COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I

F.R. Cohen^{∗}

Introduction:

The purpose of this article is to analyze properties of “universal examples” for certain groups which appear in the classical distributivity law in non-stable homotopy theory. The groups themselves appear as a subgroup of units in non-abelian analogues of exterior algebras and are described in detail below. Furthermore, these groups are given by iterated central extensions of modules Lie(n) which are free abelian groups of rank (n-1)! and which are also modules over the integral group ring of the symmetric group on n-letters. These representations of the symmetric groups are obtained from a natural action on a free Lie algebra and can be found in [C, Theorem 12.3] via the action of the symmetric groups on the ”top” homology group of configuration spaces of points in Euclidean space. In addition, these representations are also encountered in combinatorics, cohomology of moduli spaces for certain choices of surfaces, and in stable homotopy theory. Applications to the homotopy groups of certain double suspensions will be given in the sequel.

The point of view here is to construct a universal group which side-steps direct calculations with the classical distributivity law which have proven to be quite complicated [B],[BS], and [Ba]. Instead, one might consider a universal group which codifies all of these computations in a single group. The resulting construction is an inverse limit of groups H_{n} where the natural map of H_{n} to H_{n−1} is a surjection with kernel given by the center of H_{n} where this center is isomorphic to Lie(n) the free abelian group of rank (n-1)! alluded to above.

The main information in this article is a precise identification of this extension by using the group of units in a non-abelian version of an exterior algebra. In addition, this tower obtained from the H_{n} is filtered with the associated graded module given by a direct sum of the modules Lie(n), n > 0, and where the induced structure of Lie algebra on the level of associated graded is obtained from the

Typeset by AMS-TEX 2 F.R. COHEN^{∗}

AMS subject classification Primary: 55P35,55Q35, Secondary: 14H99,20F99

* Partially supported by the NSF

classical shuffle product. These constructions admit several applications which will appear elsewhere.

One view of the information obtained here is as follows: The groups H_{n} give the ”general structure” for the group of pointed homotopy classes of self-maps for the loop space of a double suspension. Many of these maps are essential, but induce the trivial map on homotopy groups. Some of this information is given in chapter 2 of this article where the abelianization of the tower of groups H_{n} is studied. The abelianization map always has a kernel for the H_{n} with n > 1; thus one might regard this structure as informing on the group of homotopy classes of self-maps of a loop space, and where many of the maps involved induce the trivial map on homotopy groups.

Restrict to the category of compactly generated, weak Hausdorff spaces with non-degenerate basepoint and let [X, Y] denote the set of pointed homotopy classes of maps. Consider the groups [X^{n}, ΩΣX], [J_{n}X, ΩΣX], and [ΩΣX, ΩΣX] where X is assumed to be a suspension and J_{n}X denotes the nth stage of the James construction JX. It should be pointed out that the identity element in [ΩΣX, ΩΣX] is the class of the \underline{constant} map rather than the identity map for ΩΣX. The phrase “universal example” for [J_{n}X, ΩΣX] is defined in the next paragraph.

Choose elements in the group [ΩΣX, ΩΣX] as follows:

(i) the class of the kth power map k : ΩΣX → ΩΣX with k a natural number, and

(ii) the class of the composite for each natural number k

ΩΣX →^{h_{k}} ΩΣX^{(k)} →^{Ωω_{k}} ΩΣX

where h_{k} is the kth James-Hopf invariant, X^{(k)} is the k-fold smash product, and ω_{k} : ΣX^{(k)} → ΣX is a k-fold iterated Whitehead product.

Let S(X) denote the subgroup of [ΩΣX, ΩΣX] generated by the elements (i)-(ii) above. Let S_{n}(X) denote the subgroup of [J_{n}X, ΩΣX] given by the image of the restriction map [ΩΣX, ΩΣX] → [J_{n}X, ΩΣX] applied to S(X). Notice that S(X) and S_{n}(X) are quotients of the free group F generated by symbols k and (Ωω_{k})∘h_{k} one for each element listed in (i)–(ii) above. Define H_{n} to be the quotient of F modulo the normal subgroup given by the intersection of the kernels of composites F ⟶ S_{n}(X) ⟶ [J_{n}X, ΩΣX] for every space X which is a suspension. Thus H_{n} is the smallest group which admits a surjection to S_{n}(X) for all suspensions X. Define the “universal example” of the groups [J_{n}X, ΩΣX] to be the group H_{n}.

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 3

The first step in analyzing the structure of H_{n} is to analyze the analogous “universal example” for [X^{n}, ΩΣX]. Given a space X, there are n canonical choices of maps p_{i} : X^{n} → ΩΣX where p_{i} is the composite of ith coordinate projection π_{i} : X^{n} → X followed by the Freudenthal suspension map E : X → ΩΣX. Let F [p_{1}, …, p_{n}] denote the free group generated by the p_{i} and θ̄ : F [p_{1}, …, p_{n}] → [X^{n}, ΩΣX] be the induced homomorphism. Let K_{n}[x_{1}, …, x_{n}] be the smallest quotient group of F [p_{1}, …, p_{n}] such that there is an induced homomorphism θ : K_{n}[x_{1}, …, x_{n}] → [X^{n}, ΩΣX] for which θ(x_{i}) = θ̄(p_{i}) for every suspension X = Σ(X′).

The groups K_{n}[x_{1}, …, x_{n}] are closely related to a non-commutative analogue of an exterior algebra. Here, let V be a free ℤ-module of rank n with a fixed choice of basis {y_{1}, …, y_{n}}. Let J denote the two-sided ideal of the tensor algebra T [V] generated by all monomials in this choice of basis elements given by y_{i_{1}}⊗y_{i_{2}}⊗⋯⊗y_{i_{t}} where y_{i_{j}} = y_{i_{k}} for some 1 ≤ j < k ≤ n. Define

A[V] = T [V]/J.

Let π be a group with descending central series denoted by Γ^{i}(π) with Γ^{1}(π) = π and Γ^{q+1}(π) the subgroup generated by the commutators [π, Γ^{q}(π)]. Let Γ^{q} denote the q-th stage of the descending central series for the group K[x_{1}, …, x_{n}] and let E_{0}^{q} denote Γ^{q}/Γ^{q+1}.

Theorem A. If n ≥ 1, K_{n}[x_{1}, …, x_{n}] is a torsion free nilpotent group of class n. The filtration quotients E_{0}^{q} are free abelian groups of rank (q −1)!\binom{n}{q}. Furthermore, K_{n}[x_{1}, …, x_{n}] is isomorphic to the subgroup of the group of units of A[V] generated by 1 + y_{i}, 1 ≤ i ≤ n.

The underlying structure of Lie algebra for E_{0}^{∗} is given below. The centers of the groups K_{n}[x_{1}, … x_{n}], denoted by Λ_{n}, are useful here and support an action of the symmetric group on n letters Σ_{n}. These centers are closely related to certain free Lie algebras over ℤ. Namely, let L[V] be the free Lie algebra generated by V (where V, of course, is ungraded). Define Lie(n) to be the linear span of the elements

[… [y_{σ(1)}, y_{σ(2)}]y_{σ(3)}] … y_{σ(n)}]

where σ runs over the elements in the elements in the symmetric group Σ_{n}. Thus Lie(n) is isomorphic to ⨁_{(n−1)!} ℤ as a ℤ[Σ_{n}]-module. It is worth remarking that

4 F.R. COHEN^{∗}

Lie(n) has appeared in several other contexts recently in connection with combinatoris, cohomology of certain choices of moduli spaces, stable homotopy theory, algebraic K-theory and conformal field theories. It was proven in the early ’70’s that Lie(n) is isomorphic to H_{n−1}(P_{n}; ℤ) tensored with the sign representation as a ℤ[Σ_{n}]-module where P_{n} is the pure braid group for n-stranded braids [C, Theorem 12.3].

Theorem B. Λ_{n} is isomorphic to Lie(n) as a ℤ[Σ_{n}]-module.

There is a quotient map π : X^{n} → J_{n}X defined in [J]. The natural homomorphism π^{∗} : [J_{n}X, ΩY] → [X^{n}, ΩY] is a split monomorphism of sets. There exist subgroups H_{n} of K_{n}[x_{1}, …, x_{n}] which correspond to the image of π^{∗} and which were defined above in this article.

Theorem C. There is a non-split central extension

1 → Λ_{n} → H_{n} →^{q_{n}} H_{n−1} → 1.

Thus there is a tower of groups

⋮
↓
Λ_{n} ⟶ H_{n}
↓^{q_{n}}
Λ_{n−1} ⟶ H_{n−1}
↓
H_{n−2}
↓
⋮

where q_{n} is an epimorphism with kernel Λ_{n} which is isomorphic to Lie(n) as a ℤ[Σ_{n}]-module.

Define H_{∞} to be the inverse limit of the groups H_{n}, \underleftarrow{lim} H_{n}. Recall the groups S_{n}(X) defined earlier this article. Restriction induces a surjection S_{n}(X) → S_{n−1}(X). Define S_{∞}(X) to be \underleftarrow{lim} S_{n}(X).

Theorem D. There is a commutative diagram of groups

H_{n} ⟶^{θ_{n}} [J_{n}X, ΩΣX]
↓  ↓
H_{n−1} ⟶^{θ_{n}} [J_{n−1}X, ΩΣX]

for every suspension X. Furthermore the natural homomorphism θ : H_{∞} → \underleftarrow{lim}_{n}[J_{n}X, ΩΣX] has image given by S_{∞}(X).

Thus in the case that the natural homomorphism

[ΩΣX, ΩΣX] → \underleftarrow{lim}_{n}[J_{n}X, ΩΣX]

is an isomorphism of groups H_{∞} may be regarded as the universal example for [ΩΣX, ΩΣX]. It should be remarked that there are finite p-groups which are p^{r}- torsion analogues of the groups K_{n}[x_{1}, …, x_{n}] and H_{n}. The resulting groups have analogous structure where Lie(n) is replaced by Lie(n)⊗_{ℤ} ℤ/p^{r}ℤ. These groups are also analyzed in the proofs given below.

Given the filtration H_{∞}in Theorem D, there is a natural Lie algebra obtained via the filtration quotients. Furthermore, this filtration is specifically not that induced by the descending central series, but ”converges” more slowly. The associated graded is a Lie algebra which, as an abelian group,is the direct sum of all of the Lie(n). The Lie bracket is specified by the shuffle product and is given in Theorem 2.4.2 below.

The results above codify the relations between power maps together with compositions of Hopf invariants and Whitehead products. These relations have been of interest since the 1950’s; a sample is given by [B, BS]. Also focus on specific compositions is given in [S].

The main point of the analysis of the groups H_{n} is that the information obtained gives a “global” picture of relations in a simple computable form. Namely the group structure “carries” the relations. This information in turn provides data about the kernel of the looping functor

Ω_{∗} : [J_{n}X, ΩΣX] → [ΩJ_{n}X, Ω^{2}ΣX]

which is the group homomorphism sending a map f to its’ looping. In particular if Σ_{i>0}dimH̄_{i}(ΣX; 𝔽_{p}) > 1, then Ω_{∗} always has a kernel. This structure implies

6 F.R. COHEN^{∗}

information concerning essential maps in the kernel of Ω_{∗} and thus about π_{∗}ΣX. In any case, the point of this article is to give the precise extension attached to these groups. One naive reason that this kernel is non-trivial is that the source is a non-abelian group while the target is abelian.

Notice that these relations imply that certain contributions of the classical distributivity law have a universal zero effect on homotopy groups; they give non-trivial self-maps of single loop spaces, but they are null-homotopic after looping the map. A specific example is given in [C_{2}]. Some of these relations are described in chapter 2 here. One result given as Theorem H below which gives a partial analysis of these relations.

It should be remarked that these groups give no new information about the homotopy groups of spheres. However, they do give new direct information concerning double suspensions which are \underline{not} homotopy equivalent to spheres. These features are explored further in chapter 2 here.

The construction of the groups K_{n}[x_{1}, …, x_{n}] naturally give rise to the construction of a functor from the category of simplicial sets to the category of simplicial groups. If X is a reduced simplicial set, then write K[X] for this construction (which is given more precisely below). One application of K[S^{1}] is in [W] where this simplicial group is used to give an application to to an unpublished result of J. Milnor concerning the non-existence of minimal simplicial groups.

In addition, the tower H_{∞} is an example of a general construction of a functor from simplicial groups to pro-groups which assembles the chains in the Moore chain complex for the homotopy groups of a simplicial group into a single group. One of the theorems below gives that the center of the group K_{n+1}[x_{1}, …, x_{n+1}], Λ_{n+1}, is isomorphic to the group of cycles in degree n for the Moore chain complex for calculating π_{n}K̄[Δ[1]] where Δ[n] is the simplicial n-simplex. These remarks are elucidated below in Theorem K. First, some general remarks concerning simplicial sets are required.

group calculating realization omitted and

In what follows in this section, the objects and morphism sets are tacitly assumed to be sets. Some general references are given in [GZ] and [Cu] (although morphisms are not explicitly required to be sets in [GZ]). With these conventions, consider the following categories where the morphisms are the natural ones:

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 7

(i) ·Ens, the category of pointed sets, (ii) · Δ^{∘}Ens, the category of pointed simplicial sets, (iii) Grp, the category of groups, (iv) Δ^{∘}Grp, the category of simplicial groups,and

(v) Pro-Grp, the category of pro-groups with index set given by the nonnegative integers.

There are functors

F : ·Ens ⟶ Grp and

Δ^{∘}F : ·Δ^{∘}Ens ⟶ Δ^{∘}Grp

defined as follows:

(1) F is the “free group functor” which sends the base-point to the trivial group and is left adjoint to the pointed forgetful functor from Δ^{∘}Grp to ·Ens,

(2) Δ^{∘}F is Milnor’s free group functor from pointed simplicial sets to simplicial groups [Mi]. In particular if X is in ·Δ^{∘}Ens with base point b_{o}, then Δ^{∘}F [X] in dimension n is given by the free group generated by the elements of degree n in X with the single defining relation that b_{n} = s_{o}^{n}b_{o} is the identity. In Milnor’s paper [Mi], the notation for this last construction is F [X]; there is a liberty taken here of changing this notation to Δ^{∘}F to avoid confusion with the notation here and to emphasize that the value of this functor lies in the category of simplicial sets.

Next, let P : ·Ens → Grp be any functor together with a natural transformation

η : F → P

which resricts to an epimorphism when restricted to objects V in ·Ens. The next statement follows at once from the definitions (prolongation of functors) and is well-known.

Statement P. Given a natural transformation η : F → P which restricts to an epimorphism on objects η : F [V] → P [V], there is a functor

Δ^{∘}P : ·Δ^{∘}Ens → Δ^{∘}Grp

such that on objects X, Δ^{∘}P (X) in dimension n is P (X_{n}) where X_{n} is the set of elements in degree n of X. Furthermore, there is a natural transformation

Δ^{∘}η : Δ^{∘}F → Δ^{∘}P

8 F.R. COHEN^{∗}

which on objects X is given by η[X_{n}] : F [X_{n}] → P [X_{n}].

A specific example of such a natural transformation η is obtained from the construction of the groups K_{n}[x_{1}, …, x_{n}]. Namely given an object V in ·Ens define

K[V] = F [V]/N[V]

with N[V] the smallest normal subgroup of F [V] which contains (1) commutators of length at least 2 in the elements v_{λ} of V such that \underline{some} v_{μ} appears at least twice and (2) the base-point. Thus, for example, [[v_{λ}, v_{μ}]v_{λ}] is in N[V].

Write

η[V] : F [V] → K[V]

for the natural quotient map. Notice that if f : V → W is a morphism of pointed sets, then the induced homomorphism F (f) : F [V] → F [W] sends N[V] into N[W]. Thus there is a natural transformation η : F → K which restricts to an epimorphism on objects.

Two other analogous functors are given by (i) Z and (ii) Z_{p} which send V to (i) the free abelian group generated by V with the base-point as identity and (ii) the vector space over the field of p-elements generated by V with the base point as the identity. Of course there are the usual natural transformations ε : F → Z and ε_{p} : F → Z_{p}. The next theorem follows at once from the definitions together with the existence of colimits in Grp.

