PARTITION COMPLEXES, DUALITY AND

INTEGRAL TREE REPRESENTATIONS

Alan Robinson

1. Introduction

This paper reveals the geometry underlying certain integral representations of the symmetric groups. It claims few original results, but gives a unified geometrical and homological treatment and frequently strengthens the known theorems. Above all it aims to show that topological methods give efficient proofs of many of them.

Geometric realization converts the lattice of non-trivial partitions of a set with n elements into the space of fully-grown (that is, non-degenerate) trees with leaves labelled by the set {0, 1, …, n}. This space has reduced homology in degree n − 3 only. Its integral cohomology is a representation of Σ_{n} which is isomorphic to the Lie representation Lie_{n}, up to a twist by the sign character. It follows from the Σ_{n+1}-symmetry of the tree space that Lie_{n} extends to an integral representation of Σ_{n+1}.

Many mathematicians have contributed to the identification of the cohomology of the partition lattice with the twisted Lie representation: excellent summaries of the history of the theory are given in [Fre] and [Wac]. We show that the isomorphism arises at the cocycle level from a duality structure: the partition lattice has a fundamental homology cycle with coefficients in a certain Σ_{n}-module 𝔖_{n}, and the isomorphism is induced by cap product. The module 𝔖_{n} is the multilinear part of a free Lie superalgebra on n generators of odd degree: it is isomorphic to the twisted Lie representation, though not uniquely so.

In the final section we give a geometrical derivation of a short exact sequence relating these representations, due to S. Whitehouse [Whi].

The tree spaces are closely related to configuration spaces, and the occurrence of the Lie representations in this topological situation was observed by F. R. Cohen [Co1],[Co2].

2. The space of fully-grown n-trees

From [R-W] we recall topological properties of the space of n-trees, and prove a homeomorphism with the nerve of the lattice of partitions of the set {1, 2, …, n}.

A tree is a contractible 1-dimensional polyhedron X, and we shall require X to be compact. There is a unique coarsest triangulation of X, in which no vertex lies on exactly two edges. A vertex meeting more than two edges is called a node; a vertex incident upon only one edge is a free vertex.

An edge which connects two nodes is an internal edge. An edge with a free vertex is called a leaf .

2000 Mathematics Subject Classification 05E25. Keywords: Partition complex, Lie superalgebra.

Following [Bor],[B-V], we introduce a moduli space of trees with a fixed number of leaves.

2.1 Definition. An n-tree is a tree with the following extra structure.

(1) Every internal edge e is assigned a length l_{e} where 0 < l_{e} ≤ 1. All leaves are conventionally assigned a length of 1.

(2) The free vertices are labelled by a bijective correspondence with the set {0, 1, …, n}.

If n ≥ 2, then the labelling of free vertices is equivalent to a labelling of the leaves by a bijection with the same set {0, 1, …, n}. The leaf labelled 0 is also called the root. Two n-trees are regarded as equivalent if there is an isometry between them which preserves the labelling. A tree is fully grown if at least one internal edge has length 1.

2.2 Example: the vertices v_{A} of T_{n}.

Let A be any subset of the set [n] = {0, 1, …, n} of labels, such that A and its complement [n] \ A each have more than one element.

We define v_{A} to be the fully grown n-tree which has just one internal edge e_{A} (which necessarily has length 1), and which has the leaves labelled by elements of A attached at one end of this, and the leaves labelled by the elements of [n] \ A at the other:

[figure: e_{A}; A; [n] \ A]

Since trees are regarded as the same when they are related by a label-preserving homeomorphism, it follows that v_{A} = v_{[n]\A}.

In the next proposition Σ_{n+1} denotes the set of all permutations of {0, 1, …, n}, and Σ_{n} the subgroup leaving 0 fixed.

2.3 Proposition.

(1) The space T_{n} of all fully-grown n-trees is a simplicial complex of dimension n − 3, in which the vertices are the trees v_{A} of 2.2.

(2) The space T̃_{n} of all n-trees is the cone on T_{n}.

(3) The symmetric group Σ_{n+1} acts on the right upon the pair (T̃_{n}, T_{n}) by acting on the left upon the set of leaf labels. This action is simplicial.

Proof. The proof is given in [R-W]. Suffice it to say here that T̃_{n} has an evident cubical structure, with the internal edge-lengths as cubical coordinates. On the subspace T_{n} the greatest edge-length is 1, so there is a simplicial structure with barycentric coordinates proportional to the edge-lengths: the vertices are the v_{A} of 2.2, and each k-simplex is an amalgam of k + 1 of the original cubes as illustrated for k = 2 on the left in the following diagram.

