Oriented hairy graphs and moduli spaces of curves
aa r X i v : . [ m a t h . QA ] M a y ORIENTED HAIRY GRAPHS AND MODULI SPACES OF CURVES
ASSAR ANDERSSON, THOMAS WILLWACHER, AND MARKO ˇZIVKOVI ´CA bstract . We discuss a graph complex formed by directed acyclic graphs with external legs. This complex comes inparticular with a map to the ribbon graph complex computing the (compactly supported) cohomology of the modulispace of points M g , n , extending an earlier result of Merkulov-Willwacher. It is furthermore quasi-isomorphic to thehairy graph complex computing the weight 0 part of the compactly supported cohomology of M g , n according to Chan-Galatius-Payne. Hence we can naturally connect the works Chan-Galatius-Payne and of Merkulov-Willwacher andthe ribbon graph complex and obtain a fairly satisfying picture of how all the pieces and various graph complexes fittogether, at least in weight zero.
1. I ntroduction
It has been shown recently by Chan, Galatius and Payne [3, 4] that the ”commutative” Kontsevich graphcomplex computes the top weight part of the cohomology of the moduli spaces of curves. A related, albeit weakerresult had been shown by Merkulov and Willwacher [14], who constructed a map between the Kontsevich graphcomplex and the Kontsevich-Penner ribbon graph complex, which also computes the cohomology of the modulispace. The purpose of the present note is to generalize the result of Merkulov-Willwacher, slightly simplify theresult of Chan-Galatius-Payne and connect the two results.To this end, one main ingredient is the “hairy” graph complex
HOGC Sn whose elements are formal series ofisomorphism classes of directed acyclic graphs with two kinds of vertices: • Internal vertices are at least bivalent and have at least one outgoing edge. There are no passing vertices,i.e. bivalent vertices with one incoming and one outgoing edge. • External vertices are 1-valent sinks, i.e. they have one incoming edge. These vertices are “distinguishable”and identified with the elements of a given finite set S .An external vertex together with the edge attach to it can be considered as a “hair” on an internal vertex, hence thename of the complex. We also call it “leg”. Here are some typical such graphs:(1) 1 ,
12 3 ,
12 3 for S = { } and S = { , , } .The di ff erential δ on HOGC Sn is given by splitting vertices. For a more precise definition of HOGC Sn and sign anddegree conventions we refer to Section 3.4 below.The graph complex HOGC Sn can be connected to several objects and constructions in the literature. First, onemay consider the ribbon graph complex RGC S that originates from work of Penner [16] and Kontsevich [10]whose genus g piece B g RGC S computes the compactly supported cohomology of the moduli space M g , S of genus g curves with | S | marked points labelled by the set S , H • + | S | (cid:16) B g RGC S (cid:17) = H • c (cid:16) M g , S (cid:17) .We show in Section 5 below that there is a natural map of complexes HOGC S → RGC S that sends graphs of loop order g to genus g ribbon graphs. We conjecture that our map induces an injection incohomology.On the other hand, our graph complex HOGC S is closely related to its undirected analog HGC S , defined in thesame manner, except that the graphs are undirected, and all internal vertices need to be at least 3-valent. Sometypical such graphs are as follows: Mathematics Subject Classification.
Key words and phrases.
Graph complexes. , , ,
12 3 for S = { } , S = { , } and S = { , , } .Chan, Galatius and Payne have recently shown the following result. Theorem 1 (Chan, Galatius, Payne [4]) . The weight 0 summand of the compactly supported cohomology of themoduli space W H c ( M g , S ) (with g + | S | ≥ ) is computed by the g-loop part B g HGC S of the graph complex HGC S , H • (cid:16) B g HGC S (cid:17) (cid:27) W H •−| S | c (cid:16) M g , | S | (cid:17) . We will give a simplified proof of this Theorem in Section 6 below.The main technical result of this paper is then the following.
Theorem 2.
There is an explicit, combinatorially defined quasi-isomorphismF : HOGC Sn + → HGC Sn which is natural in S and respects the grading by loop order on both sides. We then conjecture that the following diagrams commute, possibly up to multiplying F by a conventionalscalar: H (cid:16) B g HGC S (cid:17) H c (cid:16) M g , S (cid:17) [ −| S | ] H (cid:16) B g HOGC S (cid:17) H (cid:16) B g RGC S (cid:17) (cid:27)(cid:27) F , Here the upper horizontal arrow is the map of Chan-Galatius-Payne.Theorem 2 above also nicely connects with various previous results on graph complexes with hairs. First,we may symmetrize or antisymmetrize over permutations of labels on external vertices to obtain the hairy graphcomplexes
HGC m , n which compute the rational homotopy groups of the space of long embeddings of R m into R n ,see [2, 7]. HGC m , n = Y r > (cid:16) HGC { ,..., r } n ⊗ Q [ − m ] ⊗ k (cid:17) S r [ m ]These complexes have a natural dg Lie algebra structure, realizing the Browder brackets on the embeddingspaces’ homotopy groups. Furthermore, for n = m there is a canonical Maurer-Cartan element m = ∈ HGC n , n , that corresponds to the identity map R n → R n . The corresponding twisted complex (cid:0) HGC n , n , δ + [ m , − ] (cid:1) isessentially quasi-isomorphic to the non-hairy graph complex GC n up to degree shift. More precisely, the map(essentially inducing a quasi-isomorphism) GC n [ − → (cid:0) HGC n , n , δ + [ m , − ] (cid:1) is obtained by attaching one hair to a non-hairy graph, see [21] for a detailed discussion.The above construction may be paralleled for the directed acyclic hairy graph complexes HOGC Sn . By (anti)-symmetrizing over hairs we obtain graph complexes HOGC m , n , see Section 3.4 for details. They also carry naturaldg Lie algebra structures. Furthermore, there is a natural Maurer-Cartan element m = X k ≥ k ! ... | {z } k × ∈ HOGC n , n + . Twisting by this element we obtain a complex which is essentially quasi-isomorphic to the non-hairy orientedgraph complex
OGC n + of [23] (see also Proposition 10 below). More precisely, one has a map of complexes OGC n + [ − → ( HOGC n , n + , δ + [ m , − ])obtained by summing over all ways of attaching hairs to a non-hairy graph, see section 3.4.6 below.Now it follows from Theorem 2 and the construction of the map therein that HGC m , n and HOGC m , n + arequasi-isomorphic as complexes. This result can be strengthened so as to cover also the dg Lie structures. heorem 3. The maps of Theorem 2 induce a quasi-isomorphism of dg Lie algebras
HOGC m , n + → HGC m , n . In the case m = n this morphism takes the canonical MC element m ∈ HOGC n , n + to the MC element m ∈ HGC n , n , and furthermore makes the following diagram of complexes commute up to homotopy in loop orders ≥ . (3) OGC n + [ −
1] (
HOGC n , n + , δ + [ m , − ]) GC n [ −
1] (
HGC n , n , δ + [ m , − ]) ≃≃ ≥ ≃≃ ≥ . The left-hand vertical map is the quasi-isomorphism of [26] (up to a conventional prefactor, see section 3.3.2 be-low), and the lower horizontal map has been described in [22, 21] (see also Proposition 7). All arrows preserve theloop order. The arrows labelled ≃ are quasi-isomorphisms, and the arrows labelled ≃ ≥ are quasi-isomorphismsin loop orders ≥ . Next, similar constructions for ribbon graphs have been described in [14]. Concretely, by antisymmetrizingover the punctures one obtains the ribbon graph complex
RGC computing the antisymmetric parts of the com-pactly supported cohomologies of the moduli spaces of curves. It has been shown in [14] that RGC also has a dgLie algebra structure, and there is also a canonical Maurer-Cartan element m = ∈ RGC . This Maurer-Cartan element gives rise to a twisted di ff erential on RGC that has first been considered by T.Bridgeland to our knowledge. A conjecture by A. Cˇaldˇararu (see Conjecture 32 below) states that the twistedcomplex ( RGC , δ + [ m , − ]) computes the compactly supported cohomology of the moduli space without markedpoints M g , up to a degree shift.Finally, there is previous work of Chan-Galatius-Payne [3] connecting the non-hairy graph cohomology H ( GC )with the cohomology of the moduli space without marked points M g . Concretely, they show the following. Theorem 4 (Chan, Galatius, Payne [3]) . The weight 0 summand of the compactly supported cohomology of themoduli space W H c ( M g ) is computed by the g-loop part of the graph complex GC ,H (cid:16) B g GC (cid:17) (cid:27) W H c ( M g ) . The above results of Chan-Galatius-Payne and Merkulov-Willwacher together can then nicely be fit into acommutative diagram, and thus one obtains a relatively satisfying picture of all objects and morphisms involvedand their relations, albeit with some conjectural components pertaining to the ribbon graphs. For the detaileddiscussion we refer to Section 6 below.
Outline of the paper.
After some preliminary recollections in section 2 we discuss the definitions of variousgraph complexes in section 3, including the new complexes
HOGC Sn . The proof of the main Theorem 2 is thengiven in section 4, see in particular section 4.1.5 for an explicit combinatorial description of the map F of Theorem2. Section 5 discusses the connection to the ribbon graph complex and the work of Merkulov-Willwacher [14].Finally section 6 is concerned with the relation to the results of Chan-Galatius-Payne [3, 4]. In particular in section6.4 we draw a picture of how everything is connected (with a conjectural component).Let us also remark that we sometimes neglect a discussion of the genus g ≤ Acknowledgements.
The authors heartily thank Sergei Merkulov for his support and valuable discussions. Wealso thank A. Cˇaldˇararu and A. Kalugin for their input.T.W. has been supported by the ERC starting grant 678156 GRAPHCPX, and the NCCR SwissMAP, fundedby the Swiss National Science Foundation. 2. P reliminaries
Notation and conventions.
We usually work over a field K of characteristic zero, so vector spaces, algebrasetc. are implicitly understood to be over K . Furthermore, the phrase ”di ff erential graded” is abbreviated dg andoften altogether omitted since most objects we study are enriched versions of di ff erential ( Z -)graded vector spaces.We generally use cohomological conventions, so that di ff erentials have degree +
1. For V a dg vector space we enote by V [ k ] the dg vector space in which all degrees have been shifted down by k units. In other words, if x ∈ V has degree d , then the corresponding element of V [ k ] has degree d − k .For brevity, we will denote by [ r ] : = { , . . . , r } the set of numbers from 1 to r .We will use standard combinatorial terms for graphs. For example, a graph is directed if a direction is assignedto each edge and directed acyclic if one cannot inscribe a (nontrivial) directed closed path in the graph, alwaysfollowing the edge directions. A vertex that has only incoming edges is called a sink (or a target), and a vertexthat has only outgoing edges is called a source. The number of edges incident at a vertex is the valency of thevertex. We call an edge between a vertex and itself a tadpole. (Generally speaking, we always allow tadpoles inour graphs.) Finally, some types of graphs may also have external legs, which we also call hairs.2.2. Modular operads and the Feynman transform.
We shall use the notion of modular operad from [5] whichwe briefly recall. A stable S -module M is a collection of right S r -modules M ( g , r ) for g , r ≥ g + r ≥ . Informally, we shall think of r as ”the number of inputs” to some operation and g as a placeholder for the genus.Instead of considering the collection of M ( g , r ) we may equivalently consider a functor S → M ( g , S ) on thegroupoid of finite sets with bijections. In other words, we label our inputs by some finite set instead of thenumbers 1 , . . . , r . We shall freely switch between both conventions.For a stable S -module M and Γ a graph with external legs indexed by S and each vertex x labelled by a number g x ≥ ⊗ Γ M = ⊗ x ∈ V ( Γ ) M ( g x , star( x )) . Here V ( Γ ) is the vertex set of Γ and star( x ) is the set of half-edges incident at x .A modular operad is then a stable S -module M together with composition morphisms ⊗ Γ M → M ( g Γ , S )for each graph Γ as above with g Γ = P x g x + b ( Γ ), and b ( Γ ) the number of loops of Γ . The compositionmorphisms are required to satisfy natural coherence (”associativity”) axioms. One of them is equivariance withrespect to isomorphisms of graphs.Furthermore, one can define certain twisted versions of modular operads, by twisting the aforementioned equiv-ariance condition (essentially) by a representation of the groupoid of graphs. For details on modular D -operads(for D a hyperoperad) we refer to [5, sections 4.1, 4.2]. Dually, one obtains the notion of a modular ( D -)cooperad.To a stable S -module M we associate a corresponding free modular operad F ( M ), or more generally the freemodular D -operad F D ( M ). In the category of dg vector spaces F ( M )( g , r ) is a dg vector space spanned by iso-morphism classes of decorated graphs Γ with r legs and each vertex x decorated by an element of M ( g x , | star( x ) | ),with g = b ( Γ ) + P x g x .The D -Feynman transform of the modular D -operad M is F D ( M ) = ( F D ∨ ( M ∗ ) , d ) , where D ∨ = D − ⊗ k and k is the hyperoperad given by the top exterior power of the vector space generated by theset of edges of graphs. (Think of each edge carrying an additional cohomological degree + ff erential produces precisely one additional edge, using a modular cocompositionof M ∗ on one vertex of our graph. For more details we refer to [5].2.3. PROPs and properads.
