Wheel graph homology classes via Lie graph homology

We give a new proof of the non-triviality of wheel graph homology classes using higher operations on Lie graph homology and a derived version of Koszul duality for modular operads.


Introduction.
In the seminal paper [Wil15], Willwacher constructed a family of non-trivial graph cohomology classes σ 2j+1 by analyzing an isomorphism between the 0 th cohomology of a certain graph complex GC 2 with the Grothendieck-Teichmüller Lie algebra grt 1 . The complex GC 2 is (up to important details) the commutative variant of a construction called the Feynman transform [GK98].
In [CHKV16], the authors study a family of group extensions of the outer automorphism groups of free groups and use the Leray-Serre spectral sequence to compute the homology of a low genus portion of the Lie variant of this construction. They show in particular that in genus 1, the virtual cohomological dimension consists of a family of classes α 2j+1 .
The purpose of this paper is to show that the families of classes σ 2j+1 and α 2j+1 correspond to each other under Koszul duality.
1.1. The correspondence: non-technical version. Let us explain the nature of this correspondence and the perspective from which such a result is expected. We omit several important technical details in this informal summary.
The statement that commutative and Lie structures are Koszul dual can be encoded in the language of operads. One first defines an operadic generalization of the bar construction of an algebra. One then proves that the homology of the bar construction of the commutative operad has the homotopy type of the Lie operad, and vice-versa.
Operads have operations parameterized by graphs of genus 0 (trees), and may be generalized by considering structures with operations parameterized by all graphs. Such structures are called modular operads [GK98], and the analog of the bar construction is called the Feynman transform which we denote ft. The commutative and Lie operads may be considered modular operads by simply declaring operations corresponding to higher genus graphs to be zero.
When viewed as modular operads the analogous Koszul duality relationship between Com and Lie no longer holds. If it did, it would imply that ft(Com) had no homology in its higher genus summands which is patently false. Indeed, both commutative graph homology H * (ft(Com)) and Lie graph homology H * (ft(Lie)) have a rich and highly non-trivial structure in higher genera.
There is, however, a more subtle Koszul duality relationship between these modular operads. The modular operad H * (ft(Lie)) encoding Lie graph homology contains a copy of the commutative modular operad Com but is Koszul dual (in a suitably derived sense) to the Lie modular operad Lie. Therefore in genus g ≥ 1, the (derived) Feynman transform of Lie graph homology is acyclic on the one hand, and contains a subcomplex computing commutative graph homology on the other.
We thus conclude that every commutative graph homology class of non-zero genus may be represented by a graph labeled by Lie graph homology classes. This correspondence is realized via the boundary operator in a derived version of the Feynman transform. In particular, the differential in this acyclic complex depends not just on the modular operad structure of Lie graph homology, but also on its higher "Massey products" which we compute here-in. Under this correspondence, the homology class σ 2j+1 corresponds to a graph whose lone non-trivial vertex label is built from α 2j+1 .
1.2. The correspondence: technical version. The graph complex (GC 2 , d GC ) of [Wil15] which we consider here-in consists of graphs with no legs and no loops (aka tadpoles). The differential is a sum of all possible edge expansions, with signs encoded by a factor of the top exterior power of the set of edges of the graph. This complex splits over genus, and the complex GC g 2 is (after a shift in degree by Σ −2g ) a sub-complex of ft(Com)(g, 0). We will suppress the degree shift notation Σ −2g in this introduction.
The modular operad structure of Lie graph homology was studied in [CHKV16]. This modular operad is not formal and so, abstractly, higher operations exist which make Lie graph homology into a modular operad up-to-homotopy (called a weak modular operad) which is homotopy equivalent to the Feynman transform of an extension by zero [War19]. The derived version of the Feynman transform ft mentioned above has a differential with additional terms taking these higher operations into account. Making these higher operations explicit in all genera is both unlikely to be possible and unnecessary for the task at hand. As we shall show, the wheel graph homology classes can be detected by higher operations on Lie graph homology landing in genus 1. With this in mind, we proceed as follows.
First we truncate Lie graph homology above genus 1 and endow the resulting semi-classical modular operad with explicit higher operations. The resulting weak modular operad is denoted H Lie . The projection H Lie → Com induces a map of K-modular operads ft(Com) → ft(H Lie ). Declare a loop to be simple if its adjacent vertex is trivalent and of genus 0. The subspaces of the Feynman transform supported on graphs with simple loops form a dg submodule. Letft denote passage to the quotient; we also writeη for the projection of a vector η to this quotient. Lifting ft(Com) → ft(H Lie ) we have the following diagram of dg S-modules: (1.1) The left hand inclusion is split; we denote the projection π :ft(Com) → GC 2 . We write ω 2j+1 ∈ GC * 2 for the wheel graph; a 2j + 1-gon along with a central vertex connected to the other vertices, tensored with a fixed element in the top exterior power of the span of the set of edges of the graph (Figure 1). The main technical result of this paper is the following construction: Theorem 1.1. There exists an element η 2j+1 ∈ ft(H Lie )(2j + 1, 0) such that d(η 2j+1 ) ∈ft(Com) and ω 2j+1 , π • d(η 2j+1 ), = 0.
The condition ω 2j+1 , π • d(η 2j+1 ) = 0 ensures that the elements π • d(η 2j+1 ) are not boundaries in the submodule GC 2 ⊂ft(H Lie ). On the other hand, the elements d(η 2j+1 ) are boundaries in the larger complex and so certainly must project to cycles in GC 2 . Thus, each π • d(η 2j+1 ) represents a non-trivial graph homology classes in GC 2 . Equivalently we may state: is non-trivial. The proof of Theorem 1.1 is an explicit construction. The element η 2j+1 has underlying graph pictured in Figure 4, and carries a vertex label given by the VCD class α 2j+1 composed with 2j − 2 copies of the commutative product. In this way, we find η 2j+1 not by contracting the exterior polygon (a logical first guess) but by contracting interior polygons (Figure 1). Contracting such interior polygons produces graphs having n = 2j − 2 adjacent loops and the construction of η 2j+1 depends crucially on the representation theory of the Weyl group of SO(2n + 1) to show that the class we construct is the unique class (up to scalar multiple) which does not vanish upon passage to coinvariants by the group of isomorphisms of such a graph.
