Infinite Loop Spaces from Group Theory

Kathryn Lesh, University of Toledo

January 6, 1996

Abstract We generalize the Barratt-Priddy-Quillen theorem ΩB(∐ BΣ_{n}) ≃ QS^{0} by using tom Dieck’s classifying spaces for a family of subgroups of a group. We show how to take a compatible choice of families F_{n} of subgroups of Σ_{n} and obtain an infinite loop space by group completing ∐ BF_{n}. The spaces QS^{0} and K(Z, 0) are recovered as extreme cases and the infinite loop spaces obtained from other families can be thought of as interpolating between stable homotopy and integral homology. We study two special cases: the family of subgroups of the alternating groups, and the family generated by elementary abelian p-subgroups whose generators are disjoint p-cycles. We compute the infinite loop spaces which are formed in these cases.

In this paper^{1} we use infinite loop space machinery to construct a collection of homology theories which interpolate between stable homotopy and integral homology. These homology theories arise as generalizations of the Barratt-Priddy-Quillen theorem [B-P] [Q], which says that the group completion of the monoid ∐ BΣ_{n} is QS^{0}, the zero space of the suspension spectrum of the zero sphere made into an omega-spectrum. To generalize this theorem, we replace the classifying space BΣ_{n} with a construction of tom Dieck that associates to a family F_{n} of subgroups of Σ_{n} a classifying space BF_{n}. The theorem can be loosely stated as follows:

Theorem A compatible choice of families for each n gives rise to a monoid ∐ BF_{n} whose group completion is an infinite loop space, and hence represents a homology theory. The construction is natural in the choice of family and recovers stable homotopy and integral homology for suitable choices of families. Every infinite loop space constructed in this manner comes with a natural “coaugmentation” from QS^{0} and “augmentation” to K(Z, 0).

1AMS 1991 Subject Classification: Primary 55P47, Secondary 55N20

The precise statement of the theorem can be found in Section 1, where we discuss the classifying space of a family of subgroups of Σ_{n} and show how a suitable choice of families for each n gives rise to a symmetric monoidal category, and hence to an infinite loop space and a homology theory by the methods of [S], [B-F], or [M3]. The spaces ∐ BΣ_{n} and Z^{+} arise as the extreme cases of the construction, and since the group completions of these monoids are respectively QS^{0} and K(Z, 0), infinite loop spaces obtained by the theorem can be thought of as interpolating between stable homotopy and integral homology.

The remainder of the paper is devoted to the consideration of examples. Section 2 considers the case of subgroups of the alternating group and proves a transfer lemma which allows one to consider to consider a family of subgroups one prime at a time. Sections 3, 4, and 5 are devoted to the proof of the following theorem. (The compatible closure mentioned in this theorem turns out to be the family containing elementary abelian p-subgroups generated by disjoint p-cycles and also containing their subgroups.)

Theorem 2.3 Let p be a prime, let {F_{n}} be the families forming the compatible closure of Z/p ⊆ Σ_{p}, and let Gp(∐ BF_{n}) be the group completion of the monoid ∐ BF_{n}. Let τ : B(Σ_{p})_{+} → S^{0} be the transfer map and C(τ) be the zero space of the stable pushout of

B(Σ_{p})_{+} ⟶^{ϵ} S^{0}
↓^{τ}
S^{0}.

Then [Gp(∐ BF_{n})]_{p}ˆ ≃ C(τ)_{p}ˆ , and for primes q ≠ p, [Gp(∐ BF_{n})]_{q}ˆ ≃ (QS^{0})_{q}ˆ .

The organization of the proof of Theorem 2.3 is as follows. Section 3 computes the homotopy type of the space ∐ BF_{n} where F_{n} is the family of Theorem 2.3 and identifies it as a certain double bar construction of monoids which themselves group complete to infinite loop spaces. Section 4 shows that a double bar construction of infinite loop spaces corresponds to a pushout diagram of spectra. Finally, Section 5 completes the proof of Theorem 2.3 by comparing the bar construction on group completions to the group completion of the bar construction.

The author notes that the proof of Theorem 2.3 goes by identifying the infinite loop space obtained by group completing ∐ BF_{n} with the zero space of a colimit of spectra by means of a rather ad hoc calculation of the classifying spaces involved. It would be very nice to find a way to go directly from the choices of families to, for example, some diagram of spectra (or stable diagram of spaces), of which the representing spectrum would be a homotopy colimit.

The author would like to acknowledge many helpful e-mail exchanges with Bill Dwyer and Jim McClure and would also like to thank Gunnar Carlsson, who first brought to her attention the idea of using families to construct homology theories.

1 Construction

In this section we show how a suitable choice of families of subgroups of Σ_{n} leads to a symmetric monoidal category, and thence to an infinite loop space. We begin by reviewing tom Dieck’s classifying space for a family of subgroups of a group.

Definition 1.1 Let G be a group and let F be a collection of subgroups of G. We call F a family if F is closed under subconjugation, i.e. if H ∈ F, σ ∈ G, and σKσ^{−1} a subgroup of H implies K ∈ F.

Let Y^{H} denote the space of fixed points of the action of a group H on a space Y.

Proposition 1.2 [tD] Given a family of subgroups of a group G, there is a G-space EF with the property that

\begin{matrix}(EF)^{H} ≃ ∗ & for H ∈ F \\ (EF)^{H} = ∅ & for H ∉ F \end{matrix}

The space EF is terminal up to homotopy among G-spaces whose isotropy groups lie in F.

In particular, EF itself is contractible, being the fixed point set of the subgroup containing only the identity element, which must be in any family. The terminal property of EF is analogous to the corresponding property for EG, which is terminal up to homotopy among free G-spaces, and indeed, we can take EF = EG if F is the family consisting of only the trivial subgroup of G.