[figure: 0; 1; 2; 4; 3; 5]

The space T̃_{n} is evidently a cone with T_{n} as its base. Its apex is a star-tree as illustrated on the right above, and the cone parameter maps a tree to its maximal internal edge-length. Therefore T̃_{n} can be triangulated as the cone on the simplicial complex T_{n}. As with T_{n}, the original cubical structure is a subdivision of this triangulation. □

From now on, we use the triangulations just constructed, unless the contrary is mentioned.

2.4 Lemma. Every (n−4)-simplex of T_{n} is a face of exactly three top-dimensional simplices.

Proof. The top-dimensional simplices correspond to binary trees in which each node is the meet of three edges. An (n − 4)-simplex of T_{n} arises from a tree with one exceptional node of order four, and adjoins three top-dimensional simplices corresponding to the three ways of resolving this node into two nodes of order three [R-W]. □

Lemma 2.4 shows that T_{n} has three-to-one incidence on codimension-one simplexes. As is well known, a triangulated closed manifold has two-to-one incidence, and consequently has a fundamental top-dimensional homology class (with integer coefficients, possibly twisted). We shall see in §4 that T_{n} too has a fundamental homology class with coefficients in a suitable coefficient module 𝔖_{n}.

2.5 Theorem. The space T_{n} of fully-grown n-trees has the homotopy type of a wedge of (n − 1)! spheres of dimension n − 3. The Σ_{n}-module H̃_{n−3}(T_{n}) restricts to the regular integral representation of the subgroup Σ_{n−1}.

Proof. It is shown in [R-W] that T_{n} is obtained from a contractible space by attaching (n − 1)! simplices of dimension n − 3 along their boundaries. These simplices correspond to the trees

[figure: 0; σ(1); σ(2); σ(3); σ(n − 1); n]

where σ is any element of the permutation group Σ_{n−1}. Thus T_{n} has the homotopy type of a wedge of spheres, as claimed; and the homology classes of these spheres are regularly permuted by the subgroup Σ_{n−1}. □

2.6 Homeomorphism with the nerve of the partition lattice Λ_{n}.

We call a partition of the set {1, 2, …, n} non-trivial if it is neither indiscrete nor discrete: that is, the number of equivalence classes is greater than 1 but less than n. The non-trivial partitions, ordered by refinement, form a lattice. We use the notation Λ_{n} both for this lattice and for the simplicial complex which is its nerve.

2.7 Proposition. The nerve Λ_{n} of the partition lattice is Σ_{n}-equivariantly homeomorphic to the space T_{n} of fully-grown n-trees.

Proof. We describe an explicit homeomorphism φ : T_{n} → Λ_{n}. A point α ∈ T_{n} is an n-tree in which each internal edge has a given length between 0 and 1.

For each 1 ≤ i ≤ n there is a unique arc γ_{i} in the tree α which starts at the root labelled 0 and ends at the leaf labelled i. We parametrize all these arcs with unit speed, and we extend γ_{i} to a map [0, ∞) → α by setting γ_{i}(t) to be constant at the vertex labelled i for all t ≫ 0.

As time increases, these arcs diverge and determine finer and finer partitions of {1, 2, …, n}. For every t > 0 we define Π_{α}(t) to be the partition of {1, 2, …, n} given by the equivalence relation

i ∼ j ⟺ γ_{i}(t) = γ_{j}(t) .

The partition Π_{α}(t) refines Π_{α}(s) if t > s. Further, Π_{α}(t) is discrete for all sufficiently large t; and it is indiscrete when t is sufficiently close to zero. Hence there is a unique affine map ρ : [0, 1] → [0, ∞) such that Π_{α}(ρ(t)) is non-trivial exactly for 0 < t < 1.

We can now define the point φ(α) in the nerve Λ_{n} of the partition lattice, as follows. The barycentric coordinate of φ(α) with respect to any non-trivial partition Π is to be the length of the interval {t ∈ [0, 1] | Π_{α}(ρ(t)) = Π}. The map φ : T_{n} → Λ_{n} so defined is a homeomorphism because the fully-grown tree α is uniquely determined by its one-parameter family of partitions Π_{α}. The map is clearly equivariant. □

2.8 Corollary. The right action of Σ_{n} on the nerve of the partition lattice Λ_{n} extends to a right action of Σ_{n+1}.

Proof. This is true of the action on T_{n}, since the action of Σ_{n} there is the restriction of the action of the group Σ_{n+1} of permutations of the full set {0, 1, …, n} of labels. □

3. The representation Lie_{n} and the fundamental cycle

3.1 The Lie representations. Let L_{n} be the free Lie ring on the set of generators {x_{i}}_{1≤i≤n}. We denote by Lie_{n} the n-linear part of L_{n}. This can be described in many different ways. First, it is the direct summand of L_{n} spanned by all Lie monomials containing each of the n generators exactly once. Second, it is isomorphic to the module of all natural transformations Φ^{⊗n} → Φ, where Φ is the forgetful functor from Lie rings to abelian groups. Third, it is the nth module in the Lie operad.

