A simplicial version of the 2-dimensional Fulton-MacPherson operad
AA SIMPLICIAL VERSION OF THE 2-DIMENSIONALFULTON–MACPHERSON OPERAD
NATHANIEL BOTTMAN
Abstract.
We define an operad in
Top , called FM W . The spaces in FM W come with CW de-compositions, such that the operad compositions are cellular. In fact, each space in FM W is therealization of a simplicial set. We expect, but do not prove here, that FM W is isomorphic to the2-dimensional Fulton–MacPherson operad FM . Our construction is connected to the author’swork on the symplectic ( A ∞ , Introduction
In 1994, Getzler and Jones [GetzlerJones] introduced the Fulton–MacPherson operadFM = (cid:0) FM ( k ) (cid:1) k ≥ , (1)where FM ( k ) is the compactification `a la Fulton–MacPherson [FultonMacPherson] of the configu-ration space of k distinct labeled points in R , modulo translations and dilations. Getzler and Jonesproposed in the same paper a collection of cellular decompositions of the spaces in FM , such thatthese decompositions are compatible with the operad maps ◦ i : FM ( k ) × FM ( (cid:96) ) → FM ( k + (cid:96) − (6), there are two disjoint open 6-cells C , C with the property that C ∩ C is nonempty, as described in [Voronov, § Top to FM . Under this expected isomorphism, our decompositions are refinements of Getzler–Jones’ attempted decompositions. The context for the current paper is the author’s programto construct Symp , the symplectic ( A ∞ , A ∞ , Symp , this suggests a strategy toward endowing symplecticcohomology with a chain-level homotopy Gerstenhaber (and eventually, homotopy BV) algebrastructure that is finite in each arity, thus answering Conjecture 2.6.1 from [Abouzaid].1.1.
Getzler–Jones’ attempted decomposition.
Getzer–Jones’ attempted decomposition is anadaptation to the case of FM of Fox–Neuwirth’s decomposition [FoxNeuwirth] of the one-pointcompactification of the configuration space ( R ) k \ ∆ of k points in R , where ∆ is the fat diagonal.A Fox–Neuwirth cell corresponds to a choice of which subsets of the points p , . . . , p k should bevertically aligned, the left-to-right order in which these subsets of points should appear, and thetop-to-bottom order in which each subset of the points should appear. For instance, the followingis a real-codimension-3 cell in (cid:0) ( R ) \ ∆ (cid:1) ∗ : a r X i v : . [ m a t h . A T ] J a n
54 6 31
Figure 1.
Getzler–Jones observed that the Fox–Neuwirth cells are invariant under translations and dilations,and moreover that one can define a similar type of cell for the boundary locus. The elements in theboundary of FM ( k ) are trees of “screens”, and these “boundary cells” are defined by partitioningand ordering the points on each of the screen in the same way as with Fox–Neuwirth cells.1.2. Tamarkin’s counterexample.
As described in [Voronov], Tamarkin observed a way inwhich Getzler–Jones’ supposed decomposition fails. Consider FM (6), the open locus of whichparametrizes configurations of six distinct points in R , up to translations and dilations. Next, weconsider the two 6-cells C and C pictured in the figure below. (We omit the numberings.)The j -th bubble in C (for j = 1 ,
2) carries a modulus λ j defined in the following way: by translatingand dilating, we can move the left resp. right lines to x = 0 and x = 1; we then denote by λ j theposition of the middle line. The intersection C ∩ C is the codimension-1 locus in C in which λ = λ . What Getzler–Jones proposed is therefore not a cellular decomposition, because theintersection of the closures of two distinct n cells should be contained in the ( n − C , C , and C ∩ C will each be a union of cells.1.3. An overview of our construction.
In this paper, we construct a collection of CW complexesFM W ( k ) and maps ◦ i : FM W ( k ) × FM W ( (cid:96) ) → FM W ( k + (cid:96) − , ≤ i ≤ k. (2)Here is our main result: Main theorem.