We will use the language of properads, and in particular deformation complexesassociated to properad maps. For an introduction we refer the reader to [13].We shall denote by
LieB the properadgoverning Lie bialgebras. A Lie bialgebra structure on a vector space V consists of a Lie algebra structure anda Lie coalgebra structure (both of degree 0) satisfying a natural compatibility relation, the Drinfeld five-termidentity. In the case that the composition of the cobracket and bracket V cobracket −−−−−−→ V ⊗ V bracket −−−−−→ V is zero, then the Lie bialgebra is called involutive. We denote the corresponding properad by ILieB . It comeswith a natural map
LieB → ILieB . Both of these properads are Koszul, and one can consider the resolutions
LieB ∞ ≃ −→ LieB and
ILieB ∞ ≃ −→ LieB , which are obtained as the properadic cobar constructions of the Koszul dualcoproperads.To simplify signs, we shall also consider a graded version, Λ LieB , for which the bracket and cobracket bothhave cohomological degree +
1, and are symmetric operations. More precisely, a Λ LieB -structure on the graded ector space V consists of a Lie algebra structure on V [1], and a Lie coalgebra structure on V [ − Λ ILieB and the corresponding resolutions Λ LieB ∞ , Λ ILieB ∞ . For more details andrecollections on these definitions we refer the reader to [14, section 2]. (There the notation LieB , is used in placeof Λ LieB .)Finally, we use the Frobenius properad
Frob and its involutive version
IFrob . They are defined such that
Frob ( r , s ) = IFrob ( r , s ) = K for all r ≥ s ≥
1. All composition morphisms of
Frob are the identity map. In
IFrob all genus zero compositionsare the identity map, while the higher genus compositions are defined to be zero. We will use that Λ LieB ∞ =Ω ( IFrob ∗ ), where Ω denotes the properadic cobar construction.3. G raph complexes In this section we recall the definition of various graph complexes. For those complexes that have already ap-peared elsewhere in the literature we just sketch the construction and provide some references. Graph complexesusually can be defined in (at least) two ways: Either completely combinatorially ”by hand” or by algebraic con-structions such as (pr)operadic deformation complexes. Both ways have their advantages and disadvantages, andwe shall provide or sketch both if available.3.1.
Complex of non-hairy undirected graphs.
Let us quickly recall the combinatorial description of the sim-plest graph complex GC n . Consider the set ¯ V v ¯ E e grac containing directed graphs that: • are connected; • have v > • have e ≥ n ∈ Z , let(5) ¯ V v ¯ E e G n : = D ¯ V v ¯ E e grac E [(1 − n ) e + nv − n ]be the vector space of formal linear combinations of ¯ V v ¯ E e ¯grac with coe ffi cients in K . It is a graded vector spacewith a non-zero term only in degree d = ( n − e − nv + n .There is a natural right action of the group S v × (cid:16) S e ⋉ S × e (cid:17) on ¯ V v ¯ E e ¯grac , where S v permutes vertices, S e permutes edges and S × e changes the direction of edges. Let sgn v , sgn e and sgn be one-dimensional representationsof S v , respectively S e , respectively S , where the odd permutation reverses the sign. They can be considered asrepresentations of the whole product S v × (cid:16) S e ⋉ S × e (cid:17) . Let us consider the space of invariants:(6) V v E e G n : = (cid:16) ¯ V v ¯ E e G n ⊗ sgn e (cid:17) S v × ( S e ⋉ S × e ) for n even, (cid:16) ¯ V v ¯ E e G n ⊗ sgn v ⊗ sgn ⊗ e (cid:17) S v × ( S e ⋉ S × e ) for n odd.Because the group is finite, the space of invariants may be replaced by the space of coinvariants. In any case theoperation of taking (co)invarints e ff ectively removes the edges directions and numberings of vertices and edges(up to sign). The underlying vector space of the graph complex is(7) G n : = M v ≥ , e ≥ V v E e G n . The di ff erential acts by edge contraction:(8) d ( Γ ) = X a ∈ E ( Γ ) Γ / a where E ( Γ ) is the set of edges of Γ and Γ / a is the graph produced from Γ by contracting edge a and merging itsend vertices.We may also define the dual complex(9) ( GC n , δ ) = ( G n , d ) ∗ . Here the di ff erential δ acts combinatorially by splitting a vertex, which is the operation dual to edge contraction.3.2. Complexes of undirected hairy graphs. .2.1. Description through modular operads and Feynman transform.
We consider the D n : = k − n -modular op-erad Com n such that Com n ( g , r ) = (sgn S ) ⊗ n [ − rn + n ] for g = r ≥
30 otherwise , where sgn S is the sign representation of the group of bijections of S . Then we define the graph complex HGC Sn = Y g F D n ( Com n )( g , S ) ⊗ sgn S [ −| S | ] . Combinatorially the elements of ths complex are series of graphs with | S | external legs (hairs) indexed by theelements of S . In the case of S = ∅ being the empty set one recovers the non-hairy graph complex of the previoussubsection GC n (cid:27) HGC ∅ n [ n ] . For n even we also consider D n -modular operads Com modn such that
Com modn ( g , r ) = (sgn S ) ⊗ n [ − rn + n − ng ] for 2 g + r ≥
30 otherwise , and define HGC nS , mod = Y g F ( Com n )( g , S ) ⊗ sgn S [ −| S | ] . Elements are now series of graphs with additional decorations of a number g x on each vertex x .We shall only need the complex HGC S , modn in the case of even n , and in fact n =
0, in this paper.
Lemma 5 ([4]) . For even n the inclusion
HGC Sn → HGC S , modn coming from the modular operad map Com modn → Com n is a quasi-isomorphism in genera ≥ , where the genus of a graph is the loop order plus the sum of thedecorations g x on vertices. In genus one the inclusion is a quasi-isomorphism if | S | ≥ .Proof. One may filter
HGC Sn and HGC S , modn by the total number of vertices in graphs and consider the correspond-ing spectral sequence. For HGC Sn the di ff erential always creates exactly one vertex, and hence the di ff erentialon the associated graded is zero. For HGC nS , mod there remains the piece of the di ff erential that introduces atadpole at a vertex x , and simultaneously reduces the decoration g x by one. One can easily check that the coho-mology is identified with graphs with no tadpoles and all decorations g x =
0. Hence on the E -pages the map HGC Sn → HGC
Sn mod becomes the map from the graph complex with tadpoles to that without tadpoles, sending alltadpoled graphs to zero. It is known that this is essentially a quasi-isomorphism. More precisely, if | S | > | S | =
1, say S = [1], the mapping cone has one-dimensional cohomology, spannedby the graph(10) ∈ HGC [1] n in genus 1. We refer to the proof of [22, Proposition 3.4] for the detailed argument. (cid:3) One also has a map of complexes in the other direction π : HGC S , modn → HGC Sn as follows: • A graph Γ ∈ HGC S , modn is sent to zero if there is a vertex x with g x ≥ | star( x ) | ≥
2, or with g x ≥ • If our graph is of the form (10), with any genus g x at the vertex, then we send it to zero. • A vertex x decorated by g x = g x = · · · 7→ · · · If no such adjacent vertex exists, the graph is sent to zero by the previous rule. • Otherwise the graph is sent to itself.
Lemma 6.
The map π : HGC S , modn → HGC Sn above (with n even) is a well defined map of complexes and aone-sided inverse to that of Lemma 5 in genera ≥ . roof sketch. It is clear that our map π is a one-sided inverse to that of Lemma 5, apart from the fact that the imageof the special graph (10) is sent to zero.It hence su ffi ces to check that π commutes with the di ff erentials, i.e., π ( δ Γ ) = δπ ( Γ ) for all graphs Γ ∈ HGC S , modn .Suppose first that Γ has a vertex x with genus decoration g x ≥
2, or with genus decoration g x = x . Then δ Γ is a linear combination of graphs with the same feature and hence ≥
2, and hence π ( δ Γ ) = δπ ( Γ ) = x of Γ has either g x = g x =
1, and in the latter case has no tadpoleat x . By the same argument, we also see that π ( δ Γ ) = δπ ( Γ ) = Γ has (at least) two distict vertices x , y with g x = g y =
1, and both having valency | star( x ) | , | star( y ) | ≥
2. If Γ has a single vertex x such that g x = | star( x ) | ≥
2, then π ( Γ ) =
0, so we need to check π ( δ Γ ) =
0. But the only terms produced by δ Γ which arepotentially not send to zero are schematically of the form g x = · · · δ −→ · · · + g x = · · · and cancel each other when mapped via π . Hence we can assume that the only vertices x of Γ that have g x = ff erentials follows from the following schematic graphical computation g x = · · · · · · + ( · · · ) · · · · · · + ( · · · ) δπ πδ g x = · · · · · · + g x = · · · + ( · · · )0 0 δπ πδ (cid:3) Remark 1.
The overall degree shifts in the definition of the above graph complexes are purely conventional. Theyindicate that we like to think of the edges as carrying degree n − Combinatorial description.
Let us quickly recall the combinatorial description of
HGC Sn . It is similar tothe combinatorial description of non-hairy graphs. Consider the set ¯ V v ¯ E e ¯ H S grac containing directed graphs that: • are connected; • have v > • have e ≥ • have | S | ≥ S .For some pictures of such graphs see (2).For n ∈ Z , let(11) ¯ V v ¯ E e ¯ H S G n : = D ¯ V v ¯ E e ¯ H S grac E [(1 − n ) e + nv ]be the vector space of degree shifted formal linear combinations. onsider again the action of the group S v × (cid:16) S e ⋉ S × e (cid:17) on ¯ V v ¯ E e ¯ H S grac . Let(12) V v E e ¯ H S G n : = (cid:16) ¯ V v ¯ E e ¯ H S G n ⊗ sgn e (cid:17) S v × ( S e ⋉ S × e ) for n even, (cid:16) ¯ V v ¯ E e ¯ H S G n ⊗ sgn v ⊗ sgn ⊗ e (cid:17) S v × ( S e ⋉ S × e ) for n odd.The underlying vector space of the hairy graph complex is(13) HG Sn : = M v ≥ , e ≥ V v E e ¯ H S G n . The di ff erential again acts by edge contraction:(14) d ( Γ ) = X a ∈ E ( Γ ) Γ / a , but here E ( Γ ) is the set of edges of Γ that are not connected to an external vertex, i.e. edges towards an externalvertex can not be contracted.The dual complex is(15) (cid:16) HGC Sn , δ (cid:17) = (cid:16) HG Sn , d (cid:17) ∗ , where the di ff erential δ acts combinatorially by splitting an internal vertex.3.2.3. Complexes with (anti-)symmetrized hairs.
We may (anti-)symmetrize external vertices, to make them (upto the sign) indistinguishable.Let S = { , . . . , s } and let S s (the group of bijections of S ) act on HG Sn by permuting external vertices. Let sgn s be one-dimensional representations of S s , where the odd permutation reverses the sign. For an integer m let(16) HG sm , n : = (cid:16) HG Sn ⊗ Q [ − m ] ⊗ k (cid:17) S s [ m ] for m even, (cid:16) HG Sn ⊗ Q [ − m ] ⊗ k ⊗ sgn s (cid:17) S s [ m ] for m odd.(17) HG m , n = M s > HG sm , n The di ff erential d is still contracting an edge. Again we also consider the dual complex(18) (cid:0) HGC m , n , δ (cid:1) = (cid:0) HG m , n , d (cid:1) ∗ . There is a dg Lie algebra structure on
HGC m , n defined as Γ ... , Γ ′ ... = X Γ ... Γ ′ ... − ( − | Γ || Γ ′ | X Γ ′ ... Γ ... , where the sum runs over all external vertices of one graph and over all ways of attaching its edge to internalvertices of another graph. | Γ | is the degree of Γ .Furthermore, for n = m there is a Maurer-Cartan element m : = ∈ HGC n , n , that can be used to twist the complex to ( HGC n , n , δ + [ m , · ]). There is morphism of complexes( GC n [ − , δ ) → (cid:0) HGC n , n , δ + [ m , − ] (cid:1) between the non-hairy graph complex GC n and the twisted complex obtained by attaching one hair to a non-hairygraph. More precisely, a graph Γ ∈ GC n [ −
1] is sent to the linear combination X v Γ v ∈ HGC n , n , where the sum is over the vertices of Γ and Γ v is obtained from Γ by attaching a hair at vertex v . The followingresult can be found in [22, 21]. Proposition 7.