It is a standard exercise in the use of the Jacobi identity to show that Lie_{n} is a free abelian group of rank (n − 1)!, and that the left-regulated Lie brackets

(3.1.1)  λ_{σ} = [x_{σ(1)}, [x_{σ(2)}, [...[x_{σ(n−1)}, x_{n}]..]]] for σ ∈ Σ_{n−1}

form a ℤ-basis for Lie_{n} (see for example [Reu] or [Whi]). The symmetric group Σ_{n} acts on Lie_{n} on the left by permuting the n generators. The (integral) representation thus afforded is known as the Lie representation. The subgroup Σ_{n−1} permutes the above standard basis simply and transitively. Thus the restricted representation Res_{Σ_{n−1}}^{Σ_{n}}Lie_{n} is the regular integral representation of Σ_{n−1}.

We denote by Lie_{n}^{∗} the dual representation Hom(Lie_{n}, ℤ). This has the same character as Lie_{n}, and therefore Lie_{n} and Lie_{n}^{∗} are isomorphic over the rationals. Over the integers, the two are distinct.

3.2 Example. Take n = 3, and consider the essentially unique Σ_{3}-invariant Euclidean metric on the two-dimensional space Lie_{3} ⊗ ℝ. Both Lie_{3} and Lie_{3}^{∗} are hexagonal lattices in this (self-dual) Euclidean space. However, in Lie_{3} the shortest vectors have the form [x_{i}, [x_{j}, x_{k}]], and so are (−1)-eigenvectors of involutions (j, k) ∈ Σ_{3}; whereas in Lie_{3}^{∗} the shortest vectors are (+1)-eigenvectors of involutions.

Let ε denote the sign character of Σ_{n}. Then we have the twisted module ε Lie_{n} which is the same abelian group Lie_{n} with the twisted action of Σ_{n} defined by

σ · f(x_{1}, …, x_{n}) = ε(σ) f(x_{σ(1)}, …, x_{σ(n)})

for every multilinear Lie monomial f and every σ ∈ Σ_{n}. We also denote the module ε Lie_{n} by ε⊗Lie_{n} when we need to stress that it is the tensor product with the sign module. We similarly have the twisted representation ε Lie_{n}^{∗}, defined analogously.

3.3 Connection with Lie superrings. There is a reinterpretation of the module ε Lie_{n} which significantly simplifies our theory. In place of the free Lie ring on the set of generators {x_{i}}_{1≤i≤n} we may consider the free Lie superring (or Lie superalgebra over ℤ) having the x_{i} as odd-degree generators. This we denote by S_{n}. It is a graded algebra S_{n}^{even} ⊕ S_{n}^{odd} freely generated by the x_{i} subject to the relations

[a, b] = (−1)^{|a||b|+1}[b, a]
[[a, b], c] = [a, [b, c]] − (−1)^{|a||b|}[b, [a, c]]

when a and b are homogeneous elements of degrees |a| and |b| respectively.

Let 𝔖_{n} denote the super-Lie representation of Σ_{n}. This we define to be the n-linear part of S_{n}: it is spanned by the super-Lie monomials containing each x_{i} exactly once, and is a subgroup of S_{n}^{even} or S_{n}^{odd} according to the parity of n. The symmetric group acts by permuting the generators, exactly as in the Lie representation. We now determine the structure of 𝔖_{n}.

3.4 Proposition. The super-Lie representation 𝔖_{n} and the twisted Lie representation ε Lie_{n} are isomorphic Σ_{n}-modules.

Proof. The module Lie_{n} is spanned by all Lie monomials g(x_{1}, x_{2}, …, x_{n}) in which each of the n generators occurs once. The n variables occur in g in a definite order: let this order, reading from left to right, be x_{γ(1)}, x_{γ(2)}, …, x_{γ(n)}. This gives a permutation γ ∈ Σ_{n} depending upon g. Every such g is the bracket [h, k] of two monomials in complementary subsets of the variables.

Let J be any Lie superring, and y_{1}, y_{2}, …, y_{n} any elements of odd degree in J. We claim that there is a unique homomorphism Θ : Lie_{n} → J of abelian groups such that for each monomial g(x_{1}, x_{2}, …, x_{n}) as above

Θ(g(x_{1}, x_{2}, …, x_{n})) = ε(γ)G(y_{1}, y_{2}, …, y_{n})

where G(y_{1}, y_{2}, …, y_{n}) is the element of J obtained by replacing all the Lie bracket operators in g by Lie superbrackets, then substituting y_{i} for x_{i}; and ε(γ) is the sign of γ(g). The uniqueness of Θ is clear.

