COMBINATORIAL MODELS FOR ITERATED LOOP SPACES

F. R. COHEN* AND R. LEVI

1. Introduction

The purpose of this note is to record natural filtered simplicial group models for iterated loop spaces. The models are derived by classical methods for simplicial groups along with the identification for certain group theoretic kernels. The main content of the article is the study of some useful properties of these models.

One feature of these models is that some of their properties follow from standard group theoretic facts, which themselves are derived from elementary covering space theory. Another feature is that within these models one can write homotopies for certain maps in a relatively straight forward way. Specific applications will not be discussed within this note, but it should be pointed out that these models were constructed as a framework in which one could deal with certain classical problems in homotopy theory.

Before presenting details, the main group theoretic input of the constructions here will be described. Restrict to the category of pointed sets. The coproduct A ∨ B in this category admits a map to A by sending B to the base-point. There is an induced map on the level of free groups F r[A ∨ B] ^{}F r[A] for which the kernel is the free group with basis given by the set of conjugates B^{F r[A]} in F r[A ∨ B]. This kernel gives a group theoretic tool for constructing simplicial group models for double loop spaces. Iteration of this construction gives methods for constructing simplicial group models for iterated loop spaces. This iteration becomes complicated, at least in terms of notation. The case of Ω^{2}Σ^{2}X, and Ω^{2}X are both clean, and useful; it is described as a special case below.

Several well known constructions in simplicial homotopy theory will be used here. Namely Milnor’s simplicial suspension functor σ, the functor F, again due to Milnor, which models the loop-suspension functor, Kan’s G functor which gives a model for the loop space functor and the loop functor Ω, due to Moore, which gives a model for the loop space functor in the case where the simplicial set under consideration is a Kan complex.

The work is carried out in the category of reduced simplicial sets, whose objects are connected simplicial sets X, such that X_{i} = b_{i}, the base point in dimension i, for i less than or equal to the connectivity of X. This condition is imposed for simplicity, but is not a substantial restriction.

Some of the results in this article require rather elaborate notation. Thus the introduction shall only contain an extract of the contents of the paper, rather than precise statements. The combinatorial constructions given here model the functors Ω^{n}Σ^{n} for arbitrary connected spaces and Ω^{n} on n-connected spaces.

The first construction models the functor Ω^{n}Σ^{n}. Thus for a connected reduced simplicial set X a model for Ω^{n}Σ^{n}|X| is given by Ω^{n−1}F Σ^{n−1}X. This object is a simplicial group, which shall be denoted by Λ_{n}X. The objective of the following theorem is to study the structure of this simplicial group. Let G be an arbitrary group and let W, B ⊆ G be two subsets. Denote by B^{W} the subset of G given by all conjugates of elements of B by elements of W. In particular if W contains the identity element then the set B^{W} contains B.

The first author is partially supported by an NSF grant.

Theorem 1.1. For each positive integer n there is a functor Λ_{n} from the category of pointed connected reduced simplicial sets to the category of free simplicial groups such that the geometric realization of Λ_{n}X is homotopy equivalent to Ω^{n}Σ^{n}|X|. Moreover,

1. For each n ≥ 1 there is a natural transformation of functors E : Λ_{n} ^{}Λ_{n+1} whose geometric realization is the Freudenthal suspension map.

2. A basis for Λ_{n}X is given by the graded set B_{n}(X)^{F r[A_{n}(X)]}, where B_{n}(X) is the image of the n-fold Freudenthal suspension E^{n} : X ^{}Λ_{n}X and A_{n}(X) is its pointed complement in the canonical generating set for F Σ^{n−1}X.

3. The simplicial operators in Λ_{n}X can be described explicitly in terms of those of X.

4. The groups Λ_{n}(X) admit a natural increasing filtration by simplicial subgroups.

The first interesting case of the theorem is n = 2. An explicit description of this case is given here. Higher values of n admit similar statements, but the notation becomes more laborious.

Proposition 1.2. A basis for the free simplicial group Λ_{2}X in dimension d is given by all elements of the form c(w)[x], where [x] = (x, 1) ∈ (ΣX)_{d+1} and w is a word in the free group on

⋁_{{1<i<d+1 \atop q+i=d+1}}(X_{q}, i).

The simplicial operators on Λ_{2}X are the unique multiplicative extension of the functions which operate on generators as follows.

d_{i}^{Λ_{2}}(c(w)[x]) = d_{i+1}^{F Σ}(c(w)[x]) = c(d_{i+1}^{F Σ}(w))[d_{i}x]
s_{i}^{Λ_{2}}(c(w)[x]) = s_{i+1}^{F Σ}(c(w)[x]) = c(s_{i+1}^{F Σ}(w))[s_{i}x].

The filtration given in Theorem 1.1 in this case is described as follows. Filter the free group on the set ⋁_{{1<i<d+1 \atop q+i=d+1}} (X_{q}, i) by the reduced word length filtration. We say that a generator c(w)[x] is in filtration n if w is in filtration n of the reduced word length filtration. Now define F_{n}Λ_{2}(X) to be the subgroup of Λ_{2}(X) generated by all elements c(w)[x] of filtration at most n. It is easy to see that F_{0}Λ_{2}(X) is the subgroup generated by the image of the Freudenthal suspension. The other filtrations are not understood at present.

The construction of a free simplicial group model for Ω^{n}X is analogous. Given an n-connected pointed reduced simplicial set X, let L_{n}(X) denote the free simplicial group given by Ω^{n−1}GX. Here in order to present a basis for this free simplicial group, one needs to make a change of basis for the free group GX. Assuming this has been done in the appropriate manner, consider the formal (n − 1)-fold desuspension of this generating set. This is a pointed graded set, which can be decomposed as a one-point union A_{n}(X) ∨ B_{n}(X).

Theorem 1.3. Let X be a pointed n-connected reduced simplicial set and let n be a non-negative integer. There is a simplicial group L_{n}X, whose geometric realization is homotopy equivalent to Ω^{n}|X|. Moreover,

1. A basis for L_{n}(X) in degree n is given by the graded set B_{n}(X)^{F [A_{n}(X)]}.

2. The simplicial operators in L_{n}(X) are described explicitly in terms of those of X in the proof of this theorem.

3. The groups L_{n}(X) admit a natural increasing filtration by simplicial subgroups.

Here too the first interesting case is n = 2. An explicit description of it is given in the following proposition. For higher values of n, the description becomes recursive and hence less explicit. The reader is referred to section 4 for details.

Proposition 1.4. Let X be a 2-connected reduced complex. Then (L_{2}X)_{n} is isomorphic as a group to the quotient of the free group on symbols ⟨w, x⟩ with w ∈ (GX)_{n} and x ∈ X_{n+2} subject to the relations ⟨w, x⟩ = 1 if x = s_{i}y for some y ∈ X_{n+1} and i ∈ {0, 1}. The simplicial operators are given by the formulas

1. s_{i}^{L_{2}}⟨w, x⟩ = ⟨s_{i}^{G}w, s_{i+2}x⟩ for i ≥ 0.

2. d_{i}^{L_{2}}⟨w, x⟩ = ⟨d_{i}^{G}w, d_{i+2}x⟩ for i ≥ 1.

3. d_{0}^{L_{2}}⟨1, x⟩ = ⟨1, d_{2}x⟩⟨d_{0}^{G}d_{0}x, d_{1}x⟩^{−1}⟨d_{0}^{G}d_{0}x, d_{0}x⟩ and d_{0}^{L_{2}}⟨w, x⟩ can be computed in terms of the generators whenever w is given as an explicit word in GX.

Viewing the last proposition as a description of pointed mapping spaces out of a 2-sphere, one is motivated to wonder about models for mapping spaces out of more general orientable surfaces. Let S_{g}^{n} denote a pointed surface of genus g and a boundary consisting of n circles wedged at the base point.

Theorem 1.5. Let X be a 1-connected complex. Then for every g, k ≥ 0 there is a Kan complex M_{g}^{k}(X) whose geometric realization is equivalent to the pointed mapping space Map_{∗}(S_{g}^{k}, X). The simplices of M_{g}^{k}(X) in dimension n are given by (2g + k + 1)-tuples (u_{1}, v_{1}, ⋯, u_{g}, v_{g}, a_{1}, ⋯ a_{k}, w), where u_{i}, v_{j}, a_{r} ∈ (GX)_{n}, and w ∈ (GX)_{n+1} are such that

[u_{1}, v_{1}] ⋯ [u_{g}, v_{g}] · (a_{1} ⋯ a_{k})^{−1} = d_{o}^{G}(w).