The above map of complexes ( GC n [ − , δ ) → (cid:0) HGC n , n , δ + [ m , − ] (cid:1) induces an isomorphism incohomology in loop orders ≥ . .3. Complexes of directed acyclic non-hairy graphs.
In this and the next subsection we define, or recall thedefinition of several complexes of directed acyclic graphs. We call these complexes ”oriented graph complexes”to comply with the notation of the literature, and also to avoid confusion with the unrelated meaning of the term”acyclic” in homological algebra.Here we will define the complex of oriented non-hairy graphs. It is considered e.g. in [23] and [26].3.3.1.
Combinatorial description.
Consider the set ¯ V v ¯ E e O grac containing directed graphs that • are connected; • have v > • have e ≥ • have no passing vertices (2-valent vertices with one incoming and one outgoing edge ); • have no closed directed path along the directed edges ( directed cycles ).For n ∈ Z , let(19) ¯ V v ¯ E e OG n : = D ¯ V v ¯ E e O grac E [( n − e − nv + n ]be the vector space of degree shifted formal linear combinations.Unlike for non-oriented graphs, here we want to keep the direction of edges, i.e. we will not take the space ofinvariants under the action of changing the direction of an edge. Therefore, let us consider the action of the group S v × S e on ¯ V v ¯ E e ¯ H S O grac . Let(20) V v E e OG n : = (cid:16) ¯ V v ¯ E e OG n ⊗ sgn e (cid:17) S v × S e for n even, (cid:16) ¯ V v ¯ E e OG n ⊗ sgn v (cid:17) S v × S e for n odd,The underlying vector space of the oriented graph complex is given by(21) OG n : = Y v ≥ , e ≥ V v E e OG n . The di ff erential again acts by edge contraction:(22) d ( Γ ) = X a ∈ E ( Γ ) Γ / a where E ( Γ ) is the set of edges of Γ . If a directed cycle is produced, we consider the result to be zero.The dual complex is(23) ( OGC n , δ ) : = ( OG n , d ) ∗ . Here di ff erential δ acts by splitting a vertex.3.3.2. Comparison of complexes of oriented and undirected non-hairy graphs.
The following results shows thatthe graph complexes G n and OG n + are homologicaly essentially the same. Theorem 8 ([26], [27]) . For every n ∈ Z there is a morphism of complexesh : ( G n , d ) → ( OG n + , d ) that respects the gradings by loop order and that induces an isomorphism on cohomology in loop orders ≥ . The map from the theorem is defined as(24) h ( Γ ) : = g X x ∈ V ( Γ ) ( v ( x ) − X τ ∈ S ( Γ ) h x ,τ ( Γ ) , where sums go through all vertices x of Γ and all spanning trees τ of Γ , v ( x ) is the valence of x and g is the looporder (the first Betti number) of the graph.The graph h x ,τ ( Γ ) is the oriented graph obtained from Γ by giving to edges of τ the direction away from thevertex x , and replacing other edges with the structure . Detailed construction can be found in [26] or [27],and it is similar to our construction from Subsection 4.1. The biggest di ff erence is that here we have the extrasummation over all vertices. We note that we have included a conventional prefactor g here, that is not present in[26, 27]. Recall that g = e − v + Complexes of oriented hairy graphs. .4.1. Combinatorial description.
Consider the set ¯ V v ¯ E e ¯ H S O grac containing directed graphs that • are connected; • have v > • have e ≥ • have | S | ≥ S that are 1-valent targets, i.e. they haveone incoming edge attached; • have no internal targets (internal vertices without outgoing edge); • have no passing vertices (2-valent vertices with one incoming and one outgoing edge ); • have no closed directed path along the directed edges ( directed cycles ).For some pictures of such graphs see (1).For n ∈ Z , let(25) ¯ V v ¯ E e ¯ H S OG n : = D ¯ V v ¯ E e ¯ H S O grac E [( n − e − nv ]be the vector space of degree shifted formal linear combinations.Consider the action of the group S v × S e on ¯ V v ¯ E e ¯ H S O grac . Let(26) V v E e ¯ H S OG n : = (cid:16) ¯ V v ¯ E e ¯ H S OG n ⊗ sgn e (cid:17) S v × S e for n even, (cid:16) ¯ V v ¯ E e ¯ H S OG n ⊗ sgn v (cid:17) S v × S e for n odd,The underlying vector space of the oriented hairy graph complex is given by(27) HOG Sn : = Y v ≥ , e ≥ V v E e ¯ H S OG n . The di ff erential again acts by edge contraction:(28) d ( Γ ) = X a ∈ E ( Γ ) Γ / a where E ( Γ ) is the set of edges of Γ that are not connected to an external vertex.If a directed cycle is produced, weconsider the result to be zero.The dual complex is(29) (cid:16) HOGC Sn , δ (cid:17) : = (cid:16) HOG Sn , d (cid:17) ∗ . Here di ff erential δ acts by splitting an internal vertex.3.4.2. A version with input hairs.
We will also need to consider a slight variant
HHOG Sn of the complex HOG Sn above, obtained by changing the definition as follows: • A graph must have in addition to the output hairs labelled by S an arbitrary (positive) number of inputhairs. • The graphs must not have sources, i.e., internal vertices with only outgoing edges.Here is an example: 1Consider the two types of ’special’ vertices v for a graph Γ ∈ HHOG Sn ,(30) Γ \ v v r ... , and Γ \ v v ... , with either one ingoing internal edge, one outgoing hair and an arbitrary number of ingoing hairs, or one internaloutgoing edge and an arbitrary number of ingoing hairs. et d v be the operation of removing such special vertices d v Γ \ v v r ... : = Γ \ v r , and d v Γ \ v v ... : = Γ \ v . The di ff erential d on HHOG Sn acts by(31) d ( Γ ) = X e ∈ E ( Γ ) Γ / e − X v ∈ V ( Γ ) d v ( Γ ) . Dually, we again define (cid:16)
HHOGC Sn , δ (cid:17) : = (cid:16) HHOG Sn , d (cid:17) ∗ . It is clear that the di ff erential cannot change the loop order in graphs. Hence we get in particular a splitting ofcomplexes (cid:16) HHOGC Sn , δ (cid:17) (cid:27) Y g (cid:16) B g HHOGC Sn , δ (cid:17) , where we denote by B g HHOGC Sn ⊂ HHOGC Sn the loop order g subcomplex.3.4.3. Description as properadic deformation complex.
The oriented graph complexes are very closely connectedto properadic deformation complexes. For example, it has been shown in [15] that the complex
OGC n computes(essentially) the homotopy derivations of a degree shifted version of the Lie bialgebra properad. We can alsoidentify the complexes HHOGC Sn above with pieces of properadic deformation complexes. For our purposes herethis reformulation has the main advantage that we do not have to pay too close attention to combinatorial signsand prefactors, which are automatically handled due to generalities on deformation complexes.For general definitions and statements about deformation complexes we refer to [13] and [15, section 3], whoseconventions we shall follow. Let us only mention that if C is a cooperad, which we assume reduced in the sensethat C (1 , = Q , one can define a properad Ω ( C ) via the properadic cobar construction. Furthermore, if P isanother properad, then we can endow the graded vector space Y r , s ≥ S r × S r ( C ( r , s ) , P ( r , s ))with a dg Lie algebra structure, in such a way that the Maurer-Cartan elements are (essentially) in one-to-onecorrespondence with dg properad maps Ω ( C ) → P . Then, given a properad map f : Ω ( C ) → P we define thedeformation complex Def( Ω ( C ) f −→ P ) as the twist of the dg Lie algebra above by the Maurer-Cartan elementassociated to the map f .Now consider a properad AC such that, for r , s ≥ AC ( r , s ) : = Assoc ( r ) ⊗ Com ( s ) (cid:27) Assoc ( r ) (cid:27) K [ S r ] . The composition morphisms ◦ j : AC ( r , s ) ⊗ AC ( r , s ) → AC ( r + r − , s + s − • If r ≥ r ≥ ◦ j above is the zero morphism. • If r = r = AC ( r , s ) = K or AC ( r , s ) = K and we define the composition ◦ j to be theidentity morphism.We furthermore define all ”higher genus” compositions to be zero.The operadic ( r = AC is the commutative operad Com . In particular, we have maps ofproperads
Com → AC → Com . We also have properad maps
IFrob ∗ −→ AC ∗ −→ IFrob actoring through Com . (I.e., the maps are zero in output arity r ≥ Λ LieB ∞ = Ω ( IFrob ∗ ) → Ω ( AC ∗ ) → Ω ( IFrob ∗ ) = Λ LieB ∞ . Consider the resulting deformation complex (disregarding the di ff erential for now)Def (cid:18) Ω ( AC ∗ ) ∗ −→ Λ LieB ∞ (cid:19) = Y r , s Hom S r × S s (cid:0) AC ∗ ( r , s ) , Λ LieB ∞ ( r , s ) (cid:1) (cid:27) Y r , s ( Λ LieB ∞ ( r , s )) S s . Elements of Λ LieB ∞ ( r , s ) can graphically be considered as linear combinations of directed acyclic graphs with s numbered inputs and r numbered outputs. Hence the ( r , s ) piece of the above product is given by linear combi-nations of drected acyclic graphs with s unlabeled inputs and r numbered outputs. Furthermore, one can checkthat the di ff erential on the deformation complex is combinatorially just the dual version of the operation (31):The internal di ff erential on Λ Lie ∞ splits vertices and is dual to the first summand of (31), and the twist by theMaurer-Cartan element corresponding to the map ∗ is dual to the second summand in (31). It is hence also clearthat there are natural gradings by loop order of the graphs, and by the output arity r , and we can hence considerthe subcomplex of loop order g with r outputs B g Def (cid:18) Ω ( AC ∗ ) ∗ −→ Λ LieB ∞ (cid:19) r ⊂ Def (cid:18) Ω ( AC ∗ ) ∗ −→ Λ LieB ∞ (cid:19) . Then for g ≥ B g Def (cid:18) Ω ( AC ∗ ) ∗ −→ Λ LieB ∞ (cid:19) r (cid:27) B g HHOGC [ r ] , . Remark 2. • One may similarly construct a definition
HHOGC [ r ] , n for n , n = • Mind that we state (32) for loop orders g ≥
1. In loop order zero, there is a minor conventional di ff erencebetween both sides, in that the unit is an element in Λ LieB ∞ (1 ,
1) whereas the corresponding graph withno internal vertex is not contained in
HHOGC [ r ] , according to our conventions.Furthermore, one cannot immediately remove the ” B g ” on both sides of (32), because HHOGC [ r ] , contains potentially infinite series of diagrams whereas Λ LieB ∞ only contains linear combinations. Thisslight technical di ffi culty could be countered by taking the completion of Λ LieB ∞ by loop order, as isdone in [15].3.4.4. Comparison of both versions.
The following result shows that the graphs complexes
HOG Sn and HHOG Sn can be considered as ”the same object” homologically. Proposition 9.
For every S , n the map HHOG Sn → HOG Sn obtained by deleting all input legs from a graph induces an isomorphism in cohomology.Proof. The result is essentially [15, Proposition 4.1.2], expcept that there the output hairs are not numbered.However, numbering the output legs does not a ff ect the proof at all, so that one can obtain our result above just bycopy-pasting the argument. (cid:3) Complexes with (anti-)symmetrized hairs.