The spaces (cid:0) FM W ( k ) (cid:1) k ≥ together with the composition operations ◦ i form anon- Σ operad, and the composition maps ◦ i : FM W ( k ) × FM W ( (cid:96) ) → FM W ( k + (cid:96) − are cellular. We will now give a brief overview of the definition of FM W ( k ). . First, we define a “W-version” W W n of the 2-associahedra by completing the following analogy: K r : W (Ass) :: W n : W W n . (4)Here K r is the ( r − W (Ass) is the Boardman–Vogt W-constructionapplied to the associative operad, which is defined in terms of metric stable trees and yields anoperad of CW complexes that is isomorphic to the associahedral operad K in Top . W n is an( | n | + r − W W n is a CW complex that we define in § W n . We then refine theCW structure on W W n to a simplicial decomposition. Toward our construction of FM W ( k ), we decompose FM ( k ) into Getzler–Jones cells, thenidentify each open Getzler–Jones cell with a product of open 2-associahedra. We then replace eachsuch product by the corresponding product of interiors of the spaces W W n described in the previousstep. This product comes with a decomposition into products of simplices, and we refine this to asimplicial structure. Finally, we attach these decomposed Getzler–Jones cells together to produceFM ( k ). This part of the construction appears in § ( k ) that we must verify is that our CW decomposition is valid. It isclear that our putative open cells disjointly decompose our space, and that they are homeomorphicto open balls. The only nontrivial check we need to make is that the n -cells are attached to the( n −
1) skeleton. This is where Getzler–Jones’ attempted decomposition fails: the 6-cell C thatwe described in § n -cell by taking a closed n -simplex, then attaching it to theexisting skeleton via quotient maps from the boundary ( n − n − n -cell is a union of cells of dimension at most n − The relationship between our construction and
Symp . The genesis of the construction ofFM W was a connection between the symplectic ( A ∞ , Symp and E suggested by JacobLurie in 2016. (The construction of Symp is a long-term project of the author, building on work ofMa’u–Wehrheim–Woodward; see [Bottman1, Bottman2, Bottman3, Bottman4, BottmanCarmeli,BottmanWehrheim, Ma’uWehrheimWoodward].) We can express this connection concretely, via acollection of maps f Wσ : W W n → FM W ( | n | ) , (5)where σ is a , as defined in § f σ forgets the data of the lines, then labels the points according to the 2-permutation σ . Then f σ extends continuously to the boundary of W n ; it is an embedding on the interior of its domain, butcontracts some boundary cells. Example 1.1.
In the following figure, we depict W and its image under an appropriate map f σ .More precisely, we depict their nets — to “assemble” both CW complexes, one would cut them out,then glue together like-numbered edges. As is evident, most of the 2-cells of W are contractedby f σ . The following result encapsulates the connection between the symplectic ( A ∞ , W . It is an immediate consequence of our construction of W W n and FM W ( k ), and it forms thecontent of Remark 3.14 below. Proposition.
Fix r ≥ , n ∈ Z r ≥ \ { } , and a 2-permutation σ of type n . Then the associatedmap f Wσ : W W n → FM W ( | n | )(6) is cellular. Future directions.
The author plans to develop several aspects of the current paper, asexplained below. • We do not address regularity in this paper. In future work with Paolo Salvatore, we planto show that the CW complexes we construct here are regular. Moreover, we plan to showthat our CW complexes are realizations of certain simplicial sets. • With Paolo Salvatore, we plan to extend this work to produce cellular decompositions ofFM Wk for all k ≥
1, and to show that FM Wk is isomorphic to FM k in Top . • This paper can be construed as a way of incorporating identity 1-morphisms into the sym-plectic ( A ∞ , A ∞ , • Finally, we plan to upgrade this work to give a cellular model for the framed analogue of theFulton–MacPherson operad. This suggests a way of endowing symplectic cohomology witha chain-level BV-algebra structure, which is the subject of Conjecture 2.6.1 from [Abouzaid]. .6. Glossary of notation.
We introduce a number of new objects in this paper, and we in-clude the following glossary to help the reader keep them straight. The objects in the entries d ( T ) , T, d (2 T ) appeared in [Bottman1], and K Wr is well-known, but most of the remaining objectshave not appeared previously.notation interpretation p. first defined K tree r associahedral poset p. 6 (cid:0) T, ( (cid:96) e ) (cid:1) metric rooted ribbon tree (metric RRT) p. 6 d ( T ) dimension of a stable RRT p. 6 C T cell in K Wr associated to T p. 6 K Wr W-version of the associahedron p. 7 ◦ i : K Wr × K Ws → K Wr + s − composition map in K W p. 82 T = T b f → T s stable tree-pair p. 8 W tree n n stable tree-pairs) p. 8 (cid:0) T, ( L e ) , ( (cid:96) e ) (cid:1) metric stable tree-pair p. 9 d (2 T ) dimension of a stable tree-pair p. 10 C T cell in W W n associated to 2 T p. 10 W W n W-version of the 2-associahedron p. 10 π tree i , π tree forgetful maps between 2-associahedral posets p. 13 q F quotient map on a cell F of W W n p. 14 σ A p. 16 T (using abusive notation) Getzler–Jones datum p. 16GJ T Getzler–Jones cell associated to the GJ datum T p. 16FM W ( k ) W-version of the arity- k space in FM p. 17 ◦ i : FM W ( k ) × FM W ( (cid:96) ) composition map in FM W p. 17 → FM W ( k + (cid:96) − Acknowledgments.