Theorem E. There is a functor K : ·Ens → Grp which on finite sets V = {b_{o}, b_{1}, …, b_{n}} with base-point b_{0} given by

K[V] = K_{n}[b_{1}, …, b_{n}].

Furthermore K commutes with colimits and there is a natural transformation η : F → K which restricts to an epimorphism η[V] : F [V] → K[V] on objects V. There are natural transformations ε̄: K → Z and ε̄_{p} : K → Z_{p} such that ε̄ ∘ η = ε and ε̄_{p} ∘ η = ε_{p}.

Corollary F. There is a functor

Δ^{∘}K : ·Δ^{∘}Ens → Δ^{∘}Grp

which on objects X in dimension n is given by the group K[X_{n}] of Theorem A. There is a natural transformation

Δ^{∘}η : Δ^{∘}F → Δ^{∘}K

which on objects is an epimorphism

Proof. Statement P and Theorem E give F.

If X is a reduced simplicial set with geometric realization |X|, then Milnor proves that there is a natural homotopy equivalence

|Δ^{∘}F X| → ΩΣ|X|.

By corollary F, there is a factorization

|Δ^{∘}F X| → |Δ^{∘}KX| → Sp^{∞}|X|

and the next corollary follows at once.

Corollary G. There is a factorization of the abelianization homomorphism ϕ : π_{q}ΩΣ|X| → π_{q}Sp^{∞}|X| through π_{q}|Δ^{∘}KX|.

Remark. Information concerning π_{q}|Δ^{∘}KX| is implicit in Theorem A.

One might wonder where there is a natural analogue of Δ^{∘}K(X) which receives a map from Kan’s G construction G(X) for a reduced simplicial set X [K]. The answer, in a naive way, is no. To see this recall that G(X) in dimension n has one generator x̄ for every x in X_{n+1} and one relation \overline{s_{o}x} = identity for every x in X_{n}. The zeroeth face operator is d_{o}(x̄) = \overline{(d_{1}x)} · \overline{(d_{o}x)}^{−1} [K, Cu p. 134]. This face operator will not, in general, preserve the subgroup N[X_{n+1}].

In what follows below let S^{n} denote the simplicial set which is the quotient Δ[n]/Δ̇[n] [Cu, p.112]. Thus S^{n} has two non-degenerate simplices, a vertex ∗ and σ_{n} in dimension n. Of course there are iterated degeneracies of ∗, and simplicies s_{i_{t}} … s_{i_{1}}σ_{n}, n + t > i_{t} > ⋯ > i_{1} ≥ 0. The following result concerning the homotopy groups of K[S^{1}] have been proven by J. Wu [W].

Theorem H. If n ≥ 1, then π_{n}K[S^{1}] is isomorphic to Lie(n).

The analysis in Theorem H follows directly from the structure of the centers of K_{n} given in Theorem B; details are given in section 1.10.

10 F.R. COHEN^{∗}

The tower of groups H_{∞} given by the inverse limit of the H_{n} in Theorem C is a special case of a functor

χ : Δ^{∘}Grp ⟶ Pro − Grp

which is applied to the specific simplicial group. This general construction assembles all of the groups of cycles in the Moore chain complex for the homtopy groups of a simplicial group into a single group. Thus the construction is listed next.

Let Γ be a simplicial group. Define χ inductively as follows: (1) χΓ_{0} = Γ_{0}, (2) Assume that χΓ_{n−1} has been defined as a subgroup of Γ_{n−1}. Then define Γ_{n} as the pull-back

χΓ_{n} ⟶  Γ_{n}
p_{i}↓ ↓^{D}
χΓ_{n−1} ⟶_{Λ_{n−1}} Γ_{n−1} × .... × Γ_{n−1}

where D(x) = (d_{0}(x), d_{1}(x), ..., d_{n}(x)), Λ_{n−1}(y) = (y, d_{1}s_{0}(y), d_{2}s_{0}(y), ..., d_{n}s_{0}(y)), and d_{i} is the i-th face operation.

The next result follows from the definitions and the calculations of chapter 1 below where Δ[n] is the simplical n-simplex.

Theorem J.

There is a functor

χ : Δ^{∘}Grp ⟶ Pro − Grp

which satisfies the following properties:

(1) χΓ_{n} ⟶ χΓ_{n−1} is a split epimorphism of groups with kernel given by

∩_{0≤i≤n}ker(d_{i})

for d_{i} : Γ_{n} ⟶ Γ_{n−1}. Thus this kernel gives the cycles in the Moore complex for the homotopy of Γ.

(2) The functor χ preserves products, and filtered colimits, and

(3) If Γ = K[Δ[1]] , then χΓ is isomorphic to the inverse limit of the H_{n}, H_{∞}.

One might wonder whether the relations above suffice to show that power maps in double loop spaces with sufficient structure are null-homotopic. The answer in a naive way is no. Namely, there are torsion analogues of the groups H_{n} which are denoted by H_{n}^{R} for R = Z/p^{r}Z, and which are described in section 1.1 as well as in 2.1. Furthermore, there is a natural quotient of these groups Δ(H_{n}^{R}) that is described in section 2.1 and which imposes the natural relations obtained by considering the loopings of Hopf invariant and Whitehead product maps. The next theorem is proven in section 2.6.

Theorem H.

If R = Z/2^{r}Z. there is a surjection of the inverse limit of the Δ(H_{n}^{R}), Δ(H_{∞}^{R}), to the inverse limit of the Z/2^{s}Z. Thus Δ(H_{∞}^{R}) does not have an exponent.

A further investigation of these questions will appear elsewhere.

We would like to thank John Moore, and Sam Gitler for numerous conversations about this article. Furthermore, we would like to thank Michael Vaughn-Lee for an introduction to combinatorial group theory.

TABLE OF CONTENTS

Chapter 1:

§1.1: The group K_{n}^{R}[x_{1}, …, x_{n}] and universal maps.

§1.2: On the relations in K_{n}^{R}[x_{1}, …, x_{n}].

§1.3: Non-commutative analogues of exterior algebras; the structure of A_{n}^{R}[y_{1}, …, y_{n}].

§1.4: Some formulae in K_{n}^{R}[x_{1}, …, x_{n}].

§1.5: A subgroup of K_{n}^{R}[x_{1}, …, x_{n}].

§1.6. The descending central series for K_{n}[x_{1}, …, x_{n}].

12 F.R. COHEN^{∗}

§1.7: The center of K_{n}^{R}. §1.8: The groups K_{n}[x_{1}, …, x_{n}] and G.W. Whitehead’s tensor product filtration of X^{n}. §1.9: On endomorphisms of K_{n}^{R}[x_{n}, …, x_{n}]. §1.10: Simplicial objects associated to K_{n}^{R};Theorems H and J.

Chapter 2

§2.1: The groups H_{n}^{R}. §2.2: The distributivity law in H_{n}^{R}. §2.3 An automorphism of H_{n}^{R}. §2.4: A filtration of H_{n}^{R}. §2.5: Some null-homotopic maps. §2.6: On exponents for Δ(H_{n}^{R}).

Chapter 1:

Recall that all spaces and maps in this article are restricted to the category τ of compactly generated, weak Hausdorff spaces with non-degenerate base-point ∗. Furthermore, [X, Y] denotes the set of pointed homotopy class of maps from X to Y, ∑( ) denotes the reduced suspension functor, and Ω denotes the pointed looping functor. Recall the isomorphism of groups

[ΣX, Y] → [X, ΩY].

§1.1: The group K_{n}^{R}[x_{1}, …, x_{n}] and universal maps.

Consider the group [X^{n}, ΩΣX]. There exist n canonical elements in this group p_{1}, …, p_{n} specified by the composites

X^{n} ⟶^{π_{i}} X
p_{i}↓ ↓^{E}
ΩΣX ⟶_{1} ΩΣX

where π_{i} denotes projection on the ith coordinate and E is the Freudenthal suspension. Thus there is a group homomorphism

θ̄: F [p_{1}, …, p_{n}] → [X^{n}, ΩΣX]

where F [p_{1}, …, p_{n}] is the free group with generators p_{1}, …, p_{n} and θ̄(p_{i}) = p_{i}. The homomorphism θ̄ is sometimes a monomorphism as one sees in

Example 1.1.1. If X = S^{0} or X = ℂP^{∞}, then θ̄ is a monomorphism.

If X is itself a suspension, it will be shown that θ̄ has a non-trivial kernel.

Definition 1.1.2. Let K_{n}[x_{1}, …, x_{n}] to be the smallest quotient of F [p_{1}, …, p_{n}] such that there exists a commutative diagram in the category of groups

F [p_{1}, …, p_{n}] ⟶^{θ̄} [X^{n}, ΩΣX]
↓  ↓^{1}
K_{n}[x_{1}, …, x_{n}] ⟶^{θ} [X^{n}, ΩΣX]

where θ(x_{i}) = p_{i} for every suspension X = Σ(X′).

There are useful torsion analogues of these groups which are obtained as follows. Let α : ΣX → Y be a map which has order p^{r} in the (possibly non-abelian) group [ΣX, Y]. There is the induced group homomorphism

(Ωα)_{∗} : [X^{n}, ΩΣX] → [X^{n}, ΩY]

obtained by composition with Ωα.

Definition 1.1.3. Let R = ℤ/p^{r}ℤ. The group K_{n}^{R}[x_{1}, …, x_{n}] is the smallest quotient of K_{n}[x_{1}, …, x_{n}] such that there exists a commutative diagram

K_{n}[x_{1}, …, x_{n}] ⟶_{θ} [X^{n}, ΩΣX]
↓  ↓^{(Ωα)_{∗}}
K_{n}^{R}[x_{1}, …, x_{n}] ⟶_{θ} [X^{n}, ΩY]

for every suspension X = Σ(X′) and every α in [ΣX, Y] of order p^{r}.

One might regard K_{n}^{R}[x_{1}, …, x_{n}] as a universal example for the groups [X^{n}, ΩY]. For brevity of notation, write K_{n}^{R} = K_{n}^{R}[x_{1}, …, x_{n}] where R = ℤ or ℤ/p^{r}ℤ with

14 F.R. COHEN^{∗}

K_{n}^{ℤ} = K_{n}[x_{1}, …, x_{n}]. A presentation for the groups K_{n}^{R} will be given in the next theorem. Let G be any group and write [g, h] = ghg^{−1}h^{−1} (the topologist’s commutator). Inductively, write

[g_{1}, …, g_{n}] = [[g_{1}, …, g_{n−1}], g_{n}]

for g_{i} in G; these elements are sometimes called either simple products or left normalized commutators.

Theorem 1.1.4. A choice of presentation for K_{n}^{R}[x_{1}, … x_{n}] with R = ℤ or ℤ/p^{r}ℤ is given as follows.

generators : x_{1}, …, x_{n}
relations : (1)[x_{i_{1}}, …, x_{i_{k}}] = 1 if x_{i_{j}} = x_{i_{ℓ}} for some j < ℓ, and
(2)[x_{i_{1}}^{n_{1}}, x_{i_{2}}^{n_{2}}, …, x_{i_{k}}^{n_{k}}] = [x_{i_{1}}, …, x_{i_{k}}]^{n}, n = n_{1}n_{2} … n_{k}.
(3)If R = ℤ/p^{r}ℤ, then x_{i}^{p^{r}} = 1 for i = 1, …, n.

That relation (1) implies relation (2) and is thus redundant follows at once by induction on k together with the equation [a, bc] = [a, b] · [a, c] · [[c, a], b] for any elements a, b, c. Notice that the presentation given in 1.1.4 is not finite. That these groups are finitely presented follows immediately from an analysis of their descending central series. Namely, given a group G, the descending central series is given by

G = Γ^{1} ≥ Γ^{2} ≥ ⋯ ≥ Γ^{q} ≥ …

where Γ^{q+1} = [G, Γ^{q}] for which Γ^{q}/Γ^{q+1} is an abelian group.

Fix ordered sequences of t distinct integers J = (j_{1}, …, j_{t}) with 1 ≤ j_{i} ≤ n for all i and 1 ≤ t ≤ n. Define the length of J, ℓ(J), to be t. The sequence J is said to be admissible if 1 ≤ j_{1} < j_{2} < ⋯ < j_{t} ≤ n. The sequences of length t are ordered lexicographically from the left. Namely, if I = (i_{1}, …, i_{t}), then J < I provided there exists an integer r such that j_{ℓ} = i_{ℓ} for all ℓ < r, and j_{r} < i_{r}.

Let J_{t} be an admissible sequence of length t. Fix an element σ in the symmetric group on (t − 1) letters acting on the set {2, …, t}. Define elements

γ(J_{t}; σ) = [x_{j_{1}}, x_{j_{σ(2)}}, …, x_{j_{σ(t)}}]

for all admissible sequences of length t, J_{t}. Write σJ_{t} for the sequence (j_{1}, j_{σ(2)}, … j_{σ(t)}). Theorem A follows at once from the next result.

Theorem 1.1.5. The descending central series for K_{n}^{R}[x_{1}, …, x_{n}], R = ℤ/p^{r}ℤ or ℤ, satisfies Γ^{n+1} = {1} and there are isomorphisms of R-modules

E_{0}^{q} → ⨁_{(q−1)!\binom{n}{q}} R

for 1 ≤ q ≤ n. A basis for E_{0}^{q} as an R-module is given by left coset representatives of the elements

γ(J_{q}; σ) = [x_{i_{1}}, x_{i_{σ(2)}}, …, x_{i_{σ(q)}}]

where σ runs over the elements in Σ_{q−1}, the symmetric group on (q − 1) letters acting on the set {2, …, q} and J_{q} runs over all admissible sequences of length q.

Corollary 1.1.6. If R = ℤ or ℤ/p^{r}ℤ, K_{n}^{R}[x_{1}, …, x_{n}] is finitely presented.

Another immediate consequence of Theorem 1.1.5 is a normal form for the elements z in the group K_{n}^{R}[x_{1}, …, x_{n}]. Fix an integer q with 1 ≤ q ≤ n and consider admissible sequences of length q with σ in Σ_{q−1}. Further, consider the words

λ_{q} = ∏_{J_{q},σ} γ(J_{q}; σ)^{m(J_{q};σ)}, m(J_{q}; σ) ∈ R,

where the order of multiplication is specified by the left lexicographical ordering of the sequences σJ_{q}.

Corollary 1.1.7. Every element z in K_{n}^{R}[x_{1}, …, x_{n}] is equal to a product

λ_{1} · λ_{2} … λ_{n}

for unique choices of exponents m(J_{q}; σ).

Proof. Existence follows from the fact that the γ(J_{q}; σ) generate E_{0}^{q} and that Γ^{n+1} = {1}. Uniqueness follows from the fact that E_{0}^{q} is a free R-module for 1 ≤ q ≤ n.

The groups K_{n}^{R} admit “natural” surjections to the group of upper triangular matrices in GL(n+1, R), T (n+1, R), as follows: Recall that T (n+1, R) is generated as a group by the matrices a_{k}, 1 ≤ k ≤ n, with entries

a_{k}(i, j) = \{ \begin{matrix}1 if i = j \\ 1 if i = k and j = k + 1, \\ 0 otherwise. \end{matrix}

16 F.R. COHEN^{∗}

Thus there is a surjection from the free group on n letters

θ̄: F [p_{1}, …, p_{n}] ⟶ T (n + 1, R)

with θ(P_{k}) = a_{k}. Since the a_{k} satisfy the relations in Theorem 1.1.4, there is an induced morphism

θ : K_{n}^{R}[x_{1}, …, x_{n}] ⟶ T (n + 1, R)

with θ(x_{k}) = a_{k}. In the special case with n = 2, θ is an isomorphism. However θ has a non-trivial kernel if n ≥ 3.

There are stabilization maps

σ : K_{n}^{R} ⟶ K_{n+1}^{R}

with σ(x_{i}) = x_{i}. These extend to give

θ : K_{∞}^{R} ⟶ GL(R)

where K_{∞}^{R} = \underrightarrow{lim} K_{n}^{R} and θ(x_{k}) = a_{k}. As an aside, if R = 𝔽_{p}, then θ^{∗} is trivial in mod-p cohomology for n = ∞ [FP].

§1.2: On the relations in K_{n}^{R}[x_{1}, …, x_{n}].