The simplicial operators of M_{g}^{k}(X) are induced in the obvious way by those of GX. Moreover, M_{0}^{0}(X) coincides with L^{2}(X).

In closing of the introduction, the reader is warned that the models constructed here are very unlikely to have any computational utility. On the other hand they do provide a context in which general homotopy theoretic phenomena can be studied.

The paper is organized as follows. In section 2, a review of some classical simplicial constructions are given. The construction Λ_{n} and L_{n} are discussed in sections 3 and 4 respectively. Section 5 is devoted to a comparison of the models via the Freudenthal suspension and evaluation maps. A sample calculation is given in this section. The filtrations for these models are introduced and discussed in section 6. A model for a pointed mapping space out of a surface. is given in section 7.

Both authors are grateful to the Centre de Recerca Matematica in Barcelona for giving them the opportunity to meet and discuss some of the work in this paper. Both authors would also like to thank J. Smith for useful discussions.

2. Review of some classical simplicial constructions

As pointed out in the introduction, some classical simplicial constructions will be used in the sequel. The purpose of this preliminary section is to recall these constructions.

The reader is assumed to be familiar with the concept of a Kan complex. These are simplicial sets with an additional property that makes it possible to combinatorially define homotopy groups. The actual definition will not be recalled here as these facts are not actually used in this article. Furthermore, any simplicial set can be made into a Kan complex without changing the homotopy type of its geometric realization, but in fact simplicial groups are automatically Kan complexes. The term ”a Kan complex” will thus be used to emphasize that the given simplicial set is assumed to be a Kan complex. A complex will mean an arbitrary simplicial set.

For a pointed, connected complex X the suspension ΣX is defined to be the complex, which in dimension r consists of a 1-point union

⋁_{{0<i<r \atop q+i=r}}(X_{q}, i),

where (X_{q}, i) is a copy of X_{q}. Ignoring the simplicial operators momentarily, notice that the construction can be easily iterated. The following proposition summarizes the structure of the iterated suspension with the simplicial operators. Notice also that the suspension of a Kan complex is not a Kan complex in general.

Proposition 2.1. Let X be a pointed connected reduced complex and let n be a positive integer. The the n-fold suspension Σ^{n}X of X is the complex, which in dimension r is given by

⋁_{{0<i_{1},i_{2},··· ,in<r \atop q+i_{1}+···+in=r}}(X_{q}, i_{1}, i_{2}, ⋯, i_{n}),

where (X_{q}, i_{1}, i_{2}, ⋯, i_{n}) is a copy of X_{q} as sets.

The simplicial operators are given inductively by the formulas

d_{i}^{Σ^{n}}(a_{q}, i_{1}, ⋯, i_{n}) = \begin{cases}(a_{q}, i_{1}, ⋯, i_{n−1}, i_{n} − 1) & if i < i_{n} \\ (d_{i−i_{n}}^{Σ^{n−1}}(a_{q}, i_{1}, ⋯, i_{n−1}), i_{n}) & if i ≥ i_{n} \end{cases}

s_{i}^{Σ^{n}}(a_{q}, i_{1}, ⋯, i_{n}) = \begin{cases}(a_{q}, i_{1}, ⋯, i_{n−1}, i_{n} + 1) & if i < i_{n} \\ (s_{i−i_{n}}^{Σ^{n−1}}(a_{q}, i_{1}, ⋯, i_{n−1}), i_{n}) & if i ≥ i_{n} \end{cases}

For a pointed simply-connected reduced complex X, the Moore path-complex construction P X is given by

(P X)_{n} = X_{n+1}.

The simplicial operators are given by d_{i}^{P X} = d_{i+1}^{X}, and s_{i}^{P X} = s_{i+1}^{X} for i ≥ 0.

Then there is a simplicial projection map from P X to X given by the leftover face d_{0} and the Moore loop space construction ΩX is defined as the set theoretic kernel of this map, i.e. the inverse image of the base-point under the projection. If X is a Kan complex then so is ΩX and in that case |ΩX| ≃ Ω|X| but not generally otherwise (for example, if S^{n} is the standard simplicial model for the n-sphere, with one non-degenerate simplex in dimension n, then ΩS^{n} = S^{n−1}).

The Moore loop-complex construction can be iterated and the resulting complex is described in the following

Proposition 2.2. Let n be a positive integer and let X be a pointed n-connected reduced complex. The n-fold Moore loop-complex Ω^{n}(X) is the complex which in dimension k consists of all simplices in X_{n+k} such that d_{i}(x) = b_{n+k−1} for all 0 ≤ i ≤ n, where b_{j} denotes the base-point simplex in dimension j. The simplicial operators in Ω^{n}(X) are given by d_{i}^{Ω^{n}}(x) = d_{i+n}(x) and s_{i}^{Ω^{n}}(x) = s_{i+n}(x).

Let X be a pointed simply-connected reduced complex. The Kan loop-group functor G associates with X a simplicial group GX which in dimension n is freely generated by the simplices of X_{n+1} subject to the relation s_{0}(x) = 1 for any x ∈ X_{n}. The simplicial operators are given on generators as follows

1. d_{0}^{G}(x) = d_{0}(x)^{−1}d_{1}(x)

2. d_{i}^{G}(x) = d_{i+1}(x) for i ≥ 1.

3. s_{i}^{G}(x) = s_{i+1}(x) for i ≥ 0. The complex GX is automatically a Kan complex by virtue of being a simplicial group. Its geometric realization is homotopy equivalent to Ω|X| [2].

Let X be a pointed, connected complex. The Milnor free group construction F X is the free simplicial group, which in dimension n is generated by X_{n} modulo the relation setting the base-point to be equal to the identity element. The face and degeneracy operators are the unique multiplicative extensions of the respective operators in X. Notice that being a simplicial group F X is automatically a Kan complex. The geometric realization of F X is homotopy equivalent to ΩΣ|X|. The standard references for these constructions are [2, 5].

Finally recall an elementary fact about free groups. Let F r[−] denote the pointed free group functor on the category of pointed sets. Namely, F r[X] is the free group generated by X with a single relation setting the base-point to be the identity. Let A and B be arbitrary pointed sets. Then there is a natural map

π_{A} : F r[A ∨ B] ^{}F r[A].

An elementary covering space argument shows that the kernel of π_{A} is freely generated by the set B^{F r[A]}, namely the set of all conjugates of non-trivial elements of B by arbitrary words in the free group on A [3]. This fact is also used in [1], page 139. A proof using covering spaces is given in the appendix.

For w ∈ F r[A] and b ∈ B let c(w)(b) = w^{−1}bw. Thus Ker(π_{A}) is freely generated by the symbols c(w)(b), for b ∈ B and w ∈ F [A].

3. The construction Λ_{n}

Simplicial group models for Ω^{n}Σ^{n}X are known in the literature. In particular there is a model for these spaces due to J. Smith [7]. Smith’s construction is elegant and ”homologically minimal”.

The constructions in this article by contrast are large, but on the other hand have the feature that writing down explicit maps out of them becomes relatively easy.

The model for Ω^{n}Σ^{n}X is based on the simplicial suspension construction Σ , Milnor’s F construction and the Moore loop complex construction Ω, all of which are reviewed above. Thus define Λ_{0}(X) = X and for each n ≥ 1 let

Λ_{n}X = Ω^{n−1}F Σ^{n−1}X.

There is a natural simplicial map

E : X ^{}Λ_{1}X,

given by taking a simplex x ∈ X to the generator in Λ_{1}X = F X represented by it. One may regard this map as a natural transformation from the identity functor Λ_{0} to the functor Λ_{1}. The geometric realization of E is the usual Freudenthal suspension on the realization of X.

The first interesting case is n = 2, which is handled separately, as its relative simplicity provides some insight. Also, examination of the resulting model in this case will enable us to define the n-fold Freudenthal suspension map on the level of simplicial groups.

Proposition 3.1. A basis for the free simplicial group Λ_{2}X in dimension d is given by all elements of the form c(w)[x], where [x] = (x, 1) ∈ (ΣX)_{d+1} and w is a word in the free group on

⋁_{{1<i<d+1 \atop q+i=d+1}}(X_{q}, i).

The simplicial operators on Λ_{2}X are the unique multiplicative extension of the functions which operate on generators as follows.

d_{i}^{Λ_{2}}(c(w)[x]) = d_{i+1}^{F Σ}(c(w)[x]) = c(d_{i+1}^{F Σ}(w))[d_{i}x]
s_{i}^{Λ_{2}}(c(w)[x]) = s_{i+1}^{F Σ}(c(w)[x]) = c(s_{i+1}^{F Σ}(w))[s_{i}x].