This paper is a solution to homework problem A ∞ , E -algebras. Alexander Voronov explained to the author the colorful historysurrounding this problem. A conversation with Naruki Masuda, Hugh Thomas, and Bruno Valletteled the author to think about replacing FM with a “W-construction version” thereof. The au-thor thanks Dean Barber, Michael Batanin, Sheel Ganatra, Ezra Getzler, Mikhail Kapranov, BenKnudsen, Paolo Salvatore, and Dev Sinha for their interest and encouragement.The author was supported by an NSF Mathematical Sciences Postdoctoral Research Fellowshipand by an NSF Standard Grant (DMS-1906220). He thanks the Institute for Advanced Study, theMathematical Sciences Research Institute, and the University of Southern California for providingexcellent working conditions during the period when this work was carried out.2. A “W-version” of the 2-associahedra
In this section, we construct a “W-version” of the 2-associahedra. (The 2-associahedra wereoriginally defined in [Bottman1].) This is an essential ingredient in our definition of FM W ( k ),which will appear in § A warm-up: K W , i.e. W (Ass) , i.e. a W-version of the associahedra. In this subsection,we recall a certain operad, which we will denote by K W = (cid:0) K Wr (cid:1) r ≥ . This is simply the Boardman–Vogt W-construction applied to the associative operad Ass. We construct only K W rather thanrecalling the general definition of the W-construction, because this one-off construction will be a seful warm-up to our construction of W W later in this section. As noted in [Barber], K W isisomorphic in Top to the associahedral operad K .The following proposition summarizes what we will prove about K W . Proposition 2.1.
The spaces (cid:0) K Wr (cid:1) r ≥ form a non- Σ operad of CW complexes, and the composi-tion maps ◦ i : K Wr × K Ws → K Wr + s − (7) defined in Def. 2.10 are cellular. We will prove Prop. 2.1 at the end of the current subsection.We begin with a definition of rooted ribbon trees. Stable rooted ribbon trees with r leaves indexthe strata of the associahedron K r , and they will be an integral part of the definition of K Wr . Definition 2.2 (Def. 2.2, [Bottman1]) . A rooted ribbon tree (RRT) is a tree T with a choice ofa root α root ∈ T and a cyclic ordering of the edges incident to each vertex; we orient such a treetoward the root. We say that a vertex α of an RRT T is interior if the set in( α ) of its incomingneighbors is nonempty, and we denote the set of interior vertices of T by T int . An RRT T is stable ifevery interior vertex has at least 2 incoming edges. We define K tree r to be the set of all isomorphismclasses of stable rooted ribbon trees with r leaves.We denote the i -th leaf of an RRT T by λ Ti . For any α, β ∈ T , T αβ denotes those vertices γ suchthat the path [ α, γ ] from α to γ passes through β . We denote T α := T α root α . (cid:52) Next, we define a version of RRTs with internal edge lengths.
Definition 2.3. A metric RRT (cid:0) T, ( (cid:96) e ) (cid:1) is the following data: • An RRT T . • For every edge e of T not incident to a leaf (but possibly incident to the root), a length (cid:96) e ∈ [0 , metric RRT of type T . (cid:52) Now we will define a “dimension” function d on stable RRTs. Definition 2.4 (Definition 2.4, [Bottman1]) . For T a stable RRT in K tree r , we define its dimension d ( T ) ∈ [0 , r −
2] like so: d ( T ) := r − T int − . (8) (cid:52) Definition 2.5.
Given a stable tree T , the cell associated to T is denoted by C T and is defined toconsist of all metric RRTs of type T . (cid:52) Note that we can canonically identify C T with the closed cube of dimension equal to the numberof internal edges of T . That is: C T ∼ = [0 , T int − = [0 , r − − d ( T ) . (9)As we will see, K Wr is ( r − d ( T ) is the codimension of C T in K Wr .(The unfortunate clash of terminology between “dimension” and “codimension” is due to the factthat in K r , the cell indexed by T has dimension d ( T ).)We now define K Wr by taking the union of the cells C T for T any stable RRT with r leaves, thencollapsing edges of length 0. efinition 2.6. Given r ≥
1, we define K Wr to be the following quotient: K Wr := (cid:16) (cid:71) T ∈ K tree r C T (cid:19)(cid:46) ∼ . (10)Here ∼ identifies (cid:0) T, ( (cid:96) e ) (cid:1) and (cid:0) T (cid:48) , ( (cid:96) (cid:48) e ) (cid:1) if, after collapsing all edges e of T with (cid:96) e = 0 and alledges e of T (cid:48) with (cid:96) (cid:48) e = 0, both metric RRTs reduce to the same metric RRT (cid:0) T (cid:48)(cid:48) , ( (cid:96) (cid:48)(cid:48) e ) (cid:1) . (cid:52) Example 2.7.