We construct Θ by induction on n. Suppose the result is known already for fewer than n generators. Then if g has n variables and g = [h, k] as above, we have Θ(g) = η·[Θ(h), Θ(k)] where [Θ(h), Θ(k)] is the superbracket and η is the sign of the inverse shuffle which places the generators occurring in h before those occurring in k. In order to show that Θ is a well-defined homomorphism in the n-variables case, we must verify that it respects the antisymmetry and Jacobi relations [h, k] = − [k, h] and [h, [k, l]] + [k, [l, h]] + [l, [h, k]] = 0 of Lie_{n}. But this is true because the shuffle signs convert these relators into the corresponding signed relators, which are zero in the Lie superring J. The induction is now complete.

It is not at all difficult to give an instance where Θ is a monomorphism: one begins by taking the y_{i} to be elementary matrices representing elements of the endomorphism superring of a superspace of large rank. Therefore Θ is also a monomorphism when J is the free superring generated by the odd-degree elements y_{1}, y_{2}, …, y_{n}. So in this case Θ is an isomorphism of abelian groups onto its image, which is 𝔖_{n}. For any σ ∈ Σ_{n} we have

Θ(σ · g) = ε(σγ)(σ · G) = ε(σ)(σ · Θ(g)) .

Hence Θ induces an isomorphism between Lie_{n} and ε ⊗ 𝔖_{n}, and between ε Lie_{n} and 𝔖_{n}. □

The construction of the isomorphism between ε Lie_{n} and 𝔖_{n} makes use of the natural ordering of the set x_{1}, x_{2}, …, x_{n} of generators. The effect of adopting a different fundamental ordering, related to the natural one by a permutation τ, is to multiply the isomorphism by the sign of τ. When the set of generators is unordered, the isomorphism is uniquely defined only up to sign.

3.5 Corollary. The left-regulated Lie superbrackets

λ_{σ} = [x_{σ(1)}, [x_{σ(2)}, [...[x_{σ(n−1)}, x_{n}]..]]] for σ ∈ Σ_{n−1}

form a basis for 𝔖_{n}.

Proof. This follows from 3.1.1 and 3.4, or by direct computation as for Lie_{n}. □

The first advantage of 𝔖_{n} over ε Lie_{n} in our homology theory is that the Lie superbracket is extremely useful in building the fundamental orientation cycle. The second advantage is that the isomorphism of 𝔖_{n} with the cohomology of T_{n}, which we shall construct in Theorem 4.1 below, is totally natural: it does not depend upon an ordering of the set of non-zero labels.

3.6 Construction of the fundamental cycle. We now describe the construction of a certain cycle F_{n} in the top-dimensional simplicial chain group C_{n−3}(T_{n}; 𝔖_{n}), which after 3.4 is isomorphic to C_{n−3}(T_{n}; ε Lie_{n}). A top-dimensional simplex X of T_{n} is an n-tree which is binary; that is, every internal node is the meet of three edges (see §2). The vertices of the simplex correspond to the internal edges of the tree X; so an orientation of the simplex may be specified by a word w_{X} = e_{0}e_{1} … e_{n−3} in which each internal edge of X appears once. We write this oriented simplex ⟨w_{X}⟩.

If the order of the edges in w_{X} is altered by a permutation σ, then ⟨w_{X}⟩ is multiplied by ε(σ). We shall choose such an ordering for each (n − 3)-simplex X of T_{n}, and we shall set

F_{n} = ∑_{X}⟨w_{X}⟩ ⊗ c_{X} ∈ C_{n−3}(T_{n}; 𝔖_{n})

where c_{X} ∈ 𝔖_{n} is a coefficient monomial which we now describe.

We use induction on n to define w_{X} and c_{X}. When n ≤ 2 the complex T_{n} is empty, as the unique n-trees X have no internal edges; but nevertheless we formally define c_{X} = x_{1} when n = 1 and c_{X} = −[x_{1}, x_{2}] when n = 2. Here, and throughout the rest of §3, the bracket is the operation in a free Lie superring. The variables x_{i} are in bijective correspondence with the non-zero tree labels. Since the x_{i} are assigned odd grading, we have [x_{1}, x_{2}] = [x_{2}, x_{1}]. The ordering on the set of labels has no significance: the non-zero labels could be replaced by any finite set.