Proof. By definition (Λ_{2}X)_{d} is equal to the kernel of the homomorphism

d_{0}^{F Σ} : (F ΣX)_{d+1} ^{}(F ΣX)_{d}.

Notice that the group (F ΣX)_{d+1} is generated freely by all elements of the form (a_{q}, i), where 0 < i < d+1 and i+q = d+1. Thus the generating set can be written as

Ā_{2}(X)_{d+1} ∨ B̄_{2}(X)_{d+1},

where Ā_{2}(X)_{d+1} is the set of all elements (a_{q}, i) with i > 1 and B̄_{2}(X)_{d+1} is the pointed complement of Ā_{2}(X)_{d+1} in (ΣX)_{d+1}. Notice that the set Ā_{2}(X)_{d+1} is exactly the image of s_{0}^{F Σ} : (ΣX)_{d} ^{}(ΣX)_{d+1}. Thus the restriction of d_{0}^{F Σ} to Ā_{2}(X)_{d+1} is an isomorphism of sets. Let A_{2}(X) denote the desuspension of Ā_{2}(X) and similarly define B_{2}(X).

The discussion above now gives that

(Λ_{2}X)_{d} ≅ F r[B_{2}(X)_{d}^{F r[A_{2}(X)_{d}]}]

as claimed. The calculation of the simplicial operators follows directly from the definitions.

There is a natural transformation of functors

ΩE_{Σ} : Λ_{1} ^{}Λ_{2},

given as the unique multiplicative extension of the function given on generators by

ΩE_{ΣX}(x) = (x, 1) = [x].

Notice that one can define B_{2}(X) as the image of X under ΩE_{ΣX} and A_{2}(X) as its pointed complement.

Lemma 3.2. The geometric realization of ΩE_{ΣX} is the loops on the Freudenthal suspension for Σ|X|,

ΩE_{Σ|X|} : ΩΣ|X| ^{}Ω^{2}Σ^{2}|X|.

Proof. Notice that for any simplicial set X, the composition GΣX is canonically isomorphic to F X. There is the simplicial Freudenthal suspension map for ΣX

E_{ΣX} : ΣX ^{}F ΣX.

Applying Kan’s G functor one obtains a multiplicative map

GE_{ΣX} : F X ^{}GF ΣX.

On the other hand there is a natural map

ι : Λ_{2}X = ΩF ΣX ^{}GF ΣX

which is a homotopy equivalence. It follows directly from the definition that the image of GE_{ΣX} is contained in the image of the map ι and since both maps are monomorphic, there is a unique factorization of GE_{ΣX} through Λ_{2}X, which is easily checked to coincide with ΩE_{ΣX}. The lemma follows.

For n ≥ 2 define natural transformations

Ω^{n}E_{Σ^{n}} : Λ_{n} ^{}Λ_{n+1}

by composing the functors Ω^{n−1} and Σ^{n−1} with the natural transformation ΩE_{Σ}. More precisely,

Ω^{n}E_{Σ^{n}} = Ω^{n−1}(ΩE_{Σ})Σ^{n−1}.

It follows at once from Lemma 3.2 that for a given simplicial set X, the geometric realization of Ω^{n}E_{Σ^{n}X} is the (n)-fold loops on the Freudenthal suspension for Σ^{n}X.

These transformations encode the n-fold Freudenthal suspension

E^{n} : Λ_{0} ^{}Λ_{n}

in the obvious way. Namely E^{2} = ΩE_{Σ} ∘ E and if E^{n−1} is defined then E^{n} = Ω^{n−1}E_{Σ^{n−1}} ∘ E^{n−1}. Notice that B_{2}(X) = E_{X}^{2}(X).

Lemma 3.1 generalizes to n > 2 as follows. Let Ā_{n}(X) denote the graded set which in dimension d + n − 1 consists of all elements in (Σ^{n−1}X)_{d+n−1} of the form (a_{q}, i_{1}, ⋯, i_{n−1}), such that i_{j} > 1 for some 1 ≤ j ≤ n − 1. Notice that by definition a_{q} ∈ X_{q} and q + i_{1} + ⋯ + i_{n−1} = d. Let B̄_{n}(X) denote the pointed complement of Ā_{n}(X) in Σ^{n−1}X. Let A_{n}(X) denote the (n − 1)-fold desuspension of Ā_{n}(X) and define B_{n}(X) in a similar fashion.

Proposition 3.3. For a simplicial set X and an integer n ≥ 2 a basis for the free simplicial group Λ_{n}X is given by the graded set B_{n}(X)^{F r[A_{n}(X)]}. Furthermore, the set B_{n}(X) ⊆ Λ_{n}X given by the image of the n-fold Freudenthal suspension E_{X}^{n} : X ^{}Λ_{n}X.

Proof. Consider the map of sets

β : (Σ^{n−1}X)_{d+n−1} ^{d_{0}⊤d_{1}···⊤d_{n−2}}^{}∏_{n−1}(Σ^{n−1}X)_{d+n−2}.

Then (Σ^{n−1}X)_{d+n−1} can be written as a wedge of the inverse image under β of the base-point and its pointed complement. Examination of the definition above gives that B̄_{n}(X)_{d+n−1} is exactly the inverse image of the base-point under β in this dimension.

Notice that (Λ_{n}X)_{d} is isomorphic to the kernel of

β : F (Σ^{n−1}X)_{d+n−1} ^{d_{0}⊤d_{1}···⊤d_{n−2}}^{}∏_{n−1} F (Σ^{n−1}X)_{d+n−2}.

By the discussion above this kernel is freely generated by B_{n}(X)_{d}^{F r[A_{n}(X)_{d}]}.

For n = 2 the discussion above gives that for every x ∈ X one has ΩE_{ΣX}[x] = (x, 1) ∈ Λ_{2}(X) where [x] represents the class E_{X}(x) of x in Λ_{1}X = F X. In fact it follows from the definition that if [x] ∈ Λ_{n−1}X denotes E_{X}^{n−1}(x) then Ω^{n−1}E_{Σ^{n−1}X}[x] = ([x], 1) ∈ Λ_{n}X. Then for each x ∈ X one has

E_{X}^{n}(x) = Ω^{n−1}E_{Σ^{n−1}X}(E^{n−1}(x)) = Ω^{n−1}E_{Σ^{n−1}X}[x] = ([x], 1) ∈ Λ_{n}X

and it follows that B_{n}(X) coincides with E_{X}^{n}(X) as a subset of Λ_{n}X.

Corollary 3.4. For every n ≥ 2 a basis for Λ_{n}X is given by homogeneous symbols c(w)[x] where x ∈ X is a non-trivial simplex, [x] = E_{X}^{n}(x) ∈ Λ_{n}X and w is a word (possibly trivial) in the free group generated by A_{n}(X).

The face and degeneracy operators on Λ_{n}X admit an analogous description to the case n = 2. The proof is also similar and is left to the reader.

Proposition 3.5. The simplicial operators on Λ_{n}X are the unique multiplicative extension of the functions which operate on generators as follows.

d_{i}^{Λ_{n}}(c(w)[x]) = d_{i+n−1}^{F Σ^{n−1}}(c(w)[x]) = c(d_{i+n−1}^{F Σ^{n−1}}(w))[d_{i}x]
s_{i}^{Λ_{n}}(c(w)[x]) = s_{i+n−1}^{F Σ^{n−1}}(c(w)[x]) = c(s_{i+n−1}^{F Σ^{n−1}}(w))[s_{i}x].

The interested reader can verify now that all simplicial operators preserve the generating set except for d_{0}^{Λ_{n}}. In fact d_{0} may take a generator to a word of arbitrary length. To see this consider the following example. In Λ_{2}X the element c((y, 2))(x, 1) is a generator. If i > 0 then

d_{i}^{Λ_{2}}(c((y, 2))(x, 1)) = c(d_{i+1}^{F Σ}(y, 2))(d_{i}x, 1) = c((d_{i−1}y, 2))(d_{i}x, 1).

But for i = 0 one gets

d_{0}^{Λ_{2}}(c((y, 2))(x, 1)) = c(d_{1}^{F Σ}(y, 2))(d_{0}x, 1) = c((y, 1))(d_{0}x, 1),

which is a product of three generators.

The next proposition records a formula the n-fold loops on the degree q map on Σ^{n}X.