In the following figure, we depict the CW complex K W . ba ba b ab a ba b b Note that this is a refinement of K , which (as a CW complex) is a pentagon. We have labeledthe open top cells by the metric stable RRTs that they parametrize, where each a and b is allowedto vary in [0 , b in both cells becomes 0. The boundaryof K Wr is the union of cells where at least one edge length is 1. (cid:52) Finally, we define a simplicial refinement of the CW structure on K Wr . To approach this, wenote that if P is the poset { , } k , where σ < σ if σ can be gotten by changing some of the 0’sof σ to 1’s, then the nerve of P is a simplicial decomposition of the cube [0 , k . More concretely,the top simplices are the sets of the form (cid:8) ( x , . . . , x k ) ∈ [0 , k | < x σ (1) < · · · < x σ ( k ) < (cid:9) , (11)where σ is a permutation on k letters. The remaining simplices are the result of replacing some ofthese inequalities by equalities. Definition 2.8.
We refine the CW structure on K Wr by decomposing each cell C T in K Wr like so:we make the identification C T ∼ = [0 , r − − d ( T ) , then perform the simplicial decomposition describedin the previous paragraph. This refinement equips K Wr with a simplicial decomposition. (cid:52) Example 2.9.
In the following figure, we depict the simplicial complex K W . bab a ba ba ba ba b a bab a ba < b a < ba < bb < a b < a b < aa < b b < aa < bb < a This is the refinement of our initial, cubical, CW decomposition of K Wr gotten by subdividing eachof the five squares into two triangles. We indicate the new edges by coloring them blue. (cid:52) Now that we have constructed the spaces K Wr , we can prove Prop. 2.1, which states that (cid:0) K Wr (cid:1) is a non-Σ operad and that the operad maps are cellular. Definition 2.10.
Fix r , s , and i ∈ [1 , r ]. We wish to define the composition map ◦ i : K Wr × K Ws → K Wr + s − . (12)We do so cell by cell. That is, fix cells C T ⊂ K Wr and C T (cid:48) ⊂ K Ws . Define T (cid:48)(cid:48) to be the result ofgrafting T (cid:48) to the i -th leaf of T . Then we define ◦ i on C T × C T (cid:48) like so: given collections of edgelengths on T and T (cid:48) , combine them to produce a collection of edge lengths on T (cid:48)(cid:48) , where we assignto the single newly-formed interior edge the length 1. (cid:52) Proof of Prop. 2.1.
Fix r , s , and i ∈ [1 , r ], and consider the composition map ◦ i : K Wr × K Ws → K Wr + s − . (13)To show that ◦ i is cellular, let’s consider the restriction of ◦ i to a product C T × C T (cid:48) of closed cubes,for T ∈ K r and T (cid:48) ∈ K s . Denote by T (cid:48)(cid:48) the tree obtained by grafting the root of T (cid:48) to the i -th leafof T . Then ◦ i includes C T × C T (cid:48) into C T (cid:48)(cid:48) as the face gotten by requiring the outgoing edge of theroot of T (cid:48) to have length 1. The CW structure of this face of C T (cid:48)(cid:48) is finer than that of C T × C T (cid:48)(cid:48) ,so ◦ i is indeed cellular. (cid:3) Metric tree-pairs and the definition of W W n . Just as we defined K Wr to be the parameterspace of metric stable RRTs, we will define W W n to parametrize metric stable tree-pairs. Thedefinition of metric stable tree-pairs is somewhat involved, so we devote the current subsection tothis definition.Before defining metric stable tree-pairs, we recall the definition of stable tree-pairs. Definition 2.11 (Def. 3.1, [Bottman1]) . A stable tree-pair of type n is a datum 2 T = T b f → T s ,with T b , T s , f described below: • The bubble tree T b is an RRT whose edges are either solid or dashed, which must satisfythese properties: The vertices of T b are partitioned as V ( T b ) = V comp (cid:116) V seam (cid:116) V mark , where: ∗ every α ∈ V comp has ≥ ∗ every α ∈ V seam has ≥ ∗ every α ∈ V mark has no incoming edges and either a dashed or no outgoing edge.We partition V comp =: V (cid:116) V ≥ according to the number of incoming edges of agiven vertex. – ( stability ) If α is a vertex in V and β is its incoming neighbor, then β ) ≥ α is a vertex in V ≥ and β , . . . , β (cid:96) are its incoming neighbors, then there exists j with β j ) ≥ • The seam tree T s is an element of K tree r . • The coherence map is a map f : T b → T s of sets having these properties: – f sends root to root, and if β ∈ in( α ) in T b , then either f ( β ) ∈ in( f ( α )) or f ( α ) = f ( β ). – f contracts all dashed edges, and every solid edge whose terminal vertex is in V . – For any α ∈ V ≥ , f maps the incoming edges of α bijectively onto the