Corollary 1.2 may also be derived from the results of [Wil15], see [CGP21, Theorem 2.6]. What's new here is the technique, as our proof completely avoids discussion of Drinfeld associators and grt 1 . The interplay between Lie and commutative graph homology is subtle, and much interesting work remains. For example how to realize Morita and Eisenstein classes via Massey products on commutative graph homology, or how to relate open conjectures on either side of 2 i -1 2i 2i-2 2i+1 Figure 1. Left: The wheel graph ω 2j+1 for j = 2 with edge order indicated. Center: The "exterior" 2j + 1-gon is the pictured (red) subgraph. Right: The "interior" 2j + 1-gon is the pictured (blue) subgraph. this correspondence. For now we simply offer the results of this paper as a polite suggestion that commutative and Lie graph homology may be effectively studied in tandem.
This paper is organized as follows. A brief overview of the prerequisites is given in Section 2. In Section 3 we compute relations in the modular operad for Lie graph homology and then construct higher operations for Lie graph homology landing in genus 1. The construction of the element η 2j+1 and verification of its properties is given in Section 4. Throughout we mostly work with the "co-Feynman transform", and then linear dualize as the very last step (Subsection 4.4) to prove the results stated in this introduction.  1.3. Conventions. Throughout we work in the category of differential graded vector spaces over Q with homological grading conventions. We write A * for the graded linear dual of A. We denote symmetric groups by S n and denote irreducible representations of S n by V α for a partition α = (α 1 , . . . , α r ) of n.

Background.
2.1. Graphs. We briefly review the standard operadic definition of an abstract graph, see [War19] for full details. A graph γ = (V, F, a, ι) consists of a finite non-empty set of vertices V , a finite set of flags (also called half-edges) F , a function a : F → V which indicates to which vertex a flag is adjacent, and an involution ι : F → F . The orbits of ι of order two are called the edges of γ and denoted E(γ). The orbits of order one are called the legs γ and denoted leg(γ). A loop is an edge {f, ι(f )} for which a(f ) = a(ι(f )). The number |a −1 (v)| is called the valence of the vertex v.
A graph determines a 1-dimensional CW complex and we say the graph is connected if this CW complex is connected. A genus labeling of a graph is a function g : V → Z ≥0 . A connected, genus labeled graph is stable if 2g(v) + |a −1 (v)| ≥ 3 for every v ∈ V . A leg labeling of a graph is a bijection {1, . . . , n} → leg(γ), for the appropriate n. A modular graph is a stable graph along with a leg labeling. The total genus of a modular graph is g(γ) := β 1 (γ) + v g(v), where β 1 denotes the first Betti number of the associated CW complex. The type of a modular graph is the pair of non-negative integers (g(γ), |leg(γ)|).
An isomorphism of abstract graphs is a pair of bijections between the respective vertices and flags which commutes with the adjacency and involution maps. An isomorphism of modular graphs is an isomorphism of abstract graphs which preserves the leg and genus labeling. If γ is a modular graph we write Aut(γ) for the group of automorphisms of γ viewed as a modular graph. For non-negative integers g and n with 2g + n ≥ 3 we fix once and for all a skeleton of the groupoid of graphs of type (g, n) and call it Gr(g, n).
A subgraph of a graph is a pair of subsets of V and F closed under a and ι. A nest N on a graph γ is a proper, connected subgraph of γ containing no legs. Given a nest N on a graph γ we define two auxiliary graphs. The modular graph γ/N is formed by contracting the edges of N and their adjacent vertices to a single vertex labeled to preserve the total genus of the graph. We call this new vertex N . The graphN is the graph formed by adding as legs all flags of F (γ) \ F (N ) which are adjacent to vertices in N . Note the legs ofN are not numerically labeled, but are in bijective correspondence with the flags adjacent to the vertex N in γ/N .
Define K * (γ) to be the top exterior power of the set E(γ). Explicitly, K * (γ) is a 1-dimensional vector space concentrated in degree |E(γ)| with carries an alternating action of the group S E(γ) . Define K(γ) to be the linear dual of K * (γ). Observe K(γ) is naturally identified with K * (γ), but is concentrated in degree −|E(γ)|. As we are using homological grading conventions, these definitions are opposite to the conventions in [GK98]. A mod 2 order on the edges of a graph is defined to be a choice of unit vector in K * (γ) or equivalently K(γ).

Weak modular operads.
A stable S-module A is a family of differential graded S n representations A(g, n), indexed over pairs of non-negative integers (g, n) satisfying 2g + n ≥ 3. Given such an A we extend A(g, −) to a functor valued in all finite sets via left Kan extension. In particular, if X is a finite set A(g, X) is non-canonically isomorphic to A(g, |X|). Given a stable S-module A and a modular graph γ with vertex v we define A(v) := A(g(v), a −1 (v)) and define A(γ) := ⊗ v∈V (γ) A(v). Note A(γ) inherits an action of the group Aut(γ).
Definition 2.1. Let A be a stable S-module. A weak modular operad structure on A is a collection of degree −1 operations (2.1) for each γ ∈ Gr(g, n) and for all (g, n) which are S n -equivariant, Aut(γ)-coinvariant and which satisfy the differential condition µ γ/N • N µN = 0.
Here µ γ/N • N µN is the composition of operations given by plugging the output of µN into the vertex N of γ/N and the sum is over all nests N on γ. The factors of K are composed by pulling back the wedge product isomorphism: which in turn encodes signs in the differential. We refer to [War19, Proposition 3.21] for additional details. Classical modular operads are weak modular operads for which µ γ = 0 if |E(γ)| > 1. In this case, the differential condition collapses to the associativity of composition of these one edged operations.
2.3. Co-Feynman transform. Let A be a weak modular operad. The co-Feynman transform of A, denoted B(A) is defined to be its bar construction viewed as an algebra over the Koszul resolution of the groupoid colored operad encoding modular operads [War19]. Explicitly this means B(A) is the dg K-modular co-operad given by: The K-modular co-operad structure is co-free and specified by decomposition maps for each γ ∈ Gr(g, n). These decomposition maps are defined on the summand of the source corresponding to a graph γ by summing over all ways to add a single layer of nests to it such that γ / N i = γ.