Proposition 3.6. Let q be a positive integer and let [q] denote the degree q map on Σ|X|. Then the n-fold loop of [q] on Σ^{n}(X) is given by

Ω^{n}[q](c(w)[x]) = c([q]w)[x]^{q},

where [q]w means - raise each letter in the word composing w to the q-th power.

Proof. For an integer q the degree q map on Σ^{n}X is adjoint to the composition

X ^{E^{n}}^{}Ω^{n}Σ^{n}X ^{⟨q⟩}^{}Ω^{n}Σ^{n}X,

where ⟨q⟩ is the q-th power map. Thus ad[q] : X ^{}Λ_{n}X has the form

ad[q](x) = [x]^{q}.

On the other hand the n-fold loop of [q] can be computed in Λ_{n}X as follows. On Λ_{1}X the single loops on [q] is given by sending a generator [x] to its q-th power [x]^{q}. Thus on Λ_{n}X the n-fold loops on [q] is given by sending a generator c(w)[x] to c([q]w)[x]^{q}, as claimed.

4. The construction L_{n}X

The model below for Ω^{n}X is constructed in a very similar fashion to Λ_{n}X. Assume that the complex X is n-connected and reduced. Replace the functor F Σ^{n−1} in the construction of Λ_{n}X by Kan’s loops group functor G. Thus define L_{n}X to be Ω^{n−1}GX.

The generating set in this case is more complicated to describe. However, once this is done L_{n}X has some similar features to those of Λ_{n}X which are described in a later section.

As before, start with an analysis of the group L_{2}X = ΩGX. As a free group, GX is generated by the set X̄ = X/s_{0}(Σ^{−1}X). In dimension n, L_{2}X is given as the kernel of the homomorphism

d_{0}^{G} : (GX)_{n+1} ^{}(GX)_{n}.

Since d_{0}^{G} is an epimorphism with right inverse s_{0}^{G} and (GX)_{n} is free, there is a group isomorphism

(GX)_{n+1} ≅ (L_{2}X)_{n} ⋊ (GX)_{n}.

The following definition is useful.

Definition 4.1. An epimorphism f : F ^{}G of free groups is called proper if it has a right inverse s satisfying the following property: there exist sets X freely generating F and Y freely generating G, such that s maps Y into a subset of X.

If f : F ^{}G is a proper epimorphism of free groups, then with the notation above, let A(X) denote the subset s(Y) of X. Let B′(X) denote the pointed complement of A(X). Let B(X) denote the subset of F given by all elements x · sf(x)^{−1} for all x ∈ B′(X).

Lemma 4.2. Let f : F ^{}G be a proper epimorphism of free groups with kernel K. Then, with the notation above, K is freely generated by the set B(X)^{F r[A(X)]}.

Proof. Let α : B′(X) ^{}B(X) be the map of sets given by α(x) = x · sf(x)^{−1}. Then α is an epimorphism by construction. Let x_{1}, x_{2} ∈ B′(X) be such that α(x_{1}) = α(x_{2}). Then

α(x_{1}) = x_{1}sf(x_{1})^{−1} = x_{2}sf(x_{2})^{−1} = α(x_{2}).

Thus x_{2}^{−1}x_{1} = s(f(x_{2})^{−1}f(x_{1})). But the left hand side of the last equation is a word in the free group on B′(X), whereas the right hand side is a word in the free group on A(X). Since these two sets intersect only on the base point, it follows that both sides are trivial so x_{1} = x_{2} and α is an isomorphism of sets.

Now write X as A(X) ∨ B′(X). There is an isomorphism of sets from X to the subset A(X) ∨ B(X) of F taking any element of A(X) to itself and any element of B′(X) to B(X) via α. It is immediate that A(X)∨B(X) freely generates F because sf(x) is in the free subgroup generated by A(X). Thus there is a commutative diagram with exact rows

[figure: 1; K; F; f; G; 1; 1; F r[B(X)^{F r[A(X)]}]; F r[A(X) ∨ B(X)]; F r[1∨α]; π_{A}; F r[A(X)]; 1; 1]

Since the right two vertical arrows are isomorphisms so is the left one and the result follows.

Definition 4.3. Define an operator κ on GX by κ(w) = s_{0}^{G}d_{0}^{G}(w).

Lemma 4.4. Let X be a simplicial set and let X̄ denote the generating set for GX as defined above. Then a point x ∈ X̄ is a fixed point of κ if and only if x = s_{1}y for some y ∈ X.

Proof. Since s_{0}^{G} is the unique multiplicative extension of s_{1}, it is clear that each fixed point x of κ in X̄ is in the image of s_{0}^{G}. But x considered as a word in GX is of length 1 (or otherwise is the base point). On the other hand

κ(x) = s_{0}^{G}d_{0}^{G}x = s_{1}(d_{1}(x)d_{0}(x)^{−1}) = s_{1}d_{1}(x) · s_{1}d_{0}(x)^{−1}

is of length at most 2. Hence one of the factors must be trivial and we have x = s_{1}d_{i}x for i = 0 or i = 1.

Conversely, if x = s_{1}y then

κ(x) = κ(s_{0}^{G}y) = s_{0}^{G}d_{0}^{G}s_{0}^{G}y = s_{0}^{G}y = x.

Let X be a 2-connected reduced complex. Let A′(X) denote the graded set given by the intersection of X̄ with the fixed point set of κ on GX. Let B′′(X) denote the graded set given the pointed complement of A′(X) in X̄. Let A(X) and B′(X) denote the double desuspension of A′(X) and B′′(X) respectively. Then

(GX)_{n+1} = F r(X̄_{n+2}) = F [A(X)_{n} ∨ B′(X)_{n}].

Let B(X) denote the graded set, which in dimension n is the subset of (GX)_{n+1} given by all elements of the form α(x) = xκ(x)^{−1} for x ∈ B′(X)_{n}. Then (GX)_{n+1} is freely generated by A(X)_{n} ∨ B(X)_{n}.

The next corollary follows from Lemma 4.2.

Corollary 4.5. The group L_{2}X is isomorphic as a graded group to the free group F r[B(X)^{F r[A(X)]}].

Thus in terms of a word in the free group on the simplices of X every n-dimensional generator in the specified generating set for L_{2}X has the form c(w)(α(x)), where w is a word in the free group on A(X)_{n} and x ∈ B′(X)_{n}. A more convenient description for the group L_{2}X, analogous to the presentation of Λ_{2}X given above, will now be derived.

Notation 4.6. For x ∈ B′(X)_{n}, denote α(x) by [x]. Then the generating set for L_{2}X defined above consists of symbols of the form c(w)[x], where w ∈ F r[A(X)].

Consider the behavior of α with respect to the simplicial operators. Notice that d_{i}^{L_{2}} and s_{i}^{L_{2}} are the multiplicative extensions of d_{i+2} and s_{i+2} respectively. Equivalently these operators can be thought of as extensions of d_{i+1}^{G} and s_{i+1}^{G} respectively. The following lemma is routine and the verification is left to the reader.

Lemma 4.7. With the notation above, the function α satisfies the following relations

1. For i > 0, d_{i}^{L_{2}}[x] = [d_{i+2}(x)].

2. For i ≥ 0, s_{i}^{L_{2}}[x] = [s_{i+2}(x)].

3. d_{0}^{L_{2}}[x] = d_{2}(x)d_{1}(x)^{−1}d_{0}(x).

For any x ∈ X define γ(x) = d_{2}(x)d_{1}(x)^{−1}d_{0}(x). Then d_{0}^{L_{2}}[x] = γ(x). The operator γ will be of some significance. Its commutation properties with respect to the simplicial operators are recorded in the following lemma. Here too, the verification is easy and is left for the reader.

Lemma 4.8. The function γ satisfies the following relations 1. d_{i}^{L_{2}}(γ(x)) = γ(d_{i+3}(x)).

2. s_{i}^{L_{2}}(γ(x)) = γ(s_{i+3}(x)).

3. γ(s_{0}^{L_{2}}(x)) = [x].

Notice that 1 and 2 in the previous lemma imply that the set of all elements of the form γ(x), x ∈ X form a simplicial subset of L_{2}(X). Furthermore, 3 implies that this simplicial subset contains all elements of the form [x] = α(x).

The function α has a useful multiplicative property which is recorded next. For any word w ∈ GX define α(w) = w · κ(w)^{−1}. By analogy to the case when w is of length 1 denote α(w) by [w].

Lemma 4.9. Let w = x_{1}x_{2} ⋯ x_{n} ∈ GX. Then

[w] = [x_{1}] · c(κ(x_{1})^{−1})([x_{2}]) ⋯ c(κ(x_{1}x_{2} ⋯ x_{n−1})^{−1})([x_{n}]).