We continue with our inductive definition. Suppose now that n > 2. The n-tree X is obtained by grafting together at the root an i-tree Y and an (n − i)-tree Z

[figure: ρ_{Y}; ρ_{Z}; ρ_{X}; Z; Y]

where 0 < i < n, since the root of X meets two other edges. The internal edges of X comprise the internal edges of Y and those of Z, together with the roots ρ_{Y} and ρ_{Z} of Y and Z. Suppose by induction that we have well-defined terms ⟨w_{Y} ⟩ ⊗ c_{Y} and ⟨w_{Z}⟩ ⊗ c_{Z} corresponding to orderings of the internal edges of Y and Z, the coefficients c_{Y} and c_{Z} being super-Lie monomials in the sets of non-zero labels of Y and Z respectively. We set w_{X} = ρ_{Y} w_{Y} ρ_{Z}w_{Z}, this being an ordering of all the internal edges of X. The set of non-zero labels of X is the disjoint union of those of Y and Z. We put c_{X} = (−1)^{|Y |}[c_{Y} , c_{Z}] where |Y | is the number i of leaves, excluding the root, in Y. This is a super-Lie monomial in the non-zero labels of X.

To justify this construction, we verify in the next two lemmas that ⟨w_{X}⟩ ⊗ c_{X} is a well-defined element of the chain group C_{n−3}(T_{n}; 𝔖_{n}), and that the sum F_{n} of these elements over all top-dimensional simplices X is a cycle.

3.7 Lemma. The term

⟨w_{X}⟩ ⊗ c_{X} ∈ C_{n−3}(T_{n}; 𝔖_{n})

is well defined for every binary n-tree X, and is Σ_{n}-invariant.

Proof. We again use induction on n. For n = 1 the result is completely trivial. When n ≥ 2 the tree X is obtained by grafting to the root an i-tree Y and an (n − i)-tree Z, where 0 < i < n and therefore both ⟨w_{Y} ⟩ ⊗ c_{Y} and ⟨w_{Z}⟩ ⊗ c_{Z} are well-defined by inductive hypothesis.

Since the oriented simplices ⟨w_{Y} ⟩ and ⟨w_{Z}⟩ are evidently well defined up to sign, the coefficients c_{Y} and c_{Z} are well defined up to sign. Therefore the total sign in the formula

⟨w_{X}⟩ ⊗ c_{X} = ⟨ρ_{Y} w_{Y} ρ_{Z}w_{Z}⟩ ⊗ (−1)^{|Y |}[c_{Y} , c_{Z}]

is independent of the orientations of the two simplices Y and Z, by bilinearity of tensor product. The only remaining ambiguity is that the sub-trees Y and Z can be listed in either order. Interchanging them may affect the orientation of the simplex X, and may alter the sign of c_{X} = (−1)^{|Y |}[c_{Y} , c_{Z}].

We compute the change in the total sign when Y and Z are interchanged. Since the number of internal edges in a binary tree equals the number of leaves (excluding the root) less two, the effect of interchanging ρ_{Y} w_{Y} and ρ_{Z}w_{Z} is to multiply ⟨w_{X}⟩ by the sign (−1)^{(i−1)(n−i−1)}. On the other hand, the replacement of (−1)^{|Y |} by (−1)^{|Z|} introduces a sign (−1)^{n}, whilst the replacement of the superbracket [c_{Y} , c_{Z}] by [c_{Z}, c_{Y}] gives a factor (−1)^{i(n−i)+1}, because c_{Y} and c_{Z} have degrees i and n − i respectively. The product of these four signs is +1, so that ⟨w_{X}⟩⊗c_{X} is independent of the order in which Y and Z are taken. The inductive step is thus complete. Finally, the Σ_{n}-invariance of ⟨w_{X}⟩ ⊗ c_{X} is obvious, since the ordering of the set of labels has nowhere been used. The lemma is proved. □

3.8 Proposition. The (n − 3)-chain

F_{n} = ∑_{X} ⟨w_{X}⟩ ⊗ c_{X} ∈ C_{n−3}(T_{n}; 𝔖_{n})

is a Σ_{n}-invariant cycle.

Proof. The Σ_{n}-invariance of F_{n} follows from Lemma 3.7.

To prove that F_{n} is a cycle, we calculate all the coefficients of the (n − 4)-chain ∂F_{n}. Every oriented (n−4)-simplex σ corresponds to a tree S which is binary apart from one exceptional node, where four edges meet. There are two cases, depending upon the position of the exceptional node.

Suppose first that the exceptional node is the root node. Then S is the result of grafting together three trees X, Y and Z, as shown on the left here. As orientation for σ we may take the edge-sequence ⟨ρ_{X}w_{X}ρ_{Y} w_{Y} ρ_{Z}w_{Z}⟩, since this lists all the internal edges (in the notation of 3.6). One of the three (n − 3)-simplices incident upon σ is obtained by grafting X to Y (which introduces a new root ρ_{XY}) and grafting the result to Z, as in the picture on the right.