The weak K-modular operad structure maps on A induce a degree −1 map of stable S-modules µ : B(A) → A. This map induces a differential ∂ : B(A)(g, n) → B(A)(g, n). To describe ∂ it is sufficient to indicate its composite with projection π γ to a summand indexed by a (g, n)-graph γ, which is defined by asserting that the following diagram commutes: Here µ is the extension of µ by the Leibniz rule, composed with − ⊗ Aut(γ) K * (γ). In particular, µ is supported on nested graphs in B(A)(γ) for which exactly one vertex of γ is not labeled by a corolla. The co-Feynman transform of a weak modular operad has a bigrading given by the number of edges r on the graph indexing a summand and s is the sum of the internal degrees of the vertex labels. With respect to this bigrading, the co-Feynman transform differential is supported on: In particular, on each bigraded component the differential splits as ⊕ e ∂ e corresponding to those terms which contract e edges, under convention that contracting 0 edges means apply the internal differential d A .
Definition 2.2. When each A(g, n) is finite dimensional in each graded component we define the (weak) Feynman transform ft to be the linear dual of the co-Feynman transform. In particular, the Feynman transform of a weak modular operad is a K-modular operad.
This definition generalizes the usual Feynman transform [GK98], under the definition that a (strict) modular operad is a weak modular operad for which µ γ = 0 whenever γ has more than one edge. Remark 2.3. Let (A, µ) be a weak modular operad with vanishing internal differentials d A = 0. Then A along with only its one-edged contractions forms a (strict) modular operad. In this case ∂ 1 is itself square zero; it is the linear dual of the differential in the usual Feynman transform of this (strict) modular operad.
2.4. ft(Com) and GC 2 . As above, Com is the operad encoding commutative algebras, viewed as a modular operad by extension to higher genus by 0. It follows that where the sum is taken over all γ ∈ Gr(g, n) for which g(v) = 0 for all v ∈ V (γ).
Since all vertices have genus 0, no differential terms expand loops, and so the sum over those graphs with no loops is a subcomplex NL(g, n) ⊂ ft(Com)(g, n). For each g ≥ 3 we define the chain complex GC g 2 = (Σ 2g NL(g, 0)). We then define GC 2 = g GC g 2 . Note that [Wil15] uses cohomological conventions, so to recover exactly his GC 2 , one must take the cochain complex associated to the chain complex which we have called GC 2 by negating the indices: In particular, a homogeneous element in GC 2 is specified by a scalar multiple of an isomorphism class of a connected graph with no loops, all of whose vertices have valence 3 or greater, along with a mod 2 order on the set of edges. The degree of such a vector is 2g −E = E −2V + 2 and the differential is given by a sum of edge expansions. The alternating action on the edges of a representative has the effect that any isomorphism class of a graph with parallel edges vanishes.
2.5. Wheel graphs. As above we define the wheel graph ω 2j+1 by connecting a new vertex to all the edges in a 2j + 1-gon. As a convention the mod 2 edge order is chosen to coincide with [Wil15, Proposition 9.1], see Figure 1.
With this convention the wheel graph ω 2j+1 may be viewed as a degree 4j + 2 element of B(Com)(2j + 1, 0), having each vertex labeled by the commutative product of suitable valence. Note that contracting any edge in a wheel graph produces parallel edges with commutative labels and hence: Lemma 2.4. The wheel graph ω 2j+1 is a cycle in the complex B(Com)(2j + 1, 0).

Higher operations on Lie graph homology.
3.1. Lie graph homology. Following [GK98], after [Kon93], we define Lie graph homology to be the modular operad H * (ft(Σs −1 Lie)), where s denotes the (cyclic) operadic suspension and Σ denotes an shift up in degree. In particular extension by zero makes Σs −1 Lie a K-modular operad, and its Feynman transform is a modular operad. Following [CHKV16] we will denote this modular operad by H * (Γ) and use the notation H * (Γ g,n ) = H * (Γ)(g, n). For our purposes however, we will only require the following partial characterization of this modular operad in genus ≤ 1.
(2) As an S n module, is injective.
We may iterate the modular operadic composition map above to form Here • e is the modular operadic composition which corresponds to gluing along a tree with one edge adjacent to vertices of type (0, t + 1) and (1, n). To be precise, this composition is only well defined after a choice of labeling of the t + n − 1 legs of the tree by the set {1, . . . , t + n − 1}.
Proof. The map • e of Equation 3.1 is S t ×S 2j equivariant, where the target carries the restricted action along the standard inclusion S t × S 2j → S t+2j . By abuse of notation we also write • e for the adjoint: To produce such a vector z we embed the problem in the group ring. That is, consider Form the Young diagram of shape t + 1, 1 2j−1 labeled numerically right to left, then down. So t + 1 is in the pivot position. Call this tableau ζ. Its associated Young symmetrizer c ζ is c ζ = (id + (1, t + 1) + (2, t + 1) + · · · + (t, t + 1))y.
Thus, this z is in the kernel of the • e in Equation 3.2 as desired. Moreover the kernel is spanned by the S t+2j orbit of this z.
3.2. The weak semi-classical modular operad H Lie . In this section we endow the genus ≤ 1 spaces of H * (Γ g,n ) with higher operations. The result will be a weak modular operad which we denote (H Lie , µ).
As stable S-modules we define The operations µ γ are defined as follows. We define µ γ = 0 unless one of the two mutually exclusive conditions is met: • γ has genus < 2 and only one edge, or • γ has genus 1, and the underlying leg free graph of γ is a 2j + 1-gon, for some j ≥ 1. v e e P 1 P 0 Figure 3. The Massey project associated to P = P 1 is defined by expanding the edge e, applying the Massey product µ P 0 and then the modular operadic composition • e . Leg labels are suppressed in the figure.
In the first case we define µ γ to be the operation induced by the modular operad structure on H * (Γ). In the second case we proceed as follows.
Let p 2j+1 be the standard trivalent 2j + 1-gon. By this we mean the modular graph formed by attaching a leg to each vertex of a 2j + 1-gon. The vertices have genus label 0. The leg labels are in the dihedral order, and we give this modular graph an edge ordering e 1 < · · · < e 2j+1 such that the i th edge connects the vertices adjacent to flags i and i + 1 (mod 2j + 1).
Observe that H Lie (p 2j+1 ) ∼ = k. We define Lemma 3.3. The above operations extend to a unique weak modular operad structure on H Lie .
