Multi-directed graph complexes and quasi-isomorphisms between them I: oriented graphs
aa r X i v : . [ m a t h . QA ] F e b MULTI-DIRECTED GRAPH COMPLEXES AND QUASI-ISOMORPHISMS BETWEEN THEM I:ORIENTED GRAPHS
MARKO ˇZIVKOVI ´CA bstract . We construct a direct quasi-isomorphism from Kontsevich’s graph complex GC n to the oriented graph com-plex OGC n + , thus providing an alternative proof that the two complexes are quasi-isomorphic. Moreover, the result isextended to the sequence of multi-oriented graph complexes, where GC n and OGC n + are the first two members. Thesecomplexes play a key role in the deformation theory of multi-oriented props recently invented by Sergei Merkulov.
1. I ntroduction
Generally speaking, graph complexes are graded vector spaces of formal linear combinations of isomorphismclasses of some kind of graphs. Each of graph complexes play a certain role in a subfield of homological algebraor algebraic topology. They have an elementary and simple combinatorial definition, yet we know very little aboutwhat their cohomology actually is.In this paper we study a sequence of graph complexes O k GC n , called k -oriented graph complexes, for k ≥ GC n equals the well known Kontsevich’s graph complex GC n introduced by M. Kontsevich in[2], [3]. Bigger number k introduces k di ff erent kinds (colours) of orientation on edges. See Section 2 below forthe strict definition.1-oriented graph complex, or simply oriented graph complex O GC n is first introduced (to the knowledge ofthe author) by Merkulov. In [9] Willwacher showed that it is quasi-isomorphic to Kontsevich’s graph complexGC n − . The same result is the consequence of the broader theory of Merkulov and Willwacher, [7, 6.3.8.]. Thepurpose of this paper is to present the third, more direct proof of the same result, and to extend it to k -orientedgraph complexes, as stated in the following theorem. Theorem 1.
For every k ≥ there is a quasi-isomorphism O k GC n → O k + GC n + . Multi-oriented graph complexes play a key role in the deformation theory of multi-oriented props recentlyinvented by Sergei Merkulov in [4] and to appear in [5]. Similarly to multi-oriented graphs, multi-oriented propshave multiple directions on each edge, without loops. Props naturally have one basic direction that goes fromthe inputs to the outputs, and that direction is clearly without loops, see e.g. [6]. In his papers Merkulov gives ameaning to the extra directions, and provides interesting applications and representations of multi-oriented props.Techniques used in this paper have recently been developed by the author to get new results about sourcedgraph complexes in [11].1.1.
Structure of the paper.
In Section 2 we define graph complexes needed in the paper. Sections 3 and 4introduce some sub-complexes that are more convenient to work with. Finally, in Section 5 we construct thequasi-isomorphism and prove Theorem 1.
Acknowledgements.
I am very grateful to Sergei Merkulov for providing the motivation for this result, especiallyto extend it for k >
1. I thank Thomas Willwacher for fruitful discussions.2. G raph complexes
In this section we define oriented graph complex O k fGC n for n ∈ Z , k ≥
0, called k-oriented graph complex .Particularly O fGC n is the Kontsevich’s graph complex fGC n defined for example in [8], cf. [10]. Strictlyspeaking, the complex we define here is the dual of the one in those papers, but it does not change the homology.Since the complex is defined as a formal vector space over the bases consisting graphs, the dual can be identifiedwith the complex as the vector space. The real di ff erence is in the di ff erential.We will work over a field K of characteristic zero. All vector spaces and di ff erential graded vector spaces areassumed to be K -vector spaces. Key words and phrases.
Graph Complexes. .1. Graphs.Definition 2.
Let v > , e ≥ and k ≥ be integers. Let V : = { , , . . . , v − } be set of vertices, E : = { , , . . . , e } set of edges and K : = { , , . . . , k } set of “colors”.A graph Γ with v vertices and e edges is a map Γ = ( Γ − , Γ + ) : E → V such that Γ − ( a ) , Γ + ( a ) for every a ∈ E.A k -oriented graph is the graph Γ together with maps o c : E → { + , −} for c ∈ K such that for every c there isno cycle a , a , . . . , a i = a such that Γ o c ( a j ) ( a j ) = Γ − o c ( a j + ) ( a j + ) for every j = , . . . , i − . The orientation of an edge a from Γ − ( a ) to Γ + ( a ) is called the intrinsic orientation of the edge a . For a color c ∈ K the orientation o c ( a ) is called the orientation of the color c of the edge a and goes from Γ − o c ( a ) ( a ) to Γ o c ( a ) ( a ).The condition in the definition of the oriented graph says that there is no oriented cycle in the graph in any color.Some examples of the graphs are drawn in Figure 1. Note that by our definition a graph has distinguishablevertices and edges. No tadpoles (i.e. edges a such that Γ − ( a ) = Γ + ( a )) are allowed. , , . F igure
1. Example of 2-oriented graphs. The intrinsic orientation is depicted by the simpleblack arrow, while the colored orientations are depicted by thick red and blue arrows. The lastdiagram does not represent an oriented graph because it contains a blue cycle.
Definition 3.