incoming edgesof f ( α ), compatibly with < α and < f ( α ) . – f sends every element of V mark to a leaf of T s , and if λ T s i is the i -th leaf of T s , then f − { λ T s i } contains n i elements of V mark , which we denote by µ T b i , . . . , µ T b in i .We denote by W tree n the set of isomorphism classes of stable tree-pairs of type n . Here an isomor-phism from T b f → T s to T (cid:48) b f (cid:48) → T (cid:48) s is a pair of maps ϕ b : T b → T (cid:48) b and ϕ s : T s → T (cid:48) s that fit into acommutative square in the obvious way and that respect all the structure of the bubble trees andseam trees. (cid:52) Next, we define metric stable tree-pairs. This notion is more subtle than that of metric stableRRTs, because we must impose conditions on the edge-lengths. (This should be compared with[BottmanOblomkov, § W n .) Definition 2.12. A metric stable tree-pair (cid:0) T, ( L e ) , ( (cid:96) e ) (cid:1) is the following data: • A stable tree-pair 2 T . • For every interior dashed edge e of T b , a length L e ∈ [0 , e of T s , a length (cid:96) e ∈ [0 , L α := L e for α ∈ V comp ( T b ) \ { α root } and e the outgoing edge of α , and similarly forthe edge-lengths in T s ): – For every α , α ∈ V ≥ ( T b ) and β ∈ V ( T b ) with π ( α ) = π ( α ) = π ( β ), werequire: max γ ∈ [ α ,β ) L γ = max γ ∈ [ α ,β ) L γ . (14) – For every ρ ∈ V int ( T s ) \ { ρ root } and α ∈ V ≥ ( T b ) with π ( α ) = ρ , we require: (cid:96) ρ = max γ ∈ [ α,β α ) L γ , (15) where we define β α to be the first element of V ≥ ( T b ) that the path from α to α root passes through. (cid:52) inally, we recall the dimension of a stable tree-pair. Similarly to the dimension of a stable RRT,this will be the codimension in W W n of the cell corresponding to the stable tree-pair in question. Definition 2.13 (Definition 3.3, [Bottman1]) . For 2 T a stable tree-pair, we define the dimension d (2 T ) ∈ [0 , | n | + r −
3] like so: d (2 T ) := | n | + r − V ( T b ) − T s ) int − . (16) (cid:52) We are now prepared to define W W n , the “W-version” of the 2-associahedron. We will define W W n by attaching together the cells C T , which consist of metric stable tree-pairs. Definition 2.14.
Given a stable tree-pair 2 T , the cell associated to T is the collection of all metricstable tree-pairs of type 2 T . We denote this cell by C T . (cid:52) Note that we can identify C T with the subset of the cube [0 , k defined by the equalities (14) and(15), where k is the number of interior dashed edges of T b plus the number of interior edges of T s . Definition 2.15.
Fix r ≥ n ∈ Z r ≥ \ { } . We define W W n similarly to how we defined K Wr in Def. 2.6: W W n := (cid:18) (cid:71) T ∈ W tree n C T (cid:19)(cid:46) ∼ . The quotient here is somewhat subtler than the quotient that appeared in Def. 2.6, specificallywhen it comes to T b . In T s , we simply contract any edges of length 0. We indicate in the followingfigure how to perform the necessary contractions in T b when some edge-lengths are 0:0 T b T pb T b T r b T p b T p r p b T q b T qr q b T pq b T pqr pq b T b T r b T q b T qr q b T p b T p r p b T pq b T pqr pq b T b T pb The reader should think of the left contraction as undoing a type-1 move (as in [Bottman1, § T b are 0. (cid:52) xample 2.16. In the following figure, we depict the CW complex W W . max( a, b ) ab b aa ba b ab a ba bb ab bb aab b b max( a, b )max( a, b )max( a, b ) aa aa aaaaa aaa a a aa a a Each of the parameters a and b lie in [0 , (cid:52) Finally, we refine the CW structure on W W n to a simplicial decomposition. Lemma 2.17.
Fix a stable tree-pair T . For every simplex S in the standard simplicial decom-position of [0 , k ⊃ C T , S is either contained in C T or disjoint from it. The collection of suchsimplices that are contained in C T form a simplicial decomposition of C T . (cid:52) Proof.
Fix a simplex S . S is defined by a collection of equalities and inequalities of the form0 ∗ x σ (1) ∗ · · · ∗ x σ ( k ) ∗ , (17)where each “ ∗ ” is either a “ < ” or an “=” and where σ is a permutation on k letters. After imposingthese (in)equalities, the left- and right-hand sides of the equalities (14) and (15) become singlevariables. This collection of equalities will either be always satisfied or never satisfied, dependingon the constraints in (17). Depending on which of these is the case, S is either contained in C T or disjoint from it.It follows immediately that the collection of simplices that are contained in C T form a simplicialdecomposition of C T . (cid:3) Example 2.18.