Proof. We refer to Definition 2.1. The S n equivariance defines µp for any other edge ordered trivalent polygonp. One easily checks that this definition is not over-prescribed, since symmetries of a 2j + 1-gon induce permutations of the edges and the legs which have matching parity.
We then want to show that if P is a non-trivalent graph whose underlying leg free graph is a 2j + 1-gon, that µ P is determined by the above operations. For this we induct on the number of non-trivalent vertices. First suppose that this number is 1, at a vertex v of P 1 := P.
Let γ be the graph formed by blowing up v to separate the two flags which belong to edges of the polygonal subgraph from the rest of the flags at v; see Figure 3. In particular γ has 2j + 2 edges, 2j + 1 of which form a polygon P 0 with trivalent vertices. Let N be a nest on γ. By the above definition, the composition µ γ/N • N µN will be zero unless N = {e} orN = P 0 (pictured blue and red in Figure 3).
Thus, applying the differential condition of Definition 2.1 with internal differential d = 0, it must be the case that Here we write • e = µ e for the modular operadic composition map which contracts the edge e. But this map µ e : H Lie (γ) → H Lie (P) is simply a composition in the commutative operad and so is an isomorphism, H Lie (γ) ∼ = → H Lie (P). Thus Equation 3.4 uniquely determines µ P 1 . For the induction step we repeat the above argument, reducing the number of non-trivalent vertices by one at each step. The fact that the operation is independent of the choice of order of the non-trivalent vertex follows from the fact that H * (Γ) is a (strong) modular operad.
Thus there is at most one weak modular structure on H Lie extending the above operations. Conversely the maps defined above have the requisite degree and equivariance, so it remains to show that the differential condition is satisfied for every modular graph γ.
If γ has total genus ≥ 2 or has fewer than 2 edges, all terms in Equation 3.5 are 0 by definition so there is nothing to check. If γ has exactly two edges, then Equation 3.5 is merely the associativity axiom for the (strict) modular operad structure on H * (Γ g,n ).
So we now assume γ has more than two edges and has total genus ≤ 1. Let N be a nest on γ. Either N or γ/N has two or more edges, thus for µ(γ/N ) • N µ(N ) to be non-zero requires that one of γ/N and N is a polygon with an odd number of sides and the other must be a lone edge. In particular γ must have an even number of edges, have first betti number 1, and hence only genus 0 vertices. There are two cases for such a γ. Either its lone cycle has an odd number of edges, in which case there is an additional edge pointing outward or its lone cycle has an even number of edges, in which case the leg free graph underlying γ must be a 2n-gon.
The first case follows as above. In particular, suppose γ has 2n edges and its lone cycle is of length 2n − 1. Let e be the unique edge of γ which is not in the cycle. This edge is connected to vertices v and w, with v belonging to the polygon and w not. If v is a trivalent vertex in γ, then the differential condition was verified above. If v is not trivalent, the differential condition is verified by a double iteration of Equation 3.4.
So we now consider the second case. Suppose γ = P t is a polygon with, 2n ≥ 4 sides whose vertices have genus 0, and which has t ≥ 2n legs labeled {1, . . . , t}, with at least one leg at each vertex. The only non-zero terms in the differential condition are given by first choosing N to be a single edge. Thus the differential condition in this case is: 0 = e∈Pt µ Pt/e • µ e . This condition can be rephrased as saying the following composite vanishes: Let us first consider the case t = 2n. Since the maps in Diagram 3.6 are S 2n -equivariant it is sufficient to consider the case that the legs of P 2n are labeled in the dihedral order. The source of this diagram is closed under the Z 2n action restricted along the standard inclusion Z 2n ∼ = σ := (12 . . . 2n) ⊂ S 2n . Denoting this 1-dimensional Z 2n representation E := Res S 2n Z 2n (H Lie (P 2n ) ⊗ K * (P 2n )), we compute its character χ E (σ) = sgn(σ) = −1. This in turn determines the isomorphism type of the irreducible Z 2n -representation E; namely the 2n cycle σ acts by −1.
To show the composite in Diagram 3.6 is 0, it thus suffices to show that there are no copies of this irreducible representation appearing in Res S 2n Z 2n (H 2n−2 (Γ 1,2n )). The irreducible representations of Z 2n over the algebraic closure Q are all 1-dimensional and are given by letting σ act by multiplication of a root of x 2n − 1. Let ω = e iπ/n and write W i for the irreducible representation corresponding to multiplication by ω i . Let V 2n ⊕ V 2n−1,1 be the permutation representation of S 2n . One easily calculates its restriction Since V 2n−1,1 ⊗ V 1 2n ∼ = V 2,1 2n−2 , the number of copies of E ∼ = W n appearing in Res S 2n Z 2n (V 2,1 2n−2 ) is the number copies of W 0 appearing in Res S 2n Z 2n (V 2n−1,1 ) which is 1 − 1 = 0, as desired. Whence the case t = 2n. Now suppose t > 2n. Choose an ordering of the vertices of P compatible with the dihedral ordering and let T i be the set of flags adjacent to the i th vertex. In particular {1, . . . , t} = T i . LetT i be the set T i along with an added basepoint, called the root.
Consider the following diagram: The right hand side of this diagram is exactly Diagram 3.6. The left hand side of this diagram is Diagram 3.6 in the prior case t = 2n, then tensored with the 1-dimensional, trivial Z 2nrepresentation 2n i=1 H Lie (0,T i ). The horizontal arrows are contractions using the modular operad structure along the graph identifying the root ofT i with leg i of P 2n .
The commutativity of the top square can be seen just by looking at each summand -both routes give the same graph with commutative labels. The commutativity of the bottom square follows immediately from the definition of the Massey product associated to each P t /e.
Since the left hand side of the diagram vanishes, by the t = 2n case considered above, and since the top horizontal arrow is an isomorphism, the right hand side of the diagram also vanishes, as desired.
Viewing Com as a modular operad, as in Subsection 2.4, we see immediately that there is a level-wise surjective morphism of weak modular operads H Lie → Com. Taking the weak Feynman transform, we have a level-wise injective morphism of K-twisted modular operads ft(Com) → ft(H Lie ).

Proof of the main results.