Consider the elements p_{i} in [X^{n}, ΩΣX] given by the composite X^{n} →^{π_{i}} X →^{E} ΩΣX which are specified in §1.1. Fix the left normalized commutators given by [p_{i_{1}}, …, p_{i_{k}}] = [… [p_{i_{1}}, p_{i_{2}}]p_{i_{3}} …]p_{i_{k}}]. As an element of the group [X^{n}, ΩΣX], this last commutator is represented by the following composite:

X^{n} →^{Δ^{k}} X^{n} × ⋯ × X^{n}_{←k→} →^{p_{i_{1}} ×···×p_{i_{k}}} ΩΣX × ⋯ × ΩΣX_{←k→} →^{c̄_{k}} ΩΣX

where Δ^{k} is the k-fold diagonal and c̄_{k} is the left normalized commutator given inductively by c̄_{k+1} = c̄_{2} ∘ (c̄_{k} × 1) with c̄_{2} the natural left normalized 2-fold commutator.

Notice that c̄_{k} factors, up to homotopy, throught the k-fold smash product to yield a homotopy commutative diagram

ΩΣX × ⋯ × ΩΣX ⟶^{c̄_{k}} ΩΣX
↓ ↓^{1}
ΩΣX ∧ ⋯ ∧ ΩΣX_{←k→} ⟶_{c_{k}} ΩΣX.

Thus, there is a homotopy commutative diagram

X^{n} ⟶^{Δ^{k}} (X^{n})^{k} ⟶^{π_{i_{1}} ×···×π_{i_{k}}} X^{k} ⟶^{E^{k}} (ΩΣX)^{k} ⟶^{c̄_{k}} ΩΣX
1↓ ↓ ↓ ↓^{1}
X^{(k)} ⟶^{c_{k}} (ΩΣX)^{(k)} ⟶^{c_{k}} ΩΣX
↓^{1}
X^{n}  ⟶^{[p_{i_{1}} ,p_{i_{2}} ,... ,p_{i_{k}}]}  ΩΣX

where X^{(k)} denotes the k-fold smash product.

Next, notice that if p_{i_{j}} = p_{i_{ℓ}} for j < ℓ, then one factor in the above composite itself factors through the diagonal map Δ : X → X × X. Thus if X is a co-H space, the commutator [p_{i_{1}}, …, p_{i_{k}}] represents 1 in [X^{n}, ΩΣX] and the next lemma follows.

Lemma 1.2.1. If X is a co-H space in τ and p_{i_{j}} = p_{i_{ℓ}} for some 1 ≤ j < ℓ ≤ k, then [p_{i_{1}}, p_{i_{2}}, …, p_{i_{k}}] = 1 in [X^{n}, ΩΣX].

Next, fix an integer q = q_{1} … q_{k} and consider [p_{i_{1}}, …, p_{i_{n}}]^{q}. Recall that if X is a co-group, then the diagram

X ⟶^{E} ΩΣX
[q]↓ ↓^{q}
X ⟶_{E} ΩΣX

homotopy commutes where q denotes the qth power map on ΩΣX and [q] : X → X is a choice of the qth power map in [X, X] if X is a cogroup. Thus the diagram

X^{(k)} ⟶ (ΩΣX)^{(k)} ⟶^{c_{k}} ΩΣX
[q_{1}]∧···∧[q_{k}]↓ ↓^{q}
X^{(k)} ⟶ (ΩΣX)^{(k)} ⟶^{c_{k}} ΩΣX

homotopy commutes. The next lemma follows.

Lemma 1.2.2. If X is a cogroup, then [p_{i_{1}}^{q_{1}}, …, p_{i_{k}}^{q_{k}}] = [p_{i_{1}}, …, p_{i_{k}}]^{q} in [X^{n}, ΩΣX] where q = q_{n} … q_{k}.

Let α : ΣX → Y with p^{r}α = 0 in [ΣX, Y]. Combining the above, one has

18 F.R. COHEN^{∗}

Theorem 1.2.3. If X is a co-group in τ, then (1) [p_{i_{1}}, …, p_{i_{k}}] = 0 in [X^{n}, ΩΣX] if p_{i_{j}} = p_{i_{ℓ}} for some 1 ≤ j < ℓ ≤ k,

(2) If q = q_{1} … q_{k}, then [p_{i_{1}}^{q_{1}}, p_{i_{2}}^{q_{k}}, …, p_{i_{k}}^{q_{k}}] = [p_{i_{1}}, …, p_{i_{k}}]^{q} in [X^{n}, ΩΣX], and

(3) If α : ΣX → Y with p^{r}α = 0 in [ΣX, Y], then p^{r}(Ωα)_{∗}(p_{i}) = 0 in [X^{n}, ΩY].

That the relations in Theorem 1.2.3 are redundant follows from the remarks after Theorem 1.1.4. In particular, relation (1) above implies relation (2).

Again recall the descending central series {Γ^{q}} of a group G where Γ^{1} = G together with the following theorem [MKS, p. 293].

Theorem 1.2.4. Let a, b, c be elements of a group G. Let k, m, n be positive integers with a ∈ Γ^{k}, b ∈ Γ^{m}, and c ∈ Γ^{n}. Then

[[ab], c] · [[ca]b] · [[bc], a] ≡ 1 mod Γ^{k+m+n+1}.

This theorem is actually stated using the commutators (a, b) = a^{−1}b^{−1}ab in [MKS], but the revision stated here is transparent. This theorem will be used when a, b, and c are specialized to commutators as in Theorem 1.2.3.

Theorem 1.2.5. Let a = [p_{i_{1}}, …, p_{i_{m}}], b = [p_{j_{1}}, …, p_{j_{n}}], and c = [p_{k_{1}}, …, p_{k_{q}}]. Then

[[ab], c] · [[ca], b] · [[bc], a] = 1

in the group [X^{n}, ΩΣX] where X is assumed to be a cogroup.

Proof. If G denotes the subgroup of [X^{n}, ΩΣX] generated by p_{i_{1}}, … p_{i_{m}}, p_{j_{1}}, …, p_{j_{n}}, p_{k_{1}}, … p_{k_{q}}, then G is nilpotent of class at most m+n+q by Lemma 1.2.1. The theorem follows at once from 1.2.4.

Notice that the relations in Theorems 1.1.4 hold in K_{n}^{R}[x_{1}, …, x_{n}] by Theorem 1.2.3. It has not yet been shown that these relations, in fact, provide a presentation of these groups. That Theorem 1.1.4 actually gives a presentation for the groups K_{n}^{A}[x_{1}, …, x_{n}] will be shown in section 1.6.

§1.3: Non-commutative analogues of exterior algebras; the structure of A_{n}^{R}[y_{1}, …, y_{n}].

The purpose of this section is to introduce a standard tool used in combinatorial group theory to study the nilpotent groups like K_{n}^{R}[x_{1}, …, x_{n}]. Let R be a commutative ring with unit. Let V be a free R-module of rank n such that {y_{1}, …, y_{n}} is a choice of basis for V.

Definition 1.3.1. The algebra A_{n}^{R}[y_{1}, … y_{n}] is the quotient of the tensor algebra T [V] modulo the two-sided ideal J generated by all monomials y_{i_{1}} ⊗ ⋯ ⊗ y_{i_{t}} where y_{i_{j}} = y_{i_{k}} for some j and k with 1 ≤ j < k ≤ t. There is an augmentation ε : A_{n}^{R}[y_{1}, …, y_{n}] → R specified by ε(1) = 1 and ε(y_{i}) = 0 for all i. The kernel of ε is denoted by IA_{n}^{R}.

Notice that A_{n}^{R}[y_{1}, …, y_{n}] is a non-commutative analogue of the exterior algebra generated by V as the natural quotient of A_{n}^{R}[y_{1}, …, y_{n}] is that exterior algebra. Also, notice that the definition of A_{n}^{R}[y_{1}, … y_{n}] depends on a choice of basis for V; write A_{n}^{R} for A_{n}^{R}[y_{1}, … y_{n}] thus suppressing the choice of basis.

Clearly V is a left R[Σ_{n}]-module where σ(v_{i}) = v_{σ(i)} for σ in Σ_{n}. Thus T [V] is a left R[Σ_{n}]-module, and, by definition 1.3.1, Σ_{n} preserves the ideal J. It follows at once that Σ_{n} is isomorphic to a subgroup of the group of algebra automorphisms of A_{n}^{R}[y_{1}, …, y_{n}] and this algebra is a left R[Σ_{n}]-module. This module structure is described below.

Given an ordered sequence of integers I = (i_{1}, …, i_{k}) with 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ n, declare that I is admissible as before when considering groups in Section 1.1. Define ℓ(I) to be k. Given an admissible sequence I, let y_{I} = y_{i_{1}} ⊗ ⋯ ⊗ y_{i_{k}} in A_{n}^{R}[y_{1}, …, y_{n}]. Thus for fixed k, there are \binom{n}{k} choices of y_{I} and the stabilizer of any y_{I} in Σ_{n} is isomorphic to Σ_{n−k}, the subgroup of Σ_{n} which fixes i_{1}, …, i_{k} pointwise. Since this stabilizer depends on the choice of I, write Σ_{n−k}(I) for it. There is a left R[Σ_{n}]-module map

θ(I) : R[Σ_{n}/Σ_{n−k}(I)] → A_{n}^{R}[y_{1}, …, y_{n}]

defined by θ(I)(σ) = σ(y_{I}) for σ in Σ_{n}. Define

θ : ⨁_{k=0}^{n} ( ⨁_{Iadmissible ; ℓ(I)=k} R[Σ_{n}/Σ_{n−k}(I)]) → A_{n}^{R}[y_{1}, …, y_{n}]

by θ(I) when restricted to R[Σ_{n}/Σ_{n−k}(I)].

Theorem 1.3.2. The algebra A_{n}^{R}[y_{1}, …, y_{n}] is a free R-module of dimension ∑_{k=0}^{n} k!\binom{n}{k} as an R-module. A basis is given by 1 and y_{i_{σ(1)}} ⊗ ⋯ ⊗ y_{i_{σ(k)}} = σ(y_{I})

20 F.R. COHEN^{∗}

for admissible sequences I = (i_{1}, … i_{k}), 1 ≤ k ≤ n, and σ in Σ_{k}. Furthermore, the map θ is an isomorphism of left R[Σ_{n}]-modules.

Proof. By definition 1.3.1, A_{n}^{R}[y_{1}, … y_{n}] is a free R-module with basis as stated in 1.3.2. Furthermore, θ is R[Σ_{n}]-linear and is an epimorphism of R-modules where the source and target are both free and of the same finite rank. The theorem follows.

Notice that there is an R-linear map

γ : ⨁_{k=1}^{n}( ⨁_{Iadmissible ; ℓ(I)=k} R[Σ_{k}]) → A_{n}^{R}[y_{1}, …, y_{n}]

defined by γ(σ) = σ(y_{I}) for I admissible and γ is restricted to Σ_{k}. The map γ ignores the R[∑_{n}]- module structure.

Corollary 1.3.3. The map γ is an isomorphism of R-modules.

It is natural to ask whether A_{n}^{R} is a Hopf algebra. The answer is no if n ≥ 2 as J is not a Hopf ideal.

Let L[V] be the free lie algebra generated by V. The algebra A_{n}^{R} = A_{n}^{R}[y_{1}, …, y_{n}] is a quotient of the tensor algebra T [V]. Define L_{n}^{R}[y_{1}, …, y_{n}] to be the image of L[V] in A_{n}^{R}. Define the Lie filtration of A_{n}^{R} in the standard way [MM] by

(1) F_{0}A_{n}^{R} = R,

(2) F_{1}A_{n}^{R} = R ⊕ L_{n}^{R}[y_{1}, …, y_{n}], and

(3) F_{q+1}A_{n}^{R} = the image of F_{q}A_{n}^{R} ⨿ (L_{n}^{R}[y_{1}, …, y_{n}] ⊗ F_{q}A_{n}^{R}) in A_{n}^{R}. Let E_{k}^{0}A_{n}^{R} = F_{k}A_{n}^{R}/F_{k−1}A_{n}^{R}.

To describe the associated graded algebra E_{∗}^{0}A_{n}^{R}, some additional information is required. Given I = (i_{1}, …, i_{k}) admissible, let τ be in the symmetric group Σ_{k−1} which acts on the set {2, … k}. Define an element which depends on τ and I as follows:

L(I, τ) = [… [y_{i_{1}}, y_{i_{τ(2)}}]y_{i_{τ(3)}}] …]y_{i_{τ(k)}}].

There are (k − 1)!\binom{n}{k} choices of L(I, τ) for any fixed k.

Next, given two admissible sequences I = (i_{1}, …, i_{k}) and J = (j_{1}, …, j_{q}), define a partial ordering by requiring that I < J if either

(1) k < q or

(2) k = q, i_{1} = j_{1}, i_{2} = j_{2}, …, i_{s} = j_{s}, and i_{s+1} < j_{s+1}.

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 21

Partially order the elements L(I, τ) by that induced by the partial ordering on the admissible sequences I. Notice that this partial ordering ignores the permutation τ and is also not defined for L(I, τ) and L(J, σ) if I = J. In addition, define

|I| = {y_{i_{1}}, …, y_{i_{k}}}, a subset of A_{n}^{R}[y_{1}, …, y_{n}].

The next theorem is the analogue of the Poincaré-Birkhoff-Witt theorem for A_{n}^{R}. However, A_{n}^{R} is not the universal enveloping algebra of L_{n}^{R}.

Theorem 1.3.4. The associated graded algebra E_{∗}^{0}A_{n}^{R} is a free R-module with basis 1 and

L(I_{α_{1}}, τ_{α_{1}}) · L(I_{α_{2}}, τ_{α_{2}}) · . . . · L(I_{α_{r}}, τ_{α_{r}})

with (1) I_{α_{1}} < I_{α_{2}} < ⋯ < I_{α_{s}}

(2) |I_{α_{i}}| ∩ |I_{α_{j}}| = ϕ if i ≠ j. A basis for E_{k}^{0}A_{n}^{R} is specified by the above with ℓ(I_{α_{1}})+⋯+ℓ(I_{α_{r}}) = k. The associated graded E_{∗}^{0}A_{n}^{R} is not an exterior algebra if n ≥ 2 even though L(I_{α_{j}}, τ_{α_{j}})^{2} = 0. In particular L(I, τ) · L(J, σ) = 0 if |I| ∩ |J| ≠ ϕ.

The algebras A_{n}^{R}[y_{1}, …, y_{n}] have additional natural structure. In particular, define

d_{i} : A_{n}^{R}[y_{1}, …, y_{n}] → A_{n−1}^{R}[y_{1}, …, ŷ_{i}, …, y_{n}]

by

d_{i}(y_{j}) = \{ \begin{matrix}y_{j} if i = j \\ 0 if i = j, \end{matrix}

and requiring d_{i} to be a morphism of algebras after noticing that the analog of the multiplication extension of d_{i} for T [V] preserves J. Consider the direct product of

algebras x_{i=1}^{n} A_{n−1}^{R}[y_{1}, …, ŷ_{i}, …, y_{n}] and notice that

d : A_{n}^{R}[y_{1}, …, y_{n}] → x_{i=1}^{n} A_{n−1}^{R}[y_{1}, …, ŷ_{i}, …, y_{n}]

given by d(x) = (d_{1}x, d_{2}x, …, d_{n}x) is a map of algebras. Filter x_{i=1}^{n} A_{n−1}^{R}[y_{1}, …, ŷ_{1}, …, y_{n}]

by the product filtration obtained from the Lie filtration on each factor. (Here F_{q}(A × B) = ∪_{i}(F_{i}A × F_{q−i}B).) By definition, d preserves filtration one of the Lie filtration and thus preserves all filtrations as d is multiplicative.

22 F.R. COHEN^{∗}

Lemma 1.3.5. The algebra morphism d is a morphism of filtered algebras. Furthermore, the kernel of d is the two sided ideal generated by y_{σ} = y_{σ(1)} ⊗ ⋯ ⊗ y_{σ(n)} for σ in Σ_{n}. Thus the kernel of d is a free R-module of rank n! with basis {y_{σ}|σ ∈ Σ_{n}}.

Proof. That d is a morphism of filtered algebras was proven above.