[figure: ρ_{X}; ρ_{Z}; ρ_{S}; ρ_{Y}; Z; Y; X; ρ_{X}; ρ_{Y}; Y; X; ρ_{XY}; ρ_{Z}; Z]

This simplex can be oriented as ⟨ρ_{XY} ρ_{X}w_{X}ρ_{Y} w_{Y} ρ_{Z}w_{Z}⟩: then its incidence index upon σ is +1, and its contribution to the coefficient of ∂F_{n} on σ is, by two applications of the construction in 3.6

(−1)^{|Y |}[[c_{X}, c_{Y}], c_{Z}] .

The other two (n − 3)-simplices incident on σ are obtained from the first by permuting X, Y and Z cyclically. The simplex ⟨ρ_{Y Z}ρ_{Y} w_{Y} ρ_{Z}w_{Z}ρ_{X}w_{X}⟩ has incidence index (−1)^{(|X|+1)(|Y |+|Z|)} upon σ (oriented as before), and the applications of the construction of 3.6 contribute altogether a sign (−1)^{|Z|}. The second simplex therefore contributes

(−1)^{|Y |+|X||Y |+|X||Z|}[[c_{Y} , c_{Z}], c_{X}] .

By cyclic permutation, the third simplex incident upon σ contributes

(−1)^{|Y |+|Z||X|+|Z||Y |}[[c_{Z}, c_{X}], c_{Y}] .

The total coefficient of ∂F_{n} on σ is thus zero in view of the Jacobi identity

(−1)^{|Y ||Z|}[[c_{Z}, c_{X}], c_{Y}] + (−1)^{|Z||X|}[[c_{X}, c_{Y}], c_{Z}] + (−1)^{|X||Y |}[[c_{Y} , c_{Z}], c_{X}] = 0

in the Lie superring. The first case is proved.

In the second case, the exceptional node (where four edges meet) of S is not the root node. Thus S is obtained by grafting together at the root a binary tree T and a tree U having one exceptional node. Let m be the number |U| of leaves of U. It follows by application of the definition in 3.6 that the coefficient of ∂F_{n} on σ is [c_{T} , e] where e is the coefficient of ∂F_{m} on an appropriate orientation of the simplex represented by U. By induction on n, we know that F_{m} is a cycle, so e is zero. This establishes the second case, and completes the proof of 3.8. □

3.9 Example. We illustrate the foregoing theory by calculating the fundamental cycle F_{5} of T_{5}. To reinforce the point that the ordering of the tree labels is nowhere used, we label the trees by {0, a, b, c, d, e} in place of {0, 1, 2, 3, 4, 5}. We use a, b, c, d, e also to denote the corresponding odd-degree generators of the Lie superring.

We enumerate the top-dimensional simplices of T_{5}. There are 105 of these, generated by the trees pictured below and the action of the symmetric group Σ_{5} on the labels a, b, c, d, e. We choose orientations for the simplices by ordering the internal edges, and then calculate the 105 corresponding terms in the cycle F_{5}. Each term is obtained by using the formula of 3.6 twice.

There are 60 terms of the form Φ = ⟨x, y, z⟩ ⊗ [a, [b, [c, [d, e]]]] corresponding to trees of the shape

[figure: x; y; z; 0; a; b; c; e; d; ;]

another 30 terms of the form Ψ = −⟨p, q, r⟩⊗[[a, b], [c, [d, e]]] corresponding to trees of the shape

[figure: p; q; r; b; a; 0; c; e; d]

and 15 terms of the form Ω = −⟨i, j, k⟩ ⊗ [a, [[b, c], [d, e]]] corresponding to trees of the shape

[figure: j; k; i; b; c; 0; a; e; d; .]

The number of terms in each case is given by dividing 5! = 120 by the order of the symmetry group of the tree fixing the root. Our construction takes care of the signs. We have

F_{5} = ∑ ⟨x, y, z⟩ ⊗ [a, [b, [c, [d, e]]]] − ∑ ⟨p, q, r⟩ ⊗ [[a, b], [c, [d, e]]]
− ∑ ⟨i, j, k⟩ ⊗ [a, [[b, c], [d, e]]]

where there are 105 terms in all, corresponding to all the labelled trees up to labelpreserving isomorphism. (The letters attached to the internal edges are bound variables over which summation is carried out: other symbols can be substituted for them at will.) Now we check that the boundary ∂F_{5} is zero. We revert to the notation Φ, Ψ, Ω for the three types of terms in F_{5}. On each type there are three face operators to consider.