For a ∈ E we say that vertices Γ − ( a ) and Γ + ( a ) are connected. We extend the notion of beingconnected by transitivity, such that it is a relation of equivalence. Equivalence classes are called connectedcomponents . A graph is connected if it has one connected component and disconnected if it has more than oneconnected component. Let(1) O k ¯V v ¯E e gracbe the set of all connected k -oriented graphs with v vertices and e edges.2.2. Group actions on the set of graphs.
There is a natural action of the symmetric group S v on O k ¯V v ¯E e gracthat permutes vertices:(2) ( σ Γ )( a ) = ( σ ( Γ − ( a )) , σ ( Γ + ( a )))for σ ∈ S v and a ∈ E .Similarly, there is an action of the symmetric group S e on O k ¯V v ¯E e grac that permutes edges:(3) ( σ Γ )( a ) = Γ ( σ − ( a )) , ( σ o c )( a ) = o c ( σ − ( a ))for σ ∈ S e , a ∈ E and c ∈ K .Finally, for every edge a ∈ E there is an actions of the symmetric group S on O k ¯V v ¯E e grac that reverses theintrinsic orientation of the edge a :(4) ( σ Γ )( a ) = ( σ + ( a ) , σ − ( a )) , ( σ o c )( a ) = − o c ( a )for c ∈ K where σ ∈ S is the non-trivial element. Note that the colored orientation is always preserved under thisaction relative to actual vertices. By this definition, the condition of having colored cycle is preserved.All this actions together define the action of the group S v × (cid:16) S e ⋉ S × e (cid:17) on O k ¯V v ¯E e grac.2.3. Graded graph space.Definition 4.
Let n be integer. Then (5) O k ¯V v ¯E e GS : = h O k ¯V v ¯E e grac i [( v − n + (1 − n ) e ] . is the vector space of formal linear combinations of graphs. It is a graded vector space with non-zero term onlyin degree d = ( v − n + (1 − n ) e. The action of the group S v × (cid:16) S e ⋉ S × e (cid:17) is by linearity extended to the space O k ¯V v ¯E e GS.Let sgn v , sgn e and sgn be one-dimensional representations of S v , respectively S e , respectively S , where theodd permutation reverses the sign. They can be considered as representations of the whole product S v × (cid:16) S e ⋉ S × e (cid:17) . efinition 5. For k ≥ the full k -oriented graph complex is (6) O k fGCc n : = M v , e (cid:16) O k ¯V v ¯E e GS ⊗ sgn e (cid:17) S v × ( S e ⋉ S e ) for n even, M v , e (cid:16) O k ¯V v ¯E e GS ⊗ sgn v ⊗ sgn e (cid:17) S v × ( S e ⋉ S e ) for n odd. The group in the subscript means taking coinvariants of the group action. The e ff ect of tensoring with one-dimensional sign representation for n even is that switching edges turns graph to its negative, while vertices andintrinsic orientation of edges are indistinguishable. We say that edges are odd, while vertices and intrinsic edgeorientations are even. For n odd edges are indistinguishable and switching vertices and intrinsic edge orientationsin O k fGCc n turns graph to its negative. We say that vertices and intrinsic edge orientations are odd, while edgesare even.With abuse of terminology, the isomorphism class of a graph will also be called a graph. For distinguishing,any linear combination of graphs will not be called a graph.Graphs will be drawn without mentioning number of vertices or edges, or without intrinsic orientation on edges,if the element is even in the complex. We still do the same if the element is odd, and the sign of the graph is notimportant. The colored orientation is always drawn since it is the essential data of the graph. Remark.
For k = n . Remark.
For k = S that reverse the orientation would not been needed, making the definition a bit simpler in this case.For k > k .2.4. The di ff erential. The di ff erential on O k fGCc n is defined for a graph Γ as follows:(7) δ Γ = X t ∈ E c t ( Γ ) − X x ∈ Vx d x ( Γ ) , where the map c t is “contracting t ” and means putting a vertex x instead of tx v and reconnecting all edgesthat were previous connected to old x and v to the new x . Before contracting we permute vertices and edges inorder to put vertex v and the edge t to be the last. If the contraction forms a cycle of any color, the resultinggraph is considered to be 0. The map d v is “deleting x and its edge” and means deleting the vertex x and one edgeadjacent to it, after permuting vertices and edges in order to put vertex x and the edge adjecent to it to be the lastand changing the intrinsic direction of the edge towards x . Unless t connects two 1-valent vertices, d x will cancelcontracting the edge adjacent to x .One can check that the di ff erential is well defined and that δ = k fGCc ≥ n . The definition of that complexes is straightforward, we just start from theset of particular graphs instead of O k ¯V v ¯E e grac.2.5. Splitting complexes and convergence of spectral sequence.
Spectral sequences will be used a lot in thepaper. One wants that spectral sequence converges to the homology of the starting complex. In that case we saythat spectral sequence converges correctly .For ensuring the correct convergence of a spectral sequence standard arguments are used, such as those from[1, Appendix C]. E.g. we want spectral sequence to be bounded in each degree.The di ff erential does not change the loop number b : = e − v , so the defined complexes split as the direct sum:(8) O k fGC n = M b ∈ Z B b O k fGC n , where B b O k fGC n is the sub-complex with fixed b = e − v . This will be the case also for the other complexesdefined later.To show that a spectral sequence of the complex that is equal to the direct sum of simpler complexes convergescorrectly it is enough to show the statement for the complexes in the sum. Sub-complexes with fixed loop numbermentioned above will often have bounded spectral sequences. That easily implies their correct convergence, andhence the correct convergence of the whole complex. .6. Distinguishable vertices.