Definition 1.3 The classifying space of the family F, written BF, is the orbit space of the G action on EF.

Proposition 1.4 The space BF is homotopy equivalent to the nerve of the category whose objects are transitive G-sets with isotropy in the family F and whose morphisms are equivariant maps.

A proof of this proposition can be found in [E].

To construct homology theories interpolating between stable homotopy and integral homology, we choose suitably compatible families F_{n} of subgroups of Σ_{n} for each n. Then we use these choices to construct a symmetric monoidal category, which in turn leads to an infinite loop space by methods of [S], [B-F], and [M3]. In fact, the infinite loop space in question is the group completion of the space ∐_{n} BF_{n}.

To describe the construction in detail, we first explain the compatibility condition required on the families. Consider the inclusion ι : Σ_{n}×Σ_{k} ↪ Σ_{n+k} obtained by letting Σ_{n} act on {1, ..., n} and letting Σ_{k} act on {n+1, ..., n+k}.

Definition 1.5 For each n, let F_{n} be a family of subgroups of Σ_{n}. We call the set {F_{n}}_{n∈Z^{+}} compatible if H ∈ F_{n} and K ∈ F_{k} implies ι(H × K) ∈ F_{n+k}. If {H_{α}}_{α∈I} is a collection of subgroups of finite symmetric groups, we define the compatible closure of {H_{α}}_{α∈I} to be the set of the smallest families {F_{n}}_{n∈Z^{+}} which are compatible and have the property that {H_{α}}_{α∈I} ⊆ ⋃_{n} F_{n}.

Given a compatible choice of families {F_{n}}, consider the category O_{F_{∗}} defined as follows. The objects are ordered pairs (n, S) where n is a nonnegative integer and S is a transitive Σ_{n} set with isotropy in F_{n}. There are no morphisms (n, S) → (n′, S′) if n ≠ n′, and morphisms (n, S) → (n, S′) are Σ_{n}-equivariant maps S → S′.

Proposition 1.6 The nerve of O_{F_{∗}} is homotopy equivalent to ∐_{n} BF_{n}.

Proof: This follows from Proposition 1.4.

□

We now give O_{F_{∗}} the structure of a symmetric monoidal category [Mac]. Recall that a symmetric monoidal category is a category C which comes equipped with a bifunctor ⊗ : C × C → C which is commutative and associative up to natural coherence [M2]. In the case of O_{F_{∗}}, we use the bifunctor

(n, S) ⊗ (k, T) = (n + k, Σ_{n+k} ×_{(Σ_{n}×Σ_{k})} S × T)

where the action of Σ_{n} × Σ_{k} on the right of Σ_{n+k} is given by using the above inclusion ι : Σ_{n} × Σ_{k} ↪ Σ_{n+k} followed by right multiplication on Σ_{n+k}. It is easy to check that the axioms of a symmetric monoidal category are satisfied by this bifunctor.

Theorem 1.7 If {F_{n}}_{n∈Z^{+}} is a compatible choice of families of subgroups of Σ_{n}, then ∐ BF_{n} has a monoid structure whose group completion Gp(∐ BF_{n}) is an infinite loop space which is natural in the choice of families. Any such infinite loop space comes with a natural coaugmentation and augmentation QS^{0} → Gp(∐ BF_{n}) → K(Z, 0).

Proof: The fact that Gp(∐ BF_{n}) is an infinite loop space is a consequence of the machinery of Segal [S], Bousfield and Friedlander [B-F], or May [M3]. Naturality in the choice of families is a consequence of the fact that if F′_{n} ⊆ F_{n} for each n, then O_{F′_{∗}} is a full subcategory of O_{F_{∗}} and that the inclusion is compatible with ⊗. If we take F′_{n} to be the family consisting only of the trivial subgroup of Σ_{n}, then we can take EF′_{n} = EΣ_{n} and BF′_{n} = BΣ_{n}. Since F′_{n} ⊆ F_{n} for any F_{n}, this gives a natural map of monoids ∐ BΣ_{n} → ∐ BF_{n} which becomes the coaugmentation after group completion. Similarly, taking F′_{n} to be the family consisting of all subgroups of Σ_{n}, we can take EF′_{n} = ∗ and BF′_{n} = ∗. Since for any family F_{n} we have F_{n} ⊆ F′_{n}, we obtain a natural map of monoids ∐ BF_{n} → Z^{+}, which group completes to the augmentation.

□

2 Examples and a Transfer Lemma

In this section we compute the infinite loop space obtained by group completing ∐ BF_{n} where F_{n} consists of all subgroups of the alternating subgroup of Σ_{n} (Proposition 2.1). We give a lemma allowing us to study ∐ BF_{n} one prime at a time (Lemma 2.4). Finally, we state the most difficult example of a group completion of ∐ BF_{n} that we have been able to compute to date, namely the case where the families F_{n} form the compatible closure of Z/p ⊆ Σ_{p} (Theorem 2.3).

Before beginning these tasks, we note that information about the monoid structure of ∐ BF_{n} is built into the symmetric monoidal structure of the category structure of O_{F_{∗}}. If we take the nerve of the source and target of ⊗ : O_{F_{∗}} × O_{F_{∗}} → O_{F_{∗}}, we obtain the monoid structure on ∐ BF_{n}.

An example that turns out to be easy to calculate is contained in the following proposition.

Proposition 2.1 Let F_{n} be the family of all subgroups of the alternating group A_{n}. The spectrum determined by this choice of families is the fiber of Sq^{2} : HZ → Σ^{2}HZ/2.

