WWHEEL GRAPH HOMOLOGY CLASSES VIA LIE GRAPH HOMOLOGY
BENJAMIN C. WARD
Abstract.
We give a new proof of the non-triviality of wheel graph homology classes usinghigher operations on Lie graph homology and a derived version of Koszul duality for modularoperads. Introduction.
In the seminal paper [Wil15], Willwacher constructed a family of non-trivial graph cohomol-ogy classes σ j +1 by analyzing an isomorphism between the 0 th cohomology of a certain graphcomplex GC with the Grothendieck-Teichm¨uller Lie algebra grt . The complex GC is (up toimportant 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 automorphismgroups of free groups and use the Leray-Serre spectral sequence to compute the homology of alow genus portion of the Lie variant of this construction. They show in particular that in genus1, the virtual cohomological dimension consists of a family of classes α j +1 .The purpose of this paper is to show that the families of classes σ j +1 and α j +1 correspondto each other under Koszul duality.1.1. The correspondence: non-technical version.
Let us explain the nature of this corre-spondence and the perspective from which such a result is expected. We omit several importanttechnical details in this informal summary.The statement that commutative and Lie structures are Koszul dual can be encoded in thelanguage of operads. One first defines an operadic generalization of the bar construction of analgebra. One then proves that the homology of the bar construction of the commutative operadhas the homotopy type of the Lie operad, and vice-versa.Operads have operations parameterized by graphs of genus 0 (trees), and may be generalizedby considering structures with operations parameterized by all graphs. Such structures arecalled modular operads [GK98], and the analog of the bar construction is called the Feynmantransform which we denote ft . The commutative and Lie operads may be considered modularoperads 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 genussummands which is patently false. Indeed, both commutative graph homology H ∗ ( ft ( Com )) andLie 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 commu-tative modular operad
Com but is Koszul dual (in a suitably derived sense) to the Lie modularoperad
Lie . Therefore in genus g ≥
1, the (derived) Feynman transform of Lie graph homologyis acyclic on the one hand, and contains a subcomplex computing commutative graph homologyon the other.We thus conclude that every commutative graph homology class of non-zero genus may berepresented by a graph labeled by Lie graph homology classes . This correspondence is realizedvia the boundary operator in a derived version of the Feynman transform. In particular, thedifferential in this acyclic complex depends not just on the modular operad structure of Liegraph homology, but also on its higher “Massey products” which we compute here-in. Under a r X i v : . [ m a t h . A T ] F e b his correspondence, the homology class σ j +1 corresponds to a graph whose lone non-trivialvertex label is built from α j +1 .1.2. The correspondence: technical version.
The graph complex ( GC , d GC ) of [Wil15]which we consider here-in consists of graphs with no legs and no loops (aka tadpoles). Thedifferential is a sum of all possible edge expansions, with signs encoded by a factor of the topexterior power of the set of edges of the graph. This complex splits over genus, and the complex GC g is (after a shift in degree by Σ − g ) a sub-complex of ft ( Com )( g, − g in this introduction.The modular operad structure of Lie graph homology was studied in [CHKV16]. This modularoperad is not formal and so, abstractly, higher operations exist which make Lie graph homologyinto a modular operad up-to-homotopy (called a weak modular operad) which is homotopyequivalent to the Feynman transform of an extension by zero [War19]. The derived version ofthe Feynman transform ft mentioned above has a differential with additional terms taking thesehigher operations into account. Making these higher operations explicit in all genera is bothunlikely to be possible and unnecessary for the task at hand. As we shall show, the wheel graphhomology classes can be detected by higher operations on Lie graph homology landing in genus1. With this in mind, we proceed as follows.First we truncate Lie graph homology above genus 1 and endow the resulting semi-classicalmodular 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 ) (cid:44) → ft ( H Lie ).Declare a loop to be simple if its adjacent vertex is trivalent and of genus 0. The subspaces ofthe Feynman transform supported on graphs with simple loops form a dg submodule. Let ¯ ft denote passage to the quotient; we also write ¯ η for the projection of a vector η to this quotient.Lifting ft ( Com ) (cid:44) → ft ( H Lie ) we have the following diagram of dg S -modules: GC (cid:44) → ¯ ft ( Com ) (cid:44) → ¯ ft ( H Lie ) . (1.1)The left hand inclusion is split; we denote the projection π : ¯ ft ( Com ) → GC .We write ω j +1 ∈ GC ∗ for the wheel graph; a 2 j + 1-gon along with a central vertex connectedto the other vertices, tensored with a fixed element in the top exterior power of the span of theset of edges of the graph (Figure 1). The main technical result of this paper is the followingconstruction: Theorem 1.1.
There exists an element η j +1 ∈ ft ( H Lie )(2 j + 1 , such that d (¯ η j +1 ) ∈ ¯ ft ( Com ) and (cid:104) ω j +1 , π ◦ d (¯ η j +1 ) , (cid:105) (cid:54) = 0 . The condition (cid:104) ω j +1 , π ◦ d (¯ η j +1 ) (cid:105) (cid:54) = 0 ensures that the elements π ◦ d (¯ η j +1 ) are not bound-aries in the submodule GC ⊂ ¯ ft ( H Lie ). On the other hand, the elements d (¯ η j +1 ) are boundariesin the larger complex and so certainly must project to cycles in GC . Thus, each π ◦ d (¯ η j +1 )represents a non-trivial graph homology classes in GC . Equivalently we may state: Corollary 1.2.
The homology class [ ω j +1 ] ∈ H ( GC ∗ ) is non-trivial. The proof of Theorem 1 . η j +1 has underlyinggraph pictured in Figure 4, and carries a vertex label given by the VCD class α j +1 composedwith 2 j − η j +1 not by contractingthe exterior polygon (a logical first guess) but by contracting interior polygons (Figure 1).Contracting such interior polygons produces graphs having n = 2 j − η j +1 depends crucially on the representation theory of the Weyl group of SO (2 n + 1) to show that the class we construct is the unique class (up to scalar multiple) whichdoes not vanish upon passage to coinvariants by the group of isomorphisms of such a graph.Corollary 1 . grt . The interplay between Lie and commutative graph homology is subtle, and muchinteresting work remains. For example how to realize Morita and Eisenstein classes via Masseyproducts on commutative graph homology, or how to relate open conjectures on either side of i -
2i 2i-2 i + Figure 1.
Left: The wheel graph ω j +1 for j = 2 with edge order indicated.Center: The “exterior” 2 j + 1-gon is the pictured (red) subgraph. Right: The“interior” 2 j + 1-gon is the pictured (blue) subgraph.this correspondence. For now we simply offer the results of this paper as a polite suggestionthat 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 Section2. In Section 3 we compute relations in the modular operad for Lie graph homology and thenconstruct higher operations for Lie graph homology landing in genus 1. The construction of theelement η j +1 and verification of its properties is given in Section 4. Throughout we mostly workwith the “co-Feynman transform”, and then linear dualize as the very last step (Subsection 4 . Contents
1. Introduction. 12. Background. 33. Higher operations on Lie graph homology. 64. Proof of the main results. 10References 181.3.