As in the non-directed case, we may (anti-)symmetrize externalvertices. Let S = { , . . . , s } and let S s act on HOG Sn by permuting external vertices. For an integer m let(33) HOG sm , n : = (cid:16) HOG Sn ⊗ Q [ − m ] ⊗ s (cid:17) S s [ m ] for m even, (cid:16) HOG Sn ⊗ Q [ − m ] ⊗ s ⊗ sgn s (cid:17) S s [ m ] for m odd.(34) HOG m , n = M s > HOG sm , n The di ff erential d is still contracting an edge. There is a dual(35) (cid:0) HOGC m , n , δ (cid:1) = (cid:0) HOG m , n , d (cid:1) ∗ . here is a similar Lie algebra structure on HOGC m , n defined as Γ ... , Γ ′ ... = X Γ ... Γ ′ ... − ( − | Γ || Γ ′ | X Γ ′ ... Γ ... , where the sum runs over all external vertices of one graph and over all ways of attaching its edge to internalvertices of another graph.Here, for m = n − m = X k ≥ k ! ... | {z } k × ∈ HOGC n , n + . that can be used to twist the complex to ( HOGC n , n + , δ + [ m , · ]).3.4.6. Comparison of complexes of oriented hairy and non-hairy graphs.
We recall from [15] that one can definea map of complexes between the hairy and nonhairy oriented graph complexes(
OGC n , δ ) → (cid:0) HOGC n − , n , δ + [ m , · ] (cid:1) by sending a graph Γ ∈ OGC n to the infinite series of graphs obtained by attaching hairs to vertices in all possibleways, schematically Γ X k ≥ k ! Γ ... | {z } k × ∈ HOGC n , n + If the graphs produced on the right have sinks, thus violating the conditions of section 3.4.1, they are dropped.So at least one hair needs to be added to each sink of Γ . Furthermore, it is clear that the map respects the looporder gradings on both sides. From [15] we cite the following result. Proposition 10 (Proposition 3 of [23] or Proposition 4.1.1 of [15]) . The map ( OGC n , δ ) → ( HOGC n − , n , δ + [ m , · ]) above is a quasi-isomorphism in loop orders ≥ . We shall elaborate a bit further on the result. Every oriented non-hairy graph needs to have a target, and wemay construct a filtration of the complex on the number of targets. The di ff erential graded complex, called fixedtarget graph complex , is ( OGC n , δ ) where δ is the part of the di ff erential that does not change the number oftargets (sinks). It, or rather its isomorphic version with sources instead of targets, is considered in detail in [1].There associated graded of the above map is then(36) ( OGC n [ − , δ ) ֒ → ( HOGC n − , n , δ )where a hair is attached to every target. Proposition 10 can then be seen as an immediate Corollary of the followingresult. Proposition 11.
For every n the inclusion ( OGC n [ − , δ ) ֒ → ( HOGC n − , n , δ ) is a quasi-isomorphism.Proof. Let Γ ∈ HOGC n − , n . An internal vertex in Γ is called “bad vertex” if it shares an edge with an externalvertex and it has more than one outgoing edges. The one going to the external vertex has to be outgoing.The image of the inclusion is exactly the sub-complex spanned by graphs with no bad vertices. It is enough toprove that the quotient spanned by graphs with at least one bad vertex is acyclic.On that quotient let us make a spectral sequence on the number of internal vertices that are not 2-valent badvertices. Those vertices are 2-valent sources with one edge heading towards an external vertex. On thefirst page of the spectral sequence there is the di ff erential δ that produces such a vertex. It is produced by splittinga bad vertex that was more than 2-valent.There is a homotopy h that contracts the edge of a 2-valent bad vertex that does not head towards the externalvertex. One easily checks that h δ + δ h = c id where c is the number of bad vertices. This implies that thecohomology on the first page of the spectral sequence is zero. After splitting the complexes into complexes withfixed loop order, standard spectral sequence arguments imply that the spectral sequence converges correctly, sothe result follows. (cid:3) .5. Skeleton version of
HOG . For later proofs it will be convenient to consider an alternative definition of thecomplex of directed acyclic graphs (cid:16)
HOG Sn , d (cid:17) introduced above. More precisely, let us define the graph complex (cid:16) HO sk G Sn , d (cid:17) (cid:27) (cid:16) HOG Sn , d (cid:17) as follows, using results from from [27, Section 2]:Recall the set ¯ V v ¯ E e ¯ H S grac from Subsection 3.2.2. In this context, those graphs are called core graphs . Thedirection of an edge in a core graph is called core direction . To each edge of a core graph we attach an “edge type”from σ : = { [ n − , [ n − , [ n − } . The direction of elements of σ are called type directions . Forand they go along core direction, and for it is the opposite of core direction. Admissible graphs arethose graphs with edge types such that: • every internal vertex has an attached edge of type or with type direction away from the vertex(they are not type-targets); • an edge adjacent to an external vertex is not of type or with type direction going away from theexternal vertex (external vertices are type-targets); • there are no closed paths along edges of type or along type directions ( type-directed cycles ).The set of all admissible graphs is denoted by ¯ V v ¯ E e ¯ H S O sk grac ⊂ ¯ V v ¯ E e ¯ H S grac × σ × e . For n ∈ Z , let(37) ¯ V v ¯ E e ¯ H S O sk G n : = D ¯ V v ¯ E e ¯ H S O sk grac E [ − nv ] . Note that admissible graphs already have degrees that come from degree of edge types in σ .Similarly as before, there is a natural right action of the group S v × (cid:16) S e ⋉ S × e (cid:17) on ¯ V v ¯ E e ¯ H S O sk G n , where S v permutes vertices, S e permutes edges, and S × e changes the core direction of edges and changes edge types as(38) ↔ ,
7→ − ( − n . Recall that sgn v and sgn e are one-dimensional representations of S v , respectively S e , where the odd permutationreverses the sign. They can be considered as representations of the whole product S v × (cid:16) S e ⋉ S × e (cid:17) . Let(39) V v E e ¯ H S O sk G n : = (cid:16) ¯ V v ¯ E e ¯ H S O sk G n ⊗ sgn e (cid:17) S v × ( S e ⋉ S × e ) for n even, (cid:16) ¯ V v ¯ E e ¯ H S O sk G n ⊗ sgn v (cid:17) S v × ( S e ⋉ S × e ) for n odd,(40) HO sk G Sn : = Y v ≥ , e ≥ V v E e ¯ H S O sk G n . There are two di ff erentials. The core di ff erential d C contracts edges of type and that connect twointernal vertices. The edge di ff erential d E changes type of edges as(41)
7→ − ( − n , summed over all edges, including those connecting external vertices. If such operation produces a type-directedcycle or makes an external edge a type-target, we consider the result to be zero. The total di ff erential is(42) d : = d C + ( − n deg d E . This complex (cid:16) HO sk G Sn , d (cid:17) is designed to be isomorphic to the original version (cid:16) HOG Sn , d (cid:17) . In short, at least3-valent vertices and 2-valent sources in a graph in HOG Sn are called skeleton internal vertices . Strings of edgesand vertices between two skeleton internal vertices or external vertices have to be in the set { , , } ,and they are called skeleton edges . A corresponding graph in HO sk G Sn is the one with skeleton internal vertices asinternal vertices, external vertices remaining the same, and skeleton edges as edges, where is mapped to. One can check that the degrees and parities are correctly defined, and obtain the following result. Proposition 12.
There is an isomorphism of complexes κ : (cid:16) HOG Sn , d (cid:17) → (cid:18) HO sk G Sn , d (cid:19) . It is probably easiest to illustrate the above correspondence by an example. ∈ HOG [1]2 κ −→ ± ∈ HO sk G [1]2 On the right-hand side we have drawn the crossed edges without directions, since these directions are identified,up to sign, by taking the S -invariants. In words we can say that the map κ just replaces all patterns in he graph by a crossed edge and otherwise leaves the graph the same. It is clear that this is an isomorphism,the inverse map just performs that replacement in the opposite direction. Note however that the resulting graphs’internal vertices are always at least 3-valent.4. T he relation of oriented and undirected graph complexes In this section we show our main technical Theorems 2 and 3.4.1.
The definition of the map.
In this subsection we are going to define the map Φ : (cid:16) HG Sn , d (cid:17) → (cid:16) HOG Sn + , d (cid:17) .We follow the methods from [27] and [1]. Thanks to Proposition 12 we may consider as well the ”skeletonversion” (cid:16) HO sk G Sn + , d (cid:17) (cid:27) (cid:16) HOG Sn + , d (cid:17) of the oriented graph complex and define our map as(43) Φ : (cid:16) HG Sn , d (cid:17) → (cid:18) HO sk G Sn + , d (cid:19) . The map F : (cid:16) HOGC Sn + , δ (cid:17) → (cid:16) HGC Sn , δ (cid:17) , in Theorem 2 is then the map dual to Φ .4.1.1. Forests.
Let us pick the number of internal vertices v , the number of edges e and the set of external vertices S . Let Γ ∈ ¯ V v ¯ E e ¯ H S G n be a graph with said numbers of vertices, edges and hairs.Let a forest be any subgraph of Γ that contains all its external vertices, that does not contain cycles (of anyorientation), and all of whose connected components contain exactly one external vertex each. Let a spanningforest be a forest that contains all vertices. Let F ( Γ ) be the set of all spanning forests of Γ . An example of aspanning forest is given in Figure 1. 1 2 34F igure
1. An example of a hairy graph Γ for S = { , , , } and a spanning forest τ ∈ F ( Γ ).Edges of the forest are red, while other edges are dotted.4.1.2. Model pairs.
Let τ ∈ F ( Γ ) be a spanning forest of Γ . Also recall that Γ comes with a numbering of edgesand vertices. (These numberings are later removed in the definition of the graph complex, but we need to considerthem here to define signs properly.) We say that the pair ( Γ , τ ) is a model if the following conditions are satisfied: • All edges of a connected component in τ are directed towards the external vertex of that connected com-ponent. • An edge in τ has the same label as the vertex on its tail, labels being in the set { , . . . , v } ;An example of model is given in Figure 2. igure
2. An example of model with the spanning forest from Figure 1. Edges of the forest arered, while other edges are dotted. Labels of internal vertices are thick.It is clear that every pair ( Γ , τ ) with τ ∈ F ( Γ ) can be mapped to a model by renumbering the edges and verticesand changing edge directions, i.e., by the action of an element of the group S v × (cid:16) S e ⋉ S × e (cid:17) .4.1.3. Defining the map for a model.
Let us now pick up a model ( Γ , τ ), with a (single term) graph Γ ∈ ¯ V v ¯ E e ¯ H S G n and a spanning forest τ ∈ F ( Γ ). The graph Γ after ignoring the degree can be considered as a core graph in¯ V v ¯ E e ¯ H S grac . To all of its edges that belong to the spanning forest (i.e., that are in E ( τ )) we attach an edge type, and to those that are not in the spanning forest we attach edge type to get an element of ¯ V v ¯ E e ¯ H S O sk grac.Then after taking coinvariants and adding the degrees we get a skeleton graph(44) Φ τ ( Γ ) ∈ V v E e ¯ H S O sk G n + . It is straightforward to check the following: • the result Φ τ ( Γ ) is an admissible type oriented graph; • the map is well defined in a sense that if there is an element of S v × (cid:16) S e ⋉ S × e (cid:17) that sends one model toanother model, the same result in V v E e D sk GC n + is obtained; • Φ τ ( Γ ) and Γ are of the same degree.An example of Φ τ ( Γ ) is given in Figure 3.1 2 34F igure
3. Hairy oriented graph Φ τ ( Γ ) for the graph Γ and spanning forest τ from Figure 1.4.1.4. The final map.
The map is now extended to all pairs ( Γ , τ ) by invariance under the action of S v × (cid:16) S e ⋉ S × e (cid:17) .Then let us define(45) Φ : ¯ V v ¯ E e ¯ H S G n → V v E e ¯ H S O sk G n + , Γ X τ ∈ F ( Γ ) Φ τ ( Γ ) . Mind that elements of the graph complex are linear combinations of combinatorial graphs, and we consider here a single combinatorialgraph, as an element of the graph complex. he invariance under all actions implies that the induced map Φ : V v E e ¯ H S G n → V v E e ¯ H S O sk G n + is well defined.It is then extended to a map of graded vector spaces(46) Φ : HGC Sn → HO sk G Sn + . Rewording the above construction up to signs, this map is defined on some (undirected hairy) graph Γ ∈ HGC Sn as follows: We sum over all spanning forests of Γ . For each such forest we build a directed acyclic graph by thefollowing procedure: • We direct each edge in the spanning forest towards the unique external vertex in its tree. • We replace all other edges by a crossed edge . Recall also that these crossed edges in the ”skele-ton” graph complex HO sk G Sn + are just placeholders for zigzags in graphs in the graph complex HOG Sn + . Proposition 13.