Let us regard the wreath product S 2 S n as follows. Its underlying set is S n 2 × S n . To an element ((α 1 , . . . , α n ), τ ) in this set we associate a permutation in S 2n by first acting by τ on the ordered set {1, 2}, {3, 4}, . . . , {2n − 1, 2n} of size n and then acting by α i on the ordered set {2τ (i) − 1, 2τ (i)} for each i. This defines an injective map of sets S ×n 2 × S n → S 2n , and S 2 S n carries the unique group structure for which this map is a homomorphism. The wreath product S 2 S n has a 1-dimensional representation given by letting an element ((α 1 , . . . , α n ), τ ) act by multiplication by sgn(τ ). We call this representation L n (L stands for loops).
In what follows, we abuse notation by regarding the sequence of injections as a sequence of subgroups. We write Res G H for the restriction of a representation of a group G to a representation of a subgroup H. Lemma 4.1. Let 2 ≤ r < n be integers.
• The irreducible decomposition of Res S n+r (S 2 S r−1 )×S n−r+2 (V r,1 n ) contains a unique summand of the form L r−1 V β . It is of the form L r−1 V 1 n−r+2 .
• The irreducible decomposition of Res S n+r (S 2 Sq)×S n−2q+r (V r,1 n ) has no summand of the form L q − for q ≥ r.
Proof. Fix q ≥ r − 1. The number of copies of a summand V α V β appearing in is computed via the Littlewood-Richardson rule [FH91]. In this case, because V r,1 n is a hook, each α and β appearing with non-zero coefficient must also be hooks. If q > r − 1, the number of summands of the from V q+1,1 q−1 V β appearing in the decomposition of the representation in Equation 4.1 is zero. If q = r − 1, there is a unique summand of the from V q+1,1 q−1 V β appearing in the decomposition of the representation in Equation 4.1 is zero. It is of the form V r,1 r−2 V 1 n−r+2 .
It now remains to analyze the irreducible decomposition of Res S 2q S 2 Sq of a hook. The needed calculation, modulo Frobenius reciprocity, is explicitly presented in [KT87a, Proposition 2.3' (iv)] (see also [KT87b]) which says that the number of copies of L q appearing in the restriction of a hook Res S 2q S 2 Sq (V x,1 y ) is 0 unless x = q +1 and y = q −1, in which case it is 1. This completes the proof.
While it was convenient that the calculation we needed was available in the literature, there is an argument internal to this article which may also be used to prove this. One first uses the Pieri rule to see that a summand V α V β appearing in Equation 4.1 has an (S 2 ) q ⊂ S 2 S q ⊂ S 2q invariant subspace if and only if q = r − 1, α = (r, 2q − r) and β = (1 n+r−2q ). Any summand of type L q − would restrict to an (S 2 ) q invariant subspace, which establishes the second statement. On the other hand when q = r − 1 there is a unique (S 2 ) r−1 invariant subspace in V r,1 r−2 . Since it is unique, it must contain the image of the gluing operation which grafts H 0 (Γ 0,3 ) onto each input of H r−2 (Γ 1,r−1 ) ∼ = V 1 r−1 , landing in H r−2 (Γ 1,2r−2 ) ∼ = V r,1 r−2 . This image is nonzero by Lemma 3.1. Since we're gluing on to the alternating representation H r−2 (Γ 1,r−1 ), it must be the case that this (S 2 ) r−1 invariant subspace lifts to a S 2 S r−1 representation which is alternating with respect to the S r−1 factor, i.e. to a copy of L r−1 , which establishes the first statement. Proof. Let q be the number of loops adjacent to v. Then Aut(γ) contains a subgroup isomorphic to S 2 S q generated by transposing the pair of flags in a loop and permuting the set of loops. This subgroup acts on H i (Γ 1,a −1 (v) ) ⊗ K * (γ) and since H i (Γ 1,a −1 (v) ) ∼ = V m−i,1 i , the invariants of this action correspond to the copies of L q appearing in Res S 2q S 2 Sq (V m−i,1 i ). From the proof of Lemma 4.1 we see that the number of such copies is 0 when q ≥ m − i. Any such Aut(γ)invariant element [x] would require an S 2 S q invariant element of H i (Γ 1,a −1 (v) ) ⊗ K * (γ) labeling v. Since there are no such elements when q ≥ m − i, the Aut(γ)-coinvariants vanish.
Proof. Observe that Aut(θ 2j+1 ) ∼ = S 3 ×(S 2 S 2j−2 ); the S 3 permutes the non-loop edges, the factors of S 2 transpose the flags in a loop and the S 2j−2 permutes the loop edges. As an Aut(θ 2j+1 )module, the one dimensional vector space K * (θ 2j+1 ) has representation type isomorphic to Figure 4. The case j = 3. The graph θ 7 (pictured) underlying the element β 7 ∈ B(H Lie )(7, 0). The signs indicate the effect of transposing at the vertices. The bracket labeled by a minus sign indicates that the block permutation exchanging pairs of flags on two loops gives a minus sign. The label of the genus 1 vertex is (up to scalar multiple) the unique class for which the Aut(θ 7 ) acts by such signed multiples.
4.1. Definition of β. We now construct a canonical basis vector spanning (K * (θ 2j+1 )⊗ Aut(θ 2j+1 ) H Lie (θ 2j+1 )) 2j+1,2j . Define the set X := a −1 (v 1 ). This set is partitioned by the edges of θ 2j+1 into three blocks of size 1 and 2j − 2 blocks of size 2. Choose an auxiliary order on the set of edges of θ such that the three non-loop edges are in the first three positions, and choose an order on the flags within each block. This fixes a total order on the set X and hence an isomorphism: Define x 2j+1 ∈ H 2j (Γ 1,X ) to be the composition composed with this isomorphism. Here m 2 is the generator of H 0 (Γ 0,3 ) = Com(2), and x 2j+1 = 0 by Lemma 3.1. Observe that permuting the two flags on m 2 acts by +, where-as permuting blocks of the permutation on X by σ is the same as composing with σα 2j+1 = sgn(σ)α 2j+1 (by equivariance of the operadic compositions). Therefore the element x 2j+1 spans the unique copy of V 1,1,1 L 2j−2 in Res S 4j−1 S 3 ×(S 2 S 2j−2 ) H 2j (Γ 1,X ), where the S 4j−1 action is inherited from the isomorphism in Equation 4.2. The class x 2j+1 ∈ H 2j (Γ 1,X ) depends on the choice of isomorphism in Equation 4.2, but only up to sign. Define to be the element formed by labeling v 1 with x 2j+1 and v 0 by m 2 and using the mod 2 edge order induced by the choice above. Observe that β 2j+1 is independent of the choices made. If we had picked a different edge order the result would differ by two factors of the sign of the corresponding permutation; if we had picked a different loop orientation the result would differ by a transposition of the commutative product.