Proof: We can take EF_{n} = EZ/2 where Σ_{n} acts by the sign of the permutation. Then

∐_{n} BF_{n} = ∐_{n} BZ/2

and the question is what the monoid structure is with respect to which group completion takes place. Now the construction is natural in the families and the map between constructions that we get for different families will be a map of monoids. If we consider the map

∐_{n} BΣ_{n}⟶ ∐_{n} BZ/2

which comes from including the family of the trivial subgroup into the family of subgroups of the alternating group, it is easy to see that it is induced by the reduction maps Σ_{n} → Z/2 for each n. This means that the multiplication on ∐ BZ/2 is given by (BZ/2)_{n} × (BZ/2)_{k} → (BZ/2)_{n+k} by the usual multiplication on BZ/2. Then it follows from the proposition below that the group completion of ∐_{n} BZ/2 has homology Z × H_{∗}BZ/2, i.e. the group completion of ∐_{n} BZ/2 is Z × BZ/2.

Hence the group completion has π_{0} = Z and π_{1} = Z/2, and there is no higher homotopy. Further, the map ∐_{n} BΣ_{n} → ∐_{n} BZ/2 is easily seen to be onto on π_{1}, which tells us that π_{0} and π_{1} are connected by a Sq^{2}.

□

The following proposition, useful for calculating homology of group completions, is specialized from [A, Thm 3.2.1].

Proposition 2.2 Let Z be a homotopy commutative topological monoid with π_{0}Z ≅ Z^{+} and let ι ∈ H_{0}Z be the basepoint of a nonidentity component of Z. Then the homology of the group completion of Z can be computed as the ring-theoretic localization

H_{∗}ΩBZ ≅ ι^{−1}H_{∗}Z.

The most complicated example that we have calculated to date is the infinite loop space obtained by taking the compatible closure of Z/p ⊆ Σ_{p}. These families can be described as elementary abelian p-subgroups which are generated by disjoint p-cycles, together with their subgroups.

Theorem 2.3 Let p be a prime, let {F_{n}} be the families forming the compatible closure of Z/p ⊆ Σ_{p}, and let Gp(∐ BF_{n}) be the group completion of the monoid ∐ BF_{n}. Let τ : B(Σ_{p})_{+} → S^{0} be the transfer map and C(τ) be the zero space of the stable pushout of

B(Σ_{p})_{+} ⟶^{ϵ} S^{0}
↓^{τ}
S^{0}.

Then [Gp(∐ BF_{n})]_{p}ˆ ≃ C(τ)_{p}ˆ , and for primes q ≠ p, [Gp(∐ BF_{n})]_{q}ˆ ≃ (QS^{0})_{q}ˆ .

The proof is found in Section 5. The result at primes other than p will be a consequence of the following transfer lemma.

Lemma 2.4 Let F be a choice of family of subgroups of a group G, and for a fixed prime p let _{p}F be the family of all p-groups which appear in F. Then the map B_{p}F → BF is an isomorphism on mod p homology.

In the case of studying the families forming the compatible closure of Z/p ⊆ Σ_{p} at a prime q not equal to p, it says that as far as the prime q is concerned we might as well have taken the family to consist just of the trivial subgroup, which would give us QS^{0}.

Proof: Let E_{p}G denote the universal G-space corresponding to the family of all p-subgroups of G. We will prove that if X is a G-CW complex, then the natural map

γ : (X × E_{p}G)/G → X/G

is an isomorphism on mod p homology. The lemma then follows by taking X = EF and noticing that E_{p}F is G-equivariantly homotopy equivalent to EF × E_{p}G. (Check the fixed point sets and use the uniqueness property of E_{p}F.) Our plan is to induct over the G-cell structure of X using the Mayer-Vietoris sequence in homology.

First we show that γ is an isomorphism on mod p homology if X is a point, i.e., that (E_{p}G)/G has the homology of a point. If P is a p-Sylow subgroup of G, then we claim (E_{p}G)/P is contractible, for considered as a P -space, E_{p}G is the universal space for the family of all subgroups of P, which we know is P -equivariantly contractible. It follows that (E_{p}G)/G has the mod p homology of a point if we can show that the mod p homology of (E_{p}G)/G is a retract of that of (E_{p}G)/P. We consider cellular chains and use the following isomorphisms:

Z/p ⊗_{Z/p[G]} C_{∗}(E_{p}G) ≅ C_{∗}(E_{p}G/G)
Z/p[G/P] ⊗_{Z/p[G]} C_{∗}(E_{p}G) ≅ C_{∗}(E_{p}G/P).

Now we look at the composite Z/p → Z/p[G/P] → Z/p where the first map sends 1 to the sum of the cosets, and the second is the augmentation. The composite is multiplication by |G/P |, an isomorphism, and tensoring over Z/p[G] with C_{∗}(E_{p}G) gives the desired retraction, in view of the above isomorphisms.

It follows easily from the above that γ is an isomorphism for X = G/H for a subgroup H of G, for

(G/H × E_{p}G)/G ≅ (E_{p}G)/H ≃ (E_{p}H)/H.

Finally, for a general X, the Mayer-Vietoris sequence in homology allows us to induct over the cells to prove the lemma.

□

3 Homotopy type of ∐ BF_{n}

In the next three sections, we let F_{n} be the family of elementary abelian psubgroups of Σ_{n} which are generated by disjoint p-cycles together with their subgroups, and our goal is Theorem 3.7, which identifies the homotopy type of ∐ BF_{n}. Recall from Proposition 1.4 that BF_{n} is homotopy equivalent to the classifying space of a category we will call O(F_{n}), whose objects are transitive G-sets with isotropy in the family F_{n} and whose morphisms are equivariant maps. The symmetric monoidal category O_{F_{∗}} giving the monoid structure on ∐ BF_{n} is just a disjoint union of the categories O(F_{n}).

We begin with a general lemma in the style of Dwyer and Kan [D-K] on calculating the classifying space of a category.