The map Φ : (cid:16) HGC Sn , d (cid:17) → (cid:16) HO sk G Sn + , d (cid:17) is a map of complexes of degree zero, i.e. (47) Φ ( d Γ ) = d Φ ( Γ ) for every Γ ∈ HGC Sn .Proof. We have already checked that the degree of Φ is zero. The proof of the other claim is similar to the proofsof [27, Proposition 4.4] and [1, Proposition 3.4]. Nevertheless, let us quickly go through the argument.Let Γ ∈ ¯ V v ¯ E e ¯ H S G n . It then holds that Φ ( d Γ ) = Φ X a ∈ E ( Γ ) Γ / a = X a ∈ E ( Γ ) Φ ( Γ / a ) = X a ∈ E ( Γ ) X τ ∈ F ( Γ / a ) Φ τ ( Γ / a )where Γ / a is the graph obtained by contracting the edge a in Γ . Spanning forests of Γ / a are in natural bijectionwith spanning forests of Γ that contain a , τ/ a ↔ τ , so we can write Φ ( d Γ ) = X a ∈ E ( Γ ) X τ ∈ F ( Γ ) a ∈ E ( τ ) Φ τ/ a ( Γ / a ) = X τ ∈ F ( Γ ) X a ∈ E ( τ ) Φ τ/ a ( Γ / a ) . Lemma 14.
Let Γ ∈ ¯ V v ¯ E e ¯ H S G n , τ ∈ F ( Γ ) and a ∈ E ( τ ) . Then (48) Φ τ/ a ( Γ / a ) ∼ Φ τ ( Γ ) / a , where ∼ means that they are in the same class of coinvariants under the action of S v × S e .Proof. It is clear that one side is ± the other side. Careful calculation of the sign is left to the reader. (cid:3) The lemma implies that(49) Φ ( d Γ ) ∼ X τ ∈ F ( Γ ) X a ∈ E ( τ ) Φ τ ( Γ ) / a . Next we consider the right-hand side of (47), i.e., d Φ ( Γ ). There the di ff erential decomposes as d = d C ± d E ,in a component d C that contracts (”core”) internal directed edges and d E that replaces crossed edges by directededges, see (41). Note also that the edges of Φ τ ( Γ ) are in 1-1-correspondence to those in Γ . Given the spanningforest τ , edges of Φ τ ( Γ ), i.e. of Γ , e ff ected by the di ff erential d = d C ± d E can be partitioned as E ( τ ) ⊔ ED ( τ ) ⊔ EC ( τ ) , where E ( τ ) is the set of edges between internal vertices in the forest τ and we define: • ED ( τ ) is the set of edges that connect two connected components of τ , including those attached to anexternal vertex if that vertex alone forms a connected component of τ ; • EC ( τ ) is the set of edges that make a cycle in a connected component of τ .Edges from E ( τ ) are e ff ected by core di ff erential d C , and edges from ED ( τ ) and EC ( τ ) are e ff ected by edgedi ff erential d E .Note that edges adjacent to external vertices can not be contracted, so they are not included in E ( τ ) if they arein the forest. But if they are not in the forest they are included in ED ( τ ) because edge di ff erential can act on them.Edges from E ( τ ) can be contracted by d C , so(50) d C Φ ( Γ ) = d C X τ ∈ F ( Γ ) Φ τ ( Γ ) = X τ ∈ F ( Γ ) d C ( Φ τ ( Γ )) = X τ ∈ F ( Γ ) X a ∈ E ( τ ) Φ τ ( Γ ) / a ∼ Φ ( d Γ ) . he edge di ff erential d E acts on edges of type , which are those in the sets ED ( τ ) and EC ( τ ). We thensplit(51) d E ( Φ τ ( Γ )) = d ED ( Φ τ ( Γ )) + d EC ( Φ τ ( Γ )) , where(52) d ED ( Φ τ ( Γ )) = X a ∈ ED ( τ ) d ( a ) E ( Φ τ ( Γ )) , d EC ( Φ τ ( Γ )) = X a ∈ EC ( τ ) d ( a ) E ( Φ τ ( Γ )) , where d ( a ) E maps edge a as = − ( − n + . Lemma 15.
Let Γ ∈ ¯ V v ¯ E e ¯ H S G n . Then X τ ∈ F ( Γ ) d EC ( Φ τ ( Γ )) ∼ . Proof.
Let N ( Γ ) : = X τ ∈ F ( Γ ) d EC ( Φ τ ( Γ )) = X τ ∈ F ( Γ ) X a ∈ EC ( τ ) d ( a ) E ( Φ τ ( Γ )) . Terms in the above relation can be summed in another order. Let FC ( Γ ) ( cycled forests ) be the set of all sub-graphs ρ of Γ that contain all internal and external vertices, have v + C ( ρ ) be the set of edgesin the cycle of ρ . Clearly, ρ \ { a } for a ∈ C ( ρ ) is a spanning forest of Γ and sets { ( τ, a ) | τ ∈ F ( Γ ) , a ∈ EC ( τ ) } and { ( ρ, a ) | ρ ∈ FC ( Γ ) , a ∈ C ( ρ ) } are bijective, so N ( Γ ) = X ρ ∈ FC ( Γ ) X a ∈ C ( ρ ) d ( a ) E (cid:16) Φ ρ \{ a } ( Γ ) (cid:17) . It is now enough to show that X a ∈ C ( ρ ) d ( a ) E (cid:16) Φ ρ \{ a } ( Γ ) (cid:17) ∼ ρ ∈ FC ( Γ ). Let y ∈ V ( Γ ) be the internal vertex in the cycle of ρ closest to the external vertex of itsconnected component (along ρ ). After choosing a ∈ C ( ρ ), the cycle in Φ ρ \{ a } ( Γ ) has the edge a of type , andother edges of type or with direction from y to the edge a , such as in the following diagram. y After acting by d ( a ) E this is replaced by + ( − n , such as in the following diagram. y + ( − n y Careful calculation of the sign shows that those two terms are cancelled with terms given from choosing neigh-boring edges in C ( ρ ), and two last terms which do not have a corresponding neighbor are indeed 0 as they have atype-cycle. This concludes the proof that N ( Γ ) ∼ (cid:3) The similar study of the action on edges from ED ( τ ) leads to the following lemma. Lemma 16.
Let Γ ∈ ¯ V v ¯ E e ¯ H S G n . Then X τ ∈ F ( Γ ) d ED ( Φ τ ( Γ )) ∼ . Proof.
It holds that X τ ∈ F ( Γ ) d ED ( Φ τ ( Γ )) = X τ ∈ F ( Γ ) X a ∈ ED ( τ ) d ( a ) E ( Φ τ ( Γ )) . Let FD ( Γ ) ( double-hair forests ) be the set of all sub-graphs λ of Γ that contain all internal and external vertices,have no cycles, whose one connected component has exactly two external vertices and whose other connectedcomponents have exactly one external vertex. Let those two external vertices be j ( λ ) , k ( λ ) ∈ S . or λ ∈ FD ( Γ ) let P ( λ ) be the set of edges in the path from j ( λ ) to k ( λ ). Clearly, λ \{ a } for a ∈ P ( λ ) is a spanningforest of Γ and a is in ED ( Γ ) for that spanning forest. One can easily see that sets { ( τ, a ) | τ ∈ F ( Γ ) , a ∈ ED ( τ ) } and { ( λ, a ) | λ ∈ FD ( Γ ) , a ∈ P ( λ ) } are bijective, so X τ ∈ F ( Γ ) d ED ( Φ τ ( Γ )) = X λ ∈ FD ( Γ ) X a ∈ P ( λ ) d ( a ) E (cid:0) Φ λ \{ a } ( Γ ) (cid:1) . To finish the proof it is enough to show that X a ∈ P ( λ ) d ( a ) E (cid:0) Φ λ \{ a } ( Γ ) (cid:1) ∼ λ ∈ FD ( Γ ). After choosing a ∈ P ( λ ) the path from j ( λ ) to k ( λ ) along λ in Φ λ \{ a } ( Γ ) has the edge a oftype , and the other edges of type or with direction from j ( λ ) or k ( λ ) to the edge a , such as in thefollowing diagram. k ( λ ) j ( λ ) After acting by d ( a ) E this is replaced by + ( − n , such as in the following diagram. k ( λ ) j ( λ ) + ( − n k ( λ ) j ( λ ) Careful calculation of the sign shows that those two terms are cancelled with terms given from choosing neighbor-ing edges in P ( λ ). The two last terms which does not have corresponding neighbour are zero because they haveexternal vertex which is not target. (cid:3) Equation (50), and Lemmas 15 and 16 imply that Φ ( d ( Γ )) ∼ d C ( Φ ( Γ )) ∼ d C ( Φ ( Γ )) + X τ ∈ F ( Γ ) d EC ( Φ τ ( Γ )) + X τ ∈ F ( Γ ) d ED ( Φ τ ( Γ )) = d C ( Φ ( Γ )) + d E ( Φ ( Γ )) = d ( Φ ( Γ )) . After taking coinvariants this implies that Φ ( d ( Γ )) + Φ ( χ ( Γ )) = d ( Φ ( Γ ))for each Γ ∈ HG Sn . Hence, Φ : ( HG Sn , d ) → ( HOG Sn + , d ) is a map of complexes. (cid:3) After (anti-)symmetrizing external vertices this map induces the map Φ : (cid:0) HG m , n , d (cid:1) → (cid:0) HOG m , n + , d (cid:1) . The dual map.
The dual of Φ is F : (cid:16) HOGC Sn + , δ (cid:17) → (cid:16) HGC Sn , δ (cid:17) , and the dual of its version for (anti-)symmetrized external vertices is F : (cid:0) HOGC m , n , δ (cid:1) → (cid:0) HGC m , n + , δ (cid:1) . Let us describe these maps combinatorially, to see that they are relatively simple and straightforward to compute.
Definition 17.
Let Γ ∈ HOGC m , n be a single term graph. We call it a forest graph if all its internal vertices thatare at least 3-valent have exactly 1 outgoing edge. Lemma 18.
Let Γ ∈ HOGC m , n be a (single term) graph that is not a forest graph. Then F ( Γ ) = .Proof. In the dual picture one easily checks that every graph in the linear combination Φ ( γ ) for γ ∈ HG m , n + is aforest graph. This leads to the result. (cid:3) If Γ ∈ HOGC m , n is a forest graph, F ( Γ ) is a graph in HGC m , n + up to the sign obtained from Γ by replacing eachoccurrence of with a single edge and ignoring the edge directions. Here are some examples: F −→ F −→ .2. The proof of Theorem 2.
In this subsection we prove Theorem 2 by proving its dual version:
Proposition 19.
The map Φ : (cid:16) HG Sn , d (cid:17) → (cid:16) HOG Sn + , d (cid:17) is a quasi-isomorphism.Proof. Using Proposition 12 it is enough to prove that Φ : (cid:16) HG Sn , d (cid:17) → (cid:18) HO sk G Sn + , d (cid:19) is a quasi-isomorphism.On its mapping cone we set up the spectral sequence on the number of vertices. Our complexes split into finitedimensional subcomplexes according to the loop number, hence the spectral sequence converges. It is thereforeenough to prove the claim for the first di ff erential of the spectral sequence.Since the pieces of the di ff erentials that contract edges lower the number of vertices, it is clear that on the firstpage of the spectral sequences there is the mapping cone of the map Φ : (cid:16) HG Sn , (cid:17) → (cid:18) HO sk G Sn + , d E (cid:19) . These complexes are now direct sums of subcomplexes spanned by graphs with a fixed number of vertices andedges, so it is enough to show the claim for Φ : (cid:16) V v E e ¯ H S G n , (cid:17) → (cid:16) V v E e ¯ H S O sk G n + , d E (cid:17) . Recall from (12) and (39) that both V v E e ¯ H S G n and the skeleton complex V v E e ¯ H S O sk G n + are spaces of invari-ants of the action of S v × (cid:16) S e ⋉ S × e (cid:17) . The action clearly commutes with the map Φ . Since the edge di ff erential d E does not change the number of vertices and edges, taking homology commutes with taking coinvariants of thataction. Therefore, it is now enough to show the claim for Φ : (cid:16) ¯ V v ¯ E e ¯ H S G n , (cid:17) → (cid:16) ¯ V v ¯ E e ¯ H S O sk G n + , d E (cid:17) . Let us pick up a particular (single term) graph Γ ∈ ¯ V v ¯ E e ¯ H S G n . Let h O Γ i be the subspace of ¯ V v ¯ E e ¯ H S O sk G n + spanned by skeleton graphs with the core graph Γ .The map Φ is defined such that Φ ( Γ ) ∈ h O Γ i . Also, di ff erential d E acts within particular subspace h O Γ i .Therefore, we can split the map as a direct sum and it is enough to prove the clam for(53) Φ : ( h Γ i , → ( h O Γ i , d E ) , for every Γ ∈ ¯ V v ¯ E e ¯ H S G n .In order to prove that, let us choose v edges in Γ , say a , . . . , a v . Let F ( a , . . . , a i ) be the sub-graph of Γ thatincludes those edges, all external vertices and all necessary internal vertices. We require that for every i = , . . . , v the sub-graph F ( a , . . . , a i ) is a forest. Recall that in a forest, every connected component has exactly one externalvertex. Clearly, F ( a , . . . , a v ) is a spanning forest.For every i = , . . . , v , we form a graph complex h O Γ i i as follows: it is spanned by graphs with a core graph Γ with attached edge types from ¯ σ : = { [ n ] , [ n ] , [ n − , [ n ] } such that: • edges a , . . . , a i have type , and other edges have other types; • no (internal or external) vertex in the forest F ( a , . . . , a i ) has a neighbouring edge of type orheading away from it; • every internal vertex outside the forest F ( a , . . . , a i ) has a neighbouring edge of type or headingaway from it (it is not a sink); • there are no cycles along arrows on edges of type and .Examples of graphs in h O Γ i i are shown in Figure 4. igure
4. An example of graph in h O Γ i i , with Γ as in Figure 1. The forest F ( a , a , a , a , a , a ) is depicted red.The di ff erential on h O Γ i i is the edge di ff erential d E induced by
7→ − ( − n , as usual. If a resulting graph does not fulfil the conditions above, it is considered zero. Note that thick edges arenot e ff ected by the di ff erential.It is straightforward to check that(54) (cid:16) h O Γ i , d E (cid:17) = ( h O Γ i , d E ) . Also, it holds that h O Γ v i is one dimensional, spanned by the graph with edges a , . . . , a v of type and otheredges of type .For every i = , . . . , v , there is a map(55) f i : h O Γ i − i → h O Γ i i that only change the type of the edge a i as(56) , , ( − n + , where forbidden graphs are considered zero. Lemma 20.
