Generalized representation stability for disks in a strip and no-k-equal spaces
aa r X i v : . [ m a t h . A T ] J un GENERALIZED REPRESENTATION STABILITY FOR DISKS INA STRIP AND NO- k -EQUAL SPACES HANNAH ALPERT
Abstract.
For fixed j and w , we study the j th homology of the configurationspace of n labeled disks of width 1 in an infinite strip of width w . As n grows,the homology groups grow exponentially in rank, suggesting a generalized rep-resentation stability as defined by Church–Ellenberg–Farb and Ramos. Weprove this generalized representation stability for the strip of width 2, leavingopen the case of w >
2. We also prove it for the configuration space of n labeled points in the line, of which no k are equal. Introduction
The configuration space of n labeled unit-diameter disks in an infinite stripof width w is denoted config( n, w ); Figure 1 depicts an example configuration.Specifically, parametrizing the configurations in terms of the centers of the disks,config( n, w ) is the set of points ( x , y , . . . , x n , y n ) ∈ R , such that ( x i − x j ) + ( y i − y j ) ≥ i and j , and such that ≤ y i ≤ w − for all i . We would like todescribe the topology of config( n, w ). w n · · · Figure 1.
The configuration space config( n, w ) is the set of waysto arrange n disjoint labeled disks of width 1 in R × [0 , w ].The topological study of disk configuration spaces was initiated by Baryshnikov,Bubenik, and Kahle in [BBK13]. It is closely related to topological robotics andmotion planning, described in Farber’s survey [Far08]. Earlier, disk configurationspaces were studied probabilistically, in the context of the hard spheres gas model.In the survey [Dia09], Diaconis describes that work in statistical mechanics byphysicists and materials scientists, and Carlsson et al. review the physics literaturein [CGKM12].In addition to the study of disks in [BBK13], others have studied the topologyof configuration spaces of various identical rigid objects in various shapes of con-tainer, such as in [Alp17], [Dee11], and [KKLS18]. The choice of disks in a strip Mathematics Subject Classification.
Key words and phrases.
Representation stability, FI-modules, configuration spaces, discreteMorse theory.
Figure 2.
A single homology class in the plane can correspond toseveral homology classes in the strip. Pictured are three maps S × S → config(5 ,
2) that give distinct (indeed, linearly independent)classes in H (config(5 , n la-beled unit-diameter disks in the plane, which we denote by config( n ), is homotopyequivalent to the configuration space of points in the plane, which is well under-stood (see, for instance, [Arn69] or [Sin13]). In fact, if the strip is wide comparedto the number of disks, specifically if w ≥ n , then config( n, w ) and config( n ) arehomotopy equivalent. For w < n , though, the strip shrinks the configuration spacein a way that adds topology.The paper [AKM19] introduces the spaces config( n, w ) and asks, for fixed j and w , how does H j (config( n, w )) depend on n ? That paper estimates the dimensionof H j (config( n, w )) up to a constant factor; it turns out to be exponential in n unless the strip is wide compared to j . The present paper continues the study ofhow H j (config( n, w )) depends on n , putting it into the framework of generalizedrepresentation stability as introduced by Ramos in [Ram17]. The goal is to givealgebraic relationships between the various homology groups in a way that recoversthe asymptotic results about their dimension growth.Why does dim H j (config( n, w )) grow exponentially in n ? We know that whenthe strip is replaced by the plane, dim H j (config( n )) grows polynomially in n . Whydoes the subspace config( n, w ) have so much more homology? As depicted in Fig-ure 2, cycles that are homologous in H j (config( n )) may not be homologous in H j (config( n, w )), because the strip is too narrow to let the various clusters of diskshomotope past each other.The rough idea of the exponential growth in dim H j (config( n, w )) is as follows.In config( n, w ), it is possible for w disks to revolve around each other to make a( w − w = 2, thebarriers would be the circling pairs shown in Figure 2.) Very broadly, the generatorsof H j (config( n, w )) look like sequences of barriers with smaller clusters of disks inbetween; if there are b barriers, they divide the strip into b + 1 intervals, so theremaining disks each have b + 1 choices for which interval to be in. This givesroughly ( b + 1) n linearly independent homology classes in H j (config( n, w )).In some sense, once n is large enough, incrementing n by 1 does not meaning-fully change the structure of H j (config( n, w ))—the extra disk has a choice of b + 1intervals to be placed in, and nothing else happens. The framework of represen-tation stability, first introduced in [CF13], is well suited to situations such as thisone. In fact, one of the favorite examples of representation stability is the sequence H j (config( n )), as n varies and j stays fixed. Each space config( n ) has an action of S n by permuting the disks, so each H j (config( n )) is a representation of S n . Repre-sentation stability, very broadly, says that for sufficiently large n , incrementing n by 1 changes the S n –representation H j (config( n )) in the most trivial way to give EPRESENTATION STABILITY FOR DISKS IN A STRIP 3
12 43 532 45 7 1 61 2 3 4 51 2 3 4 5 6 7
Figure 3.