Lemma 3.1 Let C be a small category. Let E(C) be the bisimplicial set whose (p, q) simplices are commuting diagrams in C of the form

C_{0,0} ⟶ C_{0,1} ... ⟶ C_{0,q}
↓  ↓  ↓
C_{1,0} ⟶ C_{1,1} ... ⟶ C_{1,q}
↓  ↓  ↓
⋮  ⋮  ⋮
↓  ↓  ↓
C_{p,0} ⟶ C_{p,1} ... ⟶ C_{p,q}

where each of the vertical maps is an isomorphism. Then diag(E(C)) ≃ BC.

Proof: We show that for each p, the simplicial set E(C)_{p,∗} ≃ BC. The result then follows from [B-K] XII, 4.3 and 4.2.

Let < p > denote the category 0 → 1 → ... → p, and define I_{p} to be the category whose objects are functors < p >→ C that take each morphism in < p > to an isomorphism in C. Morphisms in I_{p} are natural transformations of functors. Notice that the simplicial set E(C)_{p,∗} can be regarded as the classifying space of the category I_{p}. Further, there is an adjoint pair of functors F : I_{p} → C by evaluation at 0, and G : C → I_{p} by assigning to C ∈ C the functor < p >→ C which takes each i to C and each morphism to the identity. Thus E(C)_{p,∗} = BI_{p} ≃ BC, and it is easy to check that G in fact induces a map from the constant simplicial space BC to E(C) that is a levelwise homotopy equivalence, giving the data to apply [B-K] XII, 4.3 and 4.2.

□

Now that we have identified the bisimplicial set E(C) by considering E(C)_{p,∗}, we compute it by considering E(C)_{∗,q}. Call a functor F :< q >→ C a q-chain (notice we do not require morphisms in < q > to go to isomorphisms in C). Call two chains equivalent if there is a natural isomorphism between them, and let Aut(F), the automorphisms of the chain F, be the monoid of equivalences of F to itself. Then Lemma 3.1 has the following corollary.

Corollary 3.2 BC is homotopy equivalent to the diagonal of the bisimplicial set ∐_{<F >} BAut(F), where the coproduct is taken over equivalence classes of chains F.

Note that ∐_{<F >} BAut(F) is a bisimplicial set, and hence has face and degeneracy maps between the factors.

Proof: Let J_{q} be the category whose objects are q-chains in C and whose morphisms are equivalences of chains, and note that the (p, q) simplices in E(C) are the p-simplices in BJ_{q}, i.e. E(C)_{∗,q} ≃ BJ_{q}. However, the nerve of J_{q} is homotopy equivalent to the nerve of the full subcategory of J_{q} containing one object from each isomorphism class, i.e. one q-chain from each equivalence class of chains. The only morphisms in the subcategory are automorphisms of chains, and once again the result follows from [B-K] XII, 4.3 and 4.2.

□

Recall that we are trying to calculate the homotopy type of ∐_{n} BF_{n} where the families F_{n} consist of elementary abelian subgroups of Σ_{n} generated by disjoint p-cycles, together with their subgroups. Further, BF_{n} can be computed as the nerve of the category O(F_{n}) whose objects are transitive Σ_{n}-sets with isotropy groups in F_{n} and whose morphisms are Σ_{n}-equivariant maps. Our plan is first to give a smaller category than O(F_{n}) with which to calculate, and then to use Corollary 3.2 to calculate the homotopy type of BF_{n}. The following definition specifies which Σ_{n}-orbits will be part of the smaller category.

Definition 3.3 Let H ⊆ Σ_{n} and let S_{1}, ..., S_{k} be the orbits of H acting on the set {1, 2, ..., n}. Regarding H as a subgroup of the product Sym(S_{1}) × ... × Sym(S_{k}) of the symmetry groups of the S_{i}, we call H maximal if the inclusion H ↪ Sym(S_{1}) × ... × Sym(S_{k}) is an isomorphism of H with a cartesian product of subgroups H_{i} ⊆ Sym(S_{i}). We call a Σ_{n}-orbit B maximal if the isotropy group I_{b} of any element b ∈ B is maximal.

In the family F_{n}, the maximal subgroups are the ones generated by disjoint p-cycles.

Lemma 3.4 The classifying space BF_{n} is homotopy equivalent to the classifying space of the full subcategory of O(F_{n}) whose objects are maximal orbits.

Proof: Let O_{max}(F_{n}) denote the full subcategory of O(F_{n}) whose objects are maximal orbits. Let I : O_{max}(F_{n}) → O(F_{n}) be the inclusion functor. It is sufficient to give a functor M : O(F_{n}) → O_{max}(F_{n}) with natural transformations between MI and IM and the respective identity functors. Now any element of F_{n} is contained in a unique smallest maximal element. Let A ∈ O(F_{n}) and a ∈ A, and let P_{a} be the smallest maximal subgroup containing I_{a}. There is a natural map ψ_{A} : A → Σ_{n}/P_{a} which takes a to [P_{a}]. The map ψ_{A} is independent of the choice of a and defines a functor O(F_{n}) → O_{max}(F_{n}). The composite MI is the identity, and the natural maps ψ_{A} give a natural transformation of the identity to IM.

□

Thus to calculate BF_{n} we need only consider orbits of the form

Σ_{n}/(Z/p × ... × Z/p)

where the factors Z/p are generated by disjoint p-cycles of Σ_{n}. Let O(i) denote the orbit obtained by taking the quotient of Σ_{n} by the p-cycles (1, 2, ...p), (p + 1, ..., 2p) , ...., ((i − 1)p + 1, ..., ip). The orbits O(i) are a complete set of representatives of the isomorphism classes of O_{max}(F_{n}), and hence by Lemma 3.4 and Corollary 3.2, we can compute BF_{n} by considering equivalence classes of chains of the orbits O(i) and calculating the automorphism group of each chain.