For every i = , . . . , v the map f i : h O Γ i − i → h O Γ i i is a quasi-isomorphism.Proof. The essential di ff erence between h O Γ i − i and h O Γ i i is in the edge a i , it has to be of type in h O Γ i i , andit is of another type in h O Γ i − i . Since f i does not change types of other edges, it splits as a direct sum of mapsbetween complexes with fixed types of other edges f ifix : h O Γ i − fix i → h O Γ ifix i where h O Γ i − fix i and h O Γ ifix i are sub-complexes spanned by graphs with fixed types of all edges other than a i . It isenough to show that each f ifix is a quasi-isomorphism.Here, depending on the choice of fixed edge types, the conditions of the complex can disallow some possibilitiesfor the edge a i in both h O Γ i − fix i and h O Γ ifix i . We list all cases, showing that the map is a quasi-isomorphism in allof them. Let the vertex that is in the forest F ( a , . . . , a i ) but not in the forest F ( a , . . . , a i − ) be called x i .(1) If there is a vertex in the forest F ( a , . . . , a i − ) that has a neighbouring edge of type or headingaway from it, or there is a vertex outside the forest F ( a , . . . , a i ) that does not have a neighbouring edgeof type or heading away from it, or there is a cycle along arrows on edges of type oroutside the forest F ( a , . . . , a i ), both h O Γ i − fix i and h O Γ ifix i are zero complexes and the map is clearly aquasi-isomorphism.(2) If the conditions of (1) do not hold and x i has a neighbouring edge of type or heading away fromit, the edge a i (that goes from a vertex in the forest F ( a , . . . , a i − ) towards x i ) can have types orin h O Γ i − fix i , making the complex acyclic. In h O Γ ifix i , no type is allowed, so it is again the zero complex.Therefore, the map is again a quasi-isomorphism.(3) If the conditions of (1) do not hold and x i does not have a neighbouring edge of type or headingaway from it, the edge a i must have type in h O Γ i − fix i . In h O Γ ifix i that edge must have type , makingthe map an isomorphism. Thus, it is also a quasi-isomorphism. (cid:3) he lemma implies that(57) f : = f v ◦ · · · ◦ f : h O Γ i → h O Γ v i is a quasi-isomorphism. Lemma 21.
The map f ◦ Φ : h Γ i → h O Γ v i is a quasi-isomorphism.Proof. Both complexes are one-dimensional, so we only need to check that f ◦ Φ ,
0. The left-hand side complexhas a generator Γ . It holds that f ◦ Φ ( Γ ) = f X τ ∈ F ( Γ ) Φ τ ( Γ ) . The map Φ τ gives edges in E ( τ ) type or , and type to the other edges. After that, the map f = f ◦ · · · ◦ f v kills all graphs with any of edges a , . . . , a v being of type . Therefore, f ◦ h x ,τ is non-zero onlyif the forest τ consist exactly of the edges a , . . . , a v . Let us call this forest T . So(58) f ◦ Φ ( Γ ) = f ( Φ T ( Γ )) . It is clearly the generator of h O Γ v i , and therefore non-zero. (cid:3) Since f and f ◦ Φ are quasi-isomorphism, it follows that Φ is also a quasi isomorphism, what was to bedemonstrated. (cid:3) The following corollary is now straightforward.
Corollary 22.
The induced map Φ : (cid:0) HG m , n , d (cid:1) → (cid:0) HOG m , n + , d (cid:1) is a quasi-isomorphism. Using Proposition 11 we get the following corollary. It has already been shown as part of [1, Theorem 1.1.].
Corollary 23.
There is an explicit quasi-isomorphism (cid:0) HG n , n , d (cid:1) → ( OG n + , d ) . The proof of Theorem 3.
Corollary 22 implies that also the dual map F : (cid:0) HOGC m , n + , δ (cid:1) → (cid:0) HGC m , n , δ (cid:1) is a quasi-isomorphism. On this complexes we have Lie algebra structures. Proposition 24.
The map F : (cid:0) HOGC m , n + , δ (cid:1) → (cid:0) HGC m , n , δ (cid:1) respects the Lie algebra structures, i.e.F ([ Γ , Γ ′ ]) = [ F ( Γ ) , F ( Γ ′ )] for every Γ , Γ ′ ∈ HOGC m , n + .Proof. It is enough the check the relation for single term graphs Γ and Γ ′ . Recall from Lemma 18 that F ( Γ ) = Γ is a forest graph. It is easy to see that if either of Γ or Γ ′ is not a forest graph, neither is its Lie bracket[ Γ , Γ ′ ]. So it is enough to check the relation for forest graphs Γ and Γ ′ .In constructing [ Γ , Γ ′ ] a hair from one graph can connect to any vertex from another graph. But if the hair isconnected to a 2-valent vertex of the form that comes from a skeleton edge , the resulting graph isnot a forest graph, so it is sent to zero after acting by F . Therefore, to prove the relation, we need to consider onlycases where a hair is connected to at least 3-valent vertices. They come from skeleton vertices.It is now clear that connecting hairs before and after the action of F yields the same result. Careful calculationof the sign is left to the reader. (cid:3) For the next assertion of Theorem 3 we need to check that F ( m ) = m . The only term in the series of graphs m = X k ≥ k ! ... | {z } k × ∈ HOGC n , n + , that is a forest graph is , and it is sent to m = . inally we need to check that the diagram of complexes (3) homotopy commutes. We will do this by consider-ing a one sided inverse φ to the lower horizontal arrow, that has first been introduced in [21]. OGC n + [ −
1] (
HOGC n , n + , δ + [ m , − ]) GC n [ −
1] (
HGC n , n , δ + [ m , − ]) ≃≃ ≥ ≃≃ ≥ φ Concretely, for a graph Γ ∈ HGC n , n of loop order g we set φ ( Γ ) = Γ has ≥ Γ has a single hair connected to a trivalent vertex ± ( neighbors ) − g ( Γ − hair) if Γ has a single hair , where ( Γ − hair) is the graph Γ with the single hair removed and ( neighbors ) is the number of neighbors of thevertex the hair connects to (not counting the hair as a neighbor). The verification that this is indeed a one sidedinverse to the map GC n [ − → HGC n , n we refer to [21], or leave it to the reader as an exercise. Finally, one justhas to note that the inner square in the above diagram commutes in loop orders ≥
2, using the description of themap
OGC n + → GC n of section 3.3.2. Mind that the factor g appearing in the definition of φ is the reason for usintroducing a similar (conventional) factor in (24), that was absent in [26, 27] This finishes the proof of Theorem3. (cid:3)
5. T he map from the oriented to the ribbon graph complex
In this section we shall discuss the connection of the graph complexes of the previous sections to the ribbongraph complex, introduced by Penner and Kontsevich. We will start by recalling some definitions and construc-tions from [14].5.1.
Recollections from [14] . The main player in [14] is the ribbon graph properad
RGra . The space of opera-tions RGra ( r , s ) with s inputs and r outputs is the space of linear combinations of connected ribbon graphs withthe set of vertices identified with [ r ] and the set of boundary components identified with [ s ]. A ribbon graph (or fatgraph) is a graph with a prescripition of a cyclic ordering of the incident half-edges at each vertex. Thickening thegraph, one obtains an oriented surface with some disks removed. Here is an example ribbon graph in RGra (3 , Here one should think of the edges being thickened to ribbons, which connect at the vertices in the indicatedcyclic order. The properadic compositions are obtained by ”connecting a vertex and a boundary component” inthe sense that a vertex is deleted, and the incident edges are distributed along the boundary component in all planarpossible ways. We refer to [14, section 4] for more combinatorial details, and also for the precise sign and degreeconventions. We just remark that a ribbon graph Γ ∈ RGra ( r , s ) with k edges has cohomological degree + k bycovention, i.e., formally each edge has degree + Λ LieB → RGra , from the (degree shifted) Lie bialgebra properad, defined on the bracket and cobracket generator by the followingformulas:(59) bracket: ∈ RGra (1 ,
11 2 ∈ RGra (2 , ff erent properad map we enote by ∗ : Λ LieB → RGra , simply by sending the cobracket generator to zero,(60) ILieB ∗ −→ RGra bracket: ∈ RGra (1 , ∈ RGra (2 , Λ LieB ∞ → Λ LieB . Then one canconsider the properadic deformation complexDef( Λ LieB ∞ ∗ −→ RGra )(see [14, 15] for the definition and conventions), whose elements are essentially series of ribbon graphs withun-labelled vertices and boundary components. The subcomplex RGC ⊂ Def( Λ LieB ∞ ∗ −→ RGra )consisting of graphs with all vertices of valence ≥ H ( RGC ) (cid:27) Y g ≥ , n ≥ g + n ≥ ( H c ( M g , n ) ⊗ Q [ − ⊗ n ) S n Combinatorially, elements of
RGC can be seen as series of ribbon graphs, with unidentifiable vertices and bound-ary components of degree 0 and edges of degree 1.The above properadic definition of the ribbon graph complex as a deformation complex has three interestingconsequences explored in [14]:(1) Since deformation complexes are dg Lie algebras, one finds that RGC carries a dg Lie structure.(2) Instead of deforming the map ∗ of (60) we can as well deform the map (59) and consider the complexDef( Λ LieB ∞ −→ RGra ) . The former complex in particular is a deformation of the deformation complex of the map ∗ , and from thisone in particular obtains an additional di ff erential ∆ on the ribbon graph complex RGC . Combinatorially,one can check that this di ff erential acts on a ribbon graph by spitting a boundary component in two byadding one edge across the component, in all possible ways, schematically:(61) ∆ : X This operation has been first considered by T. Bridgeland to our knowledge. A. Cˇaldˇararu’s Conjecture(see Conjecture 32 below) states that the complex (
RGC , δ + ∆ ) computes the compactly supportedcohomology of the moduli spaces without maked points M g .(3) Since derivations of the properad Λ LieB ∞ obviously map into the deformation complex above, and sincethose derivations can (homotopically) be identified with the graph complex HOGC , we obtain the fol-lowing maps of complexes(62) Def( Λ LieB ∞ id −→ Λ LieB ∞ ) Def( LieB ∞ −→ RGra ) GC [ − OGC [ −
1] (
RGC , δ + ∆ ) ≃ ≥ ≃ ≥ ≃ ≥ relating the ”commutative” graph complex GC to the ribbon graph complex. All arrows respect a naturalgrading, which is given on the ribon graphs by the genus, and on the ”ordinary” graphs by the loop order.The arrows labelled ” ≃ ≥ ” are quasi-isomorphisms in loop orders (resp. genus) ≥ In fact, the trivalence condition is not very important, and H (Def( Λ LieB ∞ ∗ −→ RGra )) is only slightly larger than H ( RGC ).