First we look at the terms ∂_{2}Φ. We have ∂_{2}Φ = ⟨x, y⟩ ⊗ [[a, b], [c, [d, e]]] where ⟨x, y⟩ is an orientation of the tree

[figure: x; y; 0; a; b; c; e; d]

in which the leaves c, d and e are attached at the same node. Therefore the 60 terms ∂_{2}Φ cancel among themselves in threes by virtue of the relation

[c, [d, e]] + [d, [e, c]] + [e, [c, d]] = 0

in the coefficient module 𝔖_{5}. In the same way, the 30 terms ∂_{2}Ψ have sum zero, and the 60 terms ∂_{0}Φ cancel the 30 terms ∂_{0}Ψ by the supercommutator formula

[[a, b], [c, [d, e]]] − [a, [b, [c, [d, e]]]] − [b, [a, [c, [d, e]]]] = 0 .

Finally each of the 15 terms ∂_{0}Ω cancels with a pair of terms ∂_{1}Ψ through

[a, [[b, c], [d, e]]] = [[b, c], [a, [d, e]]] − [[d, e], [a, [b, c]]]

and each of the terms in ∂_{1}Ω or ∂_{2}Ω likewise cancels a pair of terms in ∂_{1}Φ. This accounts for all 315 terms in ∂F_{5}, and verifies that F_{5} is a cycle.

4. The homology and cohomology as integral representations of Σ_{n}

We showed in §2 that the space T_{n} has reduced homology only in dimension n − 3. We now determine the action of Σ_{n} on this finite ℤ-module. The following theorem is equivalent to a result proved by M. Wachs [Wac] for the cohomology of the poset of partitions.

4.1 Theorem. There is an isomorphism of left Σ_{n}-modules

H̃^{∗}(T_{n}; ℤ) ≈ 𝔖_{n} ≈ ε Lie_{n}

and of right Σ_{n}-modules

H̃_{∗}(T_{n}; ℤ) ≈ 𝔖_{n}^{∗} ≈ ε Lie_{n}^{∗} .

Proof. As T_{n} is a finite complex with torsion-free homology, there is for every coefficient group G a natural universal coefficient isomorphism

(4.1.1)  H̃_{∗}(T_{n}; G) ≈ Hom_{ℤ}(H̃^{∗}(T_{n}), G) .

In particular, the integral homology and integral cohomology are dual ℤ-modules. In (4.1.1) we take G to be 𝔖_{n} or the isomorphic module ε Lie_{n}. The homology class of the invariant cycle F_{n} constructed in 3.6 must correspond under the isomorphism (4.1.1) to some homomorphism θ_{n} : H̃^{∗}(T_{n}) → 𝔖_{n}, which is indeed Σ_{n}-equivariant because (4.1.1) is natural and F_{n} is Σ_{n}-invariant.

Let f_{σ} be the cocycle of T_{n} which takes the value 1 on the simplex γ_{σ} depicted in 2.5, and is zero on all other (n−3)-simplices of T_{n}. Then {[f_{σ}] | σ ∈ Σ_{n−1}} is a basis for H̃^{∗}(T_{n}), by 2.5. The value of θ_{n} on the class [f_{σ}] is ±λ_{σ}, by 3.6 and induction on n, where (as in 3.5) λ_{σ} is the basis element [x_{σ(1)}, [x_{σ(2)}, [...[x_{σ(n−1)}, x_{n}]..]]] of the ℤ-module 𝔖_{n}. Therefore we have shown that the Σ_{n}-homomorphism θ_{n} carries a basis for H̃^{∗}(T_{n}) into a basis for 𝔖_{n}, which is isomorphic to ε Lie_{n} by 3.4. That proves the first assertion in the theorem. The second assertion follows by taking ℤ-duals. □

4.2 Corollary. The Σ_{n}-module Lie_{n} is the restriction of a left Σ_{n+1}-module.

Proof. We have seen that Σ_{n+1} acts on T_{n}, so it acts upon εH̃^{∗}(T_{n}), which is isomorphic to Lie_{n}. □

4.3 Definition. We denote by L̂ie_{n} the left Σ_{n+1}-module so defined, and by L̂ie_{n}^{∗} the dual right Σ_{n+1}-module. Evidently Res_{Σ_{n}}^{Σ_{n+1}} L̂ie_{n} ≈ Lie_{n}.

5. The Whitehouse extension

Let H be a subgroup of the finite group G. Since the functor Ind_{H}^{G} is left adjoint to Res_{H}^{G}, there is an adjunction morphism Ind_{H}^{G} Res_{H}^{G} M ⟶ M for every G-module M, and it is always surjective.

The following theorem was proved algebraically by Whitehouse [Whi]; it also follows from work of Sundaram [Sun].