Proposition 3.5 Let 0 ≤ i_{0} ≤ i_{1} ≤ ... ≤ i_{k} ≤ [n/p] be integers. Then all chains O(i_{0}) → ... → O(i_{k}) are equivalent.

Proof: The calculation uses elementary theory of permutation groups. We do not give the calculation in detail, but only mention a few pertinent facts. First, a Σ_{n}-equivariant map Σ_{n}/H → Σ_{n}/K comes from an element σ which conjugates H into K. To prove the proposition, it is sufficient to show that any chain

O(i_{0}) → O(i_{1}) → .... → O(i_{k})

is equivalent to the chain whose maps are given by conjugation with the identity element of Σ_{n}, i.e. by projection. In other words, we must show if the chain corresponds to conjugation by elements σ_{0}, σ_{1}, ...σ_{k−1} ∈ Σ_{n}, then we can find elements τ_{0}, ..., τ_{k} ∈ Σ_{n} which give equivariant maps making the following diagram commute:

O(i_{0}) ⟶^{id} O(i_{1}) ⟶^{id} ... ⟶^{id} O(i_{k})
↓^{τ_{0}} ↓^{τ_{1}} .... ↓^{τ_{k}}
O(i_{0}) ⟶^{σ_{0}} O(i_{1}) ⟶^{σ_{1}} ... ⟶^{σ_{k−1}} O(i_{k}).

We leave the rest of the calculation to the reader with the note that if each element τ is written as a product τ_{j} = g_{j}h_{j} where g_{i} moves only the elements {1, 2, ..., i_{j}p} and h_{i} moves the remaining elements {i_{j}p + 1, ..., n}, then the calculation goes by first determining the elements g_{j} from left to right (start with g_{0} = id), and then the elements h_{j} from right to left.

□

Let NZ/p denote the normalizer of Z/p in Σ_{p}. Recall that NZ/p is isomorphic to the semidirect product of Z/p with Aut(Z/p) ≅ Z/(p − 1).

Proposition 3.6 Let 0 ≤ i_{0} ≤ i_{1} ≤ ... ≤ i_{k} ≤ [n/p] be integers. The automorphism group of a chain

O(i_{0}) → ... → O(i_{k})

is

(Σ_{i_{0}} ∫ Aut(Z/p)) × (Σ_{i_{1}−i_{0}} ∫ NZ/p) × ... × (Σ_{i_{k}−i_{k−1}} ∫ NZ/p) × Σ_{n−pi_{k}}.

Proof: More elementary group theory.

□

When we put all this together, we get the following theorem.

Theorem 3.7

∐_{n} BF_{n} ≃ Bar(∐ B(Σ_{n} ∫ Aut(Z/p)), ∐ B(Σ_{n} ∫ NZ/p), ∐ BΣ_{n}).

The face maps in the bar construction are given on the middle factors by multiplication. On the left we multiply B(Σ_{n} ∫ Aut(Z/p))×B(Σ_{k} ∫ NZ/p) → B(Σ_{n+k} ∫ Aut(Z/p)) by projecting NZ/p to Aut(Z/p) and multiplying using wreath product. On the right we map B(Σ_{k} ∫ NZ/p) × BΣ_{n} → BΣ_{n+pk} by including (Σ_{k} ∫ NZ/p) ↪ Σ_{pk} and then multiplying in the usual way.

Proof: Given the preceding calculations, the theorem is a consequence of Corollary 3.2 and Proposition 3.6. The face maps in the bar construction can be determined by inspection.

□

4 Homotopy Pushout Diagrams of Spectra

In this section we show that a double bar construction of infinite loop spaces corresponds to a pushout diagram on the level of spectra (Proposition 4.1). The relevance of such a proposition is that the monoids in the double bar construction of Theorem 3.7 group complete to infinite loop spaces, and we wish to show that the group completion of the double bar construction is the zero space of a stable pushout.

Proposition 4.1 Let the diagram

\begin{matrix}A & ⟶^{f} & B \\ ↓^{g} & & ↓^{h} \\ C & ⟶^{k} & D \end{matrix}

be a homotopy pushout diagram of −1-connected spectra, and let A, B, C, and D be the zero spaces of the corresponding Ω-spectra. Then

Bar(B, A, C) ≃ D.

We note that to make sense of the statement of the proposition (for example, to make the double bar construction) we must assume that the diagram of zero spaces,

A ⟶^{f} B
↓^{g}  ↓^{h}
C ⟶^{k} D,

commutes on the nose, and that A, B, C, and D are strictly associative monoids. This is easy to arrange by choosing the appropriate models. The strategy of the proof of the proposition is to show that in the above diagram, Bar(B, A, C) can replace D without losing the homotopy pullback property of the square. Then we give a map Bar(B, A, C) → D and use the homotopy long exact sequence of a pullback square to show isomorphisms on homotopy groups.

We will use the following two facts about bar constructions freely in this section:

1. [M2, Prop. 9.8] Bar(A, A, C) ≃ C;

2. [M1, Thm 7.6] If X is a space with a right action of A, then there is a fibration sequence

X → Bar(X, A, C) → Bar(∗, A, C).

Lemma 4.2 Using the notation above, there is a homotopy pullback square

\begin{matrix}A & ⟶^{f} & B \\ ↓^{g} & & ↓ \\ C & ⟶ & Bar(B, A, C) \end{matrix} (**)

where the maps from B and C to Bar(B, A, C) are inclusion as the left- and right-hand factors, respectively.

Proof:

We consider the two-by-two square

Bar(A, A, C) ⟶ Bar(∗, A, C)
↓  ↓
Bar(B, A, C) ⟶ Bar(∗, A, C)

and, by taking fibers in both directions, extend to the three-by-three square

F ⟶  F  ⟶  ∗
↓  ↓  ↓
A ⟶ Bar(A, A, C) ⟶ Bar(∗, A, C)
↓  ↓  ↓
B ⟶ Bar(B, A, C) ⟶ Bar(∗, A, C).

