_{τ(0)}, x_{τ(1)}, ⋯, x_{τ(m)}] ∈ π_{m+1}(K(S^{1})) ≅ Lie(m + 1)

where

(I, J) = (i_{0}, i_{1}, ⋯, i_{n}, j_{0}, j_{1}, ⋯, j_{m})

runs over all shuffles of (0, 1, ⋯, m+n+1) so that i_{0} < i_{1} < ⋯ < i_{n}, j_{0} < j_{1} < ⋯ < j_{m}, sign(I, J) is the sign of the permutation (I, J), σ ∈ Σ_{n+1} acts on {0, 1, ⋯, n} and τ ∈ Σ_{m+1} acts on {0, 1, ⋯, m}

Proof: Notice that {x_{0}, ⋯, x̂_{j_{0}}, ⋯, x̂_{j_{m}}, ⋯, x_{n+m+1}} = {x_{i_{0}}, ⋯, x_{i_{n}}} and s̄_{J} : {x_{0}, ⋯, x_{n}} → {x_{i_{0}}, ⋯, x_{i_{n}}} is an ordered isomorphism.

s_{J}([x_{σ(0)}, x_{σ(1)}, ⋯, x_{σ(n)}]) = [x_{i_{σ(0)}}, ⋯, x_{i_{σ(n)}}]

and

s_{I}([x_{τ(0)}, x_{τ(1)}, ⋯, x_{τ(m)}]) = [x_{j_{τ(0)}}, ⋯, x_{j_{τ(m)}}].

The assertion follows.

Definition 6.11. A simplicial group is minimal if it is also a minimal simplicial set.

Recall that a simplicial group G is minimal if and only if the Moore chain complex NG is minimal [C2].

Theorem 6.12. The simplicial group K(S^{1}) is the universal minimal simplicial quotient simplicial group of F (S^{1}) in the following sense: (1). K(S^{1}) is a minimal simplicial group. (2). Let G be a minimal simplicial group. Then every simplicial homomorphism f : F (S^{1}) → G factors through K(S^{1}).

Proof: By inspecting the proof of Theorem 6.7, K(S^{1}) is a minimal simplicial group. The assertion (2) follows from the following statement.

Statement: K(S^{1}) is the quotient simplicial group of F (S^{1}) modulo the normal subsimplicial group generated by the boundaries.

Let H denote the kernel of the quotient map p : F (S^{1}) → K(S^{1}) and let \overline{B} denote the normal subsimplicial group of F (S^{1}) generated by the boundaries BF (S^{1}). Notice that K(S^{1}) is a minimal simplicial group. Thus \overline{B} is contained in H. Let Q denote the quotient simplicial group F (S^{1})/\overline{B}. Then Q is a minimal simplicial group. By Proposition 2.6, there is a central extension

0 → K(π_{n+1}Q, n + 1) → P_{n+1}Q → P_{n}Q → 0,

where P_{n}Q is the n-th Moore-Postnikov system of Q. Notice that P_{1}Q = K(π_{Q}, 1) = K(Z, 1). Thus Γ^{n+2}P_{n+1}Q = 1 by induction on n. Notice that Q_{n+1} ≅ (P_{n+1}Q)_{n+1}. Thus Γ^{n+2}Q_{n+1} = 1. Now we show that H is contained in \overline{B} by induction on the dimension starting with H_{1} = \overline{B}_{1} = 1. Suppose that H_{n} ⊆ \overline{B}_{n} with n > 0. Notice that F (S^{1})_{n+1} = F (x_{0}, ⋯, x_{n}) and K(S^{1})_{n+1} = K(x_{0}, ⋯, x_{n}). Thus H_{n+1} is a normal subgroup of F (x_{0}, ⋯, x_{n}) generated by the commutators

[[x_{i_{1}}, x_{i_{2}}], ⋯, x_{i_{t}}], x_{i_{t}}]

such that i_{p} ≠ i_{q} for p < q. Now consider W = [[x_{i_{1}}, x_{i_{2}}], ⋯, x_{i_{t}}], x_{i_{t}}]. If t ≥ n + 1, then W ∈ Γ^{n+2}F (x_{0}, ⋯, x_{n}). Thus W ∈ \overline{B}_{n+1} since Γ^{n+2}Q_{n+1} = 1.