In the following figure, we illustrate the closed cell in W W associated to theunderlying tree-pair of the (top-dimensional) metric tree-pair shown on the right: < c < b = dc < a < b = dc < b < a = d b < c < a = d b < d < a = c d < b < a = cd < a < b = ca < d < b = c d db ba c The restriction on the lengths a, b, c, d ∈ [0 ,
1] is that they must satisfy max( a, b ) = max( c, d ); as aresult, this cell has the CW type of a square pyramid.We indicate the simplicial refinement of this cell: the square pyramid is subdivided into eight3-simplices, which are defined by imposing inequalities and equalities as shown in this figure. (cid:52) The construction of FM W In this section, the d´enouement of this paper, we will construct a collection of CW complexes (cid:0) FM W ( k ) (cid:1) k ≥ and a collection of operations ◦ i : FM W ( k ) × FM W ( (cid:96) ) → FM W ( k + (cid:96) − , (18)such that these data form an operad.We will now give an overview of our construction of FM W ( k ). This is an expansion of Step 2 inthe overview we gave in § Each open Getzler–Jones cell in FM ( k ) can be identified with a product of open 2-associahedra,i.e. a product of the form ˚ W m × · · · × ˚ W m a (where “ ˚ X ” is our notation for the interior of a space X ). For each such open cell, we replace these 2-associahedra by their W-construction equivalents,thusly: ˚ W W m × · · · × ˚ W W m a . This product comes with the product CW structure, and we refine thisin a way that endows ˚ W W m × · · · × ˚ W W m a with the structure of a simplicial complex. While an open Getzler–Jones cell can be identified with a product ˚ W m × · · · × ˚ W m a of 2-associahedra, their compactifications (in FM ( k ) and W m × · · · × W m a , respectively) are different:the compactification of the former is smaller than the compactification of the latter. This is reflectedin how we glue our products ˚ W W m × · · · × ˚ W W m a together. Specifically, we perform this gluing byapplying a quotient map to each simplex in the boundary of W W m × · · · × W W m a . This quotientmap is closely related to the maps f σ : W n → FM ( k ) that we described in § W n allows lines with no marked points, whereas thecompactification of a Getzler–Jones cell does not allow this.The following is the main result of this section, which we stated in the introduction and recordagain here: Main theorem.
The spaces (cid:0) FM W ( k ) (cid:1) k ≥ together with the composition operations ◦ i defined inDef. 3.11 form a non- Σ operad, and the composition maps ◦ i : FM W ( k ) × FM W ( (cid:96) ) → FM W ( k + (cid:96) − are cellular. roof. Combine Lemmata 3.12 and 3.13 below. (cid:3)
Quotient maps on 2-associahedra.
Before we can define the quotient involved in (24), wewill define for every cell F in ∂W W n a map q F from F to a certain product of 2-associahedra, wherethis target will vary for difference choices of F . We begin with two preliminary definitions. Definition 3.1.
Fix r ≥ n ∈ Z r ≥ \ { } , and fix i ∈ [1 , r ] such that n i = 0. Define (cid:101) n := ( n , . . . , n i − , n i +1 , . . . , n r ). We then define a map of posets π tree i : W tree n → W tree (cid:101) n by applyingthe following procedure to 2 T = T b f → T s ∈ W tree n :1. Denote by e the edge in T s incident to the i -th leaf λ T s i . If e is a solid edge in T b that ismapped identically under f to e , then we delete e . Next, we delete e . We modify f inthe obvious way.2. After performing these deletions, our tree-pair may no longer be stable. We rectify this in T b resp. T s by performing the contractions indicated on the left resp. right:More specifically, we perform these contractions as many times as necessary for the tree-pairto be stable.Denoting the end result of this procedure by (cid:102) T , we define π tree i (2 T ) := (cid:102) T .Next, we define another map of posets. Fix r ≥ n ∈ Z r ≥ \ { } . Denote by (cid:101) n the result ofdeleting all the zeroes from n , and set (cid:101) r to be the length of (cid:101) n . We define π tree : W tree n → W tree (cid:101) n byapplying the map π tree i once for each i with n i = 0. (cid:52) It is not hard to check that the choices implicit in this definition do not matter, and that theresulting maps are indeed maps of posets.
Definition 3.2.
Fix r ≥ n ∈ Z r ≥ \ { } . We define a map π W : W W n → W W (cid:101) n in the samefashion as π tree , with the provision that when we contract adjacent edges of lengths (cid:96) and (cid:96) (whether in T b or T s ), we equip the resulting edge with length max( (cid:96) , (cid:96) ).Next, we recall a W-version analogue of two properties of the 2-associahedra. W-version analogue of (forgetful) property of Theorem 4.1, [Bottman1] . Fix r ≥ n ∈ Z r ≥ \ { } . There is a surjection W W n → K Wr , which sends a metric stable tree-pair (cid:0) T b f (cid:55)→ T s , ( L e ) , ( (cid:96) e ) (cid:1) to the metric stable RRT (cid:0) T s , ( (cid:96) e ) (cid:1) . (cid:52) W-version analogue of (recursive) property of Theorem 4.1, [Bottman1] . Fix a stabletree-pair 2 T = T b f → T s ∈ W tree n . There is an inclusion of CW complexesΓ T : (cid:89) α ∈ V Tb ) , in( α )=( β ) W W β ) × (cid:89) ρ ∈ V int ( T s ) K W ρ ) (cid:89) α ∈ V ≥ Tb ) ∩ f − { ρ } , in( α )=( β ,...,β ρ )) W W β ) ,..., β α ) ) (cid:44) → W W n , (20)where the superscript on one of the product symbols indicates that it is a fiber product with respectto the maps described in (forgetful) .The map Γ T defined in [Bottman1], which is defined for the posets W tree n , is defined by attachingstable tree-pairs together in a way specified by the stable tree-pair 2 T . This map is similar, but we re attaching together metric stable tree-pairs. We assign the length 1 to the edges along whichwe attach the trees. (The image of Γ T is a union of cells in ∂W W n .) (cid:52) We can now define the quotient maps q F on W W n . Definition 3.3.