Hence the maps A → B and C → Bar(B, A, C) have the same fiber, establishing that (**) is a homotopy pullback square.

□

Lemma 4.3 All components of Bar(B, A, C) are homotopy equivalent.

Proof: There is a left action (respectively, a right action) of B (resp., C) on Bar(B, A, C), and the map B × C → Bar(B, A, C) by inclusion as the left and right factor is a map which respects these actions. Hence there is a map π_{0}B × π_{0}C → π_{0}Bar(B, A, C) which is a map of π_{0}B × π_{0}C-sets. (Note that π_{0}B and π_{0}C are commutative since B and C are infinite loop spaces.) We will show that this map is an epimorphism, which is sufficient to establish the lemma since it shows that the actions of B and C on Bar(B, A, C), when considered together, are transitive on the components.

Consider the zero- and one-simplices in the bar construction. We write A_{n}, B_{n}, and C_{n} for the n simplices of A, B, and C. The zero-simplices of Bar(B, A, C) are B_{0} × C_{0}. We want to show that if b, b′ ∈ B_{0} are in the same component of B and c, c′ ∈ C_{0} are in the same component of C, then (b, c) and (b′, c′) are in the same component of Bar(B, A, C). Suppose that x ∈ B_{1} with d_{0}x = b and d_{1}x = b′, and y ∈ C_{1} with d_{0}y = c and d_{1}y = c′. Then (x, ∗, y) is a one-simplex between (b, c) and (b′, c′).

□

Proof of Proposition 4.1: Since the square

\begin{matrix}A & ⟶^{f} & B \\ ↓^{g} & & ↓^{h} \\ C & ⟶^{k} & D \end{matrix}

is known to be a pullback square (it comes from a stable homotopy pushout), and since there is a map from (**) to this square obtained by mapping Bar(B, A, C) → Bar(D, D, D) ≃ D, the long exact sequence in homotopy for the square tells us that Bar(B, A, C) → D is an isomorphism on homotopy groups, and hence is a weak equivalence on the basepoint components. Further, since D is a loop space, all its components are homotopy equivalent, and in Lemma 4.3 we have shown that the components of Bar(B, A, C) are also all homotopy equivalent. Hence to finish the proof of the proposition, we must show that Bar(B, A, C) → D gives a one-to-one correspondence on components. The main problem to get around is that π_{0}Bar(B, A, C) is not known to be a group.

Consider the end of the long exact sequence in homotopy for the above pullback square:

π_{0}A ⟶^{(f_{#},(−g)_{#})} π_{0}B × π_{0}C ⟶ π_{0}Bar(D, D, D) ≅ π_{0}D ⟶ 0.

The last map is an epimorphism because the spectrum A is assumed to be −1connected, and the next term would be π_{−1}A. Now we have a commutative diagram

π_{0}B × π_{0}C ⟶  π_{0}Bar(B, A, C)  ⟶ 0
↓^{id}  ↓^{(h,hf,k)}
π_{0}B × π_{0}C ⟶ π_{0}Bar(D, D, D) ≅ π_{0}D ⟶ 0

and to finish the proof, we have only to show that the map π_{0}B × π_{0}C → π_{0}Bar(B, A, C) factors through (π_{0}B × π_{0}C)/(im π_{0}A). To see that this is so, write s for the degeneracy map from zero-simplices to one-simplices, and let a, b, c be elements of A_{0}, B_{0}, and C_{0}, respectively. Observe that the onesimplex (sb, sa, sc − sa) has vertices (b, c) and (b + a, c − a), which therefore lie in the same component.

□

5 Group Completion of ∐ BF_{n}

In this section we give the proof of Theorem 2.3 by computing the mod p homology of the group completion of ∐_{n} BF_{n}, where {F_{n}} is the compatible closure of Z/p ⊆ Σ_{p}. We have already shown in Section 3 that ∐_{n} BF_{n} ≃ Bar(∐ B(Σ_{n} ∫ Aut(Z/p)), ∐ B(Σ_{n} ∫ NZ/p), ∐ BΣ_{n}). One of our tools will be Proposition 2.2, which says how the homology of a monoid is related to the homology of its group completion. We will also need a tool for computing the homology of a bar construction from the homology of its factors, for which we use the following theorem.

Theorem 5.1 [M1, Theorem 13.10] Let G be a topological monoid whose identity element is a strongly nondegenerate basepoint, and let W and Z be left and right G-spaces, and let homology be taken with field coefficients. Then there is a natural spectral sequence of differential coalgebras which converges from the coalgebra E^{2} = Tor_{H_{∗}G} (H_{∗}Z, H_{∗}W) to the coalgebra H_{∗}Bar(Z, G, W).

Because the projection B(Σ_{n} ∫ Aut(Z/p)) → BΣ_{n} is an isomorphism on mod p homology, we can replace ∐_{n} BF_{n} with

Bar(∐ BΣ_{n}, ∐ B(Σ_{n} ∫ NZ/p), ∐ BΣ_{n})

up to mod p homology isomorphism. The group completion of ∐ BΣ_{n} is QS^{0} and that of ∐ B(Σ_{n} ∫ NZ/p) is Q(BNZ/p_{+}), as can be seen by applying, for example, [B-E, Theorem B] with X = BNZ/p. By Proposition 4.1, we know that the zero space of a homotopy pushout diagram

BNZ/p_{+} ⟶^{τ} S^{0}
↓^{ϵ}
S^{0}

is a double bar construction Bar(QS^{0}, Q(BNZ/p)_{+}, QS^{0}). (See Proposition 5.2, below.) We will show that the mod p homology of this double bar construction is isomorphic to that of the group completion of ∐_{n} BF_{n} by using Proposition 2.2 applied to Bar(∐ BΣ_{n}, ∐ B(Σ_{n} ∫ NZ/p), ∐ BΣ_{n}). Note that although we do not know that Bar(∐ BΣ_{n}, ∐ B(Σ_{n} ∫ NZ/p), ∐ BΣ_{n}) is a topological monoid, its mod p homology is isomorphic to that of ∐_{n} BF_{n} and hence is an algebra. Therefore computing the ring theoretic localization of Proposition 2.2 makes sense.

