THE HOMOLOGY OF CERTAIN SUBGROUPS OF THE SYMMETRIC GROUP WITH COEFFICIENTS IN Lie(n)

GREG ARONE AND MARJA KANKAANRINTA

September 10, 1996

Abstract. The authors compute the mod p homology of groups of the form Σ_{n_{1}} × ⋯ × Σ_{n_{k}} with coefficients in Lie(n), where n_{1} + ⋯ + n_{k} = n.

0. introduction

For a commutative ring R and a positive integer n, let Lie_{R}(x_{1}, …, x_{n}) be the free Lie algebra over R generated by x_{1}, …, x_{n}. Let Lie_{R}(n) ⊂ Lie_{R}(x_{1}, …, x_{n}) be the submodule spanned by all bracket monomials containing each x_{i} exactly once. In what follows, R will almost always be ℤ or ℤ/p for some prime p, and we often supress R from the notation. Whenever we perform explicit calculations, we assume R = ℤ/p, otherwise, R = ℤ. The submodule Lie(n) is invariant under the obvious action of the symmetric group Σ_{n} on Lie(x_{1}, …, x_{n}). We consider Lie(n) as a representation of Σ_{n}. It is well-known that as an R-module

Lie_{R}(n) ≅ ⨁_{(n−1)!} R.

As a Σ_{n}-representation, Lie(n) can not, in general, be described so trivially. It is known (see, for instance, [9]) that if R = ℂ then

Lie(n) ≅ hom(Ind_{ℤ/n}^{Σ_{n}}Ψ, ℂ[−1])

where Ψ is the representation of ℤ/n sending the generator to e^{\frac{2πi}{n}} and ℂ[−1] is the sign representation. However, if the ground ring is ℤ or ℤ/p then it is easy to check directly that Lie(n) is not closely related to any representation induced from a one-dimensional representation of ℤ/n. Following a suggestion of the referee, we also remark that if p|n then Lie(n) can not be projective in characteristic p since projectivity in characteristic p would imply freeness over the p-Sylow subgroup of Σ_{n} which can not happen since the order of the p-Sylow subgroup does not divide (n − 1)! when p divides n.

1991 Mathematics Subject Classification. 55P99 .

Key words and phrases. Goodwillie calculus, Hilton-Milnor theorem, group homology.

The first author was partially supported by the Alexander von Humboldt foundation.

The second author was supported by the Emil Aaltonen foundation and the Academy of Finland.

Let n_{1}, …, n_{k} be positive integers such that n = n_{1} + ⋯ + n_{k}. Consider the group Σ_{n_{1}}×⋯×Σ_{n_{k}} as a subgroup of Σ_{n}. The homology groups H_{∗} (Σ_{n_{1}} × ⋯ × Σ_{n_{k}}; Lie(n)) arise in various contexts in mathematics. In particular, they arise as basic building blocks for homology of spaces of unordered configurations of points in manifolds and for homology of Lie algebras (see, for instance, [2, 3, 4]). The homology groups H_{∗} (Σ_{n}; Lie(n)) (with the ground ring ℤ/p) were computed in [1] (in a certain guise, as will be explained below). The main purpose of this paper is to use this computation to describe

H_{∗} (Σ_{n_{1}} × ⋯ × Σ_{n_{k}}; Lie(n))

(with the same ground ring).

Thus the main result of this paper is a recursive formula (theorem 0.2) which reduces computation of H_{∗} (Σ_{n_{1}} × ⋯ × Σ_{n_{k}}; Lie(n)) to computation of H_{∗} (Σ_{d}; Lie(d)) for various values of d. Our methods involve some homotopy theory, namely the Hilton-Milnor theorem and Goodwillie’s calculus of functors ([5, 6, 7]). In fact we do a little more than prove theorem 0.2 in the sense that rather than prove that two graded vector spaces are isomorphic, we prove that two spectra are homotopy equivalent and then obtain the result on vector spaces by passing to homology. We use that the n-th derivative (in the sense of Goodwillie) of the functor ΩS (loop - suspension) is a spectrum with an action of Σ_{n}, whose homology is concentrated in one degree, and in this degree it is precisely Lie(n). In more detail, we consider the Goodwillie tower (the Taylor tower of [7]) of the multi-variable functor ΩS(X_{1} ∨ X_{2} ∨ … ∨ X_{k}) and we evaluate its differentials in two different ways. The first way is to consider the Goodwillie tower of the functor ΩS and evaluate it at X_{1}∨X_{2}∨…∨X_{k}. The n-th layer is given, as a spectrum, by the following formula (we denote W_{hΣ_{n}} = W ∧_{Σ_{n}} EΣ_{n}_{+} for any space or spectrum W):

Map_{∗} (S^{2}K_{n}, Σ^{∞} (SX_{1} ∨ SX_{2} ∨ … ∨ SX_{k})^{∧n})_{hΣ_{n}}

where K_{n} is a complex with an action of Σ_{n}, that is non-equivariantly homotopy equivalent to a wedge of (n − 1)! copies of the (n − 2)-dimensional sphere, and whose only non-trivial reduced homology group satisfies

hom(H_{n−2}(K_{n}), ℤ[−1]) ≅_{ℤ[Σ_{n}]} Lie(n).