Conventions.
Throughout we work in the category of differential graded vector spacesover 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 apartition α = ( α , . . . , α r ) of n . 2. Background.
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 whichvertex a flag is adjacent, and an involution ι : F → F . The orbits of ι of order two are calledthe edges of γ and denoted E ( γ ). The orbits of order one are called the legs γ and denotedleg( γ ). A loop is an edge { f, ι ( f ) } for which a ( f ) = a ( ι ( f )). The number | a − ( v ) | is called thevalence of the vertex v .A graph determines a 1-dimensional CW complex and we say the graph is connected if thisCW complex is connected. A genus labeling of a graph is a function g : V → Z ≥ . A connected,genus labeled graph is stable if 2 g ( v ) + | a − ( v ) | ≥ v ∈ V . A leg labeling of a graph isa bijection { , . . . , n } → leg( γ ), for the appropriate n . A modular graph is a stable graph alongwith a leg labeling. The total genus of a modular graph is g ( γ ) := β ( γ ) + (cid:80) v g ( v ), where β denotes the first Betti number of the associated CW complex. The type of a modular graph isthe pair of non-negative integers ( g ( γ ) , | leg( γ ) | ).An isomorphism of abstract graphs is a pair of bijections between the respective vertices andflags which commutes with the adjacency and involution maps. An isomorphism of modular Figure 2.
A modular graph of type (10 ,
3) having 3 vertices, 11 flags, 4 edges,1 loop and 3 legs.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 modulargraph. For non-negative integers g and n with 2 g + n ≥ 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 agraph 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 edgesof N and their adjacent vertices to a single vertex labeled to preserve the total genus of thegraph. We call this new vertex N . The graph ˆ N is the graph formed by adding as legs all flagsof F ( γ ) \ F ( N ) which are adjacent to vertices in N . Note the legs of ˆ N are not numericallylabeled, 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-dimensionalvector 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, thesedefinitions are opposite to the conventions in [GK98]. A mod 2 order on the edges of a graphis defined to be a choice of unit vector in K ∗ ( γ ) or equivalently K ( γ ).2.2. 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 2 g + n ≥ 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 astable S -module A and a modular graph γ with vertex v we define A ( v ) := A ( g ( v ) , a − ( v )) anddefine 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 acollection of degree − K ( γ ) ∗ ⊗ A ( γ ) µ γ → A ( g, n ) , (2.1)for each γ ∈ Gr ( g, n ) and for all ( g, n ) which are S n -equivariant, Aut ( γ )-coinvariant and whichsatisfy the differential condition (cid:80) µ γ/N ◦ N µ ˆ N = 0.Here µ γ/N ◦ N µ ˆ N is the composition of operations given by plugging the output of µ ˆ N intothe vertex N of γ/N and the sum is over all nests N on γ . The factors of K are composed bypulling back the wedge product isomorphism: K ( γ/N ) ⊗ K ( ˆ N ) ∧ → K ( γ ) , which in turn encodes signs in the differential. We refer to [War19, Proposition 3.21] for ad-ditional 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 compositionof these one edged operations.2.3.
Co-Feynman transform.
Let A be a weak modular operad. The co-Feynman transformof A , denoted B ( A ) is defined to be its bar construction viewed as an algebra over the Koszul esolution of the groupoid colored operad encoding modular operads [War19]. Explicitly thismeans B ( A ) is the dg K -modular co-operad given by: B ( A )( g, n ) = (cid:77) γ ∈ Gr ( g,n ) K ∗ ( γ ) ⊗ Aut ( γ ) A ( γ ) . The K -modular co-operad structure is co-free and specified by decomposition maps ν γ : B ( A )( g, n ) → K ∗ ( γ ) ⊗ Aut ( γ ) B ( A )( γ )for each γ ∈ Gr ( g, n ). These decomposition maps are defined on the summand of the sourcecorresponding to a graph γ (cid:48) by summing over all ways to add a single layer of nests to it suchthat γ (cid:48) / (cid:116) N i = γ .The weak K -modular operad structure maps on A induce a degree − S -modules µ : B ( A ) → A . This map induces a differential ∂ : B ( A )( g, n ) → B ( A )( g, n ). To describe ∂ it issufficient 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: B ( A )( g, n ) ν γ (cid:15) (cid:15) ∂ (cid:47) (cid:47) B ( A )( g, n ) π γ (cid:15) (cid:15) (cid:15) (cid:15) B ( A )( γ ) ⊗ Aut ( γ ) K ∗ ( γ ) µ (cid:47) (cid:47) A ( γ ) ⊗ Aut ( γ ) K ∗ ( γ ) . (2.2)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 bya corolla.The co-Feynman transform of a weak modular operad has a bigrading B ( A )( g, n ) = (cid:77) r ≥ ,s ∈ Z B ( A )( g, n ) r,s (2.3)given by the number of edges r on the graph indexing a summand and s is the sum of theinternal degrees of the vertex labels. With respect to this bigrading, the co-Feynman transformdifferential is supported on: B ( A )( g, n ) r,s ∂ → (cid:77) ≤ e ≤ r B ( A )( g, n ) r − e,s + e − . (2.4)In particular, on each bigraded component the differential splits as ⊕ e ∂ e corresponding to thoseterms which contract e edges, under convention that contracting 0 edges means apply theinternal 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 thanone 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 ∂ is itself square zero; it is the linear dual of the differential in the usual Feynman transformof this (strict) modular operad.2.4. ft ( Com ) and GC . As above,
Com is the operad encoding commutative algebras, viewedas a modular operad by extension to higher genus by 0. It follows that ft ( Com )( g, n ) ∼ = (cid:77) K ( γ ) Aut ( γ ) , where the sum is taken over all γ ∈ Gr ( g, n ) for which g ( v ) = 0 for all v ∈ V ( γ ). ince all vertices have genus 0, no differential terms expand loops, and so the sum over thosegraphs with no loops is a subcomplex NL ( g, n ) ⊂ ft ( Com )( g, n ). For each g ≥ GC g = (Σ g NL ( g, GC = (cid:81) g GC g . Note that [Wil15] usescohomological conventions, so to recover exactly his GC , one must take the cochain complexassociated to the chain complex which we have called GC by negating the indices: ( V i ) op = V − i .In particular, a homogeneous element in GC is specified by a scalar multiple of an isomor-phism 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 2 g − E = E − V + 2and the differential is given by a sum of edge expansions. The alternating action on the edgesof a representative has the effect that any isomorphism class of a graph with parallel edgesvanishes.2.5. Wheel graphs.
As above we define the wheel graph ω j +1 by connecting a new vertex toall the edges in a 2 j + 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 j +1 may be viewed as a degree 4 j + 2 element of B ( Com )(2 j + 1 , Lemma 2.4.
The wheel graph ω j +1 is a cycle in the complex B ( Com )(2 j + 1 , . We remark that the vector spaces B ( Com )(2 j + 1 ,
0) and ft ( Com )(2 j + 1 ,
0) are canonicallyisomorphic, via
Com ( g, n ) ∼ = Com ( g, n ) ∗ , and so the wheel graph with the above conventionsalso specifies an element ω ∗ j +1 ∈ ft ( Com )(2 j + 1 , . ω ∗ j +1 appears with non-zero coefficient is not a boundary in ft ( Com )(2 j + 1 , Higher operations on Lie graph homology.