4) It is easy to check that the map (59) in fact factors through the involutive Lie bialgebra properad
ILieB ,and one can hence consider the deformation complexesDef( Λ ILieB ∞ −→ RGra ) . In this way one can obtain further algebraic structures on the Kontsevich-Penner ribbon graph complex,see [14, section 4.3], but this is not important for us in the present paper.5.2.
Ribbon graph complex with labelled punctures.
The Merkulov-Willwacher construction recalled in theprevious subsection a priori only considers the version of the ribbon graph complex for the moduli space with”antisymmetrized” marked points. Here we now upgrade the properadic definition of the ribbon graph complex tothe case of labelled boundary components. Concretely, we will copy the version of the definition of the orientedhairy graph complex
HHOGC S via properadic deformation complexes as in section 3.4.3. To this end, we definethe map of properads ∗ : Ω ( AC ∗ ) → RGra as the composition of properad maps (see section 3.4.3 for the first map) Ω ( AC ∗ ) → Λ LieB ∞ → Λ LieB → RGra . We then consider the properadic deformation complex of that map, which readsDef (cid:18) Ω ( AC ∗ ) ∗ −→ RGra (cid:19) = Y r , s Hom S r × S s (cid:0) AC ∗ ( r , s ) , RGra ( r , s ) (cid:1) (cid:27) Y r , s RGra ( r , s ) S s , disregarding the di ff erential for now. Elements of this complex can be understood as formal series of ribbon graphswith unidentifiable vertices, and boundary components labelled by numbers 1 , . . . , r . The di ff erential δ can be seento be acting by vertex splitting. There are again two gradings that are preserved by the di ff erential, namely thegrading by the number r of boundary components and the genus of the ribbon graph. (I.e., the genus of the surfaceobtained by slightly thickening the graph.) We will denote the subcomplex of genus g and with r components by B g Def (cid:18) Ω ( AC ∗ ) ∗ −→ RGra (cid:19) r ⊂ Def (cid:18) Ω ( AC ∗ ) ∗ −→ RGra (cid:19) . Then the version of the Kontsevich-Penner ribbon graph complex for genus g surface with r labelled points (with2 g + r ≥
3) is the subcomplex B g RGC S ⊂ B g Def (cid:18) Ω ( AC ∗ ) ∗ −→ RGra (cid:19) r consisting of the ribbon graphs all of whose vertices are at least trivalent. More concretely, one has that H ( B g RGC [ r ] ) (cid:27) H c ( M g , r ) ⊗ Q [ − ⊗ r . Furthermore, we will again consider the pieces of various genera together and define the subcomplex
RGC [ r ] : = Y g B g RGC [ r ] ⊂ Def (cid:18) Ω ( AC ∗ ) ∗ −→ RGra (cid:19) r . Finally, we shall take the notational liberty to label the r boundary components by the elements of any finite set S with r elements instead, and use the the notation RGC S (cid:27) RGC [ r ] , for the resulting complex, where we have fixed some bijection S (cid:27) [ r ].5.3. An extension of the map of Merkulov-Willwacher.
The properadic definition of the ribbon graph complexwith labelled punctures makes it easy to extend the maps of diagram (62) to this case. More concretely, just byfunctoriality of the properadic deformation complexes we have the map of complexesDef( Ω ( AC ∗ ) ∗ −→ Λ LieB ∞ ) → Def( Ω ( AC ∗ ) ∗ −→ RGra )induced by the composition with the map (59). This map induces maps on the subcomplexes we have consideredabove: Def( Ω ( AC ∗ ) ∗ −→ Λ LieB ∞ ) Def( Ω ( AC ∗ ) ∗ −→ RGra ) HOGC [ r ]1 HHOGC [ r ]1 RGC [ r ]induced . y Theorem 2 we furthermore have a quasi-isomorphism HOGC [ r ]1 ≃ −→ HGC [ r ]0 . Hence we obtain in particular a map(63) H ( HGC S ) → H ( RGC S ) . It is natural to raise the following conjecture.
Conjecture 25.
After identifying the genus g part of H ( RGC S , ) with H c ( M g , S )[ −| S | ] the map (63) agrees with theone obtained by Chan-Galatius-Payne [4] (see Theorem 1), possibly up to an overall conventional multiplicativeconstant. Simplest nontrivial example (genus case). To illustrate that the map (63) is fairly explicit, let us work itout in the genus 1 situation. The nontrivial cohomology classes in H ( HGC [ r ]0 ) are represented by linear combina-tions of ”loop” graphs of the form W r = ··· r , or graphs obtained from W r by permuting the r labels. Hence is su ffi ces to compute the image of W r in H ( RGC [ r ]0 )for our purposes. First it is easy to check that under the explicit map of Theorem 2 the graph W is in the image of W ′ r = ··· ··· r ··· ∈ HOGC [ r ]1 . Mapping this to
HHOGC [ r ]1 ⊂ Def(
LieB ∞ → RGra ) we hence obtain W ′′ r = ··· ··· r ··· + ( · · · ) , where ( · · · ) is a linear combination of graphs that contain vertices of valence ≥
4. Under the map to
RGC [ r ]0 thosegraphs are sent to zero by construction, hence only the leading term of W ′′ r is relevant. To compute its imageribbon graph(s) we have to replace the trivalent vertices by ribbon ”pairs of pants” as in (59), and then apply the roperadic compositions in RGra . One quickly checks that this yields the ribbon graph W ′′′ r = r ··· ∈ RGC [ r ] , as our final result. 6. O n the work of C han -G alatius -P ayne The goal of this section is to describe an independent, shortened proof of Theorems 1 and 4, and connect thoseTheorems to the results of the previous sections.6.1.
Getzler-Kapranov graph complex.
Let us recall some operadic facts about the moduli spaces of curvesunderstood by Getzler-Kapranov [5]. First they note that the collection of the Deligne-Mumford compactifiedmoduli spaces of stable curves M = n M g , n o g , n forms a modular operad. Hence the same is true for the corre-sponding chains operad. Similarly, they show that a version of the di ff erential forms on the open moduli spaces M = n M g , n o g , n can be made into a (topological) modular cooperad, up to certain degree shifts. Finally, they show(see [5, Proposition 6.11]) that both modular (co)operads are ”Koszul dual” to each other, in the sense that theyare related via the Feynman transform. Furthermore, it was shown in [6] that the modular operad M is formal sothat one can replace its (co)chains (co)operad by the (co)homology. Combining both results hence motivates thefollowing definition. Definition 26.
The Getzler-Kapranov graph complex GK = F (cid:16) H • (cid:16) M (cid:17)(cid:17) is the Feynman transform of the homology operad of the Deligne-Mumford compactified moduli spaces of stablecurves. Let us spell out the definition more explicitly. The ( k -)modular operad GK is a collection of dg vector spaces GK ( g , n ) of genus g operations with n inputs. We will sometimes also index our inputs with some set S (sothat n = | S | ) and write GK ( g , S ) accordingly. The elements of GK ( g , S ) are series of (isomorphism classes of) H • ( M )-decorated genus g oriented graphs. Explicitly, these are triples Γ = ( γ, D , o ) as follows: • γ is an undirected graph with external legs (or hairs) indexed by S . • Each vertex x of γ is decorated by an element of H • (cid:16) M g x , star( x ) (cid:17) , with g x a non-negative integer associatedto x , and star( x ) being the set of half-edges incident at x . We may collect all these decorations into oneelement D ∈ ⊗ G H • (cid:16) M (cid:17) of the graph-wise tensor product of H • (cid:16) M (cid:17) . • The orientation o is an ordering of the set of edges of Γ . • We identify two such triples up to sign if they can be transformed into each other by applying an isomor-phism of graphs, and by changing the orientations. The sign is obtained from the one on decorations andthe permutation in the orientation change in the natural way. • The cohomological degree is(64) | D | + e ( γ )with e ( γ ) being the number of edges of γ . • The genus of the graph is g = P x g x + ( • We require that for each vertex x we have2 g x + | star( x ) | ≥ . he di ff erential on Γ is given by splitting vertices and producing one tadpole δ Γ = X x ∈ V ( Γ ) split x ( Γ ) + tad pole x ( Γ ) , with split x ( Γ ) being obtained by replacing x by 2 vertices and applying the cooperadic cocomposition to thedecoration at x , see [5], and tad pole x ( Γ ) is obtained by using the modular cocomposition to produce a tadpole at x . (This latter operation reduces g x by one.) It is clear from (64) that this operation has cohomological degree + Definition 27.
We define the weight grading on GK to be the grading by the total degree on the decorations D ,i.e., w ( Γ ) : = | D | . It is clear that the weight grading is untouched by the di ff erential and hence indeed defines a grading on GK .As outlined in the beginning of this subsection one may then extract the following result from the literature. Theorem 28 ([5],[6]) . The Getzler-Kapranov graph complex computes the compactly supported cohomology ofthe open moduli spaces, H • ( GK ) (cid:27) H • c ( M ) . In fact, one should rather consider the right-hand side as the weight associated graded of the compactly sup-ported cohomology. On the level of (degree-)graded vector spaces that we consider here this is irrelevant, however.
Proof.
We recall the following notation and results from Getzler-Kapranov [5]. First they describe a modularoperad in nuclear Fr´echet (NF) spaces C • ( M , D ). They also describe a k -modular operad in nuclear DF-spaces p Grav modeling the (degree shifted) chains on M . More concretely, H d ( p Grav ( g , r )) = H d − g − − r ( M g , r ) . They furthermore extend the Feynman transform F to a topological version F top defined on NF and nuclearDF spaces by replacing tensor products by their projectively completed versions and dually by strong duals. (Thispresents no major problem, since the Feynman transform only involves finite direct sums and finite tensor productsof duals of the argument.)Our starting point is then the result [5, Proposition 6.11] that one has an isomorphism F top k p Grav (cid:27) C ( M , D ) , Next, we know from [6] that H • ( M ) has a minimal model (of finite type) M , and furthermore that M is a formalmodular operad. This allows us to extend the above isomorphism to a zigzag of (quasi-)isomorphisms F top k p Grav (cid:27) C ( M , D ) ← M → H • ( M ) . All objects can be considered as modular operads in NF spaces and the maps are continuous – mind that the twoobjects on the right are of finite type. Applying the topological Feynman transform again allows us to write thefollowing zigzag of quasi-isomorphisms p Grav ∼ ←− F top F top k p Grav (cid:27) F top C ( M , D ) ← F top M = F → F H • ( M ) . This shows our result. Here we used the fact (see [5, section 5]) that the Feynman transform is a homotopy functorand that F ( top ) is the homotopy inverse to F ( top ) k . (cid:3) An independent proof of Theorems 1 and 4.
We may show Theorems 4 and 1 together. We use the complex GK to compute H ( M ) via Theorem 28 First it is clear that the weight 0 part is a direct summand of GK . But since H ( M ) = Com mod the weight zero part is given by the commutative graph complexes
HGC S , mod of section 3.2.1.This is in turn quasi-isomorphic (in the stable situation) to the graph complex HGC S by Lemma 5. (cid:3) Furthermore, we note that via the modular operad maps
Com mod → H • ( M ) → Com one has maps of complexes B g HGC S → W GK ( g , S ) → B g HGC S , mod → B g HGC S see also section 3.2.1. .3. Antisymmetrized Getzler-Kapranov complex and extra di ff erential. We may mimic the constructions ofthe hairy graph complexes
HGC Sn and HGC m , n of section 3.2 and define the following objects: HGK S : = Y g ≥ GK ( g , S ) ⊗ sgn S [ −| S | ] HGK : = Y r ≥ ( HGK r ) S r . As before we will denote the genus g pieces by B g HGK S = GK ( g , S ) ⊗ sgn S [ −| S | ] and B g HGK . Obviously, thecohomology of HGK may be identified with the compactly supported cohomology of the moduli spaces M g , r with antisymmetrized punctures, up to conventional degree shifts. We call the number r of marked points also thenumber of hairs to unify the notation with the other graph complexes considered.Now, forgetting one marked point on the compactified moduli spaces induces a natural map π ∗ : H • ( M g , n ) → H • ( M g , n + ) . We may apply this to every vertex to obtain a degree + ∆ : HGK → HGK Γ X x ∈ V ( Γ ) ( addhair ) x ( Γ )where ( addhair ) x ( Γ ) applies π ∗ to the decoration at vertex x . To fix the signs, we declare that the newly added hairbecomes the first in the ordering. This increases the number of hairs r by one. Lemma 29.