H_{∗} (Σ_{n_{1}} × ⋯ × Σ_{n_{l}}; Lie(n)) 3

This is proved as fact 2.3 below. K_{n} is essentially the geometric realization of the poset of partitions of the set with n elements (the precise definition is given on page 9). By using the binomial expansion, it should be intuitively clear (to anyone familiar with the Goodwille calculus) that the (n_{1}, …, n_{k})-differential of the functor ΩS(X_{1} ∨ X_{2} ∨ … ∨ X_{k}), which is the counterpart of

\frac{∂^{n}f}{∂x_{1}^{n_{1}} ⋯ ∂x_{k}^{n_{k}}} (0, …, 0)\frac{x_{1}^{n_{1}} ⋯ x_{k}^{n_{k}}}{n_{1}! ⋯ n_{k}!}

is given by

Map_{∗} (S^{2}K_{n}, Σ^{∞} ((SX_{1})^{∧n_{1}} ∧ ⋯ ∧ (SX_{k})^{∧n_{k}}))_{h(Σ_{n_{1}}×···×Σ_{n_{k}})}

where the action of Σ_{n_{1}} × ⋯ × Σ_{n_{k}} on K_{n} is given by restriction from the action of Σ_{n} and the action on (SX_{1})^{∧n_{1}} ∧ ⋯ ∧ (SX_{k})^{∧n_{k}} is the obvious one (this analogy with ordinary calculus is addressed in lemma 1.3 and the explanation following it). The second way is to use the Hilton-Milnor theorem. According to this theorem, ΩS(X_{1} ∨ X_{2} ∨ … ∨ X_{k}) is homotopy equivalent to a weak infinite product ∏_{i}ΩSY_{i},

where Y_{i} are smash products of some of the X_{j}s. Evaluating the (n_{1}, …, n_{k})-th differential of ∏_{i}ΩSY_{i} and comparing the two outcomes yields the following theorem.

Theorem 0.1. For any connected based spaces X_{1}, …, X_{k}, there is an equivalence of spectra, which is natural in X_{1}, …, X_{k}

Map_{∗} (S^{2}K_{n}, Σ^{∞} ((SX_{1})^{∧n_{1}} ∧ ⋯ ∧ (SX_{k})^{∧n_{k}}))_{h(Σ_{n_{1}}×···×Σ_{n_{k}})}

≃ ⋁_{d|n_{0}} ( ⋁_{B( \frac{n_{1}}{d} ,... , \frac{n_{k}}{d})} Map_{∗} (S^{2}K_{d}, Σ^{∞} (S(X_{1})^{∧ \frac{n_{1}}{d}} ∧ ⋯ ∧ (X_{k})^{∧ \frac{n_{k}}{d}} )^{∧d})_{hΣ_{d}} ) .

(Here n_{0} = gcd(n_{1}, …, n_{k}) and the definition of B(\frac{n_{1}}{d} , …, \frac{n_{k}}{d}) is given in formula 3 on page 12).

Next we take each one of the spaces X_{j} to be a wedge of i_{j} copies of S^{2m}, an even-dimensional sphere (m is arbitrary, but it is the same for all j). Thus, for

j = 1, …, k we take X_{j} = ⋁_{j=1}^{i_{j}}S^{2m} where i_{j} is some positive integer. Passing to

homology we obtain the following theorem as a corollary

Theorem 0.2. Let n_{0} and B(\frac{n_{1}}{d} , …, \frac{n_{k}}{d}) be as in the previous theorem. For any ground ring R, and any finitely generated free R-modules N_{1}, …, N_{k}

\begin{matrix}H_{∗} (Σ_{n_{1}} × ⋯ × Σ_{n_{k}}; Lie_{R}(n) ⊗_{R} N_{1}^{⊗n_{1}} ⊗_{R} ⋯ ⊗_{R} N_{k}^{⊗n_{k}}) ≅ \\ ⨁_{d|n_{0}} ( ⨁_{B( \frac{n_{1}}{d} ,... , \frac{n_{k}}{d})} H_{∗} (Σ_{d}; Lie_{R}(d) ⊗_{R} (N_{1}^{⊗ \frac{n_{1}}{d}} ⊗_{R} ⋯ ⊗_{R} N_{k}^{⊗ \frac{n_{k}}{d}} )^{⊗d})). \end{matrix}

Now take the ground ring to be ℤ/p for some prime p, and let N_{1} ≅ ⋯ ≅ N_{k} ≅ R (which corresponds to taking all the spaces X_{j} in theorem 0.1 to be S^{2m}). The groups H_{∗} (Σ_{d}; Lie(d)) have been computed in [1]. In particular, if d is not a power of p, then H_{∗} (Σ_{d}; Lie(d)) ≅ 0. If d = p^{i} for some non-negative integer i, then let us denote M_{i} = H_{∗} (Σ_{p^{i}}; Lie(p^{i})). We will describe M_{i} explicitly in section 3. Recall that n_{0} = gcd(n_{1}, …, n_{k}) and let us write n_{0} = mp^{j} where p does not divide m. Then the only values of d in theorem 0.2 that we need to consider are those that are powers of p. Therefore

H_{∗} (Σ_{n_{1}} × ⋯ × Σ_{n_{k}}; Lie(n)) ≅ ⨁_{i=0}^{j} M \frac{p^{i}}{n} ∑_{e|mp^{j−i}}μ(e)\frac{(\frac{n}{ep^{i}})!}{(\frac{n_{1}}{ep^{i}})! ⋯ (\frac{n_{k}}{ep^{i}})!}_{i} .

Here μ denotes the number-theoretic Möbius function. Because of the properties of μ, the only values of e we need to consider in the exponent of the right hand side are m′ and (if j − i > 0) m′p where m′|m. We obtain the following theorem

Theorem 0.3. Let n = n_{1} +⋯+n_{k}. Let n_{0} = gcd(n_{1}, …, n_{k}). Let the ground ring be ℤ/p. Write n_{0} = mp^{j} where p does not divide m. Then

H_{∗} (Σ_{n_{1}} × ⋯ × Σ_{n_{k}}; Lie(n)) ≅

⨁_{i=0}^{j−1}M \frac{p^{i}}{n} ∑_{m′|m}μ(m′) (\frac{(\frac{n}{m′p^{i}})!}{(\frac{n_{1}}{m′p^{i}})! ⋯ (\frac{n_{k}}{m′p^{i}})!} − \frac{(\frac{n}{m′p^{i+1}} )!}{(\frac{n_{1}}{m′p^{i+1}} )! ⋯ (\frac{n_{k}}{m′p^{i+1}} )!})_{i}

⨁ M \frac{p^{j}}{n} ∑_{m′|m}μ(m′)\frac{(\frac{n}{m′p^{j}})!}{(\frac{n_{1}}{m′p^{j}})! ⋯ (\frac{n_{k}}{m′p^{j}})!}_{j} .