Lie graph homology.
Following [GK98], after [Kon93], we define Lie graph homology tobe the modular operad H ∗ ( ft (Σ s − Lie )), where s denotes the (cyclic) operadic suspension andΣ denotes an shift up in degree. In particular extension by zero makes Σ s − Lie a K -modularoperad, and its Feynman transform is a modular operad. Following [CHKV16] we will denotethis 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 thismodular operad in genus ≤ Lemma 3.1. [CHKV16]
The graded vector spaces H ∗ (Γ g,n ) form a modular operad with thefollowing properties:(1) The underlying cyclic operad H ∗ (Γ , − ) is canonically isomorphic to the commutative(cyclic) operad.(2) As an S n module, H i (Γ ,n ) ∼ = (cid:40) V n − i, i if i is even and ≤ i ≤ n − else(3) The modular operadic composition map H (Γ , ) ⊗ H d (Γ ,n ) → H d (Γ ,n +1 ) is injective. Notice that H j (Γ , j +1 ) is the alternating representation. We fix generators α j +1 ∈ H j (Γ , j +1 )once and for all. We will also write m t ∈ Com ( t ) = H (Γ ,t +1 ) for the commutative product.We may iterate the modular operadic composition map above to form H (Γ ,t +1 ) ⊗ H d (Γ ,n ) ◦ e −→ H d (Γ ,n + t − ) . (3.1) ere ◦ e is the modular operadic composition which corresponds to gluing along a tree with oneedge adjacent to vertices of type (0 , t + 1) and (1 , n ). To be precise, this composition is onlywell defined after a choice of labeling of the t + n − { , . . . , t + n − } .We fix the convention that ◦ e corresponds to labeling the genus 0 vertex by { , . . . , t } and thegenus 1 vertex by { t + 1 , . . . , t + n − } . Observe that repeated application of Lemma 3 . m t ◦ e α j +1 (cid:54) = 0.We now calculate relations between compositions in the modular operad H ∗ (Γ). Lemma 3.2.
The non-zero homology class m t ◦ e α j +1 ∈ H j (Γ ,t +2 j ) satisfies t +1 (cid:88) i =1 ( i, t + 1)( m t ◦ e α j +1 ) = 0 , where ( i, t + 1) ∈ S t +2 j denotes a transposition.Proof. The map ◦ e of Equation 3 . S t × S j equivariant, where the target carries the restrictedaction along the standard inclusion S t × S j (cid:44) → S t +2 j . By abuse of notation we also write ◦ e for the adjoint: Ind S t +2 j S t × S j ( H (Γ ,t +1 ) ⊗ H j (Γ , j +1 )) ◦ e → H j (Γ , j + t ) . (3.2)Using the Littlewood-Richardson rule ([FH91, p.456]) we compute the irreducible decompositionof the source of Equation 3 . V t +1 , j − ⊕ V t, j .By Lemma 3 .
1, the target of ◦ e is an irreducible S t +2 j -representation of type V t, j . Thus,any vector z for which Q [ S t +2 j ] · z ∼ = V t +1 , j − must be in the kernel of ◦ e . To produce sucha vector z we embed the problem in the group ring. That is, consider the S t × S j equivariantmap H (Γ ,t +1 ) ⊗ H j (Γ , j +1 ) → Q [ S t +2 j ] defined by m t ⊗ α j +1 (cid:55)→ y := ( (cid:88) σ ∈ S t σ )( (cid:88) σ ∈ S { t +1 ,...t +2 j } sgn ( σ ) σ ) . (3.3)Form the Young diagram of shape t + 1 , j − 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. Since y is in the image of the map defined in Equation 3 . c ζ is in the image of the adjointmorphism from the induced representation. By construction c ζ generates a copy of V t +1 , j − under left multiplication by Q [ S t +2 j ], hence so does z := ( id + (1 , t + 1) + (2 , t + 1) + · · · + ( t, t + 1))( m t ⊗ α j +1 ) . Thus, this z is in the kernel of the ◦ e in Equation 3 . S t +2 j orbit of this z . (cid:3) The weak semi-classical modular operad H Lie . In this section we endow the genus ≤ H ∗ (Γ g,n ) with higher operations. The result will be a weak modular operad whichwe denote ( H Lie , µ ).As stable S -modules we define H Lie ( g, n ) = (cid:40) H ∗ (Γ g,n ) if g <
20 if g ≥ µ γ are defined as follows. We define µ γ = 0 unless one of the two mutuallyexclusive conditions is met: • γ has genus < • γ has genus 1, and the underlying leg free graph of γ is a 2 j + 1-gon, for some j ≥ e e P P Figure 3.
The Massey project associated to P = P is defined by expandingthe edge e , applying the Massey product µ P and then the modular operadiccomposition ◦ 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 structureon H ∗ (Γ). In the second case we proceed as follows.Let p j +1 be the standard trivalent 2 j + 1-gon. By this we mean the modular graph formedby attaching a leg to each vertex of a 2 j + 1-gon. The vertices have genus label 0. The leg labelsare in the dihedral order, and we give this modular graph an edge ordering e < · · · < e j +1 such that the i th edge connects the vertices adjacent to flags i and i + 1 (mod 2 j + 1).Observe that H Lie ( p j +1 ) ∼ = k . We define µ p j +1 : H Lie ( p j +1 ) ⊗ K ( p j +1 ) → H Lie (1 , j + 1)by µ p j +1 (1 ⊗ e ∧ · · · ∧ e j +1 ) = α j +1 . 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 µ ˆ p for any other edge orderedtrivalent polygon ˆ p . One easily checks that this definition is not over-prescribed, since sym-metries of a 2 j + 1-gon induce permutations of the edges and the legs which have matchingparity.We then want to show that if P is a non-trivalent graph whose underlying leg free graph is a2 j + 1-gon, that µ P is determined by the above operations. For this we induct on the numberof non-trivalent vertices. First suppose that this number is 1, at a vertex v of P := P .Let γ be the graph formed by blowing up v to separate the two flags which belong to edges ofthe polygonal subgraph from the rest of the flags at v ; see Figure 3. In particular γ has 2 j + 2edges, 2 j + 1 of which form a polygon P with trivalent vertices. Let N be a nest on γ . By theabove definition, the composition µ γ/N ◦ N µ ˆ N will be zero unless N = { e } or ˆ N = P (picturedblue and red in Figure 3).Thus, applying the differential condition of Definition 2 . d = 0, itmust be the case that 0 = µ P ◦ µ e + µ γ/ P ◦ µ P . (3.4)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 andso is an isomorphism, H Lie ( γ ) ∼ = → H Lie ( P ). Thus Equation 3 . µ P .For the induction step we repeat the above argument, reducing the number of non-trivalentvertices by one at each step. The fact that the operation is independent of the choice of orderof 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 toshow that the differential condition (cid:88) N on γ µ γ/N ◦ N µ ˆ N = 0 (3.5)is satisfied for every modular graph γ . f γ has total genus ≥ . γ has exactly two edges, then Equation 3 . 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 ( ˆ N ) to be non-zero requires thatone 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 onlygenus 0 vertices. There are two cases for such a γ . Either its lone cycle has an odd number ofedges, in which case there is an additional edge pointing outward or its lone cycle has an evennumber of edges, in which case the leg free graph underlying γ must be a 2 n -gon.The first case follows as above. In particular, suppose γ has 2 n edges and its lone cycle is oflength 2 n −