4.2.
Analysis of ∂ −1 (β). We define a loop in a stable graph γ to be simple if the vertex v to which the loop is adjacent satisfies |a −1 (v)| = 3 and g(v) = 0.
Definition 4.5. Define Gr(g, n) ⊂ Gr(g, n) to be the subset of graphs which do not contain simple loops. For a weak modular operad A we then define: n). Proof. Given a modular graph γ with no simple loops, terms in the weak co-Feynman transform differential are indexed by contractions of subgraphs of γ. If such a differential term has a simple loop it must have been created by contracting a subgraph of type (0, 3). However Gr(0, 3) consists only of the (0, 3)-corolla, so no such contraction is possible.
We remark that since the weak modular operad H Lie has internal differential 0, the summand of ∂ which contracts just 1 edge, call it ∂ 1 , is itself a differential (Remark 2.3). In this case Lemma 4.6 also shows ( B(H Lie )(g, n), ∂ 1 ) is a subcomplex of (B(H Lie )(g, n), ∂ 1 ).
Proof. As above let X := a −1 (v 1 ). The set X is partitioned by the edges of θ 2j+1 into 3 blocks of size one and 2j − 2 blocks of size 2. Label the blocks of size 1 by a 1 , a 2 , a 3 and label the elements of each block of size two by l i 1 , l i 2 label, where i indexes the 2j − 2 loops of θ 2j−2 . As above we write x 2j+1 ∈ H 2j (Γ 1,X ) for a basis vector spanning the unique invariant subspace of Res S X Aut(θ 2j+1 ) isomorphic to V 1,1,1 L 2j−2 . The S {a 1 ,a 2 ,a 3 } action on x 2j+1 is alternating while each S {l i 1 ,l i 2 } action on x 2j+1 is the identity. Write ∂ θ 1 for the composition of ∂ 1 with projection to the θ 2j+1 summand of B(H Lie )(2j +1, 0). Define Θ to be the set of graphs γ for which the following composite is non-zero: To prove the claim it is sufficient to show that the set Θ is empty. By way of contradiction, suppose γ ∈ Θ. Then γ has an edge e = {r, s} such that γ/e ∼ = θ.
Note that the vertex of γ/e corresponding to e must be sent to the vertex v 1 of θ 2j+1 since the vertex v 0 is of type (0,3) and hence indecomposible. Note also that the edge e of γ can not be a loop for degree reasons -such a γ would have only genus 0 vertices, and so internal degree 0 = 2j + 1.
Therefore the edge e of γ is adjacent to two vertices whose genera add to 1. Let Y ∪ {r} be the flags of γ adjacent to the genus 1 vertex and Z ∪ {s} be the flags of γ adjacent to the genus 0 vertex. The isomorphism γ/E ∼ = θ specifies a partition X = Y Z, along with a linear map This linear map is S Y × S Z equivariant. In particular, |X| = |Y | + |Z| = 4j − 1. We say a loop l i of θ is split (by e) if both {l i 1 , l i 2 } ∩ Y and {l i 1 , l i 2 } ∩ Z are nonempty. Such a differential term being nonvanishing implies the following: • |Y ∪ r| ≥ 2j + 1 and hence |Y | ≥ 2j. Thus 3 ≤ |Z ∪ s| ≤ 2j.
• |{a 1 , a 2 , a 3 } ∩ Z| < 2, since the representation type of H 0 (Γ 0,Z∪{s} ) is trivial. So we suppose without loss of generality that a 2 , a 3 ∈ Y . • At most one loop is split, since otherwise we would create parallel edges with an alternating action of {l i 1 , l h 1 } at one vertex an identity action of {l i 2 , l h 2 } at the other vertex which pass equivariantly to the identity action of both {l i 1 , l i 2 } and {l h 1 , l h 2 } on X. Let be the number of loops adjacent to Y . By Corollary 4.2 we know ≤ |Y | + 1 − 2j − 1 = |Y | − 2j, and hence 2j ≤ |Y | − .
Consider the possible cases for such a γ ∈ Θ.
Case a 1 ∈ Z and no loop is split: Then |Y | = 2 + 2 which implies 2j − 2 ≤ and hence 2j − 2 = , since 2j − 2 is the total number of loops. But then the arity of Z would be 2, contradiction.
Case a 1 ∈ Z and one loop is split: Then |Y | = 2 + 3 which implies 2j − 3 ≤ and hence 2j − 3 = , since 2j − 2 is the total number of loops and one was split so can't be adjacent to Y . This means that Y carries the maximum number of loops for its given valence, so is alternating off the loops (Lemma 4.1). Let l 1 , l 2 be the split loop with l 1 ∈ Y and l 2 ∈ Z. Thus the action of l 1 , a 2 is alternating on Y and hence on X. But the action of a 1 , a 2 is alternating whilst a 1 , l 2 is id on Z and hence X, which is a contradiction of the fact that the transpositions (a 1 a 2 ), (a 1 l 2 ) and (l 1 l 2 ) generate the symmetric group of a 1 , a 2 , l 1 , l 2 .
So we conclude a 1 ∈ Y and proceed to: Case: Suppose one loop is split. Then |Y | = 2 + 4 and so 2j − 4 ≤ < 2j − 2, but only one loop was split so by stability considerations, one must go on Z, hence = 2j − 4. Let l i be the split loop and l h be the loop on Z. Equivariance will imply that all permutations of {l i 1 , l i 2 , l h 1 , l h 2 } must act by the identity on X, (since we can switch l i 2 , l h 2 adjacent to Z). This contradicts the definition of x 2j+1 which says that (l i 1 l h 1 )(l i 2 l h 2 ) must act by −1. We thus conclude no loop is split, hence |Y | = 2 + 3 and so 2j − 3 ≤ ≤ 2j − 2. But if = 2j − 2, then the vertex adjacent to Z would be unstable. So the only remaining possibility is that = 2j − 3, which in turn implies that Z is a vertex of valence 3, genus 0 and adjacent to 1 loop. But such graphs are excluded from B(H Lie ) by definition. We thus conclude Θ is empty, hence β 2j+1 ∈ B(H Lie )(2j + 1, 0) is not a boundary.