The paper is organized as follows: in section 1 we present a very brief survey of some of Goodwillie’s theory of Taylor towers and state it in the multi-variable form that we need. All the material in this section is either quoted from [7] or is adapted from there in a straightforward way. In section 2 we recall various facts from algebraic topology that we need. In section 3, we prove theorems 0.1 and 0.2 and also describe the M_{i}s of theorem 0.3.

1. Multivariable calculus

Let U denote the category of spaces with a non-degenerate basepoint. We will be concerned with calculus of homotopy functors F : U^{×k} → U. We begin with an overview of the case k = 1, which is the subject of [5, 6, 7]. According to Goodwillie’s calculus, certain functors are analogous to polynomial functions. Namely, a functor is analogous to a polynomial of degree n if it is n-excisive in the sense of [6, definition 3.1]. One of the main theorems of [7] is that if a functor F is ρ-analytic in the sense of [6, definition 4.2], then there exists a tower of functors {P_{n}F }_{n≥0} with natural transformations ⋯ → P_{n}F → P_{n−1}F → ⋯ such that P_{n}F is of degree n and there exist natural transformations p_{n} : F → P_{n}F which are the unique best approximations of F by n-excisive functors in some precise sense. Since [7] has not been published yet, we spell out the details of the construction, but not of the proofs.

Let C(n) denote the category whose objects are subsets of \underline{n} = \{1, …, n\} and whose morphisms are inclusions. Let C_{0}(n) denote the full subcategory of C(n) whose objects are the non-empty subsets of \underline{n}. Let C be a category. An n-dimensional cubical diagram in C is a functor C(n) → C. A punctured n-dimensional cubical diagram is a functor C_{0}(n) → C. For our purposes, C will always be any well-behaved version of the category of either based spaces or spectra.

P_{n} involves the infinite iteration of another construction T_{n}. For X ∈ U the diagram U ↦ X ∗ U, U ⊆ \underline{n + 1} (here ∗ denotes join) determines a strongly cocartesian \underline{n + 1}-cube [6, definition 2.1] and a map, as in [6, definition 1.2],

F (X) → (T_{n}^{1}F)(X) =^{def.} holim {F (X ∗ U)|U ∈ C_{0}(n + 1)} .

For i > 1, define the functor T_{n}^{i}F inductively by T_{n}^{i}F = T_{n}^{1}T_{n}^{i−1}F. It is easy to see that

T_{n}^{i}F (X) ≅ holim {F (X ∗ U_{1} ∗ ⋯ ∗ U_{i})|U_{1} × ⋯ × U_{i} ∈ C_{0}(n + 1)^{i}}

and we might as well have taken this to be the definition. We will do this when we define the multi-variable analogue of T_{n}^{i}. Clearly, there are natural transformations T_{n}^{i}F → T_{n}^{i+1}F. P_{n}F is defined to be the homotopy colimit of the diagram

F = T_{n}^{0}F → T_{n}^{1}F → T_{n}^{2}F → ⋯ → T_{n}^{i}F → · · · .

It is shown in [7] that P_{n}F is n-excisive and the map p_{n} : F → P_{n}F is characterized, up to weak homotopy equivalence, among natural transformations from F to nexcisive functors by the property that p_{n}(X) : F (X) → P_{n}F (X) is (k − ρ)(n + 1) − cconnected, where k is the connectivity of X and c is constant which does not depend on X or n.

According to [7], this tower should be thought of as the Taylor expansion of F at the one-point space (but we will subsequently call it the Goodwillie tower of F). Thus, P_{n}F is the n-th “Taylor polynomial” of F and in particular P_{0}F is a (homotopy) constant functor with P_{0}F (X) ≃ F (∗) for all X. Note also that if F is reduced, i.e F (∗) ≃ ∗, then T_{1}F (X) ≃ ΩF (SX) and P_{1}F, the linearization of F, is, essentially, Ω^{∞}F (S^{∞}X). Let D_{n}F denote the homotopy fiber of the map P_{n}F → P_{n−1}F. This is a well defined functor and we call it the n-th differential of F. It can be shown that D_{n}F is homogeneous of degree n in the sense that it is excisive of degree n and P_{n−1}D_{n}F ≃ ∗. Homogeneous functors were classified in [7]: a homogeneous functor of degree n is determined, at least up to homotopy, by a spectrum A endowed with an action of the symmetric group Σ_{n} and (up to weak homotopy equivalence) it has the form Ω^{∞} ((A ∧ X^{∧n})_{hΣ_{n}}). Note the visual resemblance to the formula \frac{ax^{n}}{n!} for the

n-th summand in a Taylor expansion. If D_{n}F (X) ≃ Ω^{∞} ((A ∧ X^{∧n})_{hΣ_{n}}), then we say that the spectrum A, together with the action of Σ_{n}, is the n-th derivative of F. Note also that for n = 1 one gets the classical description of a generalized homology theory as given by the Brown representability theorem.

For a general k, we will say that a functor F : U^{×k} → U is analytic if it is analytic considered in each variable separately. For a multi-index n= (n_{1}, …, n_{k}), we say that F is n-excisive if it is n_{i}-excisive considered as a functor of X_{i} for any choice of fixed X_{1}, …, X̂_{i}, …, X_{k}. We say that F is multilinear if it is reduced and linear (1-excisive) in each variable separately. Given a subgroup H of Σ_{k} we say that F is symmetric with respect to H if it extends to a functor F : H ≀ U^{×k} → U. If H = Σ_{k}, then we simply say that F is symmetric. It is shown in [7] that homogeneous onevariable functors of degree k are equivalent to symmetric multilinear functors of k variables. In fact, up to weak homotopy equivalence, a symmetric multilinear functor has the form Ω^{∞} (A ∧ X_{1} ∧ ⋯ ∧ X_{k}), where A is a spectrum with an action of Σ_{k}.