Fix r ≥ n ∈ Z r ≥ \ { } , a stable type- n tree-pair (cid:102) T , and a face F of theassociated cell C (cid:102) T in W W n with the property that F lies in ∂W W n . (Equivalently, the metric tree-pairs in F have at least one length that is identically equal to 1.) The quotient map associated to F is a map q F from F to a product of 2-associahedra. Given a metric stable tree-pair (cid:0) T, ( L e ) , ( (cid:96) e ) (cid:1) ,we define its image under π in the following fashion:1. Break up T b and T s along the edges that are identically 1 in F . Equivalently, choose 2 T ofminimal dimension with the property that F lies in the image of Γ T , then identify F as atop cell in a product of fiber products of the following form: (cid:89) α ∈ V Tb ) , in( α )=( β ) W W β ) × (cid:89) ρ ∈ V int ( T s ) K W ρ ) (cid:89) α ∈ V ≥ Tb ) ∩ f − { ρ } , in( α )=( β ,...,β ρ )) W W β ) ,..., β α ) ) . (21) As a result, we obtain a list of metric stable tree-pairs, which we can regard as lying insidea product W W m × · · · × W W m a .2. We then apply the map π W to each of the factors in the product just recorded, henceproducing an element of W W (cid:103) m × · · · × W W (cid:103) m a . (As in Defs. 3.1 and 3.2, (cid:102) m i denotes the resultof removing the 0’s from m i .) (cid:52) Note that for two cells F , F in the boundary of W W n , the targets of q F and q F are typicallydifferent. Example 3.4.
In the following figure, we illustrate several things about W W : b b ab aa baab b ab b ab bab b b a aa a b ab b b b b b a a b b b b a b b b b a a a Initially, W W is an octagon, decomposed into eight squares; this is indicated by the black lines.The simplicial refinement divides each square into two 2-simplices. We have indicated the metrictree-pairs that correspond to each of the eight squares, as well as those corresponding to the sixteen1-simplices that comprise ∂W W . (Some dashed edges are not labeled; these should be interpretedas having length max( a, b ).)Finally, we have indicated the behavior of the quotient maps on W W . This map are the identityon every edge except for those indicated in red. Each pair of red edges is contracted to a point.One reflection of this is that in Ex. 1.1, the octagons in W are taken to the (cellular) hexagonsin the Getzler–Jones cell indicted on the right. (cid:52) The construction of FM W ( k ) . In this subsection, we tackle the construction of FM W ( k ).First, we will describe our version of the Getzler–Jones cells. Next, we will explain how to gluethese spaces together. o define the Getzler–Jones cells, we must introduce , which will allow us toenforce the alignment and ordering of special points on screens as in Fig. 1. Definition 3.5.
Fix a finite set A . A σ on A is the following data: • An ordered decomposition A = A (cid:116) · · · (cid:116) A r , (22) where A r is allowed to be empty. • For each i , a linear order on A i .We define the type of σ to be the vector n := (cid:0) | A | , . . . , | A r | (cid:1) . If σ is a 2-permutation whose type n has no zero entries, then we say that σ has no empty part . (cid:52) Remark . Note that a type-(1 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) r r letters. The same is true of a type-( n ) 2-permutation. (cid:52) Next, we define a
Getzler–Jones datum , the set of which indexes the Getzler–Jones cells in FM W ( k ). Definition 3.7.
Fix k ≥
2. A
Getzler–Jones datum consists of the following data: • A stable rooted tree T with k leaves, together with a numbering of its leaves from 1 through k . • For every interior vertex v ∈ T int , a 2-permutation σ on its incoming vertices V in ( T ) suchthat σ has no empty part.We denote the type of the 2-permutation associated to v by n ( v ). We will abuse notation anddenote the entire Getzler–Jones datum by T . (cid:52) Finally, we can define the
Getzler–Jones cells of type k . Definition 3.8.