Proof. The Lemma establishes the case s = 2j. Suppose 0 < s < 2j. Without loss of generality we may assume ξ is a homogeneous element supported on a summand index by a modular graph γ. A term in ∂(ξ) is non-zero upon projection to the θ 2j+1 summand only if it is possible to contract a subgraph γ such that γ/γ ∼ = θ where γ has 2j + 1 − s > 1 edges. As above, such an isomorphism must send the vertex corresponding to γ to v 1 , so the total genus of γ must be 1. By definition of H Lie , such an operation is non-trivial only if γ has first betti number 1. These two conditions are true simultaneously only if each vertex of γ has genus 0, which in turn implies that each vertex carries a label in some H * (Γ 0,m ), which is concentrated in internal degree 0. The only other vertex of γ has genus 0 as well, hence such an element must be supported on internal degree 0.
Proof. By Lemma 4.4, it suffices to show that the projection of ∂(ω 2j+1 ) to the θ 2j+1 summand of B(H Lie )(2j + 1, 0) 2j+1,2j is non-zero. After Diagram 2.2, it is sufficient to show that composition in the following diagram is not zero: By definition, ν θ (ω 2j+1 ) is determined by summing over ways to nest the graph ω 2j+1 such that collapsing nests gives the graph θ 2j+1 . Such a nesting specifies two induced graphs of ω 2j+1 ,N 0 andN 1 which collapse to the two vertices v 0 and v 1 of θ 2j+1 . Since the vertex v 0 is of type (0, 3), the induced stable graphN 0 must be a corolla and N 0 must consist of a lone vertex. Given such an N 0 , there is a unique such N 1 for which the composite with µ X is non-zero. It is given by the unique 2j +1-gon N 1 which is a subgraph of ω 2j+1 and which misses a distinguished vertex N 0 (right hand side of Figure 1).
When j = 1, a direct computation shows that the four choices for a lone vertex N 0 are sent by ν θ 3 to the same element, hence ν θ 3 = 0. Indeed, with the convention that the edges of the nested triangle appear last in the decomposition, one must apply the permutations (14),(25),(36) and (14)(25)(36) to the conventional edge ordering of ω 3 (Figure 1) to decompose, and these permutations are all odd. Then since α 3 = x 3 , µ X is simply the Massey product which contracts the triangle, which is not zero. Whence the case j = 1.
So we now assume j > 1. In this case there are 2j + 1 choices for such an N 0 , corresponding to the outer vertices of ω. Therefore, ν θ (ω) is a sum of 2j + 1 terms, corresponding to the choice of an outer vertex and the complimentary 2j + 1 gon. Since these terms are related by an automorphism of ω, it is enough to show that any one of them is non-zero when composed with µ X . So let us fix such an N 0 and N 1 . The term in the sum ν θ (ω) corresponding to this choice of nesting is given by choosing an isomorphism ω/N 1 ∼ = θ, which in turn specifies a labeling of the flags ofN 1 by the set X := a −1 (v 1 ). We import the notation X = a −1 (v 1 ) = {a 1 , a 2 , a 3 , i 1 , i 2 | 1 ≤ i ≤ 2j − 2} from the proof of Proposition 4.7. Since ω has a unique non-trivalent vertex, so doesN 1 . The conditions on this X-labeling of the flags ofN 1 coming from the isomorphism ω/N 1 ∼ = θ are that the non-trivalent vertex ofN 1 must be adjacent to flags labeled by exactly one of the a i , and one flag from each loop. The two vertices adjacent to the non trivalent vertex must have the other a labels ( Figure 5). Let P ⊂ B(H Lie )(1, X) be the span of such X-labeled, 2j + 1-gons. In particular dim(P ) = 3!(2j − 2)!2 2j−2 /2. We conclude that the bottom row of the diagram is supported on the restriction where Com(2) labels v 0 and P labels v 1 .
The bottom row in Diagram 4.4 is given by contracting the X-labeled graph at the vertex v 1 of θ 2j+1 , via the associated Massey product. It thus remains to show that contraction of such X-labeled polygons is non-zero upon passage to Aut(θ)-coinvariants. Since this map is Aut(θ) invariant, it is enough to know that the class x 2j+1 ∈ H Lie (1, X) is in the image of the contraction when restricted to P ⊂ B(H Lie )(1, X) → H Lie (1, X). The remainder of the proof is dedicated to this calculation.
Recall that the class x 2j+1 is defined by choosing a total order on the set X, which fixes an isomorphism H 2j (Γ 1,4j−1 ) ∼ = H 2j (Γ 1,X ) (see Equation 4.2). By abuse of notation we write x 2j+1 ∈ H 2j (Γ 1,4j−1 ) for the image of x 2j+1 under this isomorphism. Note that since x 2j+1  Figure 5. At left, the vector ρ in the case j = 3 is a graph with 11 legs whose underlying leg free graph is a 7-gon. The image of ρ under the heptagonal Massey product is determined by first contracting the heptagon in the center to the VCD class α 7 ∈ H 6 (Γ 1,Y ) and then contracting the lone edge in the graph on the right. depends on this isomorphism only up to sign, the span is independent of choice. Thus, using the bijection X ∼ = {1, . . . , 4j − 1}, we may view P as carrying numerical labels, and it suffices to show the contraction surjects onto the span of x 2j+1 . To be pedantic, the bijection X ∼ = {1, . . . , 4j − 1} sends a i → i for 1 ≤ i ≤ 3 and i r → 2(i + 1) + (r − 1). As such we call a pair of numbers 2(i + 1), 2(i + 1) + 1 with 1 ≤ i ≤ 2j − 2, a loop, called loop i or the i th loop. We say a representative of a loop is one of its entries.
The contraction map P → H 2j (Γ 1,2j−1 ) is given by the Massey product which contracts each such 2j + 1-gon. The image of this contraction is determined by expanding the distinguished (non-trivalent) vertex and its legs to a new edge, contracting the 2j+1-gon, and then contracting the expanded edge ( Figure 5). In other words, this map factors as Notice the map • e is surjective, since it is non-zero and the target is irreducible. We now analyze • −1 e (x 2j+1 ). Recall x 2j+1 spans a representation of Aut(θ) for which S {1,2,3} acts by the alternating representation, transpositions of representatives of a loop acts by the identity and where each (45)(2i, 2i + 1) with 1 < i − 1 ≤ 2j − 2 acts by −1.