By a k-index, or simply a multi-index, when k is clear from the context, we mean an ordered k-tuple of non-negative integers. We denote multi-indices with bold letters such as i, j, n. Given multi-indices i= (i_{1}, …, i_{k}) and j= (j_{1}, …, j_{k}), we say that i≤j if i_{l} ≤ j_{l} for l = 1, …, k. For a functor F : U^{×k} → U, and multi-indices i= (i_{1}, …, i_{k}) and n= (n_{1}, …, n_{k}) we define

\begin{matrix}T inF (X_{1}, …, X_{k}) = holim\{F (X_{1} ∗ U_{1}^{1} ∗ U_{2}^{1} ∗ ⋯ ∗ U_{i_{1}}^{1}, …, X_{k} ∗ U_{1}^{k} ∗ ⋯ ∗ U_{i_{k}}^{k})| \\ (U_{1}^{1}, …, U_{i_{1}}^{1}, …, U_{i_{k}}^{k}) ∈ C_{0}(n_{1} + 1)^{i_{1}} × ⋯ × C_{0}(n_{k} + 1)^{i_{k}}\}. \end{matrix}

By analogy with the one-variable case, it is clear that if i_{1} ≤ i_{2} then there is a canonical natural transformation T i_{1} nF → T i_{2} nF and if also i_{2} ≤ i_{3}, then the map T i_{1} nF → T i_{3} nF is the composition T i_{1} nF → T i_{2} nF → T i_{3} nF. We define Pn to be the homotopy direct limit of the thus obtained k-dimensional diagram of all T in. Intuitively,

P_{(n_{1},... ,n_{k})}F = T_{(n_{1},... ,n_{k})}^{(∞,... ,∞)}F.

Clearly, PnF is n-excisive. Let 1_{j} denote the multi-index (0, …, 1, …, 0) having a 1 at the j-th place and zeros elsewhere. Just as in the 1-variable case there is a map P_{n} → P_{n−1} induced by restriction of homotopy inverse limits, in the k-variables case there are maps Pn → P_{n−1_{j}}. Thus the functors PnF fit into a “k-dimensional” inverse limit system, that converges to F in an appropriate sense. When we speak of PnF, we allow that some of the indices of n are ∞, in which case we mean that we take the inverse limit with respect to these indices. For example, P_{(∞,n_{2},... ,n_{k})} =

holim_{n_{1}→∞}P_{(n_{1},n_{2},... ,n_{k})}.

In the case of a one-variable functor F, there is an important relation between the n-th differential of F and the n-th cross-effect of F. The connection is explained in [7]. We present a brief summary. Given a functor F : U → U, the n-th cross-effect of F, denoted χ_{n}F, is a functor of n variables defined as follows: For based spaces X_{1}, …, X_{n}, we define an n-dimensional cubical diagram S(X_{1}, …, X_{n}) by

\underline{n} ∖ T ↦ ⋁_{s∈T} X_{s}

with the maps in the cube being the obvious retractions. Then χ_{n}F (X_{1}, …, X_{n}) is defined to be the iterated homotopy fiber (or the total fiber, as defined in [7, definitions 1.1-1.2a]) of S(X_{1}, …, X_{n}). It is easily seen from the definitions that χ_{n}F is symmetric and reduced. It follows that P_{(1,1,... ,1)}χ_{n}F (X_{1}, …, X_{n}), the multilinearization of χ_{n}F, is given, up to homotopy, by

hoco lim_{k_{1},... ,k_{n}→∞} Ω^{k_{1}+···+k_{n}}χ_{n}F (S^{k_{1}}X_{1}, …, S^{k_{n}}X_{n}).

We denote this functor D^{(n)}F. D^{(n)}F is a symmetric multilinear functor, and as such is represented by a spectrum with an action of Σ_{n}; denote this spectrum by A_{n}. The following result is proved in [7]

Lemma 1.1. If A_{n} is as above, then A_{n} is the n-th derivative of F.

Let F : U^{k} → U be a functor of k variables. Let n= (n_{1}, …, n_{k}). We define DnF to be the iterated homotopy fiber of the k-dimensional cube given by U ↦ P_{n−1_{U}}, where 1_{U} is defined to be the multi-index which has in its j-th place 1 or 0 depending on whether j ∈ U or j ∉ U respectively. DnF is n-homogeneous in the appropriate sense. There is a natural (weak) homotopy equivalence

DnF (X_{1}, …, X_{k}) ≃ Ω^{∞} (A ∧ X_{1}^{∧n_{1}} ∧ ⋯ ∧ X_{k}^{∧n_{k}})_{h(Σ_{n_{1}}×Σ_{n_{2}}×···×Σ_{n_{k}})}

where A is a spectrum with an action of Σ_{n_{1}} × Σ_{n_{2}} × ⋯ × Σ_{n_{k}}. We may think of A, together with the action, as the n-th derivative of F. Again, we would like to establish the relation between derivatives and cross-effects. So, given F as before, we may form its n_{1}-th cross-effect with respect to the variable X_{1}, n_{2}-th cross-effect in the second variable, and so on. One can easily see that the outcome does not depend on the order of variables. The outcome is a functor of n_{1} + n_{2} + ⋯ + n_{k} variables, which is reduced and symmetric with respect to an action of Σ_{n_{1}} × Σ_{n_{2}} × ⋯ × Σ_{n_{k}}. Upon multilinearizing, we obtain a functor which is multilinear and is symmetric with respect to Σ_{n_{1}} × Σ_{n_{2}} × ⋯ × Σ_{n_{k}}. As such, it is represented by a spectrum with an action of Σ_{n_{1}} × Σ_{n_{2}} × ⋯ × Σ_{n_{k}}. Denote it by A_{n}.