Proof. For w of length 1 there is nothing to prove. Let w = x_{1}x_{2} ⋯ x_{n}. Then

α(w) = x_{1}(x_{2} ⋯ x_{n})κ(x_{2} ⋯ x_{n})^{−1}κ(x_{1})^{−1} =

x_{1}α(x_{2} ⋯ x_{n})κ(x_{1})^{−1} = α(x_{1})c(κ(x_{1})^{−1})(α(x_{2} ⋯ x_{n})).

The claim follows by induction.

Notice that it was not assumed in the lemma that the presentation of w is reduced. Thus as an immediate corollary one obtains that for every w ∈ GX,

1 = α(1) = α(ww^{−1}) = α(w)c(κ(w)^{−1})(α(w^{−1})).

So that α(w)^{−1} = c(κ(w)^{−1})(α(w^{−1})) or equivalently α(w^{−1}) = c(κ(w))(α(w)^{−1}). In the notation used above

[w^{−1}] = c(κ(w))([w]^{−1}).

Theorem 4.10. Let X be a 2-connected reduced complex. Then (L_{2}X)_{n} is isomorphic as a group to the quotient of the free group on symbols c(w)[x], with [x] ∈ B(X)_{n} and w ∈ F r(A(X)_{n}), by the relations c(w)[x] = 1 if x = 1.

The simplicial operators are given by the formulas

1. s_{i}^{L_{2}}(c(w)[x]) = c(s_{i+1}^{G}(w))[s_{i+2}(x)] for i ≥ 0.

2. d_{i}^{L_{2}}(c(w)[x]) = c(d_{i+1}^{G}(w))[d_{i+2}(x)] for i ≥ 1.

3. d_{0}^{L_{2}}([x]) = [d_{2}x]c(κd_{0}x)([d_{1}x]^{−1}[d_{0}x]) and d_{0}^{L_{2}}(c(w)[x]) = c(κd_{1}^{G}w)(c([d_{1}^{G}w])(d_{0}^{L_{2}}[x])).

Proof. The calculation of the generating set is the contents of Corollary 4.5. It remains to verify the simplicial structure of L_{2}X. By the remarks above one has d_{i}^{L_{2}}[x] = [d_{i+2}x] for i > 0 and s_{i}^{L_{2}}[x] = [s_{i+2}x] for i ≥ 0. Thus

d_{i}^{L_{2}}(c(w)[x]) = c(d_{i+1}^{G}w)[d_{i+2}x]

for i > 0 and similarly

s_{i}^{L_{2}}(c(w)[x]) = c(s_{i+1}^{G}w)[s_{i+2}x]

for i ≥ 0. This give 1 and 2 in the theorem.

For the calculation of d_{0}^{L_{2}} first recall that

d_{0}^{L_{2}}[x] = γ(x) = d_{2}x(d_{1}x)^{−1}d_{0}x.

Then observe that

[d_{2}x] · c(κd_{0}x)([d_{1}x]^{−1}[d_{0}x]) =

d_{2}x(κd_{2}x)^{−1} · c(κd_{0}x)((d_{1}x(κd_{1}x)^{−1})^{−1}d_{0}x(κd_{0}x)^{−1}) =
d_{2}x(κd_{2}x)^{−1}(κd_{0}x)^{−1}(d_{1}x(κd_{1}x)^{−1})^{−1}d_{0}x =
d_{2}x(κd_{2}x)^{−1}(κd_{0}x)^{−1}(κd_{1}x)d_{1}x^{−1}d_{0}x =
d_{2}x(κd_{1}^{G}x)^{−1}(κd_{0}^{G}x)d_{1}x^{−1}d_{0}x =
d_{2}(x)d_{1}(x)^{−1}d_{0}(x) = γ(x).

In our notation the equation above reads

d_{0}^{L_{2}}[x] = [d_{2}(x)]c(κd_{0}x)([d_{1}x]^{−1}[d_{0}x]).

More generally, notice that

d_{0}^{L_{2}}(c(w)[x]) = c(d_{1}^{G}w)(d_{0}^{L_{2}}[x]) =

c([d_{1}^{G}w]κd_{1}^{G}(w))(d_{0}^{L_{2}}[x]) =
c(κd_{1}^{G}(w))(c([d_{1}^{G}w])(d_{0}^{L_{2}}[x]).

Notice that lemma 4.9 gives a way of writing [d_{1}^{G}w] in the theorem explicitly in terms of the generators once w is specified as a word in GX. The precise formulation is omitted.

The statement of Theorem 4.10 does not generalize well to L_{m}X for m > 2 as the notation becomes quite cumbersome. Before proceeding a more uniform description for the generating set is given in the next theorem.

Theorem 4.11. Let X be a 2-connected reduced complex. Then (L_{2}X)_{n} is isomorphic as a group to the quotient of the free group on symbols ⟨w, x⟩ with w ∈ (GX)_{n} and x ∈ X_{n+2} subject to the relations ⟨w, x⟩ = 1 if x = s_{i}y for some y ∈ X_{n+1} and i ∈ {0, 1}. The simplicial operators are given by the formulas

1. s_{i}^{L_{2}}⟨w, x⟩ = ⟨s_{i}^{G}w, s_{i+2}x⟩ for i ≥ 0.

2. d_{i}^{L_{2}}⟨w, x⟩ = ⟨d_{i}^{G}w, d_{i+2}x⟩ for i ≥ 1.

3. d_{0}^{L_{2}}⟨1, x⟩ = ⟨1, d_{2}x⟩⟨d_{0}^{G}d_{0}x, d_{1}x⟩^{−1}⟨d_{0}^{G}d_{0}x, d_{0}x⟩ and d_{0}^{L_{2}}⟨w, x⟩ = c(w)(d_{0}^{L_{2}}⟨1, x⟩)

Proof. Let U_{2}X denote the graded free group generated in dimension n by the symbols described in the theorem subject to the given relations. Then there is a map

ϕ : U_{2}X ^{}L_{2}X

sending ⟨w, x⟩ to c(s_{0}^{G}w)[x].

To check that ϕ is well defined notice that if x = s_{0}y then κ(x) = 1 and so [x] = x = s_{1}y = 1 in GX. Hence generators of the form ⟨w, x⟩ with x = s_{0}y are in the kernel of ϕ. On the other hand if x = s_{1}y for some y then [x] = 1 in GX since in this case κ(x) = x. Consequently ϕ is well defined. Furthermore, since A(X) is isomorphic as a set to GX via the degeneracy operator s_{0}^{G}, it follows that ϕ is an epimorphism. To prove that ϕ is an isomorphism it remains to show that its kernel vanishes. Thus it suffices to show that if w ∈ (GX)_{n} and x ∈ X_{n+2} then c(s_{0}^{G}w)[x] = 1 if and only if x = s_{i}y for some y and i ∈ {0, 1}.

Indeed c(s_{0}^{G}w)[x] = 1 if and only if [x] = 1 as an element of GX. But [x] = xκ(x)^{−1}. Thus [x] = 1 if and only if x = κ(x). By Lemma 4.4 this holds for x ∈ X̄ if and only if x = s_{1}y for some y, whence the result.

The simplicial operators can be calculated directly from Theorem 4.10 using the identification above. Notice that d_{0}^{L_{2}}⟨w, x⟩ is not given in terms of the generating set. However, one can work out the explicit expression using Lemma 4.9 and the observation that w = [w]κ(w).

The next theorem generalizes Theorem 4.11 to n ≥ 2.

Theorem 4.12. Let X be an m-connected reduced complex for m ≥ 2. Then (L_{m}X)_{n} is isomorphic as a group to the quotient of the free group on symbols

⟨w_{m−1}, w_{m−2}, ⋯, w_{1}, x⟩, x ∈ X_{n+m} and w_{i} ∈ (L_{i}X)_{n+m−i−1},

subject to the relations ⟨w_{m−1}, w_{m−2}, ⋯, w_{1}, x⟩ = 1 if either

1. for some s < m the element ⟨w_{s−1}, ⋯ w_{1}, x⟩ represents the identity in L_{s}X or 2. there exists a generator ⟨v_{m−2}, ⋯, v_{1}, y⟩ in L_{m−1}X such that

s_{0}^{L_{m−1}}⟨v_{m−2}, ⋯, v_{1}, y⟩ = ⟨w_{m−2}, ⋯, w_{1}, x⟩.

The simplicial operators are defined recursively as follows.