In the paper we will introduce di ff erentials that does not change the number ofvertices in the complex. In that case it is reasonable to consider complexes with distinguishable vertices:(9) ¯VO k fGCc n : = M v , e (cid:16) O k ¯V v ¯E e GS ⊗ sgn e (cid:17) ( S e ⋉ S e ) for n even, M v , e (cid:16) O k ¯V v ¯E e GS ⊗ sgn e (cid:17) ( S e ⋉ S e ) for n odd.and similarly for their sub-complexes. The original complexes are now the complexes of coinvariants of thiscomplexes, possibly tensored with sgn v , under the action of S v .The homology in respect to the di ff erential that does not change the number of vertices commutes with theaction of the group, so to understand the homology of the original complex it is enough to understand the homologyof this complex with distinguishable vertices.3. S ub - complexes of full oriented graph complex In this section we investigate some simple sub-complexes of the full oriented graph complex O k fGC n neededin the paper. The results are generalization of [8, Proposition 3.4] and use essentially the same ideas for proving.3.1. At least 2-valent vertices.
Let O k fGCc ≥ in be the complex of graphs with vertices at least i -valent. We willparticularly be interested in the case when i =
2. One easily checks that the di ff erential can not produce less then2-valent vertex, so O k fGCc ≥ n is a sub-complex.We call the passing vertex a 2-valent vertex which is the head of one edge and the tail of another for everycolor. Proposition 6. H (O k fGCc n ) = H (cid:16) O k fGCc ≥ n (cid:17) Proof.
The di ff erential can neither create nor destroy 1-valent or isolated vertices. Therefore we have direct sumof complexes O k fGCc n = O k fGCc ≥ n ⊕ O k fGCc n where O k fGCc n is the sub-complex of graphs containing at least one 1-valent vertex, including the single vertexgraph. It is enough to prove that O k fGCc n is acyclic.We call the antenna a maximal connected subgraph consisting of 1-valent and passing vertices in a graph.We set up a spectral sequence on O k fGCc n on the number of edges that are not in an antenna. The spectralsequence is bounded, and hence converges correctly. The first di ff erential is the one retracting an antenna. Thereis a homotopy that extends an antenna (summed over all antennas) that leads to the conclusion that the firstdi ff erential is acyclic, and hence the whole di ff erential. (cid:3) No passing vertices.
Let O k fGCc ◦ n ⊂ O k fGCc ≥ n be spanned by the graphs with only 2-valent vertices, andlet O k fGCc ∅ n ⊂ O k fGCc ≥ n be spanned by graphs that have at least one vertex that is at least 3-valet. Clearly, theyare sub-complexes and(10) O k fGCc ≥ n = O k fGCc ◦ n ⊕ O k fGCc ∅ n . Let O k fGCc → n ⊂ O k fGCc ≥ n be the complex of graphs with at least one passing vertex. The di ff erential can notdestroy the last passing vertex, so it is indeed the sub-complex. Let O k fGCc n be the quotient O k fGCc ≥ n / O k fGCc → n .For k = fGCc n (cid:27) O fGCc ≥ n . We also have sub-complexesO k fGCc ◦ n and O k fGCc ∅9 n . We will often use the shorter notation(11) O k GC n : = O k fGCc ∅9 n . Proposition 7. (1)
For k ≥ it holds that H (cid:16) O k fGCc ∅ n (cid:17) = H (O k GC n ) , (2) For k ≥ it holds that H (cid:0) O k fGCc ◦ n (cid:1) = H (cid:0) O k fGCc ◦ n (cid:1) .Proof. In (1) it is enough to prove that O k fGCc ∅ → n is acyclic.We set up a spectral sequence on O k fGCc ∅ → n on the number of non-passing vertices. The spectral sequenceobviously converges correctly. The first di ff erential decreases the number of passing vertices by one. There is ahomotopy that extends the string of neighboring passing vertices by one, summed over all such strings, showingthe acyclicity.For (2) in O k fGCc ◦ n there is at least one (and therefore 2) non-passing vertices because there are no coloredcycles. Therefore we can again group passing vertices into the strings of neighboring ones and do the same proof. or k = (cid:3)
4. S pecial oriented graph complex
In this section we define the special oriented graph complex O k + sGC n that makes the natural codomain ofthe quasi-isomorphism O k GC n → O k + sGC n + . It is a sub-complex (or a quotient) of O k GC n , but it can also beunderstood as a special complex with additional kind of edges.Let k ≥
0. We will now work with the complexes that have k + k + k +
1. This convention is important for signs.4.1.
Skeleton graph.