Lemma 1.2. If A_{n} is as above, then A_{n} is the (n_{1}, …, n_{k})-th derivative of F.

Proof. Very similar to the proof of our lemma 1.1 as given in [7].

Lemma 1.3. Let F (X_{1}, …, X_{k}) = G ( ⋁_{i=1}^{k}X_{i}), where G is some functor of one vari-

able. Let n= (n_{1}, …, n_{k}) and let n = n_{1} + ⋯ + n_{k}. Then the n-th derivative of F is equivalent to the n-th derivative of G with the action of Σ_{n_{1}} × Σ_{n_{2}} × ⋯ × Σ_{n_{k}} obtained by restriction from the action of Σ_{n}.

Explanation. Let f(x_{1}, …, x_{k}) be an analytic function, say, from ℝ^{k} to ℝ. Assume, furthermore, that f(x_{1}, …, x_{k}) = g(x_{1} + ⋯ + x_{k}) for some g : ℝ → ℝ. Then

\frac{∂^{n}f}{∂x_{1}^{n_{1}} ⋯ ∂x_{k}^{n_{k}}} (0, …, 0)\frac{x_{1}^{n_{1}} ⋯ x_{k}^{n_{k}}}{n_{1}! ⋯ n_{k}!} = \frac{d^{n}g}{dx^{n}} (0)\frac{x_{1}^{n_{1}} ⋯ x_{k}^{n_{k}}}{n_{1}! ⋯ n_{k}!} .

Proof of lemma 1.3. Immediate from lemmas 1.1 and 1.2.

Lemma 1.4. Let F (X_{1}, …, X_{k}) = G (X_{1}^{∧m_{1}} ∧ X_{2}^{∧m_{2}} ∧ ⋯ ∧ X_{k}^{∧m_{k}}). Then DnF is trivial if \frac{n_{i}}{m_{i}} is not the same for all i = 1. Otherwise,

DnF (X_{1}, …, X_{k}) ≃ D_{l}G (X_{1}^{∧m_{1}} ∧ X_{2}^{∧m_{2}} ∧ ⋯ ∧ X_{k}^{∧m_{k}}) ,

where l = \frac{n_{i}}{m_{i}} .

Explanation. Let f(x_{1}, …, x_{k}) be an analytic function. Assume, furthermore, that f(x_{1}, …, x_{k}) = g (x_{1}^{m_{1}} · x_{2}^{m_{2}} ⋯ x_{k}^{m_{k}}) for some g. Then the Taylor series for g evaluated at x_{1}^{m_{1}} ·x_{2}^{m_{2}} ⋯ x_{k}^{m_{k}} is clearly the multi-variable Taylor expansion of f and it can be easily concluded that if \frac{n_{i}}{m_{i}} is not a constant non-negative integer independent

of i then \frac{∂^{n}f}{∂x_{1}^{n_{1}}···∂x_{k}^{n_{k}}} (0, …, 0)\frac{x_{1}^{n_{1}}···x_{k}^{n_{k}}}{n_{1}!···n_{k}!} = 0 and if \frac{n_{i}}{m_{i}} = l for all i, then

\frac{∂^{n}f}{∂x_{1}^{n_{1}} ⋯ ∂x_{k}^{n_{k}}} (0, …, 0)\frac{x_{1}^{n_{1}} ⋯ x_{k}^{n_{k}}}{n_{1}! ⋯ n_{k}!} = \frac{d^{l}g}{dx^{l}} (0)\frac{(x_{1}^{m_{1}} ⋯ x_{k}^{m_{k}})^{l}}{l!}

Proof of lemma 1.4. In fact, we want to show that for any multi-index n

PnF (X_{1}, …, X_{k}) ≃ P_{(lm_{1},... ,lm_{k})}F (X_{1}, …, X_{k}) ≃ P_{l}G (X_{1}^{∧m_{1}} ∧ X_{2}^{∧m_{2}} ∧ ⋯ ∧ X_{k}^{∧m_{k}})

where l = min\{⌊\frac{n_{i}}{m_{i}} ⌋\}. Without loss of generality, we may assume that l = ⌊\frac{n_{1}}{m_{1}} ⌋. Consider the functor P_{(n_{1},∞,... ,∞)}F, which amounts to considering F as a functor of X_{1} only, and taking the n_{1}-th Taylor polynomial in this variable. P_{(n_{1},∞,... ,∞)}F is n_{1}-excisive (in X_{1}) and the map F (X_{1}, …, X_{k}) → P_{(n_{1},∞,... ,∞)}F (X_{1}, …, X_{k}) is (n_{1} + 1)(d_{1} − ρ) − c-connected, where d_{1} is the connectivity of X_{1} and ρ and c are some numbers, which depend on X_{2}, …, X_{k}, but not on X_{1}. On the other hand, consider the functor P_{l}G (X_{1}^{∧m_{1}} ∧ X_{2}^{∧m_{2}} ∧ ⋯ ∧ X_{k}^{∧m_{k}}) as a functor of X_{1} only. It is obvious that it is lm_{1}-excisive and therefore n_{1}-excisive, since n_{1} ≥ lm_{1}. The natural map

F (X_{1}, …, X_{k}) = G (X_{1}^{∧m_{1}} ∧ ⋯ ∧ X_{k}^{∧m_{k}}) → P_{l}G (X_{1}^{∧m_{1}} ∧ ⋯ ∧ X_{k}^{∧m_{k}})

is

(l + 1)(m_{1}d_{1} + ⋯ + m_{k}d_{k} − ρ′) − c′ ≥ ((l + 1)m_{1})(d_{1} − ρ′) − c′ ≥ (n_{1} + 1)(d_{1} − ρ′) − c′

connected, for some constants ρ′ and c′. By universality properties of the Taylor approximations it follows that there is a weak homotopy equivalence