If t < n + 1, then there exists an index j ∈ {0, 1, ⋯, n} − {i_{1}, ⋯, i_{t}}. Recall that

s_{i}x_{k} = \{ \begin{matrix}x_{k} & k < i, \\ x_{k+1} & k ≥ i, \end{matrix}

for x_{k} = s_{n−1} ⋯ ŝ_{k} ⋯ s_{0}σ ∈ S_{n}^{1}. Thus

s_{j}[[x_{i′_{1}}, x_{i′_{2}}], ⋯, x_{i′_{t}}], x_{i′_{t}}] = [[x_{i_{1}}, x_{i_{2}}], ⋯, x_{i_{t}}], x_{i_{t}}],

where i′_{k} = i_{k} if i_{k} < j and i′_{k} = i_{k}−1 if i_{k} > j. By induction, [[x_{i′_{1}}, x_{i′_{2}}], ⋯, x_{i′_{t}}], x_{i′_{t}}] ∈ \overline{B}. Thus W = [[x_{i_{1}}, x_{i_{2}}], ⋯, x_{i_{t}}], x_{i_{t}}] ∈ \overline{B}. The induction is finished and the assertion follows.

The simplicial group K(S^{1}) is homotopy eqivalent to a product of the Eilenberg-Maclane spaces with a different product structure.

Proposition 6.13. ΩK(S^{1}) is an abelian simplicial group. Therefore K(S^{1}) is homotopy equivalent to a product of the Eilenberg-MacLane spaces as a simplicial set.

Proof: Consider

d_{0} : K(x_{0}, x_{1}, ⋯, x_{n}) → K(x_{0}, x_{1}, ⋯, x_{n−1})

d_{0}(x_{0}) = 1 and d_{0}(x_{j}) = x_{j−1}. Thus Kerd_{0} ∩ K_{n+1}(S^{1}) ≅< x_{0} > is the normal subgroup generated by x_{0} which is abelian by Lemma 6.4. The assertion follows.

In the end of this section, we give some applications of K(S^{1}) to minimal simplicial groups.

Proposition 6.14. Let G be a minimal simplicial group such that the abelianlizer G^{ab} is a minimal simplicial group K(π, 1) for a cyclic group π. Then G is homotopy equivalent to a product of Eilenberg-Maclane spaces.

Proof: Notice that G_{1} = π. Let x be a generator for the cyclic group π and let f_{x} : S^{1} → G be a representive map of x, i.e, f_{x}(σ) = x. Let g : F (S^{1}) → G be the simplicial homomorphism induced by f_{x}. We need a lemma.

Lemma 6.15. The simplicial homomorphism g : F (S^{1}) → G is simplicial surjection.

Proof: It suffices to show that the subsimplicial group, denote by H, of G generated by G_{1} is G itself. This is given by induction on the dimensions starting with H_{1} = G_{1}. Suppose that H_{n−1} = G_{n−1} with n > 1. By a result of Condule [see, e.g, Po, Proposition 1, pp.6], G_{n} is generated by the degenerate images of lower order Moore chain complex terms and NG_{n}. Thus G_{n} is genereted by NG_{n} and H_{n} by induction. Notice that G is a minimal simplicial group. Thus NG_{n} = ZG_{n}, the cycles. By Proposition 2.1, ZG_{n} is contained in the center of G_{n}. Thus H_{n} is a normal subgroup of G_{n} and the composite ϕ : π_{n}G ≅ ZG_{n} → G_{n} → G_{n}/H_{n} is an epimorphism. Thus G_{n}/H_{n} is an abelian group and so the quotient homomorphism G_{n} → G_{n}/H_{n} factors through G_{n}^{ab}. Notice that G^{ab} = K(π, 1). Thus G_{n}^{ab} ≅ G_{n}^{ab}/ZG_{n} and so ϕ : π_{n}G → G_{n}/H_{n} is trivial. Thus G_{n}/H_{n} is trivial and the assertion follows.

Continuation of Proof of Proposition 6.14: Notice that G is minimal. The simplicial epimorphism g : F (S^{1}) → G factors through K(S^{1}) by Proposition 6.12. By Proposition 6.13, ΩK(S^{1}) is an abelian simplicial group. Thus ΩG is also an abelian simplicial group. Thus G is homotopy equivalent to a product of Eilenberg-Maclane spaces, which is the assertion.

The following counter-example for minimal simplicial groups is due to J. W. Milnor (unpublished).

Proposition 6.16. Ω(S^{n+1}[n + 1, n + 2, n + 3]) does not have a homotopy type of a minimal simplicial group for n > 0, where S^{n+1}[n + 1, n + 2, n + 3] is the 3-stage Postnikov system by taking the first three nontrivial homotopy groups of S^{n+1}.