Next observe that the ideal generated by the elements

y_{σ} = y_{σ(1)} ⊗ y_{σ(2)} ⊗ ⋯ ⊗ y_{σ(n)}, σ ∈ Σ_{n},

is clearly in the kernel of d, ker(d), because d_{i}(y_{j}) = 0 if i ≠ j. By definition 1.3.1, y_{σ}y_{i} = 0 for all i = 1, …, n. Thus the ideal generated by {y_{σ}|σ ∈ Σ_{n}} is the linear span of the y_{σ}; call this ideal W. Notice that W has dimension n! as an R-module with basis {y_{σ}|σ ∈ Σ_{n}} by Theorem 1.3.2. Next, let x ∈ ker(d). By Theorem 1.3.2, x is a linear combination of 1 and y(I, τ) where I = (i_{1}, … i_{k}) admissible with y(I, τ) = y_{i_{τ(1)}} ⊗ y_{i_{τ(2)}} ⊗ ⋯ ⊗ y_{i_{τ(k)}} and τ ∈ Σ_{k}. Thus if x = c · 1 + Σ_{I}c(I, τ)y(I, τ) for scalars c(I, τ), then d(x) ≠ 0 if either c ≠ 0 or c(I, τ) ≠ 0 for k < n by inspection. Thus if x ∈ ker (d), then x ∈ W and the lemma follows.

Proof of Theorem 1.3.4. Induct on n with the case n = 1 being trivial. Notice that the listed elements span E_{∗}^{0}A_{n}^{R} for all n and thus it suffices to prove that they are linearly independent. If k < n, then

E^{0}(d) : E_{k}^{0}A_{n}^{R} → E_{k}^{0}( x_{i=1}^{n} A_{n−1}^{R}[y_{1} …, ŷ_{i}, …, y_{n}])

is a monomorphism. Thus it suffices to see that the elements L(i_{α_{1}}, τ_{α_{1}}) … L(I_{α_{r}}, τ_{α_{r}}) are independent when ℓ(I_{α_{1}})+⋯+ℓ(I_{α_{r}}) = n. But this is clear because the ideal W in the proof of 1.3.5 is a summand of the tensor algebra T [V] with these monomials as a basis for it. The theorem follows.

Notice that A_{n}^{R} is not, in general, the universal enveloping algebra for the Lie algebra L_{n}^{R}[y_{1}, …, y_{n}]. Thus A_{n}^{R} differs from the group algebra for K_{n}^{R} [Q].

§1.4: Some formulae in K_{n}^{R}[x_{1}, …, x_{n}].

The purpose of this section is to record some useful technical properties of K_{n}^{R}[x_{1}, …, x_{n}]. Fix an integer q, 1 ≤ q ≤ n, and let ⟨x_{q}⟩ denote the subgroup of K_{n}^{R}[x_{1}, …, x_{n}] generated by x_{q} and [z, x_{q}] for z in K_{n}^{R}[x_{1}, … x_{n}].

Lemma 1.4.1. Let a, b, c be elements in a group G. Then

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

The proof is immediate from the definitions; alternatively see [MKS, p. 290].

Corollary 1.4.2. For fixed q, ⟨x_{q}⟩ is an abelian subgroup of K_{n}^{R}[x_{1}, …, x_{n}].

Proof. Since z is in K_{n}^{R}[x_{1}, …, x_{n}], z is given by a word in x_{1}, …, x_{n}. By repeated application of Lemma 1.4.1 and the relation

[x_{i_{1}}^{n_{1}}, …, x_{i_{k}}^{n_{k}}] = [x_{i_{1}}, …, x_{i_{k}}]^{n_{1}...n_{k}},

[z, x_{1}] is a product of words

[y_{i_{1}}[y_{i_{2}}[… [y_{i_{t}}, x_{q}]]^{n_{J}}

for various choices of J = (i_{1}, …, i_{t}) and y_{i_{ℓ}} run over various choices of the generators x_{1}, …, x_{n}. Thus it suffices to check that

u = [u_{1}[u_{2}[… [u_{k}, x_{q}] …] and
v = [v_{1}[v_{2}[… [v_{ℓ}, x_{q}] …]

commute where u_{i} and v_{j} run over various choices of generators x_{1}, …, x_{n}. But [u, v] = 1 by Theorems 1.2.3 and 1.2.5. Thus the lemma follows.

Theorem 1.4.3. Fix q, 1 ≤ q ≤ n, and let M = Π[y, x_{q}]. Then

(1) [x_{1}x_{2} … x_{ℓ+1}, M] = [x_{1} … x_{ℓ}, [x_{ℓ+1}, M]] · [x_{1} … x_{ℓ}, M] · [x_{ℓ+1}, M] and

(2) [x_{1} … x_{ℓ}, M] = ∏_{1≤i_{1}<i_{2}<···<i_{k≤ℓ}}[x_{i_{1}}[x_{i_{2}}[… [x_{i_{k}}, M] …].

Before proving 1.4.3, notice that the elements [x_{i_{1}}[x_{i_{2}}[… [x_{i_{k}}, M] …] all commute by 1.4.2; thus the order of multiplication is immaterial in Theorem 1.4.3(2).

Proof of Theorem 1.4.3. Formula (1) is gotten by a direct substitution in Lemma 1.4.1. Formula (2) is correct for ℓ = 1. Assume inductively that it is correct for ℓ

24 F.R. COHEN^{∗}

and check it for ℓ + 1: Elements of the form [x_{i_{1}}, [x_{i_{2}}, …, [x_{i_{k}}, M] …] all commute by 1.4.2 and the following 2 formulae hold:

(i) [x_{1} … x_{ℓ}, [x_{ℓ+1}, M]] = ∏_{1≤i_{1}<i_{2}<···<i_{k}≤ℓ}[x_{i_{1}}[x_{i_{2}}[… [x_{i_{k}}, [x_{ℓ+1}, M] …], and

(ii) [x_{1} … x_{ℓ}, M] = ∏_{1≤i_{1}<i_{2}<···<i_{k}≤ℓ}[x_{i_{1}}[x_{i_{2}}[… [x_{i_{k}}, M] …].

The theorem follows.

Next, consider the elements

γ(k; σ) = ∏_{1≤i_{i}<···<i_{k}≤n}[… [x_{i_{1}}, x_{i_{σ(2)}}]x_{i_{σ(3)}}] …]x_{i_{σ(k)}}]

where the product is taken in the left lexicographical ordering. Thus for fixed n ≥ k, define

γ(k, n; σ) = ∏_{11≤i_{1}<···<i_{k}≤n}[… [x_{i_{1}}, x_{i_{σ(2)}}]x_{i_{σ(3)}}] …]x_{i_{σ(k)}}]

with

γ(k, k; σ) = [… [x_{1}, x_{σ(2)}]x_{σ(3)}] …]x_{σ(k)}] and

inductively

γ(k, n + 1; σ) = γ(k, n; σ) · Γ(k, n; σ)

where

Γ(k, n; σ) = ∏_{\begin{matrix}1≤i_{1}<···<i_{k}≤n+1 \\ i_{k}=n+1 \end{matrix}}[… [x_{i_{1}}, x_{i_{σ(2)}}]x_{i_{σ(3)}}] …]x_{i_{σ(k)}}].

To conveniently state the next two results let L = γ(k, n−1; σ) and Γ = Γ(k, n− 1; σ). Furthermore, if a and b are elements in a group G, define ad(x)(y) = [y, x] and inductively ad^{i+1}(x)(y) = [ad^{i}(x)(y), x].

Theorem 1.4.4. If m is a positive integer, the following formula holds in K_{n}^{R}[x_{1}, …, x_{n}]:

(L · Γ)^{m} = L^{m} · Γ^{m} · ∏_{i=1}^{m−1}[ad^{i}(L^{−1})(Γ^{−1})]^{\binom{m}{i+1}}.

Corollary 1.4.5. If R = ℤ/p^{r}ℤ, the following formula holds in K_{n}^{R}[x_{1}, … x_{n}]:

(L · Γ)^{p^{r}} = L^{p^{r}} · ∏_{\begin{matrix}i=1 \\ i+1≡0 mod (p) \end{matrix}}^{p^{r}−1} ad^{i}(L^{−1})(Γ^{−1})^{\binom{p^{r}}{i+1}}

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 25

Proof. The elements ad^{i}(L^{−1})(Γ^{−1}) lie in the group ⟨x_{n}⟩ by Corollary 1.4.2 and are products of elements [x_{i_{1}}[x_{i_{2}}[… [x_{i_{t}}, x_{n}] …] which are again in ⟨x_{n}⟩. Since ⟨x_{n}⟩ is abelian by Corollary 1.4.2, ad^{i}(L^{−1})(Γ^{−1})^{q} is equal to a product of elements

[x_{i_{1}}[x_{i_{2}}[… [x_{i_{t}}, x_{n}] …]^{q} = [x_{i_{1}}^{q}[x_{i_{2}}[… [x_{i_{t}}, x_{n}] …]

which is trivial if q ≡ 0 mod ℤ/p^{r}ℤ. Since \binom{p^{r}}{i+1} ≡ 0 mod ℤ/p^{r}ℤ if i + 1 ≢ 0 mod p, the corollary follows.

Proof of Theorem 1.4.4. The proof is by induction on m with the case m = 1 being

immediate. Thus assume that (L · Γ)^{m} = L^{m} · Γ^{m} · ∏_{i=1}^{m−1}[ad^{i}(L^{−1}(Γ^{−1})]^{\binom{m}{i+1}} and use

the formula ab = ba[a^{−1}, b^{−1}] to obtain the following:

(i) (L · Γ)^{m+1} = L^{m} · Γ^{m} · ∏_{i=1}^{m−1}[ad^{i}(L^{−1})(Γ^{−1})]^{\binom{m}{i+1}} · (L · Γ).
(ii) (L · Γ^{m+1} = L^{m+1}Γ^{m}[Γ^{−m}, L^{−1}] · ∏_{i=1}^{m−1}[ad^{i}(L^{−1})(Γ^{−1^{\binom{m}{i+1}}} · [ad^{i}(L^{−1})(Γ^{−1^{−\binom{m}{i+1}}}, L^{−1}] · Γ.

Since Γ and ad^{i}(L^{−1})(Γ^{−1}) are in the group ⟨x_{n}⟩, these elements commute by Corollary 1.4.2. Furthermore

[ad^{i}(L^{−1})(Γ^{−1})^{−q}, L^{−1}] = [ad^{i}(L^{−1})(Γ^{−1})^{q}, L^{−1}]^{−1} = [L^{−1}, ad^{i}(L^{−1})(Γ^{−1})^{q}] by

Corollary 1.4.2 as ad^{i}(L^{−1})(Γ^{−1}) is a product of elements of the form

x_{I} = [x_{i_{1}}[x_{i_{2}}[… [x_{i_{j}}, x_{n}] …]^{n_{j}} and [x_{I}^{−q}, L^{−1}] = [x_{I}^{q}, L^{−1}]^{−1}.

Thus one has

(iii) (L · Γ)^{m+1} = L^{m+1} · Γ^{m+1} · ∏_{i=0}^{m−1} ad^{i+1}(L^{−1})(Γ^{−1})^{\binom{m}{i+1}}ad^{i+1}(L^{−1})(Γ^{−1})^{\binom{m}{i+2}}

and the theorem follows from the identity for binomial coefficients \binom{m}{i+1} + \binom{m}{i+2} = \binom{m+1}{i+2}.

§1.5: A subgroup of K_{n}^{R}[x_{1}, …, x_{n}].

Fix an integer n and let Z_{n}^{R} denote the subgroup of K_{n}^{R}[x_{1}, … x_{n}] generated by the elements

γ(n; σ) = [… [x_{1}, x_{σ(2)}]x_{σ(3)}] …]x_{σ(n)}]

26 F.R. COHEN^{∗}

for all σ in ∑_{n−1}. Since the elements γ(n; σ) are in the group ⟨x_{1}⟩, they all commute by Corollary 1.4.2. Furthermore if R = ℤ/p^{r}ℤ, then γ(n; σ)^{p^{r}} = [… [x_{1}^{p^{r}}, x_{σ(2)}] …]x_{σ(n)}] = 1. Thus if R = ℤ or ℤ/p^{r}ℤ, there is an epimorphism of R-modules

λ : ⨁_{(n−1)!} R → Z_{n}^{R}

prescribed by the generators γ(n; σ). The main result in this section is

Theorem 1.5.1. If R = ℤ or ℤ/p^{r}ℤ, then λ is an isomorphism of groups.

Since λ is an epimorphism, it suffices to show that λ is also a monomorphism. Furthermore, since the groups K_{n}^{R}[x_{1}, …, x_{n}] are defined using topological properties of the group [X^{n}, ΩY], some topology will be used in the proof of Theorem 1.5.1. Here, let X = Σ(X′) and notice that θ(γ(n; σ)) in [X^{n}, ΩΣX] where θ is the natural map from K_{n}[x_{1}, …, x_{n}] to [X^{n}, ΩΣX], is given by an n-fold Samelson product and thus the next result is immediate.

Lemma 1.5.2. If X = Σ(X′), there exist maps \overline{γ(n; σ)} : X^{(n)} → ΩΣX together with a homotopy commutative diagram

X^{n} ⟶^{θ(γ(n;σ))} ΩΣX
p↓ ↓^{1}
X^{(n)} ⟶_{\overline{γ(n;σ)}} ΩΣX.

Next, observe that if X = Σ(X′), then the group [X^{(n)}, ΩY] is abelian. Furthermore, if R = ℤ/p^{r}ℤ and α : ΣX → Y has order p^{r}, then the Samelson product (Ωα)·\overline{γ(n; σ)} obtained from 1.5.2 has order p^{r} in [X^{(n)}, ΩY] and so the next lemma follows.

Lemma 1.5.3. There is a commutative diagram of groups given by

⨁_{(n−1)!} R ⟶^{δ} [X^{(n)}, ΩY] ⟶^{p^{∗}} [X^{n}, ΩY]
↓  ↓^{1}
Z_{n}^{R}  ⟶ K_{n}^{R}[x_{1}, …, x_{n}] ⟶^{θ} [X^{n}, ΩY]

where δ(γ(n; σ)) = (Ωα) · \overline{γ(n; σ)}.

Theorem 1.5.1 follows at once from the next lemma.

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 27

Lemma 1.5.4.

(i) If R = ℤ, there exist spaces X with X = Σ(X′) such that λ is a monomorphism.

(ii) If R = ℤ/p^{r}ℤ, there exist spaces X and Y with X = Σ(X′), and α : ΣX → Y where α is of order p^{r} such that λ is a monomorphism.

Proof. Since p^{∗} : [X^{(n)}, ΩY] → [X^{n}, ΩY] is a monomorphism, it suffices to show that δ is a monomorphism by Lemma 1.5.3.

If R = ℤ, then choose X to be ∨_{n}S^{ℓ} for ℓ ≥ 2. If R = ℤ/p^{r}ℤ, choose X to be ∨_{n}S^{ℓ} for ℓ ≥ 2, and α : (∨_{n}S^{ℓ}) → Y = ∨_{n}P^{ℓ+1}(p^{r}) with the restriction of α to S^{ℓ} given by the restriction of α to the ith wedge factor S^{ℓ} determined by a generator of Π_{ℓ}P^{ℓ+1}(p^{r}) for the ith wedge factor P^{ℓ+1}(p^{r}). That ℓ is chosen to be at least 2 ensures that X is a double suspension.

Let {b_{1}, … b_{n}} denote a basis for H_{ℓ}(X; R) where b_{i} is the fundamental cycle for the ith wedge factor of S^{ℓ+1}. Let {u_{1}, … u_{n}} and {v_{1}, … v_{n}} denote analogous bases for H_{ℓ}(Y; R) and H_{ℓ+1}(Y; R).

If X = ∨_{n}S^{ℓ}, then H_{∗}(ΩΣX; ℤ) is torsion free and is isomorphic to the tensor algebra T [H̄_{ℓ}(X; ℤ)] as an algebra. Since H_{∗}(ΩΣX; ℤ) is torsion free, it is also a primitively generated Hopf algebra because X is a suspension. Analogously H_{∗}(ΩΣY; R) is a Hopf algebra and Ωα induces a monomorphism in homology taken with coefficients in R.