1. Let e be the unique edge of γ which is not in the cycle. This edge is connectedto 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 conditionis verified by a double iteration of Equation 3 . γ = P t is a polygon with, 2 n ≥ t ≥ n legs labeled { , . . . , t } , with at least one leg ateach 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 = (cid:80) e ∈ P t µ P t /e ◦ µ e . Thiscondition can be rephrased as saying the following composite vanishes: B ( H Lie )(1 , t ) n, ⊃ H Lie ( P t ) ⊗ K ∗ ( P t ) ∂ (cid:15) (cid:15) B ( H Lie )(1 , t ) n − , ⊃ (cid:77) e ∈ P t H Lie ( P t /e ) ⊗ K ∗ ( P t /e ) ⊕ e µ P t/e (cid:47) (cid:47) H n − (Γ ,t ) (3.6)Let us first consider the case t = 2 n . Since the maps in Diagram 3 . S n -equivariantit is sufficient to consider the case that the legs of P n are labeled in the dihedral order.The source of this diagram is closed under the Z n action restricted along the standard in-clusion Z n ∼ = (cid:104) σ := (12 . . . n ) (cid:105) ⊂ S n . Denoting this 1-dimensional Z n representation E := Res S n Z n ( H Lie ( P n ) ⊗ K ∗ ( P n )), we compute its character χ E ( σ ) = sgn ( σ ) = −
1. Thisin turn determines the isomorphism type of the irreducible Z n -representation E ; namely the2 n cycle σ acts by − . Res S n Z n ( H n − (Γ , n )). The irreducible representa-tions of Z n over the algebraic closure Q are all 1-dimensional and are given by letting σ act bymultiplication of a root of x n −
1. Let ω = e iπ/n and write W i for the irreducible representationcorresponding to multiplication by ω i . Let V n ⊕ V n − , be the permutation representation of S n . One easily calculates its restriction Res S n Z n ( V n ⊕ V n − , ) = n − (cid:77) i =0 W i Since V n − , ⊗ V n ∼ = V , n − , the number of copies of E ∼ = W n appearing in Res S n Z n ( V , n − )is the number copies of W appearing in Res S n Z n ( V n − , ) which is 1 − t = 2 n . Now suppose t > n . 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 { , . . . , t } = (cid:116) T i . Let ˆ T i be the set T i along with an added basepoint, called theroot. onsider the following diagram:( H Lie ( P n ) ⊗ K ∗ ( P n )) ⊗ (cid:32) n (cid:79) i =1 H Lie (0 , ˆ T i ) (cid:33) ∼ = (cid:47) (cid:47) ∂ ⊗ id (cid:15) (cid:15) H Lie ( P t ) ⊗ K ∗ ( P t ) ∂ (cid:15) (cid:15) (cid:77) e ∈ P n H Lie ( P n /e ) ⊗ K ∗ ( P n /e ) ⊗ (cid:32) n (cid:79) i =1 H Lie (0 , ˆ T i ) (cid:33) ∼ = (cid:47) (cid:47) ⊕ e µ P n/e ⊗ id (cid:15) (cid:15) (cid:76) e ∈ P t H Lie ( P t /e ) ⊗ K ∗ ( P t /e ) ⊕ e µ P t/e (cid:15) (cid:15) H n − (Γ , n ) ⊗ ( ⊗ i H (Γ , ˆ T i )) (cid:80) ◦ i (cid:47) (cid:47) H n − (Γ ,t )The right hand side of this diagram is exactly Diagram 3 .
6. The left hand side of this diagramis Diagram 3 . t = 2 n , then tensored with the 1-dimensional, trivial Z n -representation (cid:78) ni =1 H Lie (0 , ˆ T i ). The horizontal arrows are contractions using the modularoperad structure along the graph identifying the root of ˆ T i with leg i of P n .The commutativity of the top square can be seen just by looking at each summand – bothroutes give the same graph with commutative labels. The commutativity of the bottom squarefollows 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 = 2 n case considered above, andsince the top horizontal arrow is an isomorphism, the right hand side of the diagram alsovanishes, as desired. (cid:3) Viewing
Com as a modular operad, as in Subsection 2 .
4, we see immediately that there isa level-wise surjective morphism of weak modular operads H Lie → Com . Taking the weakFeynman transform, we have a level-wise injective morphism of K -twisted modular operads ft ( Com ) (cid:44) → ft ( H Lie ). 4.
Proof of the main results.
Let us regard the wreath product S (cid:111) S n as follows. Its underlying set is S n × S n . To anelement (( α , . . . , α n ) , τ ) in this set we associate a permutation in S n by first acting by τ onthe ordered set { , } , { , } , . . . , { n − , n } of size n and then acting by α i on the ordered set { τ ( i ) − , τ ( i ) } for each i . This defines an injective map of sets S × n × S n (cid:44) → S n , and S (cid:111) S n carries the unique group structure for which this map is a homomorphism. The wreathproduct S (cid:111) S n has a 1-dimensional representation given by letting an element (( α , . . . , 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( S ) r − (cid:44) → S (cid:111) S r − (cid:44) → S (cid:111) S r − × S n − r +2 (cid:44) → S r − × S n − r +2 (cid:44) → S n + r as a sequence of subgroups. We write Res GH for the restriction of a representation of a group G to a representation of a subgroup H . Lemma 4.1.
Let ≤ r < n be integers. • The irreducible decomposition of
Res S n + r ( S (cid:111) S r − ) × S n − r +2 ( V r, n ) contains a unique summandof the form L r − (cid:2) V β . It is of the form L r − (cid:2) V n − r +2 . • The irreducible decomposition of
Res S n + r ( S (cid:111) S q ) × S n − q + r ( V r, n ) has no summand of the form L q (cid:2) − for q ≥ r .Proof. Fix q ≥ r −
1. The number of copies of a summand V α (cid:2) V β appearing in Res S n + r S q × S n + r − q ( V r, n ) (4.1) s computed via the Littlewood-Richardson rule [FH91]. In this case, because V r, n is a hook,each α and β appearing with non-zero coefficient must also be hooks. If q > r −
1, the numberof summands of the from V q +1 , q − (cid:2) V β appearing in the decomposition of the representationin Equation 4 . q = r −
1, there is a unique summand of the from V q +1 , q − (cid:2) V β appearing in the decomposition of the representation in Equation 4 . V r, r − (cid:2) V n − r +2 .It now remains to analyze the irreducible decomposition of Res S q S (cid:111) S q of a hook. The neededcalculation, 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 restrictionof a hook Res S q S (cid:111) S q ( V x, y ) is 0 unless x = q +1 and y = q −
1, in which case it is 1. This completesthe proof.While it was convenient that the calculation we needed was available in the literature, thereis an argument internal to this article which may also be used to prove this. One first uses thePieri rule to see that a summand V α (cid:2) V β appearing in Equation 4 . S ) q ⊂ S (cid:111) S q ⊂ S q invariant subspace if and only if q = r − α = ( r, q − r ) and β = (1 n + r − q ). Any summandof type L q (cid:2) − would restrict to an ( S ) q invariant subspace, which establishes the secondstatement. On the other hand when q = r − S ) r − invariant subspace in V r, r − . Since it is unique, it must contain the image of the gluing operation which grafts H (Γ , )onto each input of H r − (Γ ,r − ) ∼ = V r − , landing in H r − (Γ , r − ) ∼ = V r, r − . This image is non-zero by Lemma 3 .