Let Γ ∈ O k + fGCc n be a graph. Weakly passing vertex in Γ is the vertex that is passing inall colors c ≤ k and not passing in the color k +
1. Vertices in Γ that are not weakly passing are called skeletonvertices . Note that there is at least one skeleton vertex if k ≥ Γ ∈ O GC n for k =
0. If k = Γ ∈ O fGCc ◦ n does not have skeleton vertices.Now, for Γ ∈ O k + GC n or Γ ∈ O k + fGCc ◦ n if k ≥ skeleton edges , as edges. We call that graph the skeleton of Γ . Skeleton edges are the strings of edges and 2-valent weakly passing vertices, edges heading towards thesame direction in all colors c ≤ k and alternating in the color k + The construction.
In skeleton, there are two kind of edges of length 2 (2 edges in original graph) regardingthe color k +
1: and . Regarding other colors c ≤ k , there can be any orientation in them, but thesame for both edges in original graph. Instead of those two kind of edges, we will use the following two kinds ofedges, that span the same space:(12) x y : = x z ya − a − x z ya − a ! (13) x y : = x z ya − a + x z ya − a ! In the picture we drew orientations in one extra color c ≤ k as an example. So, new edges also have coloredorientation for colors c ≤ k .A convention of numbering original vertices and edges is necessary to be strict with signs. Recall that we havefixed intrinsic orientation to the orientation in the color k +
1. Let all vertices in the middle of skeleton edges( z ) come after all other vertices, all edges within them came after all other edges, and two edges in a skeletonedge come one after the other. Note that the dotted edge has a new “intrinsic” orientation heading from the edge a − a . If edges are odd (even n ) the change of the intrinsic orientation of dotted edges changesthe sign, so dotted orientation is odd, and we have to consider it for the sign. For n odd edges are even so dottedorientation is even, making it unnecessarily to draw simple arrows. Dotted edges themselves are of the same parityas vertices (even for n even and odd for n odd), i.e. switching them changes sign for n odd.It is easily seen that(14) δ ( ) = − ( − n , (15) δ ( ) = . Therefore, the complexes split as(16) O k + GC n = O k + sGC ′ n ⊕ O k + fGCc † n for k ≥ , (17) O k + fGCc ◦ n = O k + sGC ◦ n ⊕ O k + fGCc ◦† n for k ≥ , where the special oriented graph complexes O k + sGC ′ n ⊂ O k + GC n and O k + sGC ◦ n ⊂ O k + fGCc ◦ n are thesub-complexes spanned by graphs whose skeleton edges are or , and O k + fGCc † n ⊂ O k + GC n andO k + fGCc ◦† n ⊂ O k + fGCc ◦ n are the sub-complexes spanned by graphs that has at least one skeleton edge ofthe di ff erent kind. Proposition 8. (1)
For k ≥ it holds that H (O k + GC n ) = H (cid:0) O k + sGC ′ n (cid:1) , (2) For k ≥ it holds that H (cid:0) O k + fGCc ◦ n (cid:1) = H (cid:0) O k + sGC ◦ n (cid:1) . roof. It is enough to show that O k + fGCc † n for k ≥ k + fGCc ◦† n for k ≥ k + fGCc † n we first set a spectral sequence on the number of skeleton vertices, such that the first δ dif-ferential does not change the skeleton. After splitting the complex into the direct sum of complexes with fixedloop number b = e − v , the spectral sequences are finite in each degree, so they converge correctly, and hence thespectral sequence for the whole complex converges correctly too.On the first page, i.e. on the complex (O k + fGCc † n , δ ), the di ff erential deletes an edge at the end of the skeletonedge (deleting in the middle would produce a passing vertex), i.e. it decreases a length of an edge in skeleton by 1,if it is not already 1 (deleting such an edge would change the skeleton). One can check that any skeleton edge oflength 3 is mapped to ± , and it is mapped to 0. There is a homotopy that does the opposite, so the first pageis acyclic. Hence the result. (cid:3) Alternative definition.
From now on, we will often understand O k + sGC ′ n and its sub-complexes in an equiv-alent way: O k + sGC ′ n is the graph complex with two kind of edges, solid edges with orientations in colors c ≤ k + dotted edges with orientations in colors c ≤ k . Dotted edges are of the same parity as vertices(even for n even and odd for n odd) and have intrinsic orientation of the opposite parity. We write the color k in black and skip the other colors if it does not make the confusion. The intrinsic orientation is depicted by thesimple black arrow as usual. There are no cycles in any color along any kind of edges. Additionally, all verticeshave to be either at least 3-valent, or not passing in one color c ≤ k , and at least one has to be at least 3-valent.The strict definition, similar to the one from Section 2, is left to the reader. The di ff erential contracts solid edgeand maps(18) δ :
7→ − ( − n , while preserving the orientation in all other colors. There is a similar definition for O k + sGC ◦ n , but here all verticeshave to be 2-valent.4.4. Tadpoles and multiple edges.
Tadpole is an edge that starts and ends at the same vertex, and multiple edgeis the set of more than one edges that connect the same two vertices. In O k + sGC ′ n we do not care of the type ofan edge while talking about tadpoles and multiple edges.In O k + sGC ′ n there can be no tadpole of a solid edge by definition. There can not be a dotted edge if k ≥ c ≤ k and it would make a colored cycle.For n even there can be no tadpoles of dotted edge because of symmetry reasons. Multiple dotted edges, withpossibly one solid edge, are possible, but they will not be a problem later in the paper, so we define(19) O k + sGC n : = O k + sGC ′ n for n even.For n odd and k = ff erential, so the graphs withat least one dotted tadpole span a sub-complex. We define(20) O sGC n for n oddto be the quotient of the complex O sGC ′ n with the complex spanned by graphs with at least one tadpole. Proposition 9.