P_{(n_{1},∞,... ,∞)}F (X_{1}, …, X_{k}) ≃ P_{l}G (X_{1}^{∧m_{1}} ∧ X_{2}^{∧m_{2}} ∧ ⋯ ∧ X_{k}^{∧m_{k}}) . (1)

Now consider P_{(n_{1},... ,n_{k})}F. It is weakly equivalent to P_{(∞,n_{2},... ,n_{k})}P_{(n_{1},∞,... ,∞)}F. But by (1), P_{(n_{1},∞,... ,∞)}F is lm_{i}-excisive in the variable X_{i} for i = 1, 2, …, k. Since we defined l to be min\{⌊\frac{n_{i}}{m_{i}} ⌋\}, it follows that n_{i} ≥ lm_{i}, and therefore P_{(n_{1},∞,... ,∞)}F is n_{i}-excisive in the variable X_{i} for i = 2, …, k. Hence

P_{(n_{1},... ,n_{k})}F (X_{1}, …, X_{k}) ≃ P_{l}G (X_{1}^{∧m_{1}} ∧ X_{2}^{∧m_{2}} ∧ ⋯ ∧ X_{k}^{∧m_{k}}) .

The lemma readily follows.

2. Miscellaneous preliminary results

Let W be a based space or a spectrum with an action of Σ_{n}. Let R be a commutative ring. Let H̃_{∗}(−; R) denote reduced homology with coefficients in R.

Fact 2.1. Suppose that H̃_{∗}(W; R) is concentrated in degree i. Then

H̃_{∗}(W_{hΣ_{n}}; R) ≅ H_{∗+i}(Σ_{n}; H̃_{i}(W; R)).

We will use the following “geometric realization” of Lie(n). Let k_{n} be the category of unordered partitions of \underline{n}. Thus, the objects of k_{n} are unordered partitions of \underline{n}, and there is a morphism λ_{1} → λ_{2} iff λ_{2} is a refinement of λ_{1}. The category of partitions that we define here is the opposite of the category of partitions as defined in [1], but it makes no difference since we only are interested in the simplicial nerve of k_{n}. Obviously, k_{n} has an initial and a final object. Denote these ˆ0 and ˆ1 respectively. Let k̃_{n} = k_{n} ∖ {ˆ0, ˆ1}. Let K̃_{n} be the geometric realization of the simplicial nerve of k̃_{n}. Let K_{n} be the unreduced suspension of K̃_{n}. Obviously, K_{n} has an action of Σ_{n}. Let S^{n} denote the n-dimensional sphere.

Fact 2.2. Non-equivariantly

K_{n} ≃ ⋁_{(n−1)!} S^{n−2}.

In particular, the reduced homology of K_{n} is concentrated in degree n − 2.

Fact 2.3. As a Σ_{n}-representation, Lie(n) ≅ hom (H̃_{n−2}(K_{n}), Z[−1]) .

Proof. It is known ([2, theorem 12.3, page 302]) that Lie(n) is isomorphic, as a Σ_{n}-representation, to the top homology of F (R^{2k+1}; n), the space of ordered configurations of n points in an odd-dimensional Euclidean space (k is arbitrary). Now view F (R^{2k+1}; n) as a Σ_{n}-equivariant subspace of R^{(2k+1)n} and also, by adding a point at infinity, of S^{(2k+1)n}. The top homology group of S^{(2k+1)n} is isomorphic to Z[−1] as a Σ_{n}-representation. Let Δ^{n}S^{2k+1} denote the complement of F (R^{2k+1}; n) in S^{(2k+1)n}. Let H_{b} denote the bottom reduced homology of Δ^{n}S^{2k+1}. It follows by Alexander duality that H_{b} is in dimension n+2k −1 and that Lie(n) ≅_{Z[Σ_{n}]} hom(H_{b}, Z[−1]). It remains to show that H_{b} ≅ H̃_{n−2}(K_{n}) as a Σ_{n}-representation. It is enough to prove the following proposition (the first author is grateful to J. Rognes for explaining it to him)

Proposition 2.4. There is a Σ_{n}-equivariant (weak) map

Δ^{n}S^{l} → S^{l}K_{n}

which, for large enough l, induces an isomorphism on the bottom homology of Δ^{n}S^{l}.

The rest of the proof of the fact is occupied by the proof of the proposition. Consider the covering of Δ^{n}S^{l} by the subspaces U_{i,j}, 1 ≤ i < j ≤ n, where

U_{i,j} = {(s_{1} ∧ ⋯ ∧ s_{n}) ∈ Δ^{n}S^{l} | s_{i} = s_{j}} .

The point is that the intersection poset of this covering is isomorphic to the poset of unordered partitions of \underline{n} and the intersections in this covering are highly enough connected. In more detail, let A = {(i_{1}, j_{1}), (i_{2}, j_{2}), …, (i_{M}, j_{M})} be a collection of pairs 1 ≤ i_{m} < j_{m} ≤ n. Let U_{A} = ⋂_{(i_{m},j_{m})∈A} U_{i_{m},j_{m}}. We associate with A a graph on n vertices, labeled 1, …, n, as follows: There is an edge (i, j) iff (i, j) ∈ A. The connected components of this graph determine a partition of \underline{n}. Clearly, U_{A} depends only on the partition associated with A. Thus, the poset associated with the covering of Δ^{n}S^{l} by U_{i,j} is isomorphic to k_{n} ∖ ˆ1. Let J be the intersection diagram of this covering. Thus, J is a functor

k_{n} ∖ ˆ1 → Spaces

given by

λ → {(s_{1} ∧ ⋯ ∧ s_{n}) ∈ Δ^{n}S^{l} | s_{i} = s_{j} if i and j are in the same component of λ} .

Thus Δ^{n}S^{l} is the strict direct limit of J. Let Δ_{h}^{n}S^{l} be the homotopy direct limit of J. There is a canonical Σ_{n}-equivariant map

Δ_{h}^{n}S^{l} → Δ^{n}S^{l}

which is a homotopy equivalence.