1. Since we’re gluing on to the alternating representation H r − (Γ ,r − ), itmust be the case that this ( S ) r − invariant subspace lifts to a S (cid:111) S r − representation whichis alternating with respect to the S r − factor, i.e. to a copy of L r − , which establishes the firststatement. (cid:3) Corollary 4.2.
Let γ ∈ Gr ( g, n ) with a vertex v of genus g ( v ) = 1 and valence | a − ( v ) | = m .Consider a homogeneous element [ x ] ∈ H Lie ( γ ) ⊗ Aut ( γ ) K ∗ ( γ ) ⊂ B ( H Lie )( g, n ) whose vertex v carries a label in H Lie ( v ) ∼ = H ∗ (Γ ,m ) of degree i . If v is adjacent to m − i ormore loops then [ x ] = 0 .Proof. Let q be the number of loops adjacent to v . Then Aut ( γ ) contains a subgroup isomorphicto S (cid:111) 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 ,a − ( v ) ) ⊗ K ∗ ( γ ) and since H i (Γ ,a − ( v ) ) ∼ = V m − i, i , the invariantsof this action correspond to the copies of L q appearing in Res S q S (cid:111) S q ( V m − i, i ). From the proofof Lemma 4 . q ≥ m − i . Any such Aut ( γ )-invariant element [ x ] would require an S (cid:111) S q invariant element of H i (Γ ,a − ( v ) ) ⊗ K ∗ ( γ ) labeling v . Since there are no such elements when q ≥ m − i , the Aut ( γ )-coinvariants vanish. (cid:3) Definition 4.3.
For a non-negative integer j we define θ j +1 ∈ Gr (2 j + 1 ,
0) as follows. It hastwo vertices; one of genus 0, call it v , and one of genus 1, call it v . It has 2 j + 1 edges, 3 ofwhich connect the two vertices and the remaining 2 j − B ( H Lie )( g, n ) r,s denotes the bigraded component of the chain complex B ( H Lie )( g, n )having r edges and internal degree s . Lemma 4.4.
The subspace ( K ∗ ( θ j +1 ) ⊗ Aut ( θ j +1 ) H Lie ( θ j +1 )) j +1 , j ⊂ B ( H Lie )(2 j + 1 , j +1 , j is 1-dimensional.Proof. Observe that
Aut ( θ j +1 ) ∼ = S × ( S (cid:111) S j − ); the S permutes the non-loop edges, the fac-tors of S transpose the flags in a loop and the S j − permutes the loop edges. As an Aut ( θ j +1 )-module, the one dimensional vector space K ∗ ( θ j +1 ) has representation type isomorphic to - + + +++ - } - Figure 4.
The case j = 3. The graph θ (pictured) underlying the element β ∈ B ( H Lie )(7 , Aut ( θ ) acts by such signedmultiples. V , , (cid:2) L j − . Thus, the Aut ( θ j +1 ) fixed points of K ∗ ( θ j +1 ) ⊗ H Lie ( θ j +1 ) are given by thenumber of copies of V , , (cid:2) L j − in the irreducible decomposition of V j − , j ∼ = H Lie (1 , a − ( v )).Applying Lemma 4 . r = 2 j − n = 2 j , we see there is exactly one such copy. (cid:3) Definition of β . We now construct a canonical basis vector spanning ( K ∗ ( θ j +1 ) ⊗ Aut ( θ j +1 ) H Lie ( θ j +1 )) j +1 , j . Define the set X := a − ( v ). This set is partitioned by the edges of θ j +1 into three blocks of size 1 and 2 j − θ such that the three non-loop edges are in the first three positions, and choose an orderon the flags within each block. This fixes a total order on the set X and hence an isomorphism: H j (Γ , j − ) ∼ = H j (Γ ,X ) . (4.2)Define x j +1 ∈ H j (Γ ,X ) to be the composition( ... ( α j +1 ◦ j +1 m ) ◦ j m ) ... ) ◦ m ∈ H j (Γ , j − )composed with this isomorphism. Here m is the generator of H (Γ , ) = Com (2), and x j +1 (cid:54) = 0by Lemma 3 .
1. Observe that permuting the two flags on m acts by +, where-as permutingblocks of the permutation on X by σ is the same as composing with σα j +1 = sgn ( σ ) α j +1 (by equivariance of the operadic compositions). Therefore the element x j +1 spans the uniquecopy of V , , (cid:2) L j − in Res S j − S × ( S (cid:111) S j − ) H j (Γ ,X ), where the S j − action is inherited from theisomorphism in Equation 4 .
2. The class x j +1 ∈ H j (Γ ,X ) depends on the choice of isomorphismin Equation 4 .
2, but only up to sign.Define β j +1 ∈ ( K ∗ ( θ j +1 ) ⊗ Aut ( θ j +1 ) H Lie ( θ j +1 )) j +1 , j (4.3)to be the element formed by labeling v with x j +1 and v by m and using the mod 2 edgeorder induced by the choice above. Observe that β j +1 is independent of the choices made. Ifwe had picked a different edge order the result would differ by two factors of the sign of thecorresponding permutation; if we had picked a different loop orientation the result would differby a transposition of the commutative product.Here are some features of β j +1 and its underlying graph θ j +1 for particular values of j : j total genus int. deg. loops valence of v Rep. type at v V , , V , V , j j + 1 2 j j − j − V j − , j When j is fixed we may abbreviate the notation θ := θ j +1 ; β := β j +1 ; ω := ω j +1 and so on. .2. Analysis of ∂ − ( β ) . We define a loop in a stable graph γ to be simple if the vertex v towhich the loop is adjacent satisfies | a − ( v ) | = 3 and g ( v ) = 0. Definition 4.5.
Define (cid:99) Gr ( g, n ) ⊂ Gr ( g, n ) to be the subset of graphs which do not containsimple loops. For a weak modular operad A we then define: (cid:98) B ( A )( g, n ) = (cid:77) γ ∈ (cid:99) Gr ( g,n ) A ( γ ) ⊗ Aut ( γ ) K ∗ ( γ ) ⊂ B ( A )( g, n ) . Lemma 4.6.