To have an FI–module structure on H (config( n )),there must be a map from H (config(5)) to H (config(7)) for eachinjection from [5] to [7]. Pictured are one class in H (config(5)),one injection, and the class in H (config(7)) that results.the S n +1 –representation H j (config( n + 1)). The topological reason for this is that H j (config( n )) turns out to be generated by cycles in which at most 2 j of the disksmove at all. So, for n > j , the extra disks do nothing but sit on the side.The formal way to talk about extra disks sitting on the side is to say that H j (config( n )) is a finitely generated FI–module, first defined in [CEF15] by Church,Ellenberg, and Farb. The category FI is defined to have one object [ n ] = { , , . . . , n } for each natural number n , and the morphisms between these objects are the injec-tions. For instance, the set of FI–morphisms from [ n ] to [ n ] is the symmetric group S n . An FI–module M over a commutative ring k is a functor from FI to k –modules;that is, we have a k –module M n for each n , and for each injection [ n ] → [ m ] wehave a corresponding homomorphism M n → M m . In this paper we only considerthe case k = Z , where each of the modules is an abelian group. For any j , the ho-mology groups M n = H j (config( n )) form an FI–module over Z ; given an injection ϕ : [ n ] → [ m ] we have a map ϕ ∗ : H j (config( n )) → H j (config( m )) given by the mapof spaces that relabels the disks 1 , , . . . , n by ϕ (1) , ϕ (2) , . . . , ϕ ( n ) and places m − n disks with the remaining labels off to the side, as shown in Figure 3. An FI–moduleis finitely generated if there exists a finite set of elements x , . . . , x r ∈ F ∞ n =1 M n such that the only FI–submodule of M containing x , . . . , x r is M itself. Our FI-module H j (config( n )) is finitely generated by classes in H j (config(2 j )).The fact that H j (config( n )) is a finitely generated FI–module implies that itsdimension grows polynomially in n . Roughly, to find generators for H j (config( n )),we take (cid:0) n j (cid:1) copies of each generator of H j (config(2 j )), one for each choice of whichdisks do and do not move. In contrast, for disks in a strip, the homology groups H j (config( n, w )) have dimensions that grow exponentially in n and thus cannot befinitely generated FI–modules. The reason is the same as the reason for exponentialgrowth: when we add a disk there is a choice of which barriers to insert it between.The appropriate algebraic notion for H j (config( n, w )) is that of a finitely gener-ated FI d –module. The best example for understanding the idea of an FI d –moduleis the j th homology of the configuration space of n disks on the disjoint union of d planes. Each additional disk can be added to any of the d planes. HANNAH ALPERT
Figure 4.