Next consider the group homomorphism

ϕ : [X^{(n)}, ΩY] → Hom_{ℤ}(P H_{q}(X^{(n)}; R), P H_{q}(ΩY; R))

where ϕ(f) = f_{∗} and P H_{q}( ) denotes the module of primitives concentrated in degree q. Notice that the definition of a primitive is standard here as the assumptions on X and Y imply that the strong form of the Künneth theorem is satisfied when homology is taken with coefficients in R. (Recall that the analogous map

ϕ̃: [X^{(n)}, ΩY] → Hom(H_{q}(X^{(n)}; R), H_{q}(ΩY; R))

given by ϕ̃(f) = f_{∗} is not, in general, a group homomorphism. Thus one requries that the module of primitives be used here.) Next observe that b = b_{1} ⊗b_{2} ⊗⋯⊗b_{n} is primitive in H_{nℓ}(X^{(n)}; R) and

ϕ(γ(n; σ))(b) = [… [b_{1}, b_{σ(2)}]b_{σ(3)}] …]b_{σ(n)}].

28 F.R. COHEN^{∗}

Consider the free Lie algebra over R generated by the free R module spanned by {b_{j}, …, b_{n}}, L[b_{j}, …, b_{n}]. There is a morphism of Lie algebras

ζ : L[b_{1}, …, b_{n}] → L[b_{2} …, b_{n}]

where

ζ(b_{i}) = \{ \begin{matrix}0 if i = 1 \\ b_{i} if i > 1. \end{matrix}

By [CMN], the Lie algebra kernel of ζ is a free Lie algebra with generators given by b_{1} and

[… [b_{1}, b_{i_{1}}]b_{i_{2}}] …]b_{i_{k}}]

for k ≥ 1, with i_{ℓ} ≥ 2 for all ℓ. Thus ϕ(γ(n; σ))(b) for σ in Σ_{n−1} are linearly independent and the lemma follows.

A similar argument gives the next theorem.

Theorem 1.5.5. There exist spaces and maps α : Σ^{2}X → Y such that θ : K_{n}^{R}[x_{1}, …, x_{n}] → [(ΣX)^{n}, ΩY] is a monomorphism. Furthermore if R = ℤ, then ΣX may be chosen the be ∨_{n}S^{2} and if R = ℤ/p^{r}ℤ, then ΣX may be chosen to be ∨_{n}P^{3}(p^{r}).

§1.6. The descending central series for K_{n}[x_{1}, …, x_{n}].

Theorems 1.1.4 and 1.1.5 will be proven in this section. Here let R = ℤ or ℤ/p^{r}ℤ and assume that G_{n}^{R}[x_{1}, …, x_{n}] is a group with presentation given by Theorem 1.1.4. Thus, this group has generators x_{1}, …, x_{n} together with relations

(i) [x_{i_{1}}^{n_{1}}, …, x_{i_{k}}^{n_{k}}] = [x_{i_{1}}, …, x_{i_{k}}]^{n_{1}...n_{k}}, (ii) [x_{i_{1}}, …, x_{i_{k}}] = 1 if x_{i_{ℓ}} = x_{i_{t}} for some 1 ≤ i_{ℓ} < i_{t} ≤ n, and (iii) x_{i}^{p^{r}} = 1 if R = ℤ/p^{r}ℤ.

Notice that K_{n}^{R}[x_{1}, …, x_{n}], which is defined via a universal mapping property, is a quotient of G_{n}^{R}[x_{1}, …, x_{n}] by Lemma 1.2.1. It is the purpose of this section to show that this natural quotient map is an isomorphism. Let {Γ_{R}^{q}} denote the lower central series for G_{n}^{R}[x_{1}, …, x_{n}]. The associated graded {E_{0}^{q}} where E_{0}^{q}(G_{n}^{R}) = Γ_{R}^{q}/Γ_{R}^{q+1} is given in the next theorem.

Theorem 1.6.1. If 1 ≤ q ≤ n, E_{0}^{q}(G_{n}^{R}) is a free R-module of rank (q − 1)!\binom{n}{q} with basis

L(J, σ) = [… [x_{j_{1}}, x_{j_{σ(2)}}]x_{j_{σ(3)}}] … x_{j_{σ(q)}}]

for J = (j_{1}, …, j_{q}) admissible of length q. Furthermore Γ_{R}^{n+1} = {1}.

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 29

Corollary 1.6.2. If R = ℤ or ℤ/p^{r}ℤ, then G_{n}^{R}[x_{1}, …, x_{n}] is isomorphic to K_{n}^{R}[x_{1}, …, x_{n}]. Thus the filtration quotients {E_{0}^{q}} for K_{n}^{R}[x_{1}, …, x_{n}] are specified by Theorem 1.6.1.

The proof of Theorem 1.6.1 is given in two steps. In the first step, it is observed that the elements L(J, σ) project to generators of E_{0}^{q}(G_{n}^{R}) for 1 ≤ q ≤ n. The second step is to show that the L(J, σ) freely generate E_{0}^{q}(G_{n}^{R}) as an R-module. Begin by recalling the following theorem from [MKS, p. 295].

Theorem 1.6.3. Let G be a group generated by the set of elements {g_{1}, …, g_{n}} and let {Γ^{∗}} denote the descending central series of G. Then Γ^{q}/Γ^{q+1} is generated as a group by the left normalized commutators

[… [g_{j_{1}}, g_{j_{2}}]g_{j_{3}} …]g_{j_{q}}].

There is a map of R-modules

ψ_{q} : ⨁_{(q−1)!\binom{n}{q}} R → E_{0}^{q}(G_{n}^{R})

which is defined as follows: Index the R-module generators of ⨁_{(q−1)!\binom{n}{q}} R by (J, σ) where J is an admissible sequence (j_{1}, …, j_{q}) as above and σ is an element of ∑_{q−1} acting on the set {2, …, q}. Define

ψ_{q}(J, σ) = [… [x_{j_{1}}, x_{j_{σ(2)}}]x_{j_{σ(3)}}] …]x_{j_{σ(q)}}].

Corollary 1.6.4. ψ_{q} is an epimorphism of R-modules.

Proof. By 1.6.3, it suffices to check that any element [… [x_{ℓ_{1}}, x_{ℓ_{2}}] …]x_{ℓ_{q}}] is a linear combination of elements [… [x_{j_{1}}, x_{j_{σ(2)}}]x_{j_{σ(3)}}] …]x_{j_{σ(q)}}] for J = (j_{1}, …, j_{q}) admissible. This last fact is checked by induction on q with the cases q = 1 or 2 being trivial. But by the Jacobi identity in Theorem 1.2.5 with L = [… [x_{ℓ_{1}}, x_{ℓ_{2}}] …]x_{ℓ_{q−2}}], one has

[[L, x_{ℓ_{q−1}}]x_{ℓ_{q}}] = [[x_{ℓ_{q−1}}, x_{ℓ_{q}}], L]^{−1} · [[x_{ℓ_{q}}, L], x_{ℓ_{q−1}}]^{−1}

and the result follows at once.

Next recall the algebras given in section 1.3 A_{n}^{R} = A_{n}^{R}[y_{1}, …, y_{n}] with R = ℤ or ℤ/p^{r}ℤ depending on R. There is an augmentation ε : A_{n}^{R}[y_{1}, …, y_{n}] → R given

30 F.R. COHEN^{∗}

by ε(y_{i}) = 0 and ε(1) = 1. Let IA_{n}^{R} denote the augmentation ideal and filter IA_{n}^{R} by powers of the augmentation ideal. Thus

F^{q}IA_{n}^{R} = image [multiplication IA ⊗_{R} ⋯ ⊗_{R} IA_{n}^{R}_{←q→} → IA_{n}^{R}]

and

F^{n+1}IA_{n}^{R} = {0} ⊂ F^{n}IA_{n}^{R} ⊂ ⋯ ⊂ F^{1}IA_{n}^{R} = IA_{n}^{R}.

Notice that any monomial in y_{1}, …, y_{q} is nilpotent and the next lemma follows at once.

Lemma 1.6.5. If x is any element of IA_{n}^{R}, then

x^{n+1} = 0.

Since 1 + y_{i} is a unit in A_{n}^{R}[y_{1}, …, y_{n}], there is a group homomorphism from a free group into the group of units

exp : F [x_{1}, …, x_{n}] → A_{n}^{R}[y_{1}, …, y_{n}]^{×}

with exp(x_{i}) = 1 + y_{i} as (1 + y_{i})(1 − y_{i}) = 1.

Lemma 1.6.6. Let n = n_{1} … n_{q}. Then

exp[x_{j_{1}}^{n_{1}}, …, x_{j_{q}}^{n_{q}}] = 1 + n[y_{j_{1}}, …, y_{j_{q}}]

Proof. Induct on q starting with k = 1 which is immediate as y_{i}^{2} = 0. Thus let x = [x_{j_{1}}^{n_{1}} …, x_{j_{q−1}}^{n_{q−1}}] and y = [y_{j_{1}}, …, y_{j_{q−1}}] and consider exp[x, x_{j_{q}}^{n_{q}}]:

exp[x, x_{j_{q}}^{n_{q}}] = exp(xx_{j_{q}}^{n_{q}}x^{−1}x_{j_{q}}^{−n_{q}})
= exp(x) exp(x_{j_{q}}^{n_{q}}) exp(x^{−1}) exp(x_{j_{q}}^{−n_{q}})
= (1 + n_{1} … n_{q−1}y)(1 + n_{q}y_{j_{q}})(1 − n_{1} … n_{q−1}y)(1 − n_{q}y_{j_{q}})
= 1 + nyy_{j_{q}} − ny_{j_{q}}y
= 1 + n[y, y_{j_{q}}].

The Lemma follows.

By inspection of Lemma 1.6.6, the exponential map preserves the defining relations in G_{n}^{R}[x_{1}, …, x_{n}] and thus there is an induced group homomorphism

exp : G_{n}^{R}[x_{1}, …, x_{n}] → A_{n}^{R}[y_{1}, … y_{n}]^{×}.

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 31

Lemma 1.6.7. The exponential map exp : G_{n}^{R}[x_{1}, …, x_{n}] → A_{n}^{R}[y_{1}, …, y_{n}] is filtration preserving in the sense that

exp(E_{0}^{q}) ⊂ {1}xF^{q}IA_{n}^{R}.

Furthermore, the composite map of sets \overline{exp}

Γ^{q}(G_{n}^{R}) →^{exp} {1}xF^{q}IA_{n}^{R} →^{projection} F^{q}IA_{n}^{R} → E_{0}^{q}(IA_{n}^{R})

is a group homomorphism. Proof. Let

L_{1} = [x_{j_{ℓ}}, …, x_{j_{t}}] for t ≥ q and
L_{2} = [x_{i_{1}}, …, x_{i_{s}}] for s ≥ q.

Write exp(L_{i}) = 1 + \overline{L_{i}} for \overline{L_{i}} in IA_{n}^{R}. Thus

exp(L_{1}L_{2}) = (1 + \overline{L_{1}})(1 + \overline{L_{2}}) = 1 + \overline{L_{1}} + \overline{L_{2}} + \overline{L_{1}}\overline{L_{2}}.

But \overline{L_{1}} · \overline{L_{2}} is in F^{q+s}IA_{n}^{R} and thus

\overline{exp}(L_{1} · L_{2}) = \overline{L_{1}} + \overline{L_{2}} in E_{0}^{q}(IA_{n}^{R}).

The lemma follows. Corollary 1.6.8. The homomorphism \overline{exp} of 1.6.7 induces a homomorphism of abelian groups

E_{0}^{q}(G_{n}^{R}) ⟶ E_{0}^{q}(IA_{n}^{R})

Proof. If suffices to notice that \overline{exp}(Γ_{R}^{q+1}) = 0. Corollary 1.6.9. If 1 ≤ q ≤ n, then \overline{exp} · ψ_{q} is a split monomorphism of abelian groups. Proof. By Theorem 1.3.4, the set of elements

[… [y_{j_{1}}, y_{j_{σ(2)}}]y_{j_{σ(3)}}] …]y_{j_{σ(q)}}] for J = (j_{1}, …, j_{q})

admissible and σ in Σ_{q−1} extends to a basis for E_{q}^{0} (IA_{n}^{R}) and the result follows.

32 F.R. COHEN^{∗}

Corollary 1.6.10. The group homomorhism

exp : G_{n}^{R}[x_{1}, …, x_{n} → A_{n}^{R}[y_{1} …, y_{n}]^{×}

is a monomorphism.

Proof. Let x be in the kernel of exp. Thus x is Γ^{q}(G_{n}^{R}) for some q. Since x ∈ Γ_{q}^{R} and exp(x) = 1, then \overline{exp}(x) = 0 in E_{0}^{q}(IA_{n}^{R}). Thus x is in Γ^{q+1}(G_{n}^{R}) and the corollary follows at once from the next lemma.

Lemma 1.6.11. The group G_{n}^{R}[x_{1}, … x_{n}] is nilpotent of class n.

Proof. All commutators [… [x_{j_{1}}, x_{j_{2}}] …]x_{j_{n+1}}] vanish as some x_{i} must repeat. But then G_{n}^{R}[x_{1}, … x_{n}] is nilpotent of class at most n as [… [x_{j_{1}}^{ℓ_{1}}, x_{j_{2}}^{ℓ_{2}}] …]x_{j_{n+1}}^{ℓ_{n+1}}] = [… [x_{j_{1}}, x_{j_{2}}] …]x_{j_{n+1}}]^{ℓ} where ℓ = ℓ_{1} … ℓ_{n+1}. But [… [x_{1}, x_{2}]x_{3}] …]x_{n}] ≠ 1 by Theorem 1.3.4 and the lemma follows.

Proof of Theorem 1.6.1. By 1.6.4 and 1.6.9, the map ψ_{q} : ⨁_{(q−1)!\binom{n}{q}} R → E_{0}^{q}(G_{n}^{R}) is an isomorphism. The theorem follows.

There is another characterization of Γ_{R}^{q}. Let J = (j_{1}, … j_{q}) be admissible as before. Define

δ_{J} : G_{n}^{R}[x_{1}, …, x_{n}] → G_{q}^{R}[x_{j_{1}} …, x_{j_{q}}]

by

δ_{J}(x_{i}) = \{ \begin{matrix}x_{i} if i ∈ \{j_{1}, …, j_{q}\} \\ 1 if i ∉ \{j_{q}, …, j_{q}\}. \end{matrix}

Write G_{q}^{R}[J] for G_{q}^{R}[x_{j_{1}}, …, x_{j_{q}}] and define

Φ_{q} : G_{n}^{R}[x_{1}, …, x_{n}] → ∏_{J admissible ℓ(J)=q} G_{q}^{R}[J]

with projection to G_{q}^{R}[J] given by δ_{J}.

Theorem 1.6.12. The kernel of Φ_{q} is Γ_{R}^{q+1}, the qth filtration in the descending central series for G_{n}^{R}[x_{1}, …, x_{n}].

Proof. Since G_{n}^{R}[x_{1}, … x_{n}] is nilpotent of class n, Γ_{R}^{q+1} is in the kernel of each homomorphism

δ_{J} : G_{n}^{R}[x_{1}, … x_{n}] → G_{q}^{R}[J].

If x ∈ Γ_{R}^{q} − Γ_{R}^{q+1}, then δ_{J}(x) ≠ 1 for some choice of J by inspection.

Proof of Corollary 1.6.2. The relations in the presentation for G_{n}^{R}[x_{1}, …, x_{n}] are satisfied for K_{n}^{R}[x_{1}, …, x_{n}] by Lemma 1.2.1. Thus there is a group epimorphism

λ : G_{n}^{R}[x_{1}, …, x_{n}] → K_{n}^{R}[x_{1}, …, x_{n}]

defined by λ(x_{i}) = x_{i}. It suffices to show that λ is faithful and the proof of this fact will be induction on n.

The case n = 1 is trivial as both groups are isomorphic to the abelian group R. Assume that λ is an isomorphism for n < N. There are homomorphisms

δ_{i} : G_{N}^{R}[x_{1}, …, x_{N}] → G_{N−1}^{R}[x_{1} … x̂_{i}, …, x_{N}]

defined by

δ_{i}(x_{j}) = \{ \begin{matrix}x_{j} if j ≠ i \\ 1 if j = i. \end{matrix}

There are analogous homomorphsims

δ_{i} : K_{N}^{R}[x_{1}, …, x_{N}] → K_{N−1}^{R}[x_{1}, …, x̂_{i}, …, x_{N−1}]

together with a commutative diagram

G_{N}^{R}[x_{1}, …, x_{N}] ⟶^{δp_{j}} ∏_{j=1}^{N} G_{N−q}^{R}[x_{1}, …, x̂_{j}, …, x_{N}]
λ↓  ↓^{∏_{j=1}^{N}(λ)}
K_{n}^{R}[x_{1}, …, x_{N}] \underrightarrow{∏ δ_{j}} ∏_{j=1}^{N} K_{N−1}^{R}[x_{1}, …, x̂_{j}, …, x_{N}].