The space Ind 2j+1 )) has a basis given by i 1 ∧ i 2 ∧ · · · ∧ i 2j with where 1 ≤ i 1 < i 2 < ... < i 2j ≤ 4j − 1. The S 4j−1 action is by permutation, with permutation of wedge products acting by the sign of the permutation. Write a preimage of x 2j+1 in this basis: To convey additional conditions that the coefficients in Equation 4.6 must satisfy we fix some new notation. For a subset I ⊂ {1, . . . , 2j − 2} we define S I to be the set of lists of representatives of those loops not appearing in I, such that no list has two representatives of the same loop. In particular, each list has 2j − 2 − |I| entries and there are 2 (2j−2−|I|) such lists. Let S I be the formal sum of wedge products appearing in S I . For brevity we write S := S ∅ , S := S ∅ , S i := S {i} and S i := S {i} . We will denote lists of loop representatives by s ∈ S I . By abuse of notation we also write s for the associated wedge product. We write s ⊥ for the list of complimentary representatives. For example if j = 2 then S = {(4, 6), (5, 6), (4, 7), (5, 7)}, S 2 = {(4), (5)}, S = 4 ∧ 6 + 5 ∧ 6 + 4 ∧ 7 + 5 ∧ 7 and S 2 = 4 ∧ 5. If s = (4, 6) then s ⊥ = (5, 7), and by abuse of notation me may also write s = 4 ∧ 6.
With this notation we describe conditions on the coefficients in Equation 4.6. First, c(1, 2, 3, a) = (−1) i−1 c(1, 2, 3, b) if a ∈ S 1 and b ∈ S i . This condition is forced by the equivariance of • e . Transposing representatives of a loop acts by the identity, so the coefficient is independent of choice of list in S i (resp. S 1 ). On the other hand, a list in S 1 is missing one loop, namely {4, 5}, and the permutation (45)(2(i + 1), 2(i + 1) + 1) acts by −1 for any i > 1. The result may be compared with a list in S i by applying an i − 1 cycle, acting by (−1) i , hence the claim. Second, consider coefficients whose index has exactly two of 1,2,3. The condition that both representatives of a loop can't appear in a wedge, means that wedges must have one representative from each loop. This gives conditions: Define w, v ∈ Ind (4.7) The above conditions on coefficients show that a vector in • −1 e (x 2j+1 ) must be a linear combination of w and v. To find this linear combination, we invoke Lemma 3.2 which, in the present notation states (id + (1, 2j) + (2, 2j) + · · · + (2j − 1, 2j))2j ∧ 2j + 1 ∧ · · · ∧ 4j − 1 (4.8) is in the kernel of • e . Consequently any permutation of this relation is in ker(• e ) as well. Using this relation, we now calculate the linear combination of v and w which is in ker(• e ). Writing ∼ for the induced equivalence relation, the calculation follows from the following two claims: Claim 1: 1 ∧ 2 ∧ S ∼ −1 ∧ 3 ∧ S ∼ 2 ∧ 3 ∧ S ∼ −1 ∧ 2 ∧ 3 ∧ S 1 Proof: Let's prove that 2 ∧ 3 ∧ S ∼ −1 ∧ 2 ∧ 3 ∧ S 1 , with the other two following similarly. 2 ∧ 3 ∧ S is a sum of (2j − 2) 2 wedge products of length 2j. Each of the terms in the sum have 4 or 5 appearing once in the third position, and 1 does not appear at all. Hence terms appearing in this sum can be paired to write: 2 ∧ 3 ∧ S = b∈S 1 2 ∧ 3 ∧ (4 ⊕ 5) ∧ b. To each such b we apply a permutation of the relation in Equation 4.8 to 2 ∧ 3 ∧ 1 ∧ b to find: The sum over all b of −2 ∧ 3 ∧ ( b∈ b ⊥ b) ∧ b vanishes since each term has two representatives of one loop, and hence pairs with the term in which the representatives appear in the transposed order. Combining the previous two equations with this observation proves the claim.
Claim 2: (−1) i−1 1 ∧ 2 ∧ 3 ∧ S i ∼ 1 ∧ 2 ∧ 3 ∧ S 1 . Proof: The claim is vacuous for i = 1, so fix 1 < i ≤ 2j − 2. Then each term in 1 ∧ 2 ∧ 3 ∧ S i has 4 or 5 appearing in it. Apply the relations to each term with a 4 appearing: 1 ∧ 2 ∧ 3 ∧ 4 ∧ b ∼ −1 ∧ 2 ∧ 3 ∧ (5 ⊕ 2(i + 1) ⊕ (2(i + 1) + 1) ⊕ (⊕ b∈ b ⊥ b)) ∧ b for each b ∈ S {1,i} . The terms replacing 4 with 5 cancel with the terms in which 5 originally appeared. The terms where both representatives of a loop appear cancel in pairs. The remaining terms replace 4 with a representative of loop i. To compare these terms with 1 ∧ 2 ∧ 3 ∧ S 1 , it suffices to permute the order of the representatives of the loops into numerical order. This is done via a cycle of length i − 1, so produces a factor of (−1) i−2 . Combining this with the factor of −1 already appearing yields the result.
These two claims together imply (2j − 2)v − 3w ∈ ker(• e ). From this calculation we conclude that • e (v) = 0, since if it were zero, it would imply • −1 e (x 2j+1 ) = 0, but iteration of Lemma 3.1 shows this not to be the case.
Finally, it remains to observe that the vector v is in the image of P in Equation 4.5. Indeed the map P → Ind S 4j−1 S 2j−1 ×S 2j (H 0 (Γ 0,2j ) ⊗ H 2j (Γ 1,2j+1 ) sends each X-labeled 2j + 1-gon to a wedge containing two of {1, 2, 3} and exactly one representative of each loop. The vector v is defined as a linear combination of such wedge products, so it is in the image. In particular, we have shown the contraction map P → H Lie (1, X) surjects onto the span of x 2j+1 , from which the statement follows.