To simplify notation, we write X for the monoid ∐ B(Σ_{n} ∫ NZ/p) and Y for the monoid ∐ BΣ_{n}, and let X̃ and Ỹ be their respective group completions. The left action of X on Y is given by including Σ_{n} ∫ NZ/p ↪ Σ_{np} and multiplying. The right action is given by projecting Σ_{n} ∫ NZ/p → Σ_{n} and multiplying. Both actions are given by means of a homomorphism X → Y, and these homomorphisms extend to the group completions, giving actions of X̃ on Ỹ.

Proposition 5.2 Bar(Ỹ , X̃, Ỹ) is the zero space of a stable homotopoy pushout diagram

BNZ/p_{+} ⟶^{τ} S^{0}
↓^{ϵ}
S^{0}  .

Proof: This follows from Proposition 4.1. The map ∐ B(Σ_{n} ∫ NZ/p) → ∐ BΣ_{n} given by projection of the wreath product group completes to the augmentation. The map ∐ B(Σ_{n} ∫ NZ/p) → ∐ BΣ_{n} is given by the composite Σ_{n} ∫ NZ/p ↪ Σ_{n} ∫ Σ_{p} ↪ Σ_{np}. The first map in this composite group completes to the (stable) B(NZ/p)_{+} → B(Σ_{p})_{+} induced by inclusion of groups. The second group completes to the map B(Σ_{p})_{+} → S^{0} called \overline{f} in [K-P], and sometimes known as the transfer map.

□

Our strategy for proving Theorem 2.3 is to relate ι^{−1}H_{∗}Bar(Y, X, Y) to H_{∗}Bar(Ỹ , X̃, Ỹ) by means of a certain colimit which turns out to have maps to each bar construction which are homology isomorphisms. The idea in forming the colimit is to try to give a geometric construction which realizes the localization (or colimit) of Proposition 2.2. In particular, note that a consequence of Proposition 2.2 is that if we form a colimit Z → Z → Z → ... where the maps are translation by a point not in the identity component, then the colimit has the homology of the group completion of Z.

To describe the colimit we are interested in, let y_{1} be the basepoint of the one-component of Y. Left and right multiplication of Y by y_{1} (denoted, respectively, y_{1}· and ·y_{1}) are, respectively, maps of right and left X-modules, and so induce a commuting diagram

Bar(Y, X, Y) ⟶^{(y_{1}·,id,·y_{1})} Bar(Y, X, Y)
↓  ↓
Bar(Ỹ , X̃, Ỹ) ⟶^{(y_{1}·,id,·y_{1})} Bar(Ỹ , X̃, Ỹ).

The lower horizontal map is a homotopy equivalence, since translation is a homotopy equivalence on Ỹ, and hence we obtain a map lim_{⟶} Bar(Y, X, Y) →

Bar(Ỹ , X̃, Ỹ).

Proposition 5.3 The map lim_{⟶} Bar(Y, X, Y) → Bar(Ỹ , X̃, Ỹ) is an isomorphism on mod p homology.

Proof: We consider the direct limit of the spectral sequences

Tor_{H_{∗}X} (H_{∗}Y , H_{∗}Y) ⟹ H_{∗}Bar(Y, X, Y),

noting the the direct limit spectral sequence converges to the direct limit of the abutments because this is a first quadrant homology spectral sequence. Let ι_{Y} denote the class in H_{0}Y represented by y_{1}. We can dispense with keeping track of left versus right multiplication because Y is homotopy commutative, and say simply that y_{1}· and ·y_{1} each induce multiplication by ι_{Y} on H_{∗}Y. Hence the direct limit of E^{2}-terms is

Tor_{H_{∗}X} (ι_{Y}^{−1}H_{∗}Y , ι_{Y}^{−1}H_{∗}Y).

In other words, by Proposition 2.2, it is very closely related to the E^{2}-term of the spectral sequence

Tor_{H_{∗}X̃} (H_{∗}Ỹ , H_{∗}Ỹ) ⟹ H_{∗}Bar(Ỹ , X̃, Ỹ).

Now we turn our attention to X. Recall that the actions of X on Y in the bar construction are given by the homomorphisms X → Y described just before Proposition 5.2. Let x_{1} be the basepoint of the one-component of X, and let ι_{X} be the element of H_{0}X it represents. By inspection we see the homomorphisms X → Y carry ι_{X} on the one hand to ι_{Y}, and on the other to (ι_{Y})^{p}. Hence considering H_{∗}Y as a module over H_{∗}X and localizing with respect to either action of H_{∗}X, we see that

ι_{X}^{−1}H_{∗}Y ≅ ι_{Y}^{−1}H_{∗}Y.

It follows that

Tor_{H_{∗}X} (ι_{Y}^{−1}H_{∗}Y , ι_{Y}^{−1}H_{∗}Y) ≅ Tor_{H_{∗}X} (ι_{X}^{−1}H_{∗}Y , ι_{X}^{−1}H_{∗}Y).