When applying this FI –morphism to a class in H (config(5 , k labels immedi-ately after the k th circling pair.In [Ram17], Ramos introduces FI d –modules and shows that finitely generatedFI d –modules satisfy a notion of generalized representation stability, and in [Ram19]he shows that the homology groups of a certain kind of graph configuration spaceare finitely generated FI d –modules. The category FI d , like FI, has one object [ n ]for each natural number n . The morphisms are pairs ( ϕ, c ), where ϕ is an injection,say, from [ n ] to [ m ], and c is a d –coloring on the complement of the image of ϕ ; thatis, c is a map from [ m ] \ ϕ ([ n ]) to a set with d elements such as { , , . . . , d − } . AnFI d –module is a functor from FI d to modules. Figure 4 sketches the FI j +1 –modulestructure for H j (config( n, j and w , the sequence H j (config( n, w )) forms afinitely generated FI d –module for d = 1 + j jw − k , that is, d is one more than themaximum possible number of barriers. In this paper we prove the statement for w = 2; we explore in Section 7 which aspects of the proof seem harder to adapt for w > Theorem 6.1.
For any j , the homology groups H j (config( n, form a finitelygenerated FI j +1 –module over Z . The same techniques allow us to prove a similar result for a family of spaces thatare closely related to the configuration spaces of disks in a strip, but are much morewell-studied. The no– k –equal space of the line, also known as the complement of the k –equal subspace arrangement, was introduced by Bj¨orner and Welker in [BW95]and is the set of n –tuples of points in R such that no k of them are equal. Wedenote this space by no k ( n, R ) and think of it as a configuration space of points inthe line. There is a map config( n, w ) → no w +1 ( n, R ) sending each configuration tothe n –tuple of x –coordinates of the centers of the disks, and the induced map onhomology H j (config( n, w )) → H j (no w +1 ( n, R )) is projection to a direct summand,as we show in Corollary 2.4.The homology of no k ( n, R ) grows exponentially in n for the same reason that thehomology of config( n, w ) does: a cluster of k points among the n points can forma ( k − EPRESENTATION STABILITY FOR DISKS IN A STRIP 5 the left of the barrier to the right of the barrier. Unlike in the case of config( n, w ),for no k ( n, R ) we can prove for all k that the homology groups give FI d –modules.Our results recover the computation of homology of no k ( n, R ) from [BW95]. Theorem 6.2.
For any j ≥ and k ≥ , the homology groups H j (no k ( n, R )) arezero unless j is a multiple of k − . If j = b ( k − for some integer b , then thehomology groups H j ( no k ( n, R )) form a finitely generated FI b +1 –module over Z . In Section 2 we give cell complexes cell( n, w ) and desc( n, k −
1) that are homotopyequivalent to config( n, w ) and no k ( n, R ), respectively. In Section 3 we apply discreteMorse theory to the cell complexes: we construct discrete gradient vector fields thatallow us to collapse the cell complexes and eliminate most of the cells. In Section 4we construct a Z –basis for each homology group, indexed by the critical cells ofour discrete vector field. In Section 5 we prove a general lemma about how tospecify an FI d –module. In Section 6 we show that our homology groups satisfythe hypothesis of this lemma and thus form FI d –modules, and we verify that theseFI d –modules are finitely generated. In Section 7 we conclude by speculating aboutthe conjectured generalization for strips of width w > Acknowledgments.
This work was supported by the National Science Foundationunder Award No. DMS-1802914. I am very grateful to Andy Putman, Nate Har-man, Jenny Wilson, John Wiltshire-Gordon, and Eric Ramos, who all pointed metoward relevant and accessible information about representation stability; as some-one completely unfamiliar with it, I would not have known where to start otherwise.I also had many useful conversations with Matt Kahle about this material.2.