For n odd the quotient O sGC ′ n → O sGC n is a quasi-isomorphism.Proof. It is enough to show that the complex spanned by graphs with at least one tadpole is acyclic.On it, we set up a spectral sequence such that the first di ff erential is ,
7→ − , + , summed over all tadpoles. The precise set up of the filtration and the argument that the spectral sequence convergescorrectly is left to the reader.There is a homotopy mapping
7→ − , , , summed over all tadpoles, showing that the first di ff erential makes the complex acyclic. That concludes theproof. (cid:3) et O k + sGC n : = O k + sGC ′ n for k ≥
1. In O k + sGC n there are multiple solid edges, possibly with one dottededge. The di ff erential can not destroy a multiple edge, so graphs with at least one of them span the sub-complex. Remark.
For odd n the double edge can be destroyed in O sGC ′ n by making a dotted tadpole. This is thevery reason why do we switch to O sGC n first.Now we define(21) O k + sGC n for n oddto be the quotient of the complex O k + sGC n with the complex spanned by graphs with at least one multiple edge. Proposition 10.
For k ≥ and n odd the quotient O k + sGC n → O k + sGC n is a quasi-isomorphism.Proof. It is enough to show that the complex spanned by graphs with at least one multiple edge is acyclic. On itwe set up a spectral sequence on the number of skeleton vertices, such that δ is the first di ff erential. After splittingthe complex into the direct product of complexes with fixed loop number, the spectral sequences are finite in eachdegree, so they converge correctly, and hence the spectral sequence for the whole complex too.Since δ does not change the number of vertices, the homology commutes with permuting vertices and to showthat the complex is acyclic it is enough to show that the complex with distinguishable vertices is acyclic. So fromnow on we distinguish them.On the first page we set up another spectral sequences, on the number of dotted edges that are not in the multipleedge. This spectral sequence is finite and converges correctly. The first di ff erential of this spectral sequence is theone that acts only on an multiple edge.Symmetry reasons reduce possibilities of multiple edges to m , m , m − , and m − for m ≥ l represent l solid edges with the orientation of the arrow.Additionally, edges in the multiple edge have an orientation in every color c ≤ k . All edges have it the samebecause of the no-loop condition, so we can talk about the orientation of a multiple edge.The di ff erential maps: m − m , m − m , leaving the orientation of colors c ≤ k the same. The complex with this di ff erential is clearly acyclic, concludingthe proof. (cid:3)
5. T he construction of the quasi - isomorphism In this section we prove Theorem 1, that there is a quasi-isomorphism O k GC n → O k + GC n + for k ≥ The map.
Let k ≥ Γ ∈ O k GC n be a graph. We call a spanning tree of Γ a connected sub-graph withoutcycles which contains all its vertices. Let S ( Γ ) be the set of all spanning trees of Γ .For a chosen vertex x ∈ V ( Γ ) and spanning tree τ ∈ S ( Γ ) we define h x ,τ ( Γ ) ∈ O k + GC n + as follows. Theskeleton of h x ,τ ( Γ ) is isomorphic to Γ , edges which are in τ are mapped to with the orientation away fromthe vertex x in the color k + τ are mapped to , (12), with preserved intrinsicorientation. Orientations in other colors c ≤ k are preserved.In order to precisely define the sign, we make a convention that vertices of h x ,τ ( Γ ) take labels from edges in Γ and edges of h x ,τ ( Γ ) take labels from vertices in Γ in the following sense.Recall that vertices are labeled with numbers starting from 0 and edges are labeled with numbers starting from1. First we permute vertices of Γ such that the chosen vertex x is the vertex 0. That vertex in h x ,τ ( Γ ) is labeled also0. Other vertices in the skeleton of h x ,τ ( Γ ) are labeled by the former label (in Γ ) of the edge heading towards it.Vertices in the middle of a dotted skeleton edge are labeled by the former label of that edge. So, all vertices takelabel 0, and labels of all edges from Γ .Solid edges in the skeleton of h x ,τ ( Γ ) are labeled with the former label of the vertex to which they head. Thefollowing numbers are used to label remaining edges: two edges in dotted skeleton edges are labeled with twoconsecutive numbers, such that lower number is taken by the edge at the intrinsic tail of the former edge (in Γ ).The order of the pairs of edges in each dotted edge does not matter in any parity case. verything gets a pre-factor ( − nr where r is the number of edges in τ whose orientation in the color k + h x ,τ .
123 456 h , ( − n = ( − n
13 5 + . . . One can check that the map is well defined (sign change from a permutation or reversing in Γ is the same aftermapping), cycles are never formed in any color and that the degree of the map is 0.Let(22) h ( Γ ) : = X x ∈ V ( Γ ) ( v ( x ) − X τ ∈ S ( Γ ) h x ,τ ( Γ ) , where v ( x ) is the valence of the vertex x in Γ . Proposition 11.