By the inductive hypothesis, the right hand vertical arrow is an isomorphism. By Theorem 1.6.12, the kernel of ∏_{j=1}^{N}(p_{j}) is Γ_{R}^{N}. By Theorem 1.6.1, Γ_{R}^{N} is isomorphic to ⨁_{(N−1)!} R. By Theorem 1.5.1, the composite ⨁_{(N−1)!} R → G_{N}^{R}[x_{1}, …, x_{n}) →^{λ} K_{N}^{R}[x_{1}, …, x_{n}] is a monomorphism and the result follows.

§1.7: The center of K_{n}^{R}.

Consider the homomorphisms named δ_{i} in the last section

δ_{i} : K_{n}^{R}[x_{1}, …, x_{n}] → K_{n−1}^{R}[x_{1}, …, x̂_{i}, …, x_{n}], 1 ≤ i ≤ n,

obtained by setting

δ_{i}(x_{j}) = \{ \begin{matrix}1 if i = j and \\ x_{j} if i ≠ j. \end{matrix}

34 F.R. COHEN^{∗}

Indeed, the δ_{i} are obtained from well-defined homomorphisms when K_{n}^{R} is replaced by the free group F [x_{1}, …, x_{n}]. Furthermore, these analogues of δ_{i} preserve the relations in K_{n}^{R}. Thus there is a homomorphism

δ : K_{n}^{R}[x_{1}, …, x_{n}] → ∏_{i=1}^{n} K_{n−1}^{R}[x_{1}, …, x̂_{i}, … x_{n}]

where δ = ∏_{i} δ_{i}.

As in the introduction, let Λ_{n}^{R} denote the center of K_{n}^{R}. The main result of this section is the following theorem.

Theorem 1.7.1. The center of Γ_{n}^{R} is equal to Λ_{n}^{R} the nth stage of the descending central series for K_{n}^{R}.

An analogous but more useful result is

Theorem 1.7.2. The kernel of δ is the nth stage of the descending central series for K_{n}^{R}.

Proof of 1.7.2. By Corollary 1.1.7, every element z in K_{n}^{R} has an unique normal form λ_{1} … λ_{n} where λ_{q} = ∏_{J_{q_{1}} σ_{i}} γ(J_{1}; σ_{i})^{m(J_{q},σ_{i})}. Thus if δ_{j}(z) = 1 for all j, the all exponents m(J_{q}, σ_{i}) where q < n must equal 0. Thus z = λ_{n} and the theorem follows.

Proof of 1.7.1. Since K_{n}^{R}[x_{1}, …, x_{n}] is nilpotent of class n, the nth stage of the descending central series is contained in the center Λ_{n}^{R}. Let z be an element of the center with normal form given in the proof of 1.7.2. As each δ_{i} is an epimorphism, δ_{i}(z) is in the center of K_{n−1}^{R}[x_{1}, …, x̂_{i}, …, x_{n}]. Since K_{1}^{R} is abelian, it follows inductively that 1.7.1 is correct for n − 1. Thus z = λ_{n−1}λ_{n} and it suffices to check that λ_{n−1} = 1.

Restrict attention to γ(J_{n−1}; σ_{i}) where

λ_{n−1} = ∏_{J_{n−1},σ_{i}} γ(J_{n−1}; σ_{i})^{m(J_{n−1},σ_{i})}

and m(J_{n−1}, σ_{i}) ≠ 0 in R. Let t be the smallest integer, 1 ≤ t ≤ n, such that t ≠ σ_{i}(j_{k}) for j_{k} in J_{n−1}. Then the commutator [x_{t}, z] is not 1. Thus m(J_{n−1}, σ_{i}) = 0 and the theorem follows.

Notice that the center of K_{n}^{R} is isomorphic to Lie (n) as a Σ_{n}-module by Theorems 1.7.1 and 1.1.5. Thus Theorem B of the Introduction follows.

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 35

§1.8: The groups K_{n}[x_{1}, …, x_{n}] and G.W. Whitehead’s tensor product filtration of X^{n}.

Given a space (X, ∗) in τ, filter the product X^{n} according to G.W. Whitehead’s filtration given by

F_{q}X^{n} = {(x_{1}, …, x_{n}) ∈ X^{n}| At least n − q of the x_{i} = ∗}.

Thus F_{−1} = ϕ, F_{0} = {∗}, and F_{n}X^{n} = X^{n}. John Moore calls this filtration the tensor product filtration as it corresponds to a natural algebraic filtration of a tensor product of abelian groups.

The symmetric group on n letters acts on X^{n} by permutation of coordinates. The tensor product filtration of X^{n} is preserved by this action. By an evident modification of [St], one has the following result.

Lemma 1.8.1. If k ≥ j ≥ 0, then (F_{k}X^{n}, F_{j}X^{n}) is a strong NDR pair; furthermore, this pair is an equivariant NDR pair in the following sense: There are maps

u : F_{k}X^{n} → [0, 1] and H : I × F_{k}X^{n} → F_{k}X^{n}

which represent (F_{k}X^{n}, F_{j}X^{n}) as a strong NDR pair with u(σx) = u(x) and H(t, σx) = σH(t, x) for σ in Σ_{n}.

Let E_{k,j+1} denote the cofibre of the natural inclusion i : F_{j}X^{n} → F_{k}X^{n}. Since i admits a left inverse after suspension, there is an exact sequence of groups

1 → [E_{k,j+1}, ΩY] → [F_{k}X^{n}, ΩY] → [F_{j}X^{n}, ΩY] → 1

which, of course, is split as sets. Consider the natural maps

X^{n} →^{1} E_{n,1} → E_{n,2} → ⋯ → E_{n,n} →^{1} {∗}

to obtain a decreasing filtration of the group [X^{n}, ΩY] given by

F^{q}[X^{n}, ΩY] = image {[E_{n,q}, ΩY] → [X^{n}, ΩY]}.

Filter the group K_{n}[x_{1}, …, x_{n}] by the descending central series. Similarly, if α : Σ^{2}(X′) → Y has order p^{r}, there are homomorphisms K_{n}^{ℤ/p^{r}ℤ}[x_{1}, …, x_{n}] → [X^{n}, ΩY].

36 F.R. COHEN^{∗}

Theorem 1.8.2. If X = Σ(X′), the maps

θ : K_{n}[x_{1}, …, x_{n}] → [X^{n}, ΩΣX]

are filtration preserving. Furthermore, if α : Σ^{2}(X′) → Y has order p^{r}, the maps

K_{n}[x_{1}, …, x_{n}] ⟶^{θ} [X^{n}, ΩΣX]
↓ ↓^{Ωα_{∗}}
K_{n}^{ℤ/p^{r}ℤ}[x_{1}, …, x_{n}] ⟶_{θ} [X^{n}, ΩY]

are all filtration preserving.

Proof. Fix γ : [… [x_{i_{1}}, x_{i_{2}}] …]x_{i_{t}}] in K_{n}[x_{1}, …, x_{n}] with t ≥ q. Thus θ(γ) is represented by a composite X^{n} →^{π_{I}} X^{(t)} →^{f} ΩΣX where π_{I} is projection to the coordinates i_{1}, …, i_{t}. Notice that the composite F_{t−1}X^{n} →^{i} X^{n} →^{π_{I}} X^{(t)} is nullhomotopic and so there is a homotopy commutative diagram

X^{n} ⟶^{π̄_{I}} E_{n,t}
1↓ ↓
X^{n} ⟶_{π_{I}} X^{(t)}.

For fixed q with t ≥ q, observe that π̄_{I} factors through E_{n,q}. Thus if Γ^{q} is the qth stage of the lower central series for K_{n}[x_{1}, …, x_{n}], the diagram

Γ^{q}  ⟶ F^{q}[X^{n}, ΩΣX]
↓  ↓
K_{n}[x_{1}, …, x_{n}] ⟶^{θ} [X^{n}, ΩΣX]

commutes. The theorem follows by naturality.

§1.9: On endomorphisms of K_{n}^{R}[x_{n}, …, x_{n}].

There are endomorphisms of the free group F [x_{1}, …, x_{n}] given by Ψ^{q}(x_{i}) = x_{i}^{q} for all i and q in ℤ.

Lemma 1.9.1. There exist endomorphisms of K_{n}^{R}[x_{1}, …, x_{n}] specified by Ψ^{q}(x_{i}) = x_{i}^{−q} for 1 ≤ i ≤ n and q in ℤ. Furthermore the diagrams

F [x_{1}, …, x_{n}] ⟶^{Ψ^{q}} F [x_{1}, …, x_{n}]
↓  ↓
K_{n}^{R}[x_{1}, …, x_{n}] ⟶^{Ψ^{q}} K_{n}^{R}[x_{1}, …, x_{n}]

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 37

and

K_{n}^{R}[x_{1}, …, x_{n}]  ⟶^{Ψ^{q}}  K_{n}^{R}[x_{1}, …, x_{n}]
d_{i}↓  ↓^{d_{i}}
K_{n−1}^{R}[x_{1}, …, x̂_{i}, …, x_{n}] ⟶^{Ψ^{q}} K_{n−1}^{R}[x_{1}, …, x̂_{i}, …, x_{n}]

commute.

Proof. By 1.1.4, [x_{i_{1}}^{n_{1}}, x_{i_{2}}^{n_{2}}, …, x_{i_{k}}^{n_{k}}] = [x_{i_{1}}, …, x_{i_{k}}]^{n} where n = n_{1}n_{2} … n_{k}. Thus Ψ^{q} regarded as an endomorphism of F [x_{1}, …, x_{n}] preserves relation (1) in Theorem 1.1.4 as Ψ^{q}[x_{i_{1}}, …, x_{i_{k}}] = [x_{i_{1}}, …, x_{i_{k}}]^{q^{k}}. Furthermore, relation (3) is satisfied by the analogous calculation. The lemma follows.

Notice that if R = ℤ/p^{r}ℤ and (q, p) = 1, then there exist integers a and b such that aq + bp^{r} = 1. Thus Ψ^{a} · Ψ^{q}(x_{i}) = x_{i} for x_{i} in K_{n}^{ℤ/p^{r}ℤ}[x_{1}, …, x_{n}]

Corollary 1.9.2. If R = ℤ/p^{r}ℤ and (q, p) = 1, then Ψ^{q} is an automorphism of K_{n}^{R}[x_{1}, …, x_{n}].

§1.10: Simplicial objects associated to K_{n}^{R}.

The groups K_{n}^{R} = K_{n}^{R}[x_{1}, …, x_{n}], n ≥ 1, naturally assemble into a simplicial group K̄_{∗}^{R} with K̄_{q}^{R} isomorphic to K_{q+1}^{R}. Thus K̄_{∗}^{R} is not connected. The reason K̄_{∗}^{R} is chosen to be non-connected is that there is one “natural” face operation for each generator x_{i}: thus there are q face operations for K_{q}^{R}[x_{1}, …, x_{q}] and the choice K̄_{q}^{R} = K_{q+1}^{R} is given in order to encode all of these faces. In what follows, the indices for x_{i} are lowered by 1. Thus K̄_{q}^{R} = K_{q+1}^{R}[x_{0}, …, x_{q}]. The structure theorems for K_{q+1}^{R} of course remain the same with a new notation. The symbol x_{−1} denotes the identity. The next theorem follows from a standard direct calculation as d_{i} and s_{j} visibly preserve the relations given in Theorem 1.1.4.

Theorem 1.10.1. There is a simplicial group K̄_{∗}^{R} with

K̄_{n}^{R} = K_{n+1}^{R}[x_{0}, …, x_{n}]

where

d_{i}(x_{k}) = \{ \begin{matrix}x_{k} if k < i \\ x_{k−1} if k ≥ i, and \end{matrix}

s_{j}(x_{k}) = \{ \begin{matrix}x_{k} if k < j \\ x_{k+1} if k ≥ j. \end{matrix}

38 F.R. COHEN^{∗}

Each group K̄_{N}^{R} has an associated Lie algebra given by its’ descending central series with

(1) E_{0}^{q,N} = Γ^{q}(K̄_{N}^{R})/Γ^{q+1}(K̄_{N}^{R}).

(2) E_{0}^{q,N} is isomorphic to ⨁_{(q−1)!\binom{N+1}{q}} R if 1 ≤ q ≤ N + 1, and

(3) there is an R-bilinear pairing [ , ]: E_{0}^{p,N} ⊗_{R} E_{0}^{q,N} ⟶ E_{0}^{p+q,N} with [x̄, ȳ] = \overline{[x, y]} for x̄ ∈ E_{0}^{p,N}, ȳ ∈ E_{0}^{q,N} which are represented by x and y respectively.

Using the face and degeneracy operations in 1.10.1 together with 1.6.1, the next theorem is immediate.

Theorem 1.10.2. There is a bigraded simplicial Lie algebra L^{R} which is E_{0}^{q,N} in bidegree (q, N) where N is the simplicial degree. Furthermore

(i) there is an isomorphism of R-modules

E_{0}^{q,N} ⟶ ⨁_{(q−1)!\binom{N+1}{q}} R for 1 ≤ q ≤ N + 1.

(ii) a choice of basis for E_{0}^{q,N} is

[… [x_{i_{1}}, x_{i_{σ(2)}}]x_{i_{σ(3)}}] …]x_{i_{σ(4)}}]

for 1 ≤ i_{1} < i_{2} < ⋯ < i_{q} ≤ N + 1 with σ in 1 × Σ_{q−1}. (iii) the face and degeneracy operations

d_{i} : E_{0}^{q,N} ⟶ E_{0}^{q,N−1}
s_{j} : E_{0}^{q,N} ⟶ E_{0}^{q,N+1}

are specified by 10.1 together with d_{i}[x, y] = [d_{i}x, d_{i}y] and s_{j}[x, y] = [s_{j}x, s_{j}y].

Theorem 1.10.3.

(i) The group of cycles ∩_{i=0}^{N+1}ker(d_{i}) for the Moore chain complex N(K̄_{∗}^{R}) in degree N is the center of \overline{K_{N}}, Λ_{N+1}^{R}.

(ii) The group of chains in degree N is abelian and isomorphic to Λ_{N}^{R}) ⊕ Λ_{N+1}^{R}.

Furthermore, d_{0}(Λ_{N}^{R}) = Λ_{N}^{R}. (iii) The homotopy groups of K̄_{∗}^{R} are all trivial.

Proof. (i): Recall that a basis for Λ_{N+1}^{R} is given by c = [… [x_{0}, x_{σ(1)}]x_{σ(2)} …]x_{σ(N)}] for σ in Σ_{N} by 1.6.1. Furthermore d_{i}c = 1 as d_{i}x_{k} is either 1 or forces some x_{j} to repeat as all of x_{0}, …, x_{N} appear in c. Thus the center is contained in the group

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 39

of cycles. To finish the proof of (i), let z be a cycle. By 1.1.7, z is a product of λ_{1} · λ_{2} … λ_{N}. By a direct calculation of d_{i}, λ_{1} = λ_{2} = ⋯ = λ_{N−1} = 1. Thus z is in the center.

To check (ii), consider the element

z_{j} = [… [x_{0}, x_{1}]x_{2}] …]x̂_{j}]x_{j+1}] …]x_{N}], 0 ≤ j ≤ N.

Notice that if i ≥ 1, then

d_{i}z_{j} = \{ \begin{matrix}z_{0} if j = i or j = i − 1 \\ 0 otherwise. \end{matrix}

Furthermore the z_{j} all commute. Thus

z = ∑_{j=0}^{N}(−1)^{j}z_{j}

is annihilated by d_{i}, i ≥ 1, and d_{0}z = [… [x_{0}, x_{1}] …]x_{N}]. A similar formula applies with a permutation σ applied to the indices 1, …, n to obtain chains z(σ) = ∑_{j=0}^{N}(−1)^{j}σ(z_{j}).