Cells labeled by symbols of blocks
This paper is based on the technique of the paper [AKM19], which is to re-place the configuration space config( n, w ) by a homotopy-equivalent cell complexcell( n, w ) and to estimate the homology by doing combinatorics (specifically, dis-crete Morse theory) on the cell complex. We use the same cell complex cell( n, w )as in that paper, and we use the same method to find a cell complex desc( n, k − k –equal space no k ( n, R ). In the remain-der of this section we define the complexes cell( n, w ) and desc( n, w ), and we provethat desc( n, k −
1) is homotopy equivalent to no k ( n, R ) by adapting the methodof [AKM19].The cell complex cell( n, w ) is defined as a subcomplex of a cell complex cell( n )described by [BZ14]. In cell( n ), every cell is labeled by a symbol , which consists ofa string of numbers and vertical bars, such that the numbers form a permutation ofthe numbers 1 through n , and each vertical bar is both immediately preceded andimmediately followed by a number. Thinking of the numbers as the labels of thedisks in config( n ), we sometimes refer to the numbers in a symbol as labels. Eachsubstring between one vertical bar and the next (or before the first bar or after thelast bar) is called a block . We think of the elements of each block as the labels ofdisks in a vertical stack in config( n, w ), as in Figure 5.As shown in [BZ14], there is a way to form cell( n ) as a polyhedral cell complex inwhich the cells are labeled by these symbols, with the following incidence relation.Given two cells f and g , we have that f is a top-dimensional face of g if and onlyif the symbol of g can be obtained from the symbol of f by removing a bar andcombining the adjacent two blocks by a shuffle that preserves the ordering of the HANNAH ALPERT
78 6 15439 78 6 154 39
Figure 5.
We can imagine each symbol of cell( n, w ) as a config-uration in config( n, w ) where the numbers in each block are thelabels in a column of disks. Pictured are configurations represent-ing the symbol 8 7 | | | | | n − n ) in Euclidean space, only with the symbols of the cells, so in order to computethe homology with Z –coefficients, we need to specify orientations and signs.To specify the signs of the incidences in cell( n ), we use the structure of the cellsas products of permutahedra. By an injected cell we mean the result of takingany symbol in cell( n ) and any injection from [ n ] to a larger set [ m ], and applyingthe injection to every number that appears in the symbol. Given two injected cellswith disjoint sets of labels, we can take the concatenation product by writingthe two symbols next to each other with a vertical bar in between. In this way,every cell in cell( n ) is a concatenation product of smaller-dimensional injected cells,except for the top-dimensional cells in cell( n ), which have no vertical bars.The signs are defined as follows. If g is a single block—that is, an injected cellwith no vertical bars—and f is a top-dimensional face of g , then we define thecoefficient of f in ∂g to be the sign of the permutation that results from deletingthe bar in f and not reshuffling. To define signs on concatenation products, we usethe following Leibniz rule: if g and g are injected cells with disjoint sets of labels,then ∂ ( g | g ) = ∂g | g + ( − b ( g ) g | ∂g , where b ( g ) denotes the number of blocks in g .We can check that these signs are consistent by verifying that ∂ = 0. Lemma 2.1.
The differential on cell( n ) satisfies ∂ = 0 .Proof. Let g be a cell in cell( n ). First we suppose that g is a single block. Let e be a codimension–2 face of g . Then e = e | e | e , where e , e , and e areblocks. Then there are two intermediate faces between e and g . We denote themby x = x | e and y = e | y . We compare the sign of e in x times the sign of x in g , and the sign of e in y times the sign of y in g , and we show that these twoproducts are opposite signs. If we consider just the contribution from the signs ofthe permutations, both products give the sign of the permutation relating e and g ,so those contributions are equal. EPRESENTATION STABILITY FOR DISKS IN A STRIP 7
For the contribution from the Leibniz rule, only the incidence between e and y in-volves splitting a block that is not the first, so that incidence has a sign contributionof − e in g is zero,proving that ∂ g = 0 when g is a single block.If g is a concatenation product g | g , then we use induction on the number ofblocks. The Leibniz rule gives ∂ ( g | g ) = ∂ g | g + h ( − b ( g ) + ( − b ( ∂g ) i · ∂g | ∂g + g | ∂ g , which is indeed zero. (cid:3) Having described the structure of the complex cell( n ), we define cell( n, w ) to bethe subcomplex of cell( n ) consisting of all cells for which every block has at most w elements. We define desc( n, w ) to be the subcomplex of cell( n, w ) in which, inaddition, the elements of each block appear in descending order. The results inthis paper concern cell( n, w ) only for the special case of w = 2, but they addressdesc( n, w ) for all w .Theorem 3.1 of [AKM19] shows that cell( n, w ) is homotopy equivalent to theconfiguration space config( n, w ) of n disks in a strip of width w . The strategy isto find an open cover of config( n, w ) indexed by the symbols in cell( n, w ), wherethe intersections between open sets correspond to incidences in cell( n, w ); the nervetheorem then implies that config( n, w ) is homotopy equivalent to the barycentricsubdivision of cell( n, w ), and thus is also homotopy equivalent to cell( n, w ).The no– k –equal space no k ( n, R ) consists of all the elements of R n such that no k of the coordinates are equal. In the remainder of this section, we mimic the proof ofTheorem 3.1 of [AKM19], in order to verify that no k ( n, R ) is homotopy equivalentto desc( n, k − Theorem 2.2.