The map h : O k GC n → O k + GC n + is a map of complexes, i.e. δ h ( Γ ) = h ( δ Γ ) for every Γ ∈ O k GC n .Proof. It holds that h ( δ Γ ) = h X t ∈ E ( Γ ) c t ( Γ ) = X t ∈ E ( Γ ) h ( c t ( Γ )) = X t ∈ E ( Γ ) X x ∈ V ( c t ( Γ )) (cid:0) v c t ( Γ ) ( x ) − (cid:1) X τ ∈ S ( c t ( Γ )) h x ,τ ( c t ( Γ ))where c t ( Γ ) is contracting an edge t in Γ . Spanning trees of c t ( Γ ) are in natural bijection with spanning trees of Γ which contain t , c t ( τ ) ↔ τ , so we can write h ( δ Γ ) = X t ∈ E ( Γ ) X τ ∈ S ( Γ ) t ∈ E ( τ ) X x ∈ V ( c t ( Γ )) (cid:0) v c t ( Γ ) ( x ) − (cid:1) h x , c t ( τ ) ( c t ( Γ )) = X τ ∈ S ( Γ ) X t ∈ E ( τ ) X x ∈ V ( c t ( Γ )) (cid:0) v c t ( Γ ) ( x ) − (cid:1) h x , c t ( τ ) ( c t ( Γ )) . For x ∈ V ( c t ( Γ )) we have two choices, either x ∈ V ( Γ ) not adjacent to t , or x is produced by contraction of t . Itcan be seen (after being careful with signs) that in the first case h x , c t ( τ ) ( c t ( Γ )) = c t ( h x ,τ ( Γ )) and in the second case h x , c t ( τ ) ( c t ( Γ )) = c t (cid:0) h Γ − ( t ) ,τ ( Γ ) (cid:1) = c t (cid:0) h Γ + ( t ) ,τ ( Γ ) (cid:1) , where Γ − ( t ) and Γ + ( t ) are ends of the edge t . Since in the first casevertices do not change valence after applying c t , and it is v c t ( Γ ) ( x ) = v Γ ( Γ − ( t )) + v Γ ( Γ + ( t )) − h ( δ Γ ) = X τ ∈ S ( Γ ) X t ∈ E ( τ ) X x ∈ V ( Γ ) x < { Γ − ( t ) , Γ + ( t ) } ( v Γ ( x ) − c t ( h x ,τ ( Γ )) + ( v Γ ( Γ − ( t )) + v Γ ( Γ + ( t )) − c t ( h Γ − ( t ) ,τ ( Γ )) == X τ ∈ S ( Γ ) X t ∈ E ( τ ) X x ∈ V ( Γ ) ( v Γ ( x ) − c t ( h x ,τ ( Γ )) = X x ∈ V ( Γ ) ( v Γ ( x ) − X τ ∈ S ( Γ ) X t ∈ E ( τ ) c t ( h x ,τ ( Γ )) . On the other side δ h ( Γ ) = δ X x ∈ V ( Γ ) ( v Γ ( x ) − X τ ∈ S ( Γ ) h x ,τ ( Γ ) = X x ∈ V ( Γ ) ( v Γ ( x ) − X τ ∈ S ( Γ ) X t ∈ E ( h x ,τ ( Γ )) c t (cid:0) h x ,τ ( Γ ) (cid:1) . The edge t ∈ E ( h x ,τ ( Γ )) can be chosen in two ways, it is either in the spanning tree τ or one of the edges of dotedskeleton edge. In the first case the sum gives exactly h ( δ Γ ). Note that if contracting a skeleton edge makes a cyclein any color c ≤ k , it makes so also after the mapping. Therefore δ h ( Γ ) = h ( δ Γ ) + X x ∈ V ( Γ ) ( v Γ ( x ) − X τ ∈ S ( Γ ) X T ∈ D ( Γ ,τ ) C T (cid:0) h x ,τ ( Γ ) (cid:1) where D ( Γ , τ ) : = E ( Γ ) \ E ( τ ) is bijective with the set of dotted edges in skeleton of h x ,τ ( Γ ) and C T (cid:0) h x ,τ ( Γ ) (cid:1) is agraph in O k + sGC n + given from h x ,τ ( Γ ) by replacing dotted edge which come from T with − ( − n + ,and is 0 if that produces a cycle in color k +
1. It cannot produce a cycle in any other color c ≤ k .To finish the proof it is enough to show that(23) N ( Γ , x ) : = X τ ∈ S ( Γ ) X T ∈ D ( Γ ,τ ) C T (cid:0) h x ,τ ( Γ ) (cid:1) s zero for every x ∈ V ( Γ ). Terms in above relation can be summed in another order. Let CT ( Γ ) be the set of allconnected sub-graphs σ of Γ which contain all vertices and which has one more edge than spanning tree (has onecycle), and let C ( σ ) be the set of edges in the cycle of σ . Clearly, σ \ T for T ∈ C ( σ ) is a spanning tree of Γ andsets { ( τ, T ) | τ ∈ S ( Γ ) , T ∈ D ( Γ , τ ) } and { ( σ, T ) | σ ∈ CT ( Γ ) , T ∈ C ( σ ) } are bijective, so(24) N ( Γ , x ) = X σ ∈ CT ( Γ ) X T ∈ C ( σ ) C T (cid:0) h x ,σ \ T ( Γ ) (cid:1) . It is now enough to show that P T ∈ C ( σ ) C T (cid:0) h x ,σ \ T ( Γ ) (cid:1) = x ∈ V ( Γ ) and for every σ ∈ CT ( Γ ). Let y ∈ V ( Γ )be the vertex in cycle of σ closest to vertex x (along σ ). After choosing T ∈ C ( σ ), that cycle in h x ,σ \ T ( Γ ) has onedotted edge, and directions of other edges go from y to that dotted edge, such as on the diagram. y After acting by C T the dotted edge is replaced by + ( − n , like in the diagram y + ( − n y Careful calculation of the sign shows that those two terms are canceled with terms given from choosing neighbor-ing dotted edges, and two last terms which does not have corresponding neighbor are indeed 0 because they havecycle in color k +
1. Therefore(25) X T ∈ C ( σ ) C T (cid:0) h x ,σ \ T ( Γ ) (cid:1) = x ∈ V ( Γ ) and for every σ ∈ CT ( Γ ), so N ( Γ , x ) = (cid:3) Clearly, h ( Γ ) always sits in O k + sGC ′ n + , so we can define a co-restriction h : O k GC n → O k + sGC ′ n + and for n even the map to the quotient h : O k GC n → O k + sGC n + , recall (20) and (21). By the abuse of notation, we call allthese maps h . Note that for n even ( n + k GC n .5.2. The proof.Proposition 12.