Proof: Suppose that G is a minimal simplicial group such that G ≃ Ω(S^{n+1}[n + 1, n + 2, n + 3]). Let f : F (S^{1}) → Ω^{n−1}G be a simplicial homomorphism such that f(σ) is a generator of (Ω^{n−1}G)_{1} ≅ G_{n} ≅ Z. Then f_{∗} : π_{j}(F (S^{1})) → π_{j}(Ω^{n−1}G) is an isomorphism for j ≤ 3. Notice that Ω^{n−1}G is also a minimal simplicial group. The simplicial homomorphism f : F (S^{1}) → Ω^{n−1}G factors through K(S^{1}). Notice that π_{3}(F (S^{1})) ≅ π_{3}Ω^{n−1}G) ≅ Z/2 and π_{3}(K(S^{1})) ≅ Lie(3) ≅ Z ⊕ Z. There is a contradiction and the assertion follows.

More examples and counter-examples for minimal simplicial groups will be given in [Wu2]. It was know that there are still many counter-examples of two-stage Postnikov systems for minimal simplicial groups [Wu2].

References

[A] J. F. Adams, On products in minimal complexes, Trans. Amer. Math. Soc. 82 (1956),

180-189.

[AM] M. Artin and B. Mazur, On the van Kampen theorem, Topology 5(1966), 179-189.

[BK] A. K. Bousfield, D. Kan, Homotopy limits, completions and localizations, Springer Lecture Notes in Math. 304, 1972.

[Bo1] W. A. Bogley, An embedding for π_{2} of a subcomplex of a finite contractible two-complex, Glasgow Math. J. 33(1991), 365-371.

[Bo2] W. A. Bogley, J. H. C. Whitehead’s Asphericity Question, London Math. Soc. Lect. Note Series 197 (1993), 309-334.

[C1] E. B. Curtis, Some relations between homotopy and homology, Ann. of Math. 83 (1965), 386–413.

[C2] E. B. Curtis, Simplicial homotopy theory, Advances in Math, 107-209 (1971).

[Car] G. Carlsson, A simplicial group construction for balanced products , Topology, vol.23, no.1 (1985) 85–89.

[Co] F. Cohen, On the combinatorial methods in homotopy theory, To appear.

[Cond] D. Conduche, Modules croises generalises de longeur 2, J. Pure. Appl. Algebra 34

(1984), 155-178.

[K1] D. Kan, On c.s.s. complexes, Amer. J. Math. 79 (1957), 449-476.

[K2] D. Kan, A combinatorial definition of homotopy groups, Ann. of Math. 67 (1958), 288–312.

[Mi1] J. Milnor, On the construction F (K), Algebraic Topology- A Student Guide, by J.F. Adams, 119-136, Cambridge Univ. Press.

[Mi2] J. Milnor, The geometrical realization of a semi-simplicial complex, Ann. of Math.65(1957) 357-362.

[Fi] Z. Fiedorowicz, Classification spaces of topological manoids and category, Amer. J. Math. 106 (1984), 301-350.

[KT] D. Kan and W. Thurston, Every connected spaces has the homology of a K(π, 1), Topology 15 (1976), 253-258.

[Mo1] J. C. Moore, Homotopie des complexes monöideaux, Seminaire Henri Cartan, 1954-55.

[Mo2] J. C. Moore, Algebraic homotopy theory, mimeographed notes, Princeton,1956.

[Mo3] J.C. Moore, Semi-simplicial complexes and Postnikov systems, Symposium Internationale de Topologica Algebraica, pp.232-247(1958)

[Po] T. Porter, N-types of simplicial groups and crossed N-cubes, Topology, Vol.32, No.1, (1993), 5–24.

[Qu1] D.G. Quillen, The geometrical realization of a Kan fibration is a Serre fibration, Proc.

Amer. Math. Soc. 19 (1968),1499-1500.

[S] C. R. Stover, A ven Kampen spectral sequence for higher homotopy groups, Topology

29(1990), 9-26.

[Wh1] J.H.C. Whitehead, On the asphericity of regions in a 3-sphere, Fund. Math., 32(1939), 149-166.

[MKS] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Interscience

Publisher, New York, London, Sydney,1966.

[Wi] C. W. Wilkerson, Applications of minimal simplicial groups, Topology 15(1976), 111-130.

[Wu1] Jie Wu, On fibrewise simplicial groups and a generalization of Milnor’s F (K) construction, to appear.

[Wu2] Jie Wu, On minimal simplicial groups, In preparing.

MSRI, 1000 Centennial Drive, Berkeley, CA 94720-5070