The submodule (cid:98) B ( A ) ⊂ B ( A ) is closed under the co-Feynman transform differen-tial ∂ .Proof. Given a modular graph γ with no simple loops, terms in the weak co-Feynman transformdifferential are indexed by contractions of subgraphs of γ . If such a differential term has a simpleloop it must have been created by contracting a subgraph of type (0 , Gr (0 ,
3) consistsonly of the (0 , (cid:3) We remark that since the weak modular operad H Lie has internal differential 0, the summandof ∂ which contracts just 1 edge, call it ∂ , is itself a differential (Remark 2 . . (cid:98) B ( H Lie )( g, n ) , ∂ ) is a subcomplex of ( B ( H Lie )( g, n ) , ∂ ). Proposition 4.7.
The composition of ∂ with projection to the θ j +1 summand of (cid:98) B ( H Lie )(2 j +1 , j +1 , j is zero. In particular, β j +1 does not appear with non-zero coefficient in any ∂ boundary.Proof. As above let X := a − ( v ). The set X is partitioned by the edges of θ j +1 into 3 blocksof size one and 2 j − a , a , a and label theelements of each block of size two by l i , l i label, where i indexes the 2 j − θ j − . Asabove we write x j +1 ∈ H j (Γ ,X ) for a basis vector spanning the unique invariant subspace of Res S X Aut ( θ j +1 ) isomorphic to V , , (cid:2) L j − . The S { a ,a ,a } action on x j +1 is alternating whileeach S { l i ,l i } action on x j +1 is the identity.Write ∂ θ for the composition of ∂ with projection to the θ j +1 summand of (cid:98) B ( H Lie )(2 j +1 , γ for which the following composite is non-zero:( K ∗ ( γ ) ⊗ Aut ( γ ) H Lie ( γ )) j +2 , j (cid:43) (cid:43) (cid:31) (cid:127) (cid:47) (cid:47) (cid:98) B ( H Lie )(2 j + 1 , j +2 , j∂ θ (cid:15) (cid:15) ( K ∗ ( θ j +1 ) ⊗ Aut ( θ j +1 ) H Lie ( θ j +1 )) j +1 , j 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 of θ j +1 sincethe vertex v is of type (0,3) and hence indecomposible. Note also that the edge e of γ can notbe a loop for degree reasons – such a γ would have only genus 0 vertices, and so internal degree0 (cid:54) = 2 j + 1.Therefore the edge e of γ is adjacent to two vertices whose genera add to 1. Let Y ∪ { r } bethe flags of γ adjacent to the genus 1 vertex and Z ∪ { s } be the flags of γ adjacent to the genus0 vertex. The isomorphism γ/E ∼ = θ specifies a partition X = Y (cid:116) Z, along with a linear map H j (Γ ,Y ∪{ r } ) ⊗ H (Γ ,Z ∪{ s } ) ◦ e −→ H j (Γ ,X ) . This linear map is S Y × S Z equivariant. In particular, | X | = | Y | + | Z | = 4 j −
1. We say a loop l i of θ is split (by e ) if both { l i , l i } ∩ Y and { l i , l i } ∩ Z are nonempty.Such a differential term being nonvanishing implies the following: | Y ∪ r | ≥ j + 1 and hence | Y | ≥ j . Thus 3 ≤ | Z ∪ s | ≤ j . • |{ a , a , a } ∩ Z | <
2, since the representation type of H (Γ ,Z ∪{ s } ) is trivial. So wesuppose without loss of generality that a , a ∈ Y . • At most one loop is split, since otherwise we would create parallel edges with an alter-nating action of { l i , l h } at one vertex an identity action of { l i , l h } at the other vertexwhich pass equivariantly to the identity action of both { l i , l i } and { l h , l h } on X .Let (cid:96) be the number of loops adjacent to Y . By Corollary 4 . (cid:96) ≤ | Y | + 1 − j − | Y | − j , and hence 2 j ≤ | Y | − (cid:96). Consider the possible cases for such a γ ∈ Θ. Case a ∈ Z and no loop is split: Then | Y | = 2 (cid:96) + 2 which implies 2 j − ≤ (cid:96) and hence2 j − (cid:96) , since 2 j − Z would be 2,contradiction. Case a ∈ Z and one loop is split: Then | Y | = 2 (cid:96) + 3 which implies 2 j − ≤ (cid:96) and hence2 j − (cid:96) , since 2 j − Y .This means that Y carries the maximum number of loops for its given valence, so is alternatingoff the loops (Lemma 4 . l , l be the split loop with l ∈ Y and l ∈ Z . Thus the actionof l , a is alternating on Y and hence on X . But the action of a , a is alternating whilst a , l is id on Z and hence X , which is a contradiction of the fact that the transpositions ( a a ),( a l ) and ( l l ) generate the symmetric group of a , a , l , l .So we conclude a ∈ Y and proceed to: Case:
Suppose one loop is split. Then | Y | = 2 (cid:96) + 4 and so 2 j − ≤ (cid:96) < j −
2, but onlyone loop was split so by stability considerations, one must go on Z , hence (cid:96) = 2 j −
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 , l i , l h , l h } must act by the identity on X , (since we can switch l i , l h adjacent to Z ). Thiscontradicts the definition of x j +1 which says that ( l i l h )( l i l h ) must act by − | Y | = 2 (cid:96) + 3 and so 2 j − ≤ (cid:96) ≤ j −
2. But if (cid:96) = 2 j −
2, then the vertex adjacent to Z would be unstable. So the only remaining possibilityis that (cid:96) = 2 j −
3, which in turn implies that Z is a vertex of valence 3, genus 0 and adjacentto 1 loop. But such graphs are excluded from (cid:98) B ( H Lie ) by definition. We thus conclude Θ isempty, hence β j +1 ∈ (cid:98) B ( H Lie )(2 j + 1 ,
0) is not a boundary. (cid:3)
We remark that the subcomplex (cid:98) B ( H Lie )(2 j + 1 , ⊂ B ( H Lie )(2 j + 1 ,
0) does not split, and wedo not assert that β j +1 is a non-boundary when viewed in B ( H Lie )(2 j +1 , B ( H Lie )(2 j +1 , Corollary 4.8.
Let ξ ∈ (cid:98) B ( H Lie )(2 j + 1 , j +2 − s,s be a vector of positive internal degree s > .Then the projection of ∂ ( ξ ) to the θ j +1 summand is zero.Proof. The Lemma establishes the case s = 2 j . Suppose 0 < s < j . Without loss of generalitywe 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 j +1 summand only if it is possible tocontract a subgraph γ (cid:48) such that γ/γ (cid:48) ∼ = θ where γ (cid:48) has 2 j + 1 − s > γ (cid:48) to v , so the total genus of γ (cid:48) mustbe 1. By definition of H Lie , such an operation is non-trivial only if γ (cid:48) has first betti number1. These two conditions are true simultaneously only if each vertex of γ (cid:48) has genus 0, whichin turn implies that each vertex carries a label in some H ∗ (Γ ,m ), which is concentrated ininternal degree 0. The only other vertex of γ has genus 0 as well, hence such an element mustbe supported on internal degree 0. (cid:3) Co-operadic non-zero coefficient lemma.Lemma 4.9.