The operation ∆ is compatible with the splitting di ff erential δ , i.e., ( δ + ∆ ) = .Proof sketch. One can check that ∆ = ∆ δ + δ ∆ vanish separately. The first equation is immediate since weantisymmetrized over the markings. The second boils down to the commutativity of the diagrams M g , n + × M g , n + M g + g , n + n M g , n × M g , n + M g + g , n + n − and M g , n + M g + , n − M g , n M g + , n − , where the horizontal arrows are modular operadic compositions (appearing in δ ) and the vertical arrows are for-getful maps, forgetting one puncture (appearing in ∆ ). (cid:3) It is also clear that the inclusion
HGC , → HGK intertwines the operation ∆ : = [ m , − ] on HGC , of the introduction and the operation ∆ on GK just defined.We also note that ∆ is defined in the same way on HGK ∅ , giving rise to a map of complexes ∆ : HGK ∅ [ − → HGK . and this is true for both the di ff erentials δ and δ + ∆ with which we may equip the right-hand side.Altogether we obtain the following commutative diagram of complexes(65) ( HGC , , δ + ∆ ) ( HGK , δ + ∆ )( GC [ − , δ ) HGK ∅ [ − , δ ) ∆ ∆ . The left-hand colum is the weight 0 part of the right-hand column. Furthermore, note that operation ∆ preservesthe genus (in the sense of the loop order on the left), and so do the horizontal maps. We furthermore know that theleft-hand vertical arrow induces an isomorphism on cohomology in genera ≥
2. This is also true for the right-handvertical arrow: heorem 30. The right-hand vertical arrow of (65) induces an isomorphism on cohomology in genera g ≥ . Inparticular, ( HGK , δ + ∆ ) computes the compactly supported cohomology of the moduli space M g in these genera,up to a degree shift by one.Proof. We consider the mapping cone C of the map in question, i.e., C = ( HGK ∅ ⊕ HGK , δ + ∆ ) . This is a natural extension of
HGK is that one merely allows graphs without hairs.Now we filter the complex C by the number of internal (non-hair) edges in graphs. This is a bounded abovecomplete descending filtration. Hence we are done if we can show that the associated graded complex is acyclic inpositive genera. Furthermore, δ creates one internal edge, and ∆ none, hence we may take ( C , ∆ ) as the associatedgraded. Now ( C , ∆ ) is a product of direct summands of tensor products of complexes associated to single vertices.The complex associated to a single vertex with k internal legs has the form C g , k = ( Y r ≥ r ( H ( M g , k + r ) ⊗ Q [ − ⊗ r ) S r , ∆ ) , with ∆ being the pullback for the forgetful map forgetting one of the ”antisymmetric” marked points as above.The lower bound for the product is r = max(3 − g − k ,
0) and comes from the stability condition. Now if thegenus of the graph, i.e., the number of loops plus the total number of genera of vertices, is at least 2, we can alwaysfind a vertex such that 2 g + k ≥
3. Hence it su ffi ces to show that H ( C g , k , ∆ ) = ∆ . To this end let π j : M g , k + r + → M g , k be the forgetful map forgetting the location of the j -th of the k ”antisymmetric” marked points. Then ∆ = r + X j = ( − j − π ∗ j . Now let Ψ j ∈ M g , k + r + be the Ψ -class at the j -th such marking, abusively hiding the number r from the notation.We can assume that it is normalized such that ( π j ) ! Ψ j = . Then we define our homotopy h : C g , k → C g , k [ −
1] on α ∈ H ( M g , k + r ) as h ( α ) = r X j = ( − j − ( π j ) ! ( Ψ j ∧ α )if r ≥ h ( α ) = r =
0. One computes (say first for r ≥ h ( ∆ ( α )) = r + X i = r + X j = ( − i + j ( π i ) ! ( Ψ i ∧ π ∗ j α ) = X ≤ i < j ≤ r + ( − i + j ( − i + j π ∗ j − ( π i ) ! ( Ψ i ∧ α ) + r + X i = ( π i ) ! Ψ i ) | {z } = α + X ≤ j < i ≤ r + ( − i + j π ∗ j ( π i − ) ! ( Ψ i ∧ α ) = ( r + α − ∆ ( h ( α )) . Hence we see that H ( C g , k ) = (cid:3) Note that the lower right-hand complex
HGK ∅ in (65) computes the compactly supported cohomology of themoduli space of (non-pointed) curves by Theorem 28. The cohomology of the upper right-hand complex has aspectral sequence (from the filtration on the number of marked points) whose first page is Y g ≥ , n ≥ g + n ≥ (cid:16) H c ( M g , n ) ⊗ Q [ − ⊗ n (cid:17) S n ⇒ H ( HGK , δ + ∆ ) . Hence the Theorem gives a relation between the cohomology of the moduli spaces of marked and non-markedcurves. One may also give this a conjectural geometric interpretation.
Conjecture 31.
Under the identification of
HGK with (some version of) compactly di ff erential forms on themoduli spaces of curves with antisymmetrized points, the operation ∆ corresponds geometrically to the pullbackunder the forgetful map π : M g , n + → M g , n , forgetting one marked point, and the latter pullback induces a welldefined operation on the compactly supported di ff erential forms. emark 3. The main problem in showing Conjecture 31 in our framework is that we use the formality result of[6]. This covers only the modular operad structure, but not the forgetful maps, forgetting some of the markedpoints.We finally remark that Alexey Kalugin probably has a proof of the above conjecture (personal communication).
Remark 4.
By similar arguments as in the proof of Theorem 30 we may in fact compute the part of the cohomol-ogy of (
HGK , δ + ∆ ) in genus 0 and 1 as well. Concretely, following the argument and using the same notationas in that proof, we see that we need to consider only graphs all of whose vertices x satisfy 2 g + k ≤
2. Thecohomology of the complexes ( C g , k , ∆ ) is computed as follows: • In the cases g = k = g = k = H ( C g , k , ∆ ) is one-dimensional, correspondingto a single hair attached to the vertex. • In the remaining cases g = k < C g , k is 0, because there need to at least two hairs(markings) to satisfy stability, but then the corresponding classes are killed by the anti-symmetrization.Overall, one sees that one has non-trivial cohomology only in the genus 1 case, and there the remaining dia-grams are the “hedgehog” graphs of the formthat live in weight 0, and span the compactly supported cohomology of the M , n with antisymmetrized markings.6.4. Incorporating ribbon graphs, and the (conjectural) big picture.
We can now put together the maps ofsection 5 and those of the previous subsection to obtain a big commutative diagram of complexes (straight arrows)
HGK ∅ [ −
1] (
HGK , δ + ∆ ) GC [ −
1] (
HGC , , δ + [ m , − ]) OGC [ −
1] (
HOGC , , δ + [ m , − ]) ( RGC , δ + ∆ ) ≃ ≥ ≃ ? ≃ ≥ ≃ ≥ ≃ ≥ ≃ , where the symbol ≃ ≥ shall indicate that the map is a quasi-isomorphism in genera ≥
2. Recall in particular thedefinition of the ”Bridgeland” di ff erential ∆ on the ribbon graph complex RGC from above, see (61). The middlerow is the weight 0 part of the first row. Given that H ( HGK , δ ) (cid:27) H ( RGC , δ ) (cid:27) Y r ≥ , g ≥ g + ≥ ( H c ( M g , r ) ⊗ Q [ − ⊗ r ) S r it is hence natural to conjecture that there is a quasi-isomorphism (possibly a zigzag) between ( HGK , δ + ∆ ) and
RGC , δ + ∆ ) (the dashed line) that makes the right hand triangle commute.This would then in particular imply the following conjecture of A. Cˇaldˇararu (personal communication)
Conjecture 32 (Cˇaldˇararu) . The cohomology of the ribbon graph complex with altered di ff erential ( RGC , δ + ∆ ) can be naturally identified with the compactly supported cohomology of the moduli spaces of curves withoutmarked points in genera g ≥ . H ( RGC g , δ + ∆ ) (cid:27) H c ( M g )[ − . R eferences [1] Assar Andersson, Marko ˇZivkovi´c. Hairy graphs to ribbon graphs via a fixed source graph complex. preprint arXiv:1912.09438, 2019.[2] G. Arone and V. Turchin. Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots. Ann.Inst. Fourier M g , arXiv:1805.10186,2018.[4] Melody Chan, Soren Galatius and Sam Payne. Topology of moduli spaces of tropical curves with marked points arXiv:1903.07187, 2019.[5] E. Getzler and M. Kapranov. Modular operads. Compositio Math. 110 (1998), no. 1, 65–126.[6] F. Guill´en Santos, V. Navarro, P. Pascual, and A. Roig. Moduli spaces and formal operads. Duke Math. J. 129 (2005), no. 2, 291–335.[7] Benoit Fresse, Victor Turchin and Thomas Willwacher. The rational homotopy of mapping spaces of E n operads Preprint,arXiv:1703.06123, 2017.
8] Anton Khoroshkin, Thomas Willwacher and Marko ˇZivkovi´c. Di ff erentials on graph complexes. Adv. Math.
307 (2017), 1184–1214.[9] Anton Khoroshkin, Thomas Willwacher and Marko ˇZivkovi´c. Di ff erentials on graph complexes II: hairy graphs. Lett. Math. Phys.
Trans. Amer. Math. Soc. ,361(1):207–222, 2009.[12] J.-L. Loday and B. Vallette. Algebraic operads.
Grundlehren Math. Wiss. , 346, Springer, Heidelberg, 2012.[13] S. Merkulov and B. Vallette, Deformation theory of representations of prop(erad)s, J. Reine Angew. Math. (2009), 51–106. and J.Reine Angew. Math. (2009), 123–174, arXiv:0707.0889.[14] Sergei Merkulov and Thomas Willwacher. Props of ribbon graphs, involutive Lie bialgebras and moduli spaces of curves.arXiv:1511.07808, 2015.[15] Sergei Merkulov and Thomas Willwacher. Deformation Theory of Lie Bialgebra Properads. in
Geometry and Physics: Volume I: AFestschrift in honour of Nigel Hitchin , 2018, DOI: DOI:10.1093 / oso / ff erential Geom. 27 (1988), no. 1, 35–53.[17] Paul Arnaud Songhafouo Tsopm´en´e and Victor Turchin Hodge decomposition in the rational homology and homotopy of high dimen-sional string links. arXiv:1504.00896 (2015).[18] Paul Arnaud Songhafouo Tsopm´en´e and Victor Turchin Euler characteristics for the Hodge splitting in the rational homology and homo-topy of high dimensional string links. arXiv:1609.00778 (2016).[19] Victor Turchin. Hodge-type decomposition in the homology of long knots. J. Topol. , 3(3):487–534, 2010.[20] Victor Turchin and Thomas Willwacher. Relative (non-)formality of the little cubes operads and the algebraic Cerf lemma.arXiv:1409.0163, 2014.[21] Victor Turchin and Thomas Willwacher. Commutative hairy graphs and representations of
Out ( F r ). arXiv:1603.08855, to appear in J.Top.[22] Thomas Willwacher. M. Kontsevich’s graph complex and the Grothendieck-Teichm¨uller Lie algebra. Invent. Math. , 200(3): 671–760(2015).[23] Thomas Willwacher The Oriented Graph Complexes. Communications in Mathematical Physics volume 334, pages 1649–1666(2015)[24] Marko ˇZivkovi´c. Graph complexes and their cohomology.
Doctoral Thesis, University of Zurich , 2016.[25] Marko ˇZivkovi´c. Di ff erentials on Graph Complexes III - Deleting a Vertex. Lett. Math. Phys. 109 (2019), no. 4, 9751054.[26] ˇZivkovi´c, M. Multi-directed graph complexes and quasi-isomorphisms between them I: oriented graphs. High. Struct. 4(1):266283, 2020.[27] ˇZivkovi´c, M. Multi-directed Graph Complexes and Quasi-isomorphisms Between Them II: Sourced Graphs. Int. Math. Res. Not. IMRN(2019), rnz212.U niversity of L uxembourg , M aison du N ombre , 6, A venue de la F onte , L-4364 E sch - sur -A lzette , L uxembourg E-mail address : [email protected] D epartment of M athematics , ETH Z urich , R¨ amistrasse urich , S witzerland E-mail address : [email protected] U niversity of L uxembourg , M aison du N ombre , 6, A venue de la F onte , L-4364 E sch - sur -A lzette , L uxembourg E-mail address : [email protected]@uni.lu