We now define another functor J̃: k_{n} ∖ {ˆ1} → Spaces as follows:

λ → {(s_{1} ∧ ⋯ ∧ s_{n}) ∈ Δ^{n}S^{l} | s_{i} = s_{j} if i and j are in the same component of λ, ∑_{i=1}^{n} s_{i} = 0 and ∑_{i=1}^{n} ∥s_{i}∥^{2} = 1}.

Here we think of Δ^{n}S^{l} as a subspace of S^{nl}, which is the one-point compactification of ℝ^{nl}. The functor J̃ takes values in subspaces of ℝ^{nl}. Notice that J̃(ˆ0) = ∅ and therefore hocolim J̃ ≅ hocolim J̃|_{k_{n}}. In general, if λ has k components, then J̃(λ) ≅ S^{lk−l−1}. Let Δ̃_{h}^{n}S^{l} denote the homotopy direct limit of J̃. It is easy to see that

Δ_{h}^{n}S^{l} ≅ S^{l} ∗ Δ̃_{h}^{n}S^{l}

where ∗ denotes join. Next we note that since Δ̃_{h}^{n}S^{l} is the homotopy direct limit of a functor k̃_{n} → Spaces and K̃_{n} is homeomorphic to the homotopy direct limit of the constant functor k̃_{n} → ∗, there is a map

Δ̃_{h}^{n}S^{l} → K̃_{n}

which is ∼ l-connected, since the values of J̃ are all ∼ l-connected. All of the above assembles into the following chain of equivariant maps:

Δ^{n}S^{l} ←^{≃} Δ_{h}^{n}S^{l} →^{≅} S^{l} ∗ Δ̃_{h}^{n}S^{l} → S^{l} ∗ K̃_{n} →^{≃} S^{l} ∧ K_{n}.

The composed map is ∼ 2l-connected since the map S^{l} ∗Δ̃_{h}^{n}S^{l} → S^{l} ∗K̃_{n} is. It follows that the map induces an isomorphism on the bottom homology for l large enough. This completes the proof.

The space K_{n} plays an important role in calculus of functors. Consider the identity functor from based spaces to based spaces. The following is proved in [8] in a slightly different form:

Fact 2.5.

D_{n}Id(X) ≃ Ω^{∞} (Map_{∗} (SK_{n}, Σ^{∞}X^{∧n})_{hΣ_{n}}) .

Proof. As mentioned above, this is a reformulation of the main theorem of [8]. For proof of the equivalence with the result in [8] see [1, section 2].

Corollary 2.6.

D_{n}(ΩS)(X) ≃ Ω^{∞} (Map_{∗} (S^{2}K_{n}, Σ^{∞}(SX)^{∧n})_{hΣ_{n}}) .

Proof. For any functor F : U → U, consider the functor F′ : X ↦ ΩF (SX). It follows immediately from the universal properties of P_{n} that (P_{n}F′)(X) ≃ Ω(P_{n}F)(SX) and (D_{n}F′)(X) ≃ Ω(D_{n}F)(SX). The corollary follows by letting F to be the identity functor.

Remark 2.7. It may seem that there is a problem with the definition of K_{1}. In fact, there is not. Either by careful check of the definitions, or by convention, define K_{1} to be the empty set. Then SK_{1} should be interpreted as S^{0}, provided the suspension is unreduced. With these conventions fact 2.5 and corollary 2.6 hold for n = 1.

3. Proof of the main theorems

Let X_{1}, …, X_{k} be sufficiently nice topological spaces, for example connected CW - complexes with non-degenerate basepoints. By the Hilton-Milnor theorem [10] there is a natural homotopy equivalence

h: ∏_{i∈I} ΩSY_{i} → ΩS ⋁_{j=1}^{k} X_{j}, (2)

where I denotes the set of basic products on k letters. Essentially, each Y_{i} is a smash product of some of the spaces X_{1}, …, X_{k} corresponding to the basic product i. By the formula due to Witt [11] the number of basic products involving X_{j} exactly q_{j} times is

B(q_{1}, …, q_{k}) = \frac{1}{q} ∑_{d|q_{0}} μ(d)\frac{(\frac{q}{d})!}{(\frac{q_{1}}{d})! ⋯ (\frac{q_{k}}{d})!}, (3)

where q_{0} is the greatest common divisor of the numbers q_{1}, …, q_{k}, q = q_{1} + ⋯ + q_{k} and μ is the number theoretic Möbius function.

Let us denote F_{i}(X_{1}, …, X_{k}) = ΩSY_{i} and G(X_{1}, …, X_{k}) = ΩS ⋁_{j=1}^{k}X_{j}. Let F be

∏_{i∈I}F_{i}. Let n= (n_{1}, …, n_{k}), where n_{1}, …, n_{k} are positive integers and let n_{1} + ⋯ +

n_{k} = n. The natural transformation h of (2) induces an equivalence

Dnh : DnF → DnG.

Let I′ denote the set of basic products which involve X_{j} at most n_{j} times for j =

1, …, k. Clearly, I′ is finite. Let I′′ = I \ I′. Then F = ∏_{i′∈I′} F_{i′} × ∏_{i′′∈I′′} F_{i′′}.

The operator Dn commutes, up to homotopy, with finite products of functors. Therefore,

DnF ≃ Dn ∏_{i′∈I′} F_{i′} × Dn ∏_{i′′∈I′′} F_{i′′}.

Proposition 3.1.

Dn ∏_{i′′∈I′′} F_{i′′} ≃ ∗.

Proof. It follows immediately from the definition of I′′ and lemma 1.4 that for any i′′ ∈ I′′, DnF_{i′′} ≃ ∗. The proposition does not follow automatically because in general Dn commutes only with finite inverse homotopy limits. However, Dn commutes with arbitrary filtered homotopy direct limits, and since the product is a weak product, it can be written as a homotopy direct limit of finite products.