The differential of the wheel graph ∂ ( ω j +1 ) contains β j +1 with non-zero coeffi-cient. roof. By Lemma 4 .
4, it suffices to show that the projection of ∂ ( ω j +1 ) to the θ j +1 summand of B ( H Lie )(2 j + 1 , j +1 , j is non-zero. After Diagram 2 .
2, it is sufficient to show that compositionin the following diagram is not zero: ω j +1 ∈ B ( H Lie )(2 j + 1 , j +2 , ν θ (cid:15) (cid:15) ∂ (cid:47) (cid:47) B ( H Lie )(2 j + 1 , j +1 , jπ θ (cid:15) (cid:15) (cid:15) (cid:15) B ( H Lie )( θ j +1 ) j +1 ⊗ Aut ( θ ) K ∗ ( θ j +1 ) ¯ µ (cid:47) (cid:47) H Lie ( θ j +1 ) j ⊗ Aut ( θ ) K ∗ ( θ j +1 ) (4.4)By definition, ν θ ( ω j +1 ) is determined by summing over ways to nest the graph ω j +1 suchthat collapsing nests gives the graph θ j +1 . Such a nesting specifies two induced graphs of ω j +1 , ˆ N and ˆ N which collapse to the two vertices v and v of θ j +1 . Since the vertex v is oftype (0 , N must be a corolla and N must consist of a lone vertex.Given such an N , there is a unique such N for which the composite with µ X is non-zero. It isgiven by the unique 2 j +1-gon N which is a subgraph of ω j +1 and which misses a distinguishedvertex N (right hand side of Figure 1).When j = 1, a direct computation shows that the four choices for a lone vertex N are sentby ν θ to the same element, hence ν θ (cid:54) = 0. Indeed, with the convention that the edges of thenested 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 (Figure 1) to decompose, and thesepermutations are all odd. Then since α = x , µ X is simply the Massey product which contractsthe triangle, which is not zero. Whence the case j = 1.So we now assume j >
1. In this case there are 2 j + 1 choices for such an N , correspondingto the outer vertices of ω . Therefore, ν θ ( ω ) is a sum of 2 j + 1 terms, corresponding to thechoice of an outer vertex and the complimentary 2 j + 1 gon. Since these terms are related byan automorphism of ω , it is enough to show that any one of them is non-zero when composedwith µ X .So let us fix such an N and N . The term in the sum ν θ ( ω ) corresponding to this choice ofnesting is given by choosing an isomorphism ω/N ∼ = θ , which in turn specifies a labeling of theflags of ˆ N by the set X := a − ( v ). We import the notation X = a − ( v ) = { a , a , a , (cid:96) i , (cid:96) i | ≤ i ≤ j − } from the proof of Proposition 4 .
7. Since ω has a unique non-trivalent vertex, so does ˆ N . Theconditions on this X -labeling of the flags of ˆ N coming from the isomorphism ω/N ∼ = θ arethat the non-trivalent vertex of ˆ N 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 havethe other a labels (Figure 5). Let P ⊂ B ( H Lie )(1 , X ) be the span of such X -labeled, 2 j + 1-gons.In particular dim ( P ) = 3!(2 j − j − /
2. We conclude that the bottom row of the diagram issupported on the restriction ν θ ( ω j +1 ) | P ∈ ( Com (2) ⊗ P ) ⊗ Aut ( θ ) K ∗ ( θ j +1 ) ⊂ B ( H Lie )( θ j +1 ) ⊗ Aut ( θ ) K ∗ ( θ j +1 )where Com (2) labels v and P labels v .The bottom row in Diagram 4 . X -labeled graph at the vertex v of θ j +1 , via the associated Massey product. It thus remains to show that contraction of such X -labeled polygons ( Com (2) ⊗ P ) ⊗ K ∗ ( θ j +1 ) µ X → H Lie ( θ j +1 ) j ⊗ K ∗ ( θ j +1 )is non-zero upon passage to Aut ( θ )-coinvariants. Since this map is Aut ( θ ) invariant, it is enoughto know that the class x j +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 j +1 is defined by choosing a total order on the set X , which fixesan isomorphism H j (Γ , j − ) ∼ = H j (Γ ,X ) (see Equation 4 . x j +1 ∈ H j (Γ , j − ) for the image of x j +1 under this isomorphism. Note that since x j +115 a1a 3a1 23456 7 l l l l l
2a 3a1a l l l l l l l ZY {{ Figure 5.
At left, the vector ρ in the case j = 3 is a graph with 11 legs whoseunderlying leg free graph is a 7-gon. The image of ρ under the heptagonal Masseyproduct is determined by first contracting the heptagon in the center to the VCDclass α ∈ H (Γ ,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, usingthe bijection X ∼ = { , . . . , j − } , we may view P as carrying numerical labels, and it sufficesto show the contraction P ⊂ B ( H Lie )(1 , j − → H j (Γ , j − )surjects onto the span of x j +1 . To be pedantic, the bijection X ∼ = { , . . . , j − } sends a i (cid:55)→ i for 1 ≤ i ≤ (cid:96) ir (cid:55)→ i + 1) + ( r − i + 1) , i + 1) + 1with 1 ≤ i ≤ j −
2, a loop, called loop i or the i th loop. We say a representative of a loop isone of its entries.The contraction map P → H j (Γ , j − ) is given by the Massey product which contracts eachsuch 2 j + 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 2 j +1-gon, and then contractingthe expanded edge (Figure 5). In other words, this map factors as P → Ind S j − S j − × S j ( H (Γ , j ) ⊗ H j (Γ , j +1 )) ◦ e −→ H j (Γ , j − ) (4.5)Notice the map ◦ e is surjective, since it is non-zero and the target is irreducible.We now analyze ◦ − e ( x j +1 ). Recall x j +1 spans a representation of Aut ( θ ) for which S { , , } acts by the alternating representation, transpositions of representatives of a loop acts by theidentity and where each (45)(2 i, i + 1) with 1 < i − ≤ j − − Ind S j − S j − × S j ( H (Γ , j ) ⊗ H j (Γ , j +1 )) has a basis given by i ∧ i ∧ · · · ∧ i j withwhere 1 ≤ i < i < ... < i j ≤ j −
1. The S j − action is by permutation, with permutation ofwedge products acting by the sign of the permutation. Write a preimage of x j +1 in this basis: (cid:88) ≤ i <...
2. Likewise, c ( i , . . . , i j ) = 0 if it lists both representatives ofany loop since transposition of loop representatives acts by the identity.To convey additional conditions that the coefficients in Equation 4 . I ⊂ { , . . . , j − } we define S I to be the set of lists ofrepresentatives of those loops not appearing in I , such that no list has two representatives of he same loop. In particular, each list has 2 j − − | I | entries and there are 2 (2 j − −| 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 (cid:126)s ∈ S I . Byabuse of notation we also write (cid:126)s for the associated wedge product. We write (cid:126)s ⊥ for the listof complimentary representatives. For example if j = 2 then S = { (4 , , (5 , , (4 , , (5 , } , S = { (4) , (5) } , S = 4 ∧ ∧ ∧ ∧ S = 4 ∧
5. If (cid:126)s = (4 ,
6) then (cid:126)s ⊥ = (5 , (cid:126)s = 4 ∧ .