The map h : O k GC n → O k + sGC n + is a quasi-isomorphism.Proof. On the mapping cone of h we set up the standard spectral sequence, on the number of vertices (inO k + sGC n + skeleton vertices). Standard splitting of complexes as the product of complexes with fixed loopnumber implies the correct convergence.On the first page of the spectral sequence there is a mapping cone of the map h : (O k GC n , → (O k + sGC n + , δ ) . Since no di ff erential changes the number of vertices, the homology commutes with permuting vertices and to showthat this mapping cone is acyclic, it is enough to show that for distinguishable vertices, i.e. for the map h : (cid:16) ¯VO k GC n , (cid:17) → (cid:16) ¯VO k + sGC n + , δ (cid:17) . Edges are still not distinguishable. But we can talk about the positional edge , that is a pair of vertices in V ( Γ ) × V ( Γ ) for which there is at least one edge between them.For n even ( n + k GC n by symmetry nor in ¯VO k + sGC n + by definition,so positional edges are indeed edges, and they can be distinguished.For n odd ( n + k GC n and ¯VO k + sGC n + , so one positional edgecan consist of more edges and they are undistinguishable if of the same type. In ¯VO k + sGC n + symmetry reasonsallow only the following positional edges: m , m − , and m − for m ≥
1, where the thick dotted line with a number l represent l dotted lines. Additionally, a positional edge hasan orientation in colors c ≤ k both in ¯VO k GC n and ¯VO k + sGC n + .With distinguishable vertices we can also chose a preferred orientation of an edge (e.g. from the vertex with asmaller number towards the vertex with the bigger number). et the shape (26) s : ¯VO k + sGC n + → ¯VO k GC n map a graph Γ to the graph with the same vertices as the skeleton, and with the same edges, but forgetting the typeof the edge and the orientation in the color k +
1, while still remembering the orientation in other colors.The di ff erential δ does not change the shape, just the type of an edge. Therefore the mapping cone of h : (cid:16) ¯VO k GC n , (cid:17) → (cid:16) ¯VO k + sGC n + , δ (cid:17) splits as a direct product of mapping cones of maps for each shape Σ ∈ ¯VO k GC n , that is of(27) h : ( K Σ , → (cid:16) ¯VO k + sGC Σ n + , δ (cid:17) where ¯VO k + sGC Σ n + is the complex spanned by all graphs Γ ∈ ¯VO k + sGC n + such that s ( Γ ) = Σ .It is now enough to show that (27) is a quasi-isomorphism. Homology of the left-hand complex is clearlyone-dimensional, so we need to show that the homology of the right-hand side is also one dimensional, and that Σ is mapped to a representative of the class.Let us now fix a shape Σ with v vertices. In Σ we choose one vertex y and v − e , . . . , e v − ,such that for every i = , . . . , v − { e , . . . , e i } form a sub-tree of Σ that contains vertex y .Let Σ i for i = , . . . , v − Σ and threetypes of edges. There is one thick edge on the position of each positional edge { e , . . . e i } with the orientationsin colors c ≤ k , and solid and dotted edges on the other skeleton edges, solid being oriented in colors c ≤ k + c ≤ k , such that there are no cycles in any color c ≤ k + k +
1. The di ff erential δ acts on dotted edges as usual, while thethick edges stay always unchanged. The strict definition is left to the reader. An example for k = e e e F igure
2. For a skeleton Σ on the left with chosen edges in the middle, an example of a graphin Σ is drawn on the left.We define maps f i + : Σ i → Σ i + which acts on the positional edge e i + as follows(28) f i + : m , m − m − , m −
7→ − ( − n m − . The preferred direction (left to right) is the one away from the vertex y along the thick tree. It is easily seen that f i + is a map of complexes. Therefore we have defined the chain of complexes and maps between them:¯VO k + sGC Σ n + = Σ f −→ Σ f −→ . . . f v − −→ Σ v − . Lemma 13.