The no– k –equal space no k ( n, R ) is homotopy equivalent to the cellcomplex desc( n, k − . Given a symbol α of desc( n, k − U α be the subset of R n consisting ofall points ( x , . . . , x n ) with the following properties: • Whenever two numbers k and ℓ are in different blocks of α with k appearingbefore ℓ , we have x k < x ℓ . • Whenever two numbers k and ℓ are in the same block, and k ′ and ℓ ′ are indifferent blocks, we have | x k − x ℓ | < | x k ′ − x ℓ ′ | . The sets U α are open and convex in R n , and their union as α ranges over all thesymbols in desc( n, k −
1) is equal to no k ( n, R ).The nerve N of the open cover U α is the simplicial complex built by taking onevertex for each α and a simplex for each collection of open sets U α that have anonempty intersection. Because the sets U α are convex, any intersection of themis either empty or contractible. Thus, the nerve theorem says that no k ( n, R ) ishomotopy equivalent to the nerve N . The next lemma implies that N is equal tothe barycentric subdivision of desc( n, k − HANNAH ALPERT
Figure 6.
Given a configuration p in no k ( n, R ), the set of symbols α such that p ∈ U α forms a totally ordered chain in desc( n, k − U α for α = 1 | | |
5, 3 2 1 | | | Figure 7.
Given a chain of symbols in desc( n, k − | | | ≺ | | ≺ | ≺ n in R such that the intervals in R formedby the various blocks are consecutive powers of 3. The resultingelement of no k ( n, R ) is in U α for each α in the chain. Lemma 2.3.
An intersection U α ∩ U α ∩ · · · ∩ U α r is nonempty if and only if the cells corresponding to α , α , . . . , α r form a chainunder the incidence relation in desc( n, k − .Proof. Let p = ( p , . . . , p n ) be an element of U α ∩ U α ∩ · · · ∩ U α r . We can find theset of all U α containing p in the following way. Given any real number ρ , we candraw the closed interval of length ρ centered at each p , . . . , p n ∈ R , and take theunion of these intervals in R . Then we can cluster the indices 1 , . . . , n according towhich connected component of the union the points p , . . . , p n fall into. Readingoff these clusters from left to right, and ordering the indices within each cluster indescending order, we obtain a symbol α ( ρ ) associated to p and ρ , as in Figure 6.Then the symbols α ( ρ ) for various ρ form a chain under incidence in desc( n, k − p ∈ U α if and only if α = α ( ρ ) for some ρ . Thus, α , α , . . . , α r must all bepart of this chain, and so they must also form a chain.For the converse, suppose that α , α , . . . , α r form a chain in desc( n, k − p in U α ∩ · · · ∩ U α r . Without loss of generality, we assumethat the chain is maximal in desc( n ) = desc( n, n ) and that α , . . . , α r are in order,so α has only blocks of size 1, and getting to each symbol α i from the previoussymbol