6. First, c (1 , , , (cid:126)a ) = ( − i − c (1 , , ,(cid:126)b ) if (cid:126)a ∈ S and (cid:126)b ∈ S i . This condition is forced by the equivariance of ◦ e . Transposing representatives of a loop acts bythe identity, so the coefficient is independent of choice of list in S i (resp. S ). On the other hand,a list in S is missing one loop, namely { , } , and the permutation (45)(2( i + 1) , i + 1) + 1)acts by − i >
1. The result may be compared with a list in S i by applying an i − − i , hence the claim.Second, consider coefficients whose index has exactly two of 1,2,3. The condition that bothrepresentatives of a loop can’t appear in a wedge, means that wedges must have one represen-tative from each loop. This gives conditions: c (1 , , (cid:126)a ) = − c (1 , , (cid:126)a ) = c (2 , , (cid:126)a ) and c (1 , , (cid:126)a ) = c (1 , ,(cid:126)b ) for each (cid:126)a,(cid:126)b ∈ S . Define w, v ∈ Ind S j − S j − × S j ( H (Γ , j ) ⊗ H j (Γ , j +1 )) by w = (cid:88) ≤ i ≤ j − ( − i ∧ ∧ ∧ S i and v = 1 ∧ ∧ S − ∧ ∧ S + 2 ∧ ∧ S. (4.7)The above conditions on coefficients show that a vector in ◦ − e ( x j +1 ) must be a linear combi-nation of w and v . To find this linear combination, we invoke Lemma 3 . id + (1 , j ) + (2 , j ) + · · · + (2 j − , j ))2 j ∧ j + 1 ∧ · · · ∧ j − ◦ e . Consequently any permutation of this relation is in ker ( ◦ e ) as well. Usingthis 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: ∧ ∧ S ∼ − ∧ ∧ S ∼ ∧ ∧ S ∼ − ∧ ∧ ∧ S Proof:
Let’s prove that 2 ∧ ∧ S ∼ − ∧ ∧ ∧ S , with the other two following similarly.2 ∧ ∧ S is a sum of (2 j − wedge products of length 2 j . Each of the terms in the sum have 4or 5 appearing once in the third position, and 1 does not appear at all. Hence terms appearingin this sum can be paired to write: 2 ∧ ∧ S = (cid:80) (cid:126)b ∈ S ∧ ∧ (4 ⊕ ∧ (cid:126)b . To each such (cid:126)b weapply a permutation of the relation in Equation 4 . ∧ ∧ ∧ (cid:126)b to find:2 ∧ ∧ ⊕ ∧ (cid:126)b = ((14) ⊕ (15))2 ∧ ∧ ∧ (cid:126)b ∼ − ∧ ∧ ∧ (cid:126)b − ∧ ∧ ( ⊕ b ∈ (cid:126)b ⊥ b ) ∧ (cid:126)b The sum over all (cid:126)b of − ∧ ∧ ( (cid:80) b ∈ (cid:126)b ⊥ b ) ∧ (cid:126)b vanishes since each term has two representatives ofone loop, and hence pairs with the term in which the representatives appear in the transposedorder. Combining the previous two equations with this observation proves the claim. Claim 2: ( − i − ∧ ∧ ∧ S i ∼ ∧ ∧ ∧ S . Proof:
The claim is vacuous for i = 1, so fix 1 < i ≤ j −
2. Then each term in 1 ∧ ∧ ∧ S i has 4 or 5 appearing in it. Apply the relations to each term with a 4 appearing:1 ∧ ∧ ∧ ∧ (cid:126)b ∼ − ∧ ∧ ∧ (5 ⊕ i + 1) ⊕ (2( i + 1) + 1) ⊕ ( ⊕ b ∈ (cid:126)b ⊥ b )) ∧ (cid:126)b for each (cid:126)b ∈ S { ,i } . The terms replacing 4 with 5 cancel with the terms in which 5 originallyappeared. The terms where both representatives of a loop appear cancel in pairs. The remainingterms replace 4 with a representative of loop i . To compare these terms with 1 ∧ ∧ ∧ S , itsuffices to permute the order of the representatives of the loops into numerical order. This is one via a cycle of length i −
1, so produces a factor of ( − i − . Combining this with the factorof − j − v − w ∈ ker ( ◦ e ). From this calculation we concludethat ◦ e ( v ) (cid:54) = 0, since if it were zero, it would imply ◦ − e ( x j +1 ) = 0, but iteration of Lemma 3 . v is in the image of P in Equation 4 .
5. Indeedthe map P → Ind S j − S j − × S j ( H (Γ , j ) ⊗ H j (Γ , j +1 ) sends each X -labeled 2 j + 1-gon to a wedgecontaining two of { , , } and exactly one representative of each loop. The vector v is definedas a linear combination of such wedge products, so it is in the image. In particular, we haveshown the contraction map P → H Lie (1 , X ) surjects onto the span of x j +1 , from which thestatement follows. (cid:3) Proof of Main Results.
To conclude, we observe how our main results stated in theintroduction follow by linear dualizing the results of this section.
Proof of Theorem . . As above (cid:98) B denotes the subcomplex of the co-Feynman trans-form with no simple loops. Dualizing the inclusion (cid:98) B ( H Lie ) (cid:44) → B ( H Lie ), we have a projec-tion ft ( H Lie ) (cid:16) ¯ ft ( H Lie ) which quotients by simple loops. Since β j +1 spans the θ j +1 sum-mand of B ( H Lie )(2 j + 1 , j +1 , j , we may define its characteristic functional η j +1 := β ∗ j +1 ∈ ft ( H Lie )(2 j + 1 , η j +1 ∈ ¯ ft ( H Lie )(2 j + 1 ,
0) forits image under projection. Corollary 4 . ∂ (ker( (cid:98) B ( H Lie ) (cid:16) (cid:98) B ( Com )))has a non-zero coefficient of β j +1 . Therefore d (¯ η j +1 )(ker( (cid:98) B ( H Lie ) (cid:16) (cid:98) B ( Com ))) = 0, fromwhich we conclude d (¯ η j +1 ) ∈ ¯ ft ( Com ). Finally, projecting d (¯ η j +1 ) along ¯ ft ( Com )(2 j + 1 , (cid:16) Σ − j − GC j +12 , we find d (¯ η j +1 )( ω j +1 ) (cid:54) = 0 from Lemma 4 . Proof of Corollary . . The wheel graph ω j +1 ∈ Σ j +2 GC ∗ is a cycle, it remains to seethat it can’t be a boundary. Suppose that it were a boundary, so that d GC ∗ ( ξ ) = ω j +1 . Thevector ξ has a canonical pre-image ξ ∈ ˆ B ( H Lie ) for which ∂ ( ξ ) = ∂ ( ξ ) + ∂ > ( ξ ) = ω j +1 + termsof higher internal degree. Since ∂ ( ξ ) = 0, it must be the case that the non-zero scalar multipleof β j +1 appearing in ∂ ( ω j +1 ) (after Lemma 4 .
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