A direct calculation gives that the set {z(σ)|σ ∈ Σ_{n−1}} and {x(τ)|τ ∈ Σ_{N}} is a basis for NK̄_{N}^{R} where x(τ) = τ[… [x_{0}, x_{1}]x_{2}] …]x_{N}] for τ in Σ_{N}.

Part (iii) follows from (ii) together with the formula for d_{0}z. Thus d_{0}(Λ_{N}^{R}) = Λ_{N}^{R}.

Notice that Theorem J follows. In addition, the calculation which gives Theorem H is analogous, and is omitted.

Chapter 2

A subgroup H_{n}^{R} of K_{n}^{R} is constructed and analyzed in this chapter. One of the main features of H_{n}^{R} is that it is the largest subgroup of K_{n}^{R} that lifts to the group [J_{n}X, ΩY] for all X = Σ(X′) and maps α : Σ^{2}X → Y of order p^{r} when R = ℤ/p^{r}ℤ. Thus, H_{n}^{R} is the algebraic model for the group [J_{n}X, ΩY] which was described in the introduction.

Another useful feature of H_{n}^{R} is that it admits an analogue of the classical distributivity law in homotopy theory. Namely, this distributivity law arises from the difference of the ℓth power map on ΩΣX and the looping of the ℓth power map

[ℓ] : ΣX → ΣX in the group [ΣX, ΣX].

The notation [ℓ] is chosen in order to distinguish it from the ℓ-th power map in [ΩΣX, ΩΣX]. A group theoretic analogue of this difference is given here.

40 F.R. COHEN^{∗}

§2.1: The groups H_{n}^{R}.

Recall that the natural quotient map π : X^{n} → J_{n}X induces a group homomorphism π^{∗} : [J_{n}X, ΩY] → [X^{n}, ΩY] which, as a morphism of sets, is a split monomorphism. Further recall that there exist homomorphisms given by θ : K_{n}[x_{1}, …, x_{n}] → [X^{n}, ΩΣX] for all X with X = Σ(X′).

A set of generators for the group H_{n}^{R}, R = ℤ or ℤ/p^{r}ℤ, is given below. To describe these generators, some notation is required. As before, let I = (i_{1}, …, i_{k}) be admissible of weight k. Now order the admissible sequences I of weight k lexicographically from the left. Thus if J = (j_{1}, …, j_{k}) is admissible, then J ≤ I if j_{1} = i_{1}, j_{2} = i_{2}, …, j_{r} = i_{r}, and j_{r+1} < i_{r+1}. Next fix an element σ in Σ_{k}. For a fixed integer k and a fixed permutation σ in Σ_{k}, let

γ(k, σ) = ∏_{I admissible of weight k}[… [x_{i_{σ(1)}}, x_{i_{σ(2)}}]x_{i_{σ(3)}}] …]x_{i_{σ(k)}}]

where the order of the multiplication is that induced by the left lexicographical ordering of the admissible sequences of weight k. Thus, for example, γ(1, 1) = x_{1}x_{2} … x_{n} and

γ(2, 1) = [x_{1}, x_{2}][x_{1}, x_{3}][x_{1}, x_{4}] … [x_{1}, x_{n}][x_{2}, x_{3}] … [x_{n−1}, x_{n}].

Theorem 2.1.1. The elements γ(k, σ) for 1 ≤ k ≤ n and all σ in Σ_{k} with σ(1) = 1 generate a subgroup of H_{n}^{ℤ} in K_{n}^{ℤ}[x_{1}, …, x_{n}]. Thus the images of γ(k, σ) in K_{n}^{R}, 1 ≤ k ≤ n, σ(1) = 1, with R = ℤ/p^{r}ℤ generate a subgroup of H_{n}^{R} in K_{n}^{R}[x_{1}, …, x_{n}]. In either case, this subgroup is all of H_{n}^{R}.

The elements γ(k, σ) correspond to certain natural maps J_{n}X → ΩΣX. Namely, there are maps h_{k} : JY → JY^{(k)} given by the kth James-Hopf invariant where

h_{k}(y_{1}, …, y_{n}) = ∏_{1≤i_{1}<i_{2}<···<i_{k}≤n}(y_{i_{1}} ∧ y_{i_{2}}, … ∧ y_{i_{k}})

and the order of multiplication is that induced by the left lexicographical ordering. Next recall that for σ in Σ_{k} there are maps s_{k} : Y^{(k)} → ΩΣY given by a kfold Samelson product with s_{k}(y_{1}, …, y_{k}) = [… [y_{σ(1)}, y_{σ(2)}]y_{σ(3)} …]y_{σ(k)}]. Let S_{k} : ΩΣY^{(k)} → ΩΣY be the canonical multiplicative extension of s_{k}. Notice that the hypothesis that X be a suspension is unnecessary in the definitions of both s_{k} and h_{k}.

Proposition 2.1.2. The composite

J_{n}X → JX →^{h_{k}} J(X)^{(k)} →^{≃} ΩΣ(X)^{(k)} →^{S_{k}} ΩΣX

represents γ(k; σ) in [J_{n}X, ΩΣX].

Let α : ΣX → Y represent an element of order p^{r} in [ΣX, Y] and assume that X = Σ(X′). If R = ℤ/p^{r}ℤ, there is an induced homomorphism

H_{n}^{R} → [J_{n}X, ΩY]

which gives a commutative diagram

H_{n}^{ℤ} ⟶ [J_{n}X, ΩΣX]
↓ ↓^{(Ωα)_{∗}}
H_{n}^{R} ⟶ [J_{n}X, ΩY].

The next result is immediate from the definitions.

Proposition 2.1.3. If X_{i} = Σ(X′_{i}), β : X_{1} → X_{2}, and α : ΣX_{2} → Y is of order p^{r} in [ΣX_{2}, Y] with R = ℤ/p^{r}ℤ, then there exist commutative diagrams in the category of groups

H_{n}^{ℤ} ⟶^{θ} [J_{n}X_{i}, ΩΣX_{i}]
↓  ↓_{π^{∗}}
K_{n}^{ℤ} ⟶^{θ} [X_{i}^{n}, ΩΣX_{i}],
H_{n}^{ℤ} ⟶^{θ} [J_{n}X_{2}, ΩΣX_{2}]
↓  ↓^{(Ωα)_{∗}}
H_{n}^{R} ⟶^{θ} [J_{n}X_{2}, ΩY], and
H_{n}^{ℤ} ⟶^{θ} [J_{n}X_{1}, JX_{1}] ⟶^{J(β)} [J_{n}X_{2}, JX_{2}]
1↓  ↓^{(Ωα)_{∗}}
H_{n}^{ℤ} ⟶^{reduction}  H_{n}^{R}  ⟶^{θ} [J_{n}X_{2}, ΩY].

The groups H_{n}^{R} enjoy some additional features with regard to the maps δ_{i} : K_{n}^{R}[x_{1}, …, x_{n}] → K_{n−1}^{R}[x_{1}, …, x̂_{i}, …, x_{n}] with

δ_{i}(x_{j}) = \{ \begin{matrix}x_{j} if j < i, \\ 1 if j = 1, and \\ x_{j−1} if j > i. \end{matrix}

Thus the next lemma follows from the definition of the elements γ(k, σ).

42 F.R. COHEN^{∗}

Lemma 2.1.4. δ_{i}γ(k, σ) = γ(k, σ).

Thus the map δ_{n} induces an epimorphism δ_{n} : H_{n}^{R} → H_{n−1}^{R}. The main result of this section is the description of the kernel of this last homomorphism which is also given by Theorem C of the Introduction.

Theorem 2.1.5. The kernel of δ_{n} : H_{n}^{R} → H_{n−1}^{R} is the n-th stage of the descending central series for K_{n}^{R} and is thus, by Theorem 1.8.1, the center of K_{n}^{R}.

Proof. Let Γ_{R}^{n} denote the n-th stage of the descending central series for K_{n}^{R}. Then Γ_{R}^{n} is isomorphic to ⨁_{(n−1)!} R with basis given by [… [x_{1}, x_{σ(2)}]x_{σ(3)}] …]x_{σ(n)}] by Theorems 1.8.1 and 1.6.1.

Thus δ_{n}γ(n, σ) = 1 and Γ_{R}^{n} is in ker(d_{n}). It suffices to see that the inclusion is reversed.

Notice that the diagram

H_{n}^{R} ⟶  K_{n}^{R}
δ_{n}↓ ↓^{∏_{1}^{n} δ_{i}}
H_{n−1}^{R} ⟶_{Δ} ∏_{i=1}^{n} K_{n−1}^{R}

commutes where Δ(x) = (x, x, …, x) as it clearly commutes when restricted to the generators γ(k, σ). Thus the kernel of δ_{n} : H_{n}^{R} → H_{n−1}^{R} is in the kernel of ∏_{1}^{n} δ_{i}. By Theorem 1.8.2, the kernel of ∏_{1}^{n} δ_{i} is precisely Γ_{n}^{R} and the theorem follows at once.

Thus Theorems C and D of the Introduction follow.

§2.2: The distributivity law in H_{n}^{R}.

As remarked in the introduction to this chapter, the distributivity law in classical homotopy theory arises from the difference (ℓ^{−1})(Ω[ℓ]) in the group [ΩΣX, ΩΣX]. Let r : J_{n}X → ΩΣX be the composite of the restriction of J_{n}X in JX followed by the natural weak equivalence JX → ΩΣX. It will be shown that the difference r^{∗}(ℓ^{−1})(Ω[ℓ]) is in the image of the homomorphism θ : H_{n}^{ℤ} → [J_{n}X, ΩΣX]. Thus there is an analogue of the classical distributivity law given by this difference in H_{n}^{R}.

Consider the composite r∘π : X^{n} → ΩΣX. Notice that the element r∘π regarded as an element of [X^{n}, ΩΣX] is equal to θ(x_{1}, …, x_{n}). Thus r = θ(x_{1} … x_{n}) is in

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 43

the group H_{n}^{ℤ}. Similarly, Ω[ℓ] is, by definition, a loop map and r ∘ Ω[ℓ] is equal to θ(x_{1}^{ℓ}x_{2}^{ℓ} … x_{n}^{ℓ}) in [J_{n}X, ΩΣX]. Thus r^{∗}((ℓ^{−1})(Ω[ℓ])) = θ((x_{1} … x_{n})^{−ℓ}(x_{1}^{ℓ} … x_{n}^{ℓ})). It will be shown that x_{1}^{ℓ}x_{2}^{ℓ} … x_{n}^{ℓ} is an element of H_{n}^{ℤ} and so there is an equation

(x_{1} … x_{n})^{ℓ} = x_{1}^{ℓ} … x_{n}^{ℓ}Δ(ℓ, n)

for some Δ(ℓ, n) in H_{n}^{ℤ}.

The analogue of Δ(ℓ, n) where the groups H_{n}^{ℤ} are replaced by free groups has been studied in a number of papers in combinatorial group theory especially in the case n = 2 [MKS]. It is the purpose of this section to give some additional properties of Δ(ℓ, n) and to show that there is another group theoretic characterization for these words which involves the center of K_{n}^{R}.

Theorem 2.2.1. There exist words Δ(q, n) (which must be unique!) in K_{n}^{ℤ} such that

(1) (x_{1} … x_{n})^{q} = x_{1}^{q} … x_{n}^{q}Δ(q, n) and
(2) δ_{i}Δ(q, n) = Δ(q, n − 1) for 1 ≤ i ≤ n.

Furthermore

Δ(q, n) = ∏_{k=2}^{n} γ(k, σ)^{n(k,σ)}

where the product is taken in the left lexicographical ordering with a fixed choice of total ordering of the elements in Σ_{n}.

The first case of this theorem is given by the next lemma.

Lemma 2.2.2. The following formula holds in K_{2}^{R}[x_{1}, x_{2}]: If q ≥ 1, then (x_{1}x_{2})^{q} = x_{1}^{q}x_{2}^{q}[x_{2}, x_{1}]^{\binom{q}{2}}.

Proof. If q = 2, then

(x_{1}x_{2})^{2} = x_{1}x_{2}x_{1}x_{2} = x_{1}^{2}x_{2}[x_{2}^{−1}, x_{1}^{−1}]x_{2}

as ba = ab[b^{−1}, a^{−1}] for any elements a, b in a group G. But

x_{1}^{2}x_{2}[x_{2}^{−1}, x_{1}^{−1}]x_{2} = x_{1}^{2}x_{2}^{2}[x_{2}, x_{1}] in K_{2}^{R}[x_{1}, x_{2}].

44 F.R. COHEN^{∗}

Lemma 2.2.2 follows by induction assuming the formula for q:

(1) (x_{1}x_{2})^{q+1} = (x_{1}x_{2})^{q}x_{1}x_{2} = x_{1}^{q}x_{2}^{q}[x_{2}, x_{1}]^{\binom{q}{2}}x_{1}x_{2}.
(2) (x_{1}x_{2})^{q+1} = x_{1}^{q+1}x_{1}^{q}[x_{2}^{−q}, x_{1}^{−1}][x_{2}, x_{1}]^{\binom{q}{2}}x_{2} in K_{2}^{R}[x_{1}, x_{2}].
(3) (x_{1}x_{2})^{q+1} = x_{1}^{q+1}x_{2}^{q+1}[x_{2}, x_{1}]^{\binom{q}{2}+q}.

Proof of Theorem 2.2.1. The proof is by induction on n with the case n = 1 being trivial and the case n = 2 given in Lemma 2.2.2.

By the inductive hypothesis on n,

(x_{1} … x_{n})^{q} = x_{1}^{q} … x_{n}^{q} ∏_{k,σn} γ(k, σ)^{m(k,σ)}.

Next consider the difference Δ defined by

(x_{1} … x_{n+1})^{q} = x_{n}^{q} … x_{n+1}^{q}(∏ γ(k, σ)^{m(k,σ)})Δ,

where γ(k, σ) is the element in K_{n+1}^{R}[x_{1}, …, x_{n+1}] given by the combinatorial extension of the analogous element in K_{n}^{R}[x_{1}, …, x_{n}]. These elements are as defined in section 2.1. Thus by the inductive hypothesis,

δ_{i}Δ = 1

for 1 ≤ i ≤ n+1. By Theorem 1.7.2, Δ must lie in Γ_{n+1}^{R}, the center of K_{n+1}^{R}[x_{1}, …, x_{n+1}].

As γ(k, σ) is in H_{n+1}^{R} and Γ_{n+1}^{R} is in H_{n+1}^{R}, Δ is in H_{n+1}^{R}. The theorem follows.

§2.3 An automorphism of H_{n}^{R}. There is an involution χ̄ on the space (ΣX)^{n} given by

χ̄(x_{1}, …, x_{n}) = (x_{n}^{−1}, x_{n−1}^{−1}, …, x_{2}^{−1}, x_{1}^{−1})

where ( )^{−1} : ΣX → ΣX is the canonical involution. Thus there is an induced involution χ^{∗} : [(ΣX)^{n}, ΩY] → [(ΣX)^{n}, ΩY] given by χ̄^{∗}. There is an analogous involution on the free group χ : F [x_{1}, …, x_{n}] → F [x_{1}, …, x_{n}] defined by χ(x_{i}) = x_{n−i+1}^{−1}. This involution preserves the relations in K_{n}^{R}[x_{1}, …, x_{n}] and thus induces an involution χ of these groups.

Lemma 2.3.1. The diagram

K_{n}^{R}[x_{1}, …, x_{n}] ⟶^{θ} [(ΣX)^{n}, ΩY]
χ↓  ^{χ^{∗}}↓
K_{n}^{R}[x_{1}, …, x_{n}] ⟶_{θ} [(ΣX)^{n}, ΩY]

commutes.

Proof. By definition, χ^{∗}θ(x_{i}) = x_{n−i+1}^{−1}θχ(x_{i}). The lemma follows.

Next observe that χ̄ induces a map on the level of the nth stage of James construction

χ̄ : J_{n}ΣX → J_{n}ΣX

as ( )^{−1} : ΣX → ΣX is a pointed map. Thus there are maps χ̄ : JΣX → JΣX obtained via taking colimits. Let χ : ΩΣ^{2}X → ΩΣ^{2}X be the canonical loop inverse obtained by reversing the loop coordinate.