Fix k ≥ T . Then we make the following twodefinitions: GJ T := (cid:89) v ∈ T int ˚ W W n ( v ) , (cid:102) GJ T := (cid:89) v ∈ T int W W n ( v ) . (23)We call GJ T the Getzler–Jones cell GJ T associated to T , and we refer to GJ T as a type- k Getzler–Jones cell .In Lem. 2.17 we equipped W W n with the structure of a simplicial complex, which induces a CWstructure on GJ T and (cid:102) GJ T . We refine these to equip GJ T and (cid:102) GJ T with simplicial decompositions,in the fashion of Lem. 2.17. (cid:52) Remark . The reason why we do not refer to (cid:102) GJ T as a “closed Getzler–Jones cell” is because itis not the closure in FM W ( k ) of GJ T . In fact, it is larger than this closure. Our reason for makingthis second definition is that (cid:102) GJ T will be an integral part of our definition of FM W ( k ). (cid:52) We will define FM W ( k ) as a quotient of the following form, where T varies over type- k Getzler–Jones data: FM W ( k ) := (cid:16)(cid:97) T (cid:102) GJ T (cid:17)(cid:14) ∼ . (24)The remaining ingredient is the collection of maps that we will use to attach these spaces together.As a consequence of the definition of these maps, FM W ( k ) will decompose as a set into the unionof all type- k Getzler–Jones cells.Finally, we come to the definition of FM W ( k ). efinition 3.10. Fix k ≥
2. We construct FM W ( k ) via the following procedure:1. Begin with the following disjoint union, where T varies over type- k Getzler–Jones data: (cid:97) T (cid:102) GJ T . (25) 2. Fix a type- k Getzler–Jones datum T , and fix a cell F in the boundary of (cid:102) GJ T = (cid:81) v ∈ T int W W n ( v ) . F lies inside a a product of cells in the 2-associahedra that comprise (cid:102) GJ T — that is, wemay write F ⊂ (cid:81) v ∈ T int F v ⊂ (cid:81) v ∈ T int W W n ( v ) , where F v is a cell in W W n ( v ) . For every v , wehave a map q v from W W n ( v ) to a product of 2-associahedra; by combining these, we obtain amap from F to a product of 2-associahedra. In fact, we can regard the target of this mapas a Getzler–Jones cell.3. We take the quotient of the disjoint union in (25) by attaching the constituent spacestogether via the maps we defined in the previous step.We define FM W (1) to be a point. (cid:52) It is a consequence of the simplicial structure of the (cid:102) GJ T ’s that each FM W ( k ) has the structure ofa CW complex. As noted above, a result of our definition is that FM W ( k ) decomposes as a unionof Getzler–Jones cells, over all Getzler–Jones data of type k :FM W ( k ) = (cid:91) T GJ T . (26)3.3. The operad structure on FM W .Definition 3.11. Fix k , (cid:96) , and i ∈ [1 , k ]. We wish to define the composition map ◦ i : FM W ( k ) × FM W ( (cid:96) ) → FM W ( k + (cid:96) − . (27)To do so, fix Getzler–Jones data T and T (cid:48) of types k and (cid:96) , respectively, and fix cells F ⊂ GJ T and F (cid:48) ⊂ GJ T (cid:48) . We will define ◦ i on GJ T × GJ T (cid:48) = (cid:89) v ∈ T int (cid:116) T (cid:48) int W W n (v) . (28)Define T (cid:48)(cid:48) to be the result of grafting T (cid:48) to the i -th leaf of T , and completing it to a Getzler–Jonesdatum in the obvious way. We define ◦ i on GJ T × GJ T (cid:48) to be the identification of GJ T × GJ T (cid:48) withGJ T (cid:48)(cid:48) . (cid:52) Lemma 3.12.
Taken together, the spaces (cid:0) FM W ( k ) (cid:1) k ≥ together with the composition operations ◦ i form a non- Σ operad.Proof. This is immediate from the definition. (cid:3)
Lemma 3.13.
The composition maps ◦ i : FM W ( k ) × FM W ( (cid:96) ) → FM W ( k + (cid:96) − are cellular.Proof. Similar to the proof of Prop. 2.1. (cid:3) emark . Fix r ≥ n ∈ Z r ≥ \ { } , and a 2-permutation σ of type n . Then the associatedforgetful map f Wσ : W W n → FM W ( | n | )(30)is cellular. This map is defined in the obvious way: we first identify W W n with the corresponding (cid:102) GJ T , where T is a Getzler–Jones datum whose associated tree T is a corolla with | n | leaves. Then,we include (cid:102) GJ T into the disjoint union (cid:70) T (cid:102) GJ T , and finally take the quotient to land in FM W ( | n | ). (cid:52) References [Abouzaid] M. Abouzaid. Symplectic cohomology and Viterbo’s theorem. In
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Department of Mathematics, University of Southern California, 3620 S Vermont Ave, Los Angeles,CA 90089, USAMax Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
Email address : [email protected]@mpim-bonn.mpg.de