On the other hand, I′ is finite, and therefore there is a weak equivalence

D_{(n_{1},... ,n_{k})}F ≃ ∏_{i∈I′} D_{(n_{1},... ,n_{k})}F_{i} ≃ D_{(n_{1},... ,n_{k})}G. (4)

Evaluating the right hand side of (4) at (X_{1}, …, X_{k}) and using lemma 1.3, we obtain

D_{(n_{1},... ,n_{k})}G(X_{1}, …, X_{k}) (5)

= D_{(n_{1},... ,n_{k})} (ΩS ⋁_{j=1}^{k} X_{j})
= Ω^{∞}Map_{∗} (S^{2}K_{n}, Σ^{∞} ((SX_{1})^{∧n_{1}} ∧ ⋯ ∧ (SX_{k})^{∧n_{k}}))_{h(Σ_{n_{1}}×···×Σ_{n_{k}})} .

Similarly, evaluating the middle part of (4) at (X_{1}, …, X_{k}) and using lemma 1.4, we obtain

∏_{i∈I′} D_{(n_{1},... ,n_{k})}F_{i}(X_{1}, …, X_{k}) (6)

= ∏_{i∈I′} D_{(n_{1},... ,n_{k})}ΩSY_{i}

= ∏_{d|n_{0}} (Ω^{∞}Map_{∗} (S^{2}K_{d}, Σ^{∞} (S(X_{1})^{∧ \frac{n_{1}}{d}} ∧ ⋯ ∧ (X_{k})^{∧ \frac{n_{k}}{d}} )^{∧d})_{hΣ_{d}})^{B( \frac{n_{1}}{d} ,... , \frac{n_{k}}{d})} ,

where n_{0} is the greatest common divisor of the numbers n_{1}, …, n_{k}. Comparing (5) and (6), and passing from infinite loop spaces to spectra, we obtain theorem 0.1.

Let us now take X_{j} to be a wedge of copies of an even-dimensional sphere S^{2m} for every j. Then, using theorem 0.1, taking homology groups, and applying facts 2.1 and 2.3, we obtain theorem 0.2.

From here on, we assume that the ground ring is ℤ/p. It remains to describe the M_{i}s of theorem 0.3. We have indicated in the introduction that the M_{i}s were computed in [1]. To be more precise, the homology groups of the spectrum

Map_{∗} (S^{2}K_{d}, Σ^{∞}(S^{\frac{2mn}{d} +1})^{∧d})_{hΣ_{d}}

with ℤ/p coefficients have been computed in [1]. In particular, if d is not a power of p, then

H_{∗} (Map_{∗} (S^{2}K_{d}, Σ^{∞}(S^{\frac{2mn}{d} +1})^{∧d})_{hΣ_{d}}; ℤ/p) ≅ 0.

Assume therefore d = p^{i} for some non-negative integer i. The homology groups of Map_{∗} (S^{2}K_{d}, Σ^{∞}(S^{\frac{2mn}{d} +1})^{∧d})_{hΣ_{d}} are independent of m, up to a dimension shift, and we may as well take m = 0. Using facts 2.1 and 2.3 once more, it is easy to see that

M_{i} ≅ H_{∗} (Map_{∗} (S^{2}K_{d}, Σ^{∞}S^{d})_{hΣ_{d}}; ℤ/p) .

Roughly speaking, up to a suitable suspension, M_{i} has a basis consisting of the “completely inadmissible” Dyer-Lashof words of length i (which is almost the same as the set of admissible Steenrod words of length i).

Theorem 3.2. If d = p^{i}, then the following constitutes a basis for Σ^{1+i}M_{i}: if p > 2

{β^{ϵ_{1}}Q^{s_{1}} ⋯ β^{ϵ_{i}}Q^{s_{i}}u | s^{i} ≥ 1, s^{j} > ps^{j+1} − ϵ_{j+1} ∀1 ≤ j < i} ,

if p = 2

{Q^{s_{1}} ⋯ Q^{s_{i}}u | s^{i} ≥ 1, s^{j} > 2s^{j+1} ∀1 ≤ j < i} .

Here u is of dimension 1, Q^{s_{j}}s are the Dyer-Lashof operations and βs are the homology Bocksteins [2]. Thus Q^{s} increases dimension by s if p = 2 and by 2s(p − 1) if p > 2 and β decreases dimension by one.

References

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2. F.R. Cohen, T.J. Lada and J.P. May, The homology of iterated loop spaces, Lecture Notes in Math. 533 (1976).

3. F.R. Cohen and L.R. Taylor, On the representation theory associated to the cohomology of configuration spaces, Contemp. Math. 146 (1993), 91-109.

4. F.R. Cohen and L.R. Taylor, Computations of Gelfand-Fuks cohomology, the cohomology of function spaces and the cohomology of configuration spaces, in “Geometric applications of homotopy theory I”, Lecture Notes in Math. 657 (1978), 106-143.

5. T.G. Goodwillie, Calculus I: the first derivative of pseudoisotopy theory, K-Theory 4 (1990), 1-27.

6. T.G. Goodwillie, Calculus II: analytic functors, K-Theory 5 (1992), 295-332.

7. T.G. Goodwillie, Calculus III: the Taylor series of a homotopy functor, preprint.

8. B. Johnson, The derivatives of homotopy theory, Trans. Amer. Math. Soc. 347, number 4 (1995), 1295-1321.

9. R.P. Stanley, Some aspects of groups acting on finite posets, J. Combin. Theory Ser. A 32 (1982), 132-161.

10. G.W. Whitehead, Elements of homotopy theory, Springer-Verlag, New York, (1978).

11. E. Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math. 177 (1937), 152-160.

(G. Arone) The University of Chicago, Department of Mathematics, Chicago, IL 60637

E-mail address: arone@math.uchicago.edu

(M. Kankaanrinta) Department of Mathematics, P.O. Box 4 (Yliopistonkatu 5), SF-00014 University of Helsinki, Finland