Lemma 2.3.2. The diagram

(ΣX)^{n} ⟶ J_{n}ΣX ⟶ JΣX ⟶ ΩΣ^{2}X
χ̄↓ ↓χ̄ ↓χ̄ ↓^{χ}
(ΣX)^{n} ⟶ J_{n}ΣX ⟶ JΣX ⟶ ΩΣ^{2}X

commutes to homotopy.

By the definition of the maps π^{∗} and θ together with Lemma 2.3.1 and 2.3.2, there are commutative diagrams

K_{n}[x_{1}, …, x_{n}] ⟶^{θ} ∏_{x∈r}[(ΣX)^{n}, ΩΣ^{2}X] ⟵^{π^{∗}} ∏_{x∈r}[J_{n}ΣX, ΩΣ^{2}X]
χ↓  ↓^{χ^{∗}}  ↓^{}^{χ̄^{∗}}
K_{n}[x_{2}, …, x_{n}] ⟶^{θ} ∏_{x∈r}[(ΣX)^{n}, ΩΣ^{2}X] ⟵^{π^{∗}} ∏_{x∈r}[J_{n}ΣX, ΩΣ^{2}X]

Thus χ̄^{∗}, χ^{∗} and χ induce an involution on the diagrams in definition 3.1 and the next result follows.

Proposition 2.3.3. There is an involution

χ : H_{n}^{ℤ} → H_{n}^{ℤ}

together with commutative diagrams

H_{n}^{ℤ} ⟶ [J_{n}ΣX, ΩΣ^{2}X]
χ↓ ↓^{}^{χ̄^{∗}}
H_{n}^{ℤ} ⟶ [J_{n}ΣX, ΩΣ^{2}X] and

46 F.R. COHEN^{∗}

H_{n}^{ℤ} ⟶ K_{n}[x_{1}, …, x_{n}]
χ↓ ↓^{χ}
H_{n}^{ℤ} ⟶ K_{n}[x_{1}, …, x_{n}].

Notice that the involution χ̄^{∗} : [J_{n}ΣX, ΩΣ^{2}X] is not, in general, equal to the involution obtained by composition with χ : ΩΣ^{2}X → ΩΣ^{2}X. There is a group homomorphism Ω : [J_{n}ΣX, ΩΣ^{2}X] → [ΩJ_{n}ΣX, Ω^{2}Σ^{2}X] obtained by looping a map f : J_{n}ΣX → ΩΣ^{2}X. Furthermore [ΩJ_{n}ΣX, Ω^{2}Σ^{2}X] is an abelian group for which χ : Ω^{2}Σ^{2}X → Ω^{2}Σ^{2}X induces multiplication by −1. The next lemma is classical.

Lemma 2.3.4. The diagram

[J_{n}ΣX, ΩΣ^{2}X] ⟶^{Ω} [ΩJ_{n}ΣX, Ω^{2}Σ^{2}X]
χ̄^{∗}↓ ↓^{(−1)}
[J_{n}ΣX, ΩΣ^{2}X] ⟶_{Ω} [ΩJ_{n}ΣX, Ω^{2}Σ^{2}X]

commutes.

Proof. This follows immediately from 2.3.2 together with the fact that Ω(χ̄)^{∗} in [ΩJΣX, Ω^{2}Σ^{2}X] is (−1).

Thus by Proposition 2.3.3 and Lemma 2.3.4 there is a commutative diagram

H_{n}^{ℤ} ⟶ ∏_{x∈r}[J_{n}ΣX, ΩΣ^{2}X] ⟶^{Ω} ∏_{x∈r}[ΩJ_{n}ΣX, Ω^{2}Σ^{2}X]
χ↓ ↓ ↓^{(−1)}
H_{n}^{ℤ} ⟶ ∏_{x∈r}[J_{n}ΣX, ΩΣ^{2}X] ⟶_{Ω} ∏_{x∈r}[ΩJ_{n}ΣX, Ω^{2}Σ^{2}X]

Since [ΩJΣX, Ω^{2}Σ^{2}X] is an abelian group, all homomorphisms from H_{n}^{ℤ} to it factor through the first homology group of H_{n}^{ℤ}. If α : Σ^{2}X → Y has order p^{r} and R = ℤ/p^{r}ℤ there is a commutative diagram

H_{n}^{R} ⟶ [ΩJ_{n}ΣX, Ω^{2}Y]
χ↓ ↓^{−1}
H_{n}^{R} ⟶ [ΩJ_{n}ΣX, Ω^{2}Y].

These diagrams can be expressed in terms of classical group homology.

Let Π = ℤ/2ℤ. Thus Π acts on the homology of H_{n}^{R} through the involution χ. Furthermore Π acts on ℤ via the sign representation; thus χ(n) = −n for n in ℤ

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 47

where χ denotes the generators of Π. Let ℤ(−1) denote the Π-module given by sign representation.

Given these two Π-modules ℤ(−1) and the homology of H_{n}^{R}, form

H_{1}(H_{n}^{R}) ⊗_{ℤΠ} ℤ(−1) = Δ(H_{n}^{R})

and observe that this last group is a natural quotient of H_{n}^{R}.

Theorem 2.3.5. Given α : Σ^{2}X → Y of order p^{r} in [Σ^{2}X, Y], R = ℤ/p^{r}ℤ, there are commutative diagrams

H_{1}(H_{n}^{ℤ}) ⟶ [ΩJ_{n}ΣX, Ω^{2}Σ^{2}X]
↓ ↓^{1}
Δ(H_{n}^{R}) ⟶ [ΩJ_{n}ΣX, Ω^{2}Y] .

§2.4: A filtration of H_{n}^{R}.

There is a second filtration of H_{n}^{R} which converges more slowly than the descending central series for H_{n}^{R}. Furthermore, this filtration corresponds to one obtained via the natural filtrations of J_{n}X.

Definition 2.4.1. Filter H_{n}^{R} by

F^{q}H_{n}^{R} = H_{n}^{R} ∩ Γ^{q}(K_{n}^{R}).

Theorem 2.4.2. The group F^{q+1}H_{n}^{R} is normal in F^{q}H_{n}^{R} with {1} = F^{n+1}H_{n}^{R} and H_{n}^{R} = F^{1}H_{n}^{R}. The filtration quotients E_{0}^{q}(H_{n}^{R}) = F^{q}(H_{n}^{R})/F^{q+1}(H_{n}^{R}) are free R-modules of rank (q − 1)! which is isomorphic to Lie(q) ⊗_{ℤ} R = Lie^{R}(q) as an Rmodule. Furthermore, there is a graded Lie algebra L_{n}^{R} which is E_{0}^{q}(H_{n}^{R}) = Lie^{R}(q) in degree q, 1 ≤ q ≤ n. The Lie bracket is induced by forming commutators in H_{n}^{R} and is given by

[ , ] : Lie^{R}(p) ⊗_{R} Lie^{R}(q) ⟶ Lie^{R}(p + q)

where

A = [… [x_{1}, x_{σ(2)}]x_{σ(3)} …]x_{σ(p)}],
B = [… [x_{1}, x_{τ(2)}]x_{τ(3)}] …]x_{τ(q)}], and
[A, B] = Σ[[… [x_{i_{1}}, x_{i_{σ(2)}}] …]x_{i_{σ(p)}}], [… [x_{j_{1}}, x_{j_{τ(2)}}] …]x_{j_{τ(q)}}]]

where the sum is taken over all shuffles

i_{1} < ⋯ < i_{p}, j_{1} < ⋯ < j_{q}, and {i_{1}, …, i_{p}, j_{1}, …, j_{q}} = {1, …, p + q}.

48 F.R. COHEN^{∗}

§2.5: On exponents for Δ(H_{n}^{R}).

Consider H_{2^{k}}^{R} for R = ℤ/2ℤ. The purpose of this section is to study powers of α = (x_{1})(x_{2})(x_{3})(x_{4})....(x_{2^{k}}) in H_{2^{k}}^{R}.

Proposition 2.5.1. (i): The kernel of the natural epimorphism

H_{2^{k}}^{R} ⟶ H_{2^{k−1}}^{R}

is abelian and is isomorphic to the direct sum of the Lie^{R}(j) for 2^{k} ≥ j ≥ 1+2^{k−1}.

Proof. Notice that the kernel of the map is generated by the elements in H_{2^{k}}^{R} which are defined in section 1.4 for which j ≥ 1 + 2^{k−1}:

γ(j; σ) = ∏_{1≤i_{i}<···<i_{n}≤n}[… [x_{i_{1}}, x_{i_{σ(2)}}]x_{i_{σ(3)}}] …]x_{i_{σ(j)}}]

where the product is taken in the left lexicographical ordering.

These elements commute in H_{2^{k}}^{R}. That the kernel is isomorphic to the direct sum of the Lie^{R}(t) for 2^{k} ≥ t ≥ 1 + 2^{k−1} follows by counting.

Proposition 2.5.2. (i) There is a homomorphism which sends 1 to α = (x_{1})(x_{2})....(x_{2^{k}}):

ψ(k) : ℤ/2^{k+1}ℤ ⟶ H_{2^{k}}^{R}

(ii) There is a commutative diagram

ℤ/2^{k+1}ℤ ⟶^{ψ(k)} H_{2^{k}}^{R}
↓  ↓
ℤ/⊭^{k} ⟶^{ψ(k−1)} H_{2^{k−1}}^{R} .

Proof. There is a map of the integers to H_{2^{k}}^{R} which sends 1 to α. Notice this map gives the proposition at once from Proposition 2.5.1, and by induction on k starting with k = 0.

Proposition 2.5.3.

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 49

There are natural maps ρ which send α to 1:

ρ : H_{2^{k}}^{R} ⟶ ℤ/2^{k+1}ℤ

Furthermore, H_{2^{k}}^{R} is semi-direct product of ℤ/2^{k+1}ℤ and the kernel of ρ. Proof. Consider the elements γ(j; σ). Assume that j satisfies for2^{k−s+1} ≥ j ≥ 2^{k−s} + 1. Define

ρ(γ(j; σ)) = 2^{s} .

Notice that ρ extends to a group homomorphism ρ : H_{2^{k}}^{R} ⟶ ℤ/2^{k+1}ℤ. Furthermore, ρ admits a section given by ψ(k) of Proposition 2.5.2.

The next result follows from a direct calculation. Proposition 2.5.4. The 2^{k}−th power of α in H_{2^{k}}^{R} equals the sum of the generators γ(2^{k}; σ) for σ in the symmetric group on 2^{k} − 1 letters where the γ(2^{k}; σ) run over the ”canonical” basis for Lie^{R}(2^{k}). Proof. The result is correct for k = 1 and follows by induction on k, the Jacobi identity, and Proposition 2.5.1.

This last result implies consequences for the groups Δ(H_{n}^{R}). Theorem 2.5.5.

There is a commutative diagram

Δ(H_{2^{k}}^{R}) ⟶ ℤ/2^{k+1}ℤ
↓ ↓
Δ(H_{2^{k−1}}^{R}) ⟶ ℤ/2^{k}ℤ .

Thus the inverse limit of Δ(H_{n}^{R}), Δ(H_{∞}^{R}), admits a surjection to Z_{2^{∞}}, and Δ(H_{∞}^{R}) does not have an exponent. Proof.

50 F.R. COHEN^{∗}

Notice that Proposition 2.5.3 gives a surjection ρ : H_{2^{k}}^{R} ⟶ ℤ/2^{k+1}ℤ. Furthermore, the conjugation χ preserves filtration of the H_{2^{k}}^{R}. Thus ρ factors through the quotient Δ(H_{2^{k}}^{R}).

This suffices.

This last result can be interpreted as follows: Assume that the suspension order of the identity element for Σ^{2}X is of order 2^{r}. Then all relations between Hopf invariants, and Whitehead products together with null-homotopies of the diagonal map for ΣX are insufficient to deduce a (conjectured) null-homotopy of 2^{s}-power maps for Ω^{2}Σ^{2}X.

§2.6: Some null-homotopic maps.

The purpose of this section is to exhibit some sample null-homotopic maps which arise from the above group theoretic constructions. A few general examples are listed first. Explicit structures for (i) the abelianization of H_{4}^{R} as well as (ii) Δ(H_{4}^{R}) are also given, and are restricted to the case n = 4 and R = ℤ/2ℤ for brevity.

Consider the following elements in H_{n} where σ is an element of the symmetric group on k-letters which are defined in section 1.4:

γ(k; σ) = ∏_{1≤i_{i}<···<i_{k}≤n}[… [x_{i_{1}}, x_{i_{σ(2)}}]x_{i_{σ(3)}}] …]x_{i_{σ(k)}}]

where the product is taken in the left lexicographical ordering.

There are several special cases which are useful below that are given by

α = (x_{1})(x_{2})....(x_{n}), and

λ(σ) = [[...[x_{σ(1)}, x_{σ(2)}].....x_{σ(n)}].

The main calculations here center around the element γ(4; σ) for σ = 1.

Proposition 2.6.1.

Let R = ℤ/2ℤ. (i): The abelianization of H_{2^{k}}^{R} is isomorphic to the direct sum of ℤ/8ℤ, and five copies of ℤ/2ℤ. A choice of generators is listed in the proof:

COMBINATORIAL GROUP THEORY IN HOMOTOPY THEORY, I 51

α has order 8, while γ_{1}, γ_{2}, y_{5}, and y_{6} have order 2. (ii): The group Δ(H_{4}^{R}) is isomorphic to the direct sum of ℤ/8ℤ, and four copies of ℤ/2ℤ. Furthermore, Δ(H_{4}^{R}) is obtained by setting y_{5} = y_{6} in (i).

Proof. Consider the group H_{4} with generators

α = (x_{1})(x_{2})(x_{3})(x_{4}),
β = [x_{1}, x_{2}][x_{1}, x_{3}][x_{1}, x_{4}][x_{2}, x_{3}][x_{2}, x_{4}][x_{3}, x_{4}],
γ_{1} = [[x_{1}, x_{2}]x_{3}])([[x_{1}, x_{2}]x_{4}])([[x_{1}, x_{3}]x_{4}])([[x_{2}, x_{3}]x_{4}]),
γ_{2} = [[x_{1}, x_{3}]x_{2}])([[x_{1}, x_{4}]x_{2}])([[x_{1}, x_{4}]x_{3}])([[x_{2}, x_{4}]x_{3}]),
y_{1} = [[[x_{1}, x_{2}]x_{3}]x_{4}],
y_{2} = [[[x_{1}, x_{2}]x_{4}]x_{3}],
y_{3} = [[[x_{1}, x_{3}]x_{2}]x_{4}],
y_{4} = [[[x_{1}, x_{3}]x_{4}]x_{2}],
y_{5} = [[[x_{1}, x_{4}]x_{2}]x_{3}],

and

y_{6} = [[[x_{1}, x_{4}]x_{3}]x_{2}].

The y_{i} generate Lie(4) and the center. Furthermore, a direct calculation gives the following commutators:

[α, γ_{1}] = (y_{2})(y_{3})(y_{4})(y_{5})(y_{6})^{−1}.
[α, γ_{2}] = (y_{2})^{−1}(y_{3})(y_{4})^{−1}(y_{5}).
[α, β] = (γ_{2})^{2}(y_{2})^{−1}(y_{3})^{(}y_{4})^{−1}(y_{5})^{2}(y_{6})
[β, γ_{i}] = 1.

Notice that part (i) follows by reducing the above relations to H_{4}^{R}.

To derive part (ii), notice that y_{5} = y_{6} in Δ(H_{4}^{R}), and that there are no additional relations.

These observations admit extensions to the groups H_{n}. Let σ denote an element in the symmetric group on (n-1) letters, and define elements in H_{n} by λ(σ) = [[[x_{σ(1)}, x_{σ(2)}].....x_{σ(n)}].

52 F.R. COHEN^{∗}

Proposition 2.6.2.

The following equations are satisfied In the sub-group Lie^{R}(n) of Δ(H_{n}^{R}):

χ(λ(σ)) = −λ(σ)

Proof.

Notice that χ preserves filtration of H_{n}. This suffices.

Remarks: This last proposition gives choices of elements in the classical distributivity formula that are non-trivial, but which are null-homotopic after looping. These elements given by χ(λ(σ)) + λ(σ) do not contribute to homotopy groups

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Department of Mathematics University of Rochester Rochester, NY 14627 U.S.A.