But now it is an easy algebraic argument to see that in fact

Tor_{H_{∗}X} (ι_{X}^{−1}H_{∗}Y , ι_{X}^{−1}H_{∗}Y) ≅ Tor_{ι_{X}^{−1}H_{∗}X} (ι_{X}^{−1}H_{∗}Y , ι_{X}^{−1}H_{∗}Y).

Stringing these isomorphisms together tells us that the map

lim_{⟶} Bar(Y, X, Y) → Bar(Ỹ , X̃, Ỹ)

is, on the E^{2}-terms of the spectral sequences for homology,

Tor_{ι_{X}^{−1}H_{∗}X} (ι_{Y}^{−1}H_{∗}Y , ι_{Y}^{−1}H_{∗}Y) → Tor_{H_{∗}X̃} (H_{∗}Ỹ , H_{∗}Ỹ)

in other words is an isomorphism, finishing the proof of the proposition.

□

To relate the homology of the colimit to that of the group completion of ∐ BF_{n} ≃ Bar(Y, X, Y), we need the fact that the following diagram commutes:

H_{∗}Y ⊗ H_{∗}Bar(Y, X, Y)  ⟶ H_{∗}Bar(Y, X, Y)
↓  ↓
H_{∗}Bar(Y, X, Y) ⊗ H_{∗}Bar(Y, X, Y) ⟶^{μ} H_{∗}Bar(Y, X, Y).

Here μ is the Hopf algebra multiplication arising from the monoid multiplication on ∐ BF_{n}, the top horizontal map comes from the left action of Y on Bar(Y, X, Y), and the left vertical map comes from the inclusion of Y as the left factor of Bar(Y, X, Y). The commutativity of the diagram can be verified on the space level by examination of simplices using Corollary 3.2 and Proposition 3.6 once one replaces the left-hand factors of Y by ∐ B(Σ_{n} ∫ Aut(Z/p)). (Recall this does not change the homology.) A similar diagram involving H_{∗}Y acting on the right likewise commutes, again on the level of spaces, but without replacing Y.

For ease of notation, we write Z for Bar(Y, X, Y); we want to study ΩBZ and compare it to lim_{⟶} Bar(Y, X, Y).

Proposition 5.4 There is a map lim_{⟶} Bar(Y, X, Y) → ΩBZ which is a homology equivalence.

Proof: To define the map, we recall that the group completion map Z → ΩBZ is a map of monoids. Hence we have a commutative diagram

Bar(Y, X, Y) ⟶^{(y_{1}·,id,·y_{1})} Bar(Y, X, Y)
↓  ↓
ΩBZ  ⟶^{(y_{1}·)∘(·y_{1})}  ΩBZ

which passes in the limit to a map lim_{⟶} Bar(Y, X, Y) → ΩBZ. Proposition 2.2 shows that this is a homology isomorphism.

□

Proof of Theorem 2.3: Consider the diagram

Z
↓
ΩBZ ⟵ lim_{⟶} Bar(Y, X, Y) ⟶ Bar(Ỹ , X̃, Ỹ).

The maps on the bottom row are isomorphisms on mod p homology and therefore localizing the spaces with respect HZ/p turns them into homotopy equivalences. On the other hand, the outside spaces are simple (Bar(Ỹ , X̃, Ỹ) is known to be an H-space by Proposition 5.2), so localizing with respect to HZ/p is the same as p-completion. Finally, the map

Bar(Ỹ , X̃, Ỹ) → Bar(QS^{0}, QB(NZ/p)_{+}, QS^{0}) → Bar(QS^{0}, QB(Σ_{p})_{+}, QS^{0})

is also a mod p homology isomorphism, and thus an equivalence after pcompletion, finishing the proof of the theorem.

□

References

[A] J. F. Adams, Infinite Loop Spaces, Princeton University Press, 1978.

[B-E] M. Barratt and P. Eccles, Γ^{+}-structures – I: A Free Group Functor for Stable Homotopy Theory, Topology 13 (1974) 25 - 45.

[B-P] M. Barratt and S. Priddy, On the Homology of Non-Connected Monoids and Their Associated Groups, Comment. Math. Helv. 47 (1972) 1-14.

[B-F] A. Bousfield and E. Friedlander, Homotopy Theory of Γ-spaces, Spectra, and Bisimplicial Sets, in, Geometric Applications of Homotopy Theory II, M. Barratt and M. Mahowald, eds., LNM 658, Springer Verlag, 1978.

[B-K] A. Bousfield and D. Kan, Homotopy Limits, Completions, and Localizations, LNM 304, Springer Verlag, 1972.

[D-K] W. Dwyer and D. Kan, Reducing Equivariant Homotopy Theory to the Theory of Fibrations, Contemporary Mathematics Vol. 37, p 35-49, American Math. Soc., 1985.

[E] A. Elmendorf, Systems of Fixed Points, Trans. Amer. Math. Soc. 277 (1983) 275-284.

[K-P] D. Kahn and S. Priddy, The Transfer and Stable Homotopy Theory, Math. Proc. Camb. Phil. Soc. 83 (1978) 103 - 111.

[Mac] S. Mac Lane, Categories for the Working Mathematician, GTM 5, Springer Verlag, 1971.

[M1] J. Peter May, Classifying Spaces and Fibrations, Mem. Amer. Math. Soc. No. 155, Jan. 1975.

[M2] J. Peter May, The Geometry of Iterated Loops Spaces, LNM 271, Springer Verlag, 1972.

[M3] J. Peter May, E_{∞} Spaces, Group Completions, and Permutative Categories, London Math. Soc. Lecture Notes Series Vol. 11, 1974.

[Q] D. Quillen, On the Group Completion of a Simplicial Monoid, preprint.

[S] G. Segal, Categories and Cohomology Theories, Topology 13 (1974) 293-312.

[tD] T. tom Dieck, Orbittypen und äquivariante Homologie. I. Arch. Math. 23 (1972) 307-317.