1. s_{i}^{L_{m}}⟨w_{m−1}, w_{m−2}, ⋯, w_{1}, x⟩ = ⟨s_{i}^{L_{m−1}}w_{m−1}, s_{i+1}^{L_{m−2}}w_{m−2} ⋯ s_{i+m−2}^{L_{1}}w_{1}, s_{i+m}x⟩ for each i ≥ 0.

2. d_{i}^{L_{m}}⟨w_{m−1}, w_{m−2}, ⋯, w_{1}, x⟩ = ⟨d_{i}^{L_{m−1}}w_{m−1}, d_{i+1}^{L_{m−2}}w_{m−2} ⋯ d_{i+m−2}^{L_{1}}w_{1}, d_{i+m}x⟩ for each i ≥ 1.

3. The face operator d_{0}^{L_{m}} can be expanded recursively using the formula d_{0}^{L_{m}}⟨w_{m−1}, w_{m−2}, ⋯, w_{1}, x⟩ = c(w_{m−1})(d_{1}^{L_{m−1}}(u)·d_{0}^{L_{m−1}}(u)^{−1}), where u denotes the generator ⟨w_{m−2}, ⋯, w_{1}, x⟩ of L_{m−1}X.

Proof. The statement of the theorem for m = 2 is the contents of Theorem 4.11. For u ∈ L_{r}X define κ_{r}(u) = s_{0}^{L_{r}}d_{0}^{L_{r}}(u). For m > 2 assume by induction that the theorem holds for m − 1.

For each 1 ≤ s ≤ m − 1 let G_{s}X denote the subset of L_{s}X given by the symbols

⟨w_{s−1}, w_{s−2}, ⋯, w_{1}, x⟩, x ∈ X_{n+s} and w_{i} ∈ (L_{i}X)_{n+s−i−1}.

Then by induction hypothesis G_{s}X is closed under s_{0}^{L_{s}}. Let R_{s}X denote the subset of G_{s}X given by

\begin{cases}⟨w_{s−1}, w_{s−2}, ⋯, w_{1}, x⟩ & | & ∃⟨v_{s−2}, ⋯, v_{1}, y⟩ ∈ G_{s−1}X, \\ & | & s_{0}^{L_{s−1}}⟨v_{s−2}, ⋯, v_{1}, y⟩ = ⟨w_{s−2}, ⋯, w_{1}, x⟩ or \\ & | & ⟨w_{j−1}, ⋯, w_{1}, x⟩ = 1 in L_{j}X for some j < s − 1 \end{cases}\} .

Let Ḡ_{s}X denote the quotient set G_{s}X/R_{s}X. Then by induction hypothesis L_{s}X is freely generated in the pointed sense by Ḡ_{s}X, for each s ≤ m − 1.

Consider the projection

d_{0}^{L_{m−1}} : P L_{m−1}X ^{}L_{m−1}X,

whose kernel is by definition L_{m}X. The formula for s_{0}^{L_{m−1}} gives that d_{0}^{L_{m−1}} is a proper epimorphism of free groups with right inverse given by s_{0}^{L_{m−1}}.

Write

Ḡ_{m−1}X = A_{m}(X) ∨ B′_{m}(X),

where A_{m}(X) = s_{0}^{L_{m−1}}(Ḡ_{m−1}X) and B′_{m}(X) = Ḡ_{m−1}X/s_{0}^{L_{m−1}}(Ḡ_{m−1}X). Let

B_{m}(X) = {[u] := u · κ_{m−1}(u)^{−1} | u ∈ B′_{m}(X)} ⊂ L_{m}X.

Then by Lemma 4.2 there is an isomorphism

L_{m}X ≅ F r[B_{m}(X)^{F r[A_{m}(X)]}].

Notice that the group F r[A_{m}(X)] is isomorphic with the appropriate dimension shift to L_{m−1}X.

It is easy to observe, as in the proof of Lemma 4.2, that the correspondence u ↦ [u] := u · κ_{m−1}(u)^{−1} is 1 − 1. Furthermore, if u ∈ B′_{m}(X), then [u] = 1 in L_{m}X if and only if u = 1 in L_{m−1}X. To see this notice that if [u] = 1 then u = κ_{m−1}(u). Hence in this case u is in the image of s_{0}^{L_{m−1}}, which means that it is trivial as an element of B′_{m}(X). Finally notice that

B′_{m}(X) = Ḡ_{m−1}X/s_{0}^{L_{m−1}}(Ḡ_{m−1}X) = G_{m−1}X/s_{0}^{L_{m−1}}(Ḡ_{m−1}X) ∪ R_{m−1}.

Putting all this together, the presentation of L_{m}X follows.

Notice that the symbol ⟨w_{m−1}, ⋯, w_{1}, x⟩ denotes the element

c(s_{0}^{L_{m−1}}w_{m−1})([⟨w_{m−2}, ⋯, w_{1}, x⟩])

in P L_{m−1}X, where [u] := uκ_{m−1}(u)^{−1}. Thus for every i ≥ 0 one has

s_{i}^{L_{m}}⟨w_{m−1}, ⋯, w_{1}, x⟩ = s_{i+1}^{L_{m−1}}(c(s_{0}^{L_{m−1}}w_{m−1})[⟨w_{m−2}, ⋯, w_{1}, x⟩]) =

c(s_{0}^{L_{m−1}}s_{i}^{L_{m−1}}w_{m−1})[s_{i+1}^{L_{m−1}}⟨w_{m−2}, ⋯, w_{1}, x⟩],

and the formula for the degeneracy operators follows by induction. The formula for the face operators d_{i}^{L_{m}} for i ≥ 1 follows in a similar fashion.

Finally let u = ⟨w_{m−2}, ⋯, w_{1}, x⟩. Then we have

d_{0}^{L_{m}}⟨w_{m−1}, w_{m−2}, ⋯ w_{1}, x⟩ = d_{1}^{L_{m−1}}(c(s_{0}^{L_{m−1}}w_{m−1})(u · κ_{m−1}(u)^{−1}) =

c(w_{m−1})(d_{1}^{L_{m−1}}(u) · d_{0}^{L_{m−1}}(u)^{−1}).

This gives the recursive formula for d_{0}^{L_{m}} and thus completes the proof.

Example 4.13. The calculation of d_{0}^{L_{3}} on a typical generator of the form ⟨w_{2}, w_{1}, x⟩ is given next. By the theorem one has

d_{0}^{L_{3}}⟨w_{2}, w_{1}, x⟩ = c(w_{2})(d_{1}^{L_{2}}⟨w_{1}, x⟩ · d_{0}^{L_{2}}⟨w_{1}, x⟩^{−1}) =

c(w_{2})(⟨d_{1}w_{1}, d_{3}x⟩ · (c(w_{1})(d_{0}^{L_{2}}⟨1, x⟩^{−1}))).

Expanding this further can be done by using the identities w_{i} = [w_{i}]κ_{i}(w_{i}), where in each case [w_{i}] means w_{i}κ_{i}(w_{i})^{−1}. In each case one needs to know an explicit decomposition of w_{i} as a word in its respective group. The simplest instance of this occurs for w_{i} = 1 for i = 1, 2. In this case one has

d_{0}^{L_{3}}⟨1, 1, x⟩ = ⟨1, 1, d_{3}x⟩(⟨1, 1, d_{2}x⟩⟨1, d_{0}^{G}d_{0}x, d_{1}x⟩^{−1}⟨1, d_{0}^{G}d_{0}x, d_{0}x⟩)^{−1}.

5. Comparison of the models

Recall that Λ_{n}X = Ω^{n−1}F Σ^{n−1}X, whereas L_{n}X = Ω^{n−1}GX. The following theorem summarizes some obvious relationships between the two constructions.

Proposition 5.1. Let Y be connected then 1. If X = Σ^{n}Y then L_{n}X = Λ_{n}Y.

2. If Y = L_{n}Z then Y is a retract of Λ_{n}Y via explicit maps

L_{n}Z ^{E^{n}}^{}Λ_{n}L_{n}Z ^{L_{n}ev^{n}}^{}L_{n}Z.

Proof. The first statement follows at once from the definitions and the observation that the functors GΣ and F are naturally isomorphic.

For the second statement recall that the k-fold Freudenthal suspension map was described earlier. There is a natural way to write the simplicial analogue of the map Ω^{n}ev^{n}. The construction becomes a triviality once it is observed that there is a natural map ev : ΣΩW ^{}W for every simplicial set W (categorically this is just the fact that Σ is left adjoint to Ω in the category of simplicial sets). Once this map is constructed, apply its (n − 2)-fold suspension to W = Ω^{n−2}GZ to get a map

Σ^{n−2}ev : Σ^{n−1}Ω^{n−1}GZ ^{}Σ^{n−2}Ω^{n−2}GZ.