5.1 Theorem. There is a short exact sequence of Σ_{n+1}-modules

0 ⟶ Lie_{n+1}^{∗} ⟶ Ind_{Σ_{n}}^{Σ_{n+1}} Lie_{n}^{∗} ⟶ L̂ie_{n}^{∗} ⟶ 0

in which the surjection is the adjunction morphism connecting Ind_{Σ_{n}}^{Σ_{n+1}} and Res_{Σ_{n}}^{Σ_{n+1}}.

We shall give a topological proof. We recall that there is an analogue for G-spaces of the induction functor Ind_{H}^{G}, namely smash product with (G/H)_{+}. Theorem 5.1 follows immediately from the cofibration in the following theorem, by taking reduced homology.

5.2 Theorem. There is a Σ_{n+1}-equivariant cofibration

X ⊂ T_{n+1} ⟶ (Σ_{n+1}/Σ_{n})_{+} ∧ ST_{n}

where X is Σ_{n+1}-homotopy equivalent to T_{n} and S denotes suspension.

Proof. In the tree space T_{n+1} we consider for each 1 ≤ i ≤ n + 1 the star st(v_{0i}) of the vertex v_{0i} of 2.2. This consists of all the (n+1)-trees in which the leaves labelled 0 and i are separated from the rest of the tree by an internal edge of positive length. These n + 1 open stars are therefore disjoint. The closure \overline{st} (v_{0i}) is isomorphic to the contractible space T̃_{n}, by the homeomorphism which deletes the leaf 0 and then relabels the leaf n + 1 as 0. Let X be the complement T_{n+1} \ ⋃_{i=1}^{n+1} st(v_{0i}). By 2.3 the subcomplex T̃_{n} is the cone on T_{n}, so

T_{n+1}/X ≈ ⋁_{i=1}^{n+1} \overline{st} (v_{0i})/∂( \overline{st} (v_{0i})) ≈ ⋁_{i=1}^{n+1} ST_{n} .

The isomorphism \overline{st} (v_{0,n+1})/∂( \overline{st} (v_{0,n+1})) ≈ ST_{n} is Σ_{n}-equivariant, and Σ_{n+1} induces a bijection between the summands and the left cosets Σ_{n+1}/Σ_{n}, so the wedge on the right above is equivariantly isomorphic to (Σ_{n+1}/Σ_{n})_{+} ∧ ST_{n}.

We must now prove that X is Σ_{n+1}-homotopy-equivalent to T_{n}, where Σ_{n+1} acts on T_{n} by permuting the labels {0, 1, …, n}, with 0 in place of n + 1. We assume n ≥ 3: the reasoning below requires trivial modifications for n = 1, 2.

Given any tree in T_{n}, we obtain a tree in X ⊂ T_{n+1} by the following procedure: relabel the leaf 0 as n + 1, then attach a new leaf labelled 0 at any point on any internal edge. Every point of X arises once only in this way, so the reverse procedure (deleting the leaf 0 and relabelling the leaf n+1 as 0) gives a projection φ : X → T_{n}. As usual, when the leaf-deletion results in the amalgamation of two edges with lengths x and y, the new combined edge is assigned length max(x, y).

The inverse image under φ of any point t ∈ T_{n} is isomorphic to the subtree consisting of the internal edges of t. This is contractible. By the Vietoris Theorem 5.3 below, φ is a homotopy equivalence.

The map φ is Σ_{n+1}-equivariant by construction. To show that it is a equivariant homotopy equivalence, it suffices to show that, for every subgroup H of Σ_{n+1}, the restriction φ^{H} : X^{H} → T_{n}^{H} between the fixed point sets is a homotopy equivalence. If t ∈ T_{n}^{H} is any H-fixed point, then the group H acts by isometries on the tree t, permuting the labels.

Whenever a finite group acts by isometries on a finite tree, the set of invariant points is contractible. In our case the invariant points form the inverse image of the point t under φ^{H}. Thus φ^{H} satisfies the hypotheses of Theorem 5.3, and the induced map of fixed point sets is always a homotopy equivalence, which proves Theorem 5.2 and therefore Theorem 5.1. □

The Vietoris theorem used in the above proof was proved in the following strong form by M. M. Cohen ([CoM], Theorem 11.1).

5.3 Theorem. A piecewise linear map between finite polyhedra is a simple homotopy equivalence if the inverse image of every point is contractible. □

Homotopy equivalence under these hypotheses is a theorem of Smale [Sma]. The simply connected case already follows from Vietoris [Vie] by an application of Whitehead’s Theorem.

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[Vie] L. Vietoris, Über den höheren Zusammenhang kompacter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927), 454–472.

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Mathematics Institute, University of Warwick, Coventry CV4 7AL, England

   E-mail address: car@maths.warwick.ac.uk