For every i = , . . . , v − the map f i + : Σ i → Σ i + is a quasi-isomorphism.Proof. We set up a spectral sequence on the mapping cone of f i + : Σ i → Σ i + on the number of dotted edges thatare not in the positional edge e i + . The spectral sequence clearly converges correctly. On the first page there is amapping cone of the map f i + : (cid:16) Σ i , δ i + (cid:17) → (cid:16) Σ i + , (cid:17) where δ i + is the part of the di ff erential acting on the positional edge e i + . This map is clearly quasi-isomorphism,concluding the proof. (cid:3) Remark.
The good choice of maps f i + would not be possible for multiple edges in O k + sGC n + for n even. Thisis the very reason why did we get rid of those multiple edges in Subsection 4.4. Σ v − is generated by only one graph, the one without solid edges. Let us call this graph ¯ Σ . So H ( Σ v − , δ ) isone-dimensional.Let f : = f v − ◦ · · · ◦ f and let us consider the composition f ◦ h : ( K Σ , → (cid:16) Σ v − (cid:17) . Recall (22) that h ( Σ ) = X τ ∈ S ( Σ ) X x ∈ V ( Σ ) ( v ( x ) − h x ,τ ( Σ ) . ince f i sends dotted edges to 0, the term of the first sum that can survive the action of the composition f is onlythe one where the chosen sub-tree τ coincides with the positional sub-tree T : = { e , . . . , e v − } . So(29) f ◦ h ( Σ ) = M X x ∈ V ( Σ ) ± ( v ( x ) −
2) ¯ Σ , where M is a positive factor coming from multiple edges in T since, technically, if there are multiple edges in T there are more ways to choose a sub-tree of real edges.The goal is to show that Σ is not sent to 0. To do that, we need to be careful with the signs. We define the signbeing + if x = y . Let us chose x distant from y by r edges in the tree T . We will show that the signs of this term in(29) is also + .Recall the sign convention in the definition of h in Subsection 5.1. For n even pre-factor ( − nr is always + .Vertices are indistinguishable, so the sign does not depend on the choice of the vertex x . But choosing another x will change the convention of naming vertices in h x , T ( Σ ). This will give us a pre-factor ( − r as described inFigure 3. While acting by s , r edges will go in opposite direction and give a pre-factor ( − ( − n ) r = ( − r , recall(28). The overall sign of f ◦ h x , T is therefore ( − r ( − r = + .1 2 3 y = x h y x h igure
3. For n even edges in O k GC n and vertices in O k + sGC n + are odd. Choosing vertex x that is r edges away from y along the tree shifts labels of vertices between x and y . Whilerenaming them to the state of x = y we get the sign ( − r .For n odd exactly r edges from the tree will change direction while acting by h x , T , so there is a pre-factor ( − r from the definition. If we choose another vortex x , we first need to rename it to 0. We do the renaming such thatthe names of the edges at the end are the same, giving a pre-factor ( − r as described in Figure 4. The overall signof f ◦ h x , T is ( − r ( − r = + . Acting by f does not change sign in this case, recall (28), so the overall sign of f ◦ h x , T remains + .We have shown that f ◦ h maps the generator of the homology class in ( K Σ ,
0) to the generator of the homologyclass in ( Σ v − , δ ), so it is a quasi-isomorphism. Lemma 13 implies that f is quasi-isomorphism, so h from (27) isa quasi-isomorphism too. That was to be demonstrated. (cid:3) Theorem 1 now follows directly using Propositions 8, 9 and 10.R eferences [1] Anton Khoroshkin, Thomas Willwacher and Marko ˇZivkovi´c. Di ff erentials on graph complexes. Adv. Math.
In Proceedings of the I. M. Gelfand seminar 1990–1992 , 173-188.Birkhauser, 1993.[3] Maxim Kontsevich. Formality Conjecture.
Deformation Theory and Symplectic Geometry
Journal f¨ur die reine und angewandte Mathe-matik . (Qrelle) 634: 51–106 & 636: 123–174, 2009.[7] Sergei Merkulov and Thomas Willwacher. Props of ribbon graphs, involutive Lie bialgebras and moduli spaces of curves.arXiv:1511.07808.[8] Thomas Willwacher. M. Kontsevich’s graph complex and the Grothendieck-Teichm¨uller Lie algebra.
Invent. Math.
Commun. Math. Phys. = y = x
12 3 h − r ( − r = y x = h ( − r igure
4. For n odd vertices in O k GC n and edges in O k + sGC n + are odd. If we choose vertex x that is r edges away from y we need to rename it to 0 and we shift all names of the verticesin between. This gives the sign ( − r . Map h x , T changes the direction of r edges, so it givesanother sign ( − r . [10] Thomas Willwacher and Marko ˇZivkovi´c. Multiple edges in M. Kontsevich’s graph complexes and computations of the dimensions andEuler characteristics. Adv. Math. athematics R esearch U nit , U niversity of L uxembourg , G rand D uchy of L uxembourg E-mail address : [email protected]@uni.lu