Inductively one gets a map

ev^{n−1} : Σ^{n−1}Ω^{n−1}GZ ^{}GZ.

The target being a simplicial group this map extends to a simplicial homomorphism

Ωad(ev^{n−1}) : F Σ^{n−1}Ω^{n−1}GZ ^{}GZ.

This simplicial homomorphism can now be looped n − 1 times to give the desired simplicial homomorphism

Ω^{n}ev^{n} : Λ_{n}L_{n}Z ^{}L_{n}Z.

Thus define ev : ΣΩY ^{}Y as follows. Let (y, j) ∈ (ΣΩY)_{n} be any element. Hence y is a simplex of dimension n − j + 1 in Y with the additional property that d_{0}(y) = ∗. Define

ev(y, j) = s_{0}^{j−1}(y).

It is a routine verification that this map is simplicial, and natural in Y.

For an explicit example, consider Ω^{2}ev^{2} : Λ_{2}L_{2}X ^{}L_{2}X. Thus start with ev : ΣL_{2}X ^{}GX. A typical element has the form (w, j) and is sent by definition to (s_{0}^{G})^{j−1}w under ev. In particular an element of the form [w] = (w, 1) is sent to w. A generator for Λ_{2}L_{2}X has the form c((u_{1}, i_{1}) ⋯ (u_{k}, i_{k}))[w]. Thus Ω^{2}ev^{2} is the unique multiplicative extension of the map sending an element of the form above to

c((s_{0}^{G})^{i_{1}−1}(u_{1}) ⋯ (s_{0}^{G})^{i_{k}−1}(u_{k}))(w).

Notice that for any simplicial set X, the construction of the Moore loops ΩX makes no use of the simplicial operator s_{0}. Thus ΩX has an extra operator which we may denote by s_{−1}. Notice however that the use of the word “operator” here is misleading. Namely, if x ∈ X is a non-trivial simplex such that d_{0}x is the base point so that x is in fact a simplex in ΩX, then s_{−1}x is not in ΩX, since d_{0}^{Ω}s_{−1}x = d_{1}s_{0}x = x. To fix this one may think about s_{−1} as a function from ΩX to X, which is degree preserving. Notice that an i-fold iteration of s_{−1} may be thought of as a function of degree i − 1 from ΩX to X.

For every i ≥ 0 and n ≥ 2 define an operator of degree −1

ϕ_{i} : L_{n}X ^{}L_{n−1}X

by

ϕ_{i} = (s_{−1}^{L_{n}})^{i}(d_{0}^{L_{n}})^{i},

where (s_{−1}^{L_{n}})^{i} stands for (s_{0}^{L_{n−1}})^{i} considered as an operator from L_{n} to L_{n−1}. Notice that if X is n-connected and reduced and w ∈ (L_{n}X)_{k} then ϕ_{k}(w) = 1. Notice also that the operators ϕ_{i} defined above for L_{n}X are inherited by Λ_{n}X by 1 of Proposition 5.1. The following lemma summarizes the commutation features of ϕ_{i} with the simplicial operators.

Lemma 5.2. Let for i ≥ 1 and n ≥ 2, let ϕ_{i} : L_{n}X ^{}L_{n−1}X denote the operator defined above. Then

1. d_{j}^{L_{n−1}}ϕ_{i} = ϕ_{i−1}d_{j}^{L_{n}} if j ≤ i

2. d_{j}^{L_{n−1}}ϕ_{i} = ϕ_{i}d_{j−1}^{L_{n}} if j > i

3. s_{j}^{L_{n−1}}ϕ_{i} = ϕ_{i+1}s_{j}^{L_{n}} if j ≤ i

4. s_{j}^{L_{n−1}}ϕ_{i} = ϕ_{i}s_{j−1}^{L_{n}} if j > i.

Proof. We verify 1 and 3. The other two identities follow by analogy and are left to the reader. Notice that

d_{j}^{L_{n−1}}ϕ_{i} = d_{j}^{L_{n−1}}(s_{0}^{L_{n−1}})^{i}(d_{1}^{L_{n−1}})^{i}.

One then uses the simplicial identities to verify that for j < i

d_{j}s_{0}^{i}d_{1}^{i} = s_{0}^{i−1}d_{1}^{i−1}d_{j+1}.

Back in L_{n}X the right hand side is ϕ_{i}d_{j}^{L_{n}}. This gives 1. To verify 3 notice that the left hand side is

s_{j}s_{0}^{i}d_{1}^{i} = s_{0}^{i+1}d_{1}^{i}.

On the other hand the right hand side is

s_{0}^{i+1}d_{1}^{i+1}s_{j+1} = s_{0}^{i+1}d_{1}^{i−k}s_{j−k}d_{1}^{k+1} = s_{0}^{i+1}d_{1}^{i−j+1}s_{1}d_{1}^{j} = s_{0}^{i+1}d_{1}^{i−j}d_{1}^{j} = s_{0}^{i+1}d_{1}^{i}

implying 3.

The operators ϕ_{i} become useful when one writes explicit homotopies on these models. The idea of how these are used is demonstrated below in writing an explicit null-homotopy for the commutator map on an iterated loop space.

The standard 1-simplex Δ[1] is the simplicial set which in dimension n has n + 2 simplices denoted ⟨0^{i}, 1^{n+1−i}⟩ for 0 ≤ i ≤ n + 1. Faces and degeneracies are given by the formulas

d_{j}⟨0^{i}, 1^{n+1−i}⟩ = \begin{cases}⟨0^{i−1}, 1^{n+1−i}⟩ & j < i \\ ⟨0^{i}, 1^{n−i}⟩ & j ≥ i \end{cases}

s_{j}⟨0^{i}, 1^{n+1−i}⟩ = \begin{cases}⟨0^{i+1}, 1^{n+1−i}⟩ & j < i \\ ⟨0^{i}, 1^{n+2−i}⟩ & j ≥ i \end{cases}

If h : Δ[1] × X ^{}Y is a homotopy (i.e. a simplicial map), denote by h_{i} the function defined for x_{n} ∈ X_{n} by

h_{i}(x_{n}) = h(⟨0^{i}, 1^{n+1−i}⟩, x_{n}).

Notice that h_{i} is only defined on x_{n} with n ≥ i − 1.

Let X be an n-connected reduced complex, n ≥ 2, and let G denote L_{n}X. Define

τ : Δ[1] × G × G ^{}G

by

τ_{i}(u, v) = [u, ϕ_{i}(v)],

for every pair of words u, v ∈ G.

Lemma 5.3. The map τ defined above is a simplicial null-homotopy for the commutator map.

Proof. For u, v ∈ G_{k} one evidently has

τ_{k+1}(u, v) = [u, 1] = 1 and τ_{0}(u, v) = [u, v].

The verification that τ is a simplicial map is a standard exercise, using the notation given above for the 1-simplex, Lemma 5.2 and the simplicial identities, is left to the reader.

6. Filtrations for Λ_{n} and L_{n}

There is a filtration of Λ_{n}X by simplicial subgroups obtained as follows. Recall that Λ_{n}X is freely generated by elements of the form of the form c(w)[x]. The word w is an element in a free group F r[A_{n}(X)] and thus has the usual reduced word length filtration. Give c(w)[x] filtration k if w has filtration k in the word length filtration. Let F_{j}Λ_{2}X be the subgroup of Λ_{n}X generated by all c(w)[x] of filtration at most j.

Lemma 6.1. The simplicial groups F_{j}Λ_{n}X endow Λ_{n}X with the structure of a filtered simplicial group with F_{0}Λ_{n}X isomorphic to F X.

Proof. First notice that the simplicial operators preserve the generating set of the group F_{j}Λ_{n}X. This ensures that each F_{j}Λ_{n}X is indeed a simplicial subgroup of Λ_{n}X. Filtration zero, F_{0}Λ_{n}X, is exactly the subgroup generated by the image of the Freudenthal suspension E^{n} : X ^{}Λ_{n}X. Thus F_{0}Λ_{n}X = F X as claimed.

The filtration on L^{n}X arises in a similar way, but describing it requires some preparation. First, let X be a simplicial set and let Y be a subset, which is not necessarily simplicial. Define the simplicial envelope of Y in X, Env_{X}(Y) to be the simplicial subset of X given by the intersection of all simplicial subsets containing Y. Thus Env_{X}(Y) is the minimal simplicial subset of X containing Y. Obviously Env_{X}(Y) is obtained from Y by adding in all iterations of faces and degeneracies of simplices in Y.

If G is a simplicial group and X a subset of G, then the simplicial closure Env_{G}(X) is not necessarily a simplicial subgroup. However the following holds.

Lemma 6.2. Let G be a simplicial group and let X be a subset such that Env_{G}(X) is obtained from X by adjoining to X all elements of G of the form d_{0}^{i}(x), x ∈ X for some i ≥ 0. Let ⟨X⟩ denote the subgroup of G generated by X. Then Env_{G}(⟨X⟩) is a simplicial subgroup of G. Moreover

Env_{G}(⟨X⟩) = ⟨Env_{G}(X)⟩.

Proof. Since Env_{G}(X) contains X, the subgroup ⟨Env_{G}(X)⟩ contains ⟨X⟩. By definition Env_{G}(⟨X⟩) is the intersection of all simplicial subsets of G containing ⟨X⟩. Hence, ⟨Env_{G}(X)⟩ ⊇ Env_{G}(⟨X⟩). Next notice that Env_{G}(X) is obtained from X by adjoining all elements of the form d_{0}(y), y ∈ X. This follows at once from the identity d_{0}^{k} = d_{0}d_{1} ⋯ d_{k−1} and the assumption that X is closed under all simplicial operators except possibly d_{0}. Thus every element of ⟨Env_{G}(X)⟩ can be written in the form

w = x_{1}^{ϵ_{1}}d_{0}(y_{1})^{δ_{1}} ⋯ x_{r}^{ϵ_{r}}d_{0}(y_{r})^{δ_{r}},

where x_{i}, y_{j} ∈ X and ϵ_{i}, δ_{j} ∈ {−1, 0, 1}. But

x_{1}^{ϵ_{1}}d_{0}(y_{1})^{δ_{1}} ⋯ x_{r}^{ϵ_{r}}d_{0}(y_{r})^{δ_{r}} = d_{0}(s_{0}(x_{1})^{ϵ_{1}}d_{0}(y_{1})^{δ_{1}} ⋯ s_{0}(x_{r})^{ϵ_{r}}d_{0}(y_{r})^{δ_{r}})

and the right hand side is the image under d_{0} of an element in ⟨X⟩. This gives and inclusion the other way and the proof is complete.

We are now ready to define the filtration for L_{k}X. The group L_{k}X is freely generated by symbols ⟨w_{k−1}, w_{k−2}, ⋯, w_{1}, x⟩ with w_{j} ∈ L_{j}X. Let T_{r}L_{k}X denote the subset of the generating set consisting of all generators of the form above such that w_{k−1} is a word of reduced length at most r. Notice that T_{r}L_{k}X is closed under all simplicial operators with the exception of d_{0}^{L_{k}}. Thus the simplicial subset Env_{L_{k}X}(T_{r}L_{k}X) is obtained from T_{r}L_{k}X by adjoining all the images of its elements under d_{0}^{L_{k}}. Consequently Env_{L_{k}X}(⟨T_{r}L_{k}X⟩) is a simplicial subgroup of L_{k}X and is in fact equal to the simplicial subgroup generated by Env_{L_{k}X}(T_{r}L_{k}X). Define

F_{p}L_{k}X = Env_{L_{k}X}(⟨T_{p}L_{k}X⟩).
7. A simplicial model for Map_{∗}(S_{g}^{n,k}, X)

Let g be a non-negative integer and let S_{g} denote the closed Riemann surface of genus g. If k is another non-negative integer, let S_{g}^{k} denote a Riemann surface of genus g and k distinct boundary components. In particular S_{g}^{0} is identified with S_{g}. Let X be an arbitrary space. In various applications one is interested in the space of maps from S_{g}^{k} to X. In this section simplicial models for Map_{∗}(S_{g}^{k}, X) will be constructed, where here X denote an arbitrary simplicial set.

The intuition for this construction comes from the, rather simple, homotopy theory of these spaces. First notice that up to homotopy, a Riemann surface with k > 0 boundary components, is equivalent to a singular surface whose boundary consists of a wedge of k circles. Furthermore, as a cell complex the surface S_{g}^{k} can be constructed by means of the cofibration sequence

S_{z}^{1} ^{α_{g}^{k}}^{}{⋁_{i=1}^{g}(S_{x_{i}}^{1} ∨ S_{y_{i}}^{1})} ∨ ⋁_{j=1}^{k} S_{b_{j}}^{1} ^{}S_{g}^{k},

where subscripts on the circles denote the generators of the respective fundamental groups and the map α_{g}^{k} is specified up to homotopy by its effect on fundamental groups as

α_{g}^{k} = [x_{1}, y_{1}][x_{2}, y_{2}] ⋯ [x_{g}, y_{g}] · b_{j}^{−1} ⋯ b_{1}^{−1}.

For an arbitrary pointed space X, the mapping space functor Map_{∗}(−, X) turns the cofibration above into a fibration and Map_{∗}(S_{g}^{k}, X) is defined as the fibre space in this fibration, where the projection map is induced by α_{g}^{k}.

For non-negative integers g and k consider the system

(GX)^{2g+k} ^{a_{g}^{k}}^{}GX ^{}^{d_{0}} P GX.

The map a_{g}^{k} is given as follows

a_{g}^{k}(u_{1}, v_{1}, ⋯, u_{g}, v_{g}, z_{1}, ⋯, z_{k}) = [u_{1}, v_{1}][u_{2}, v_{2}] ⋯ [u_{g}, v_{g}](z_{1} ⋯ z_{k})^{−1}.

Let M_{g}^{k}(X) denote the pull-back of this system.

Theorem 7.1. If X is a simply-connected reduced complex then M_{g}^{k}(X) is a Kan complex and its geometric realization is homotopy equivalent to Map_{∗}(S_{g}^{k}, |X|). The simplicial operators in M_{g}^{k}(X) are induced from those of GX and P GX.

Proof. First notice that M_{g}^{k}(X) is a Kan complex as a pull-back space in a diagram of Kan complexes. That the maps a_{g}^{k} model the geometric maps induced by α_{g}^{k} is obvious by construction. Notice also that since the projection d_{0} : P GX ^{}GX is a Kan fibration, the homotopy type of M_{g}^{k}(X) is a homotopy invariant of X. This completes the proof.

Theorem 1.5 follows at once from Theorem 7.1. In particular for g = k = 0 the bottom left hand side corner of the pull-back square above is trivial and so M_{0}^{0}(X) is simply the kernel of d_{0}^{G}, namely the model L_{2}(X).

Notice that in general M_{g}^{k}(X) has a natural action of the simplicial group L_{2}X, with homotopy orbit space equivalent to (GX)^{2g+k}.

8. Appendix

This appendix is dedicated to the proof of the following lemma, which is well known but we are not aware of an appropriate reference.

Lemma 8.1. Let A and B be free groups generated by sets X and Y respectively. Then the kernel of the natural projection π : A∗B ^{}A is the free group generated by the set Y^{A}, namely by all conjugates of elements in Y by words in A.

Proof. Let A, and B be free groups. Let C̃ denote a tree on which A acts properly discontinuously with the orbit space C̃/A = C. Similarly, let D̃ denote a tree on which B acts properly discontinuously with the orbit space D̃/B = D. Without loss of generality we may label the edges of C and D by X and Y respectively. Then C̃ has one vertex for any element in A and one edge between two elements if their difference is a member of the generating set X. Similarly for D̃.

Fix a base-point in D, which is the image of any vertex in D̃, and attach one copy of D to each vertex in C̃ by identifying each vertex of C̃ with this fixed base-point in D. Call the resulting adjunction space E.

Then the action of A on C̃ descends to a properly discontinuous action of A on E with quotient C ∨ D. Thus there is a split short exact sequence of groups

1 → π_{1}(E) → π_{1}(C ∨ D) → A → 1.

Notice that the fundamental group π_{1}(E) is a free group which has basis given by the ordered pairs (α, β) where α runs over all vertices of C̃ and β runs over all edges of D (indexed by Y).

A choice of basis is gotten as follows: Choose a path from the vertex α to the base-point. The basis element (α, β) is represented the class on the loop α^{−1} ·β ·α in π_{1}(E) by standard covering space theory. The image of π_{1}(E) → π_{1}(C ∨ D) is thus given precisely by the subgroup freely generated by Y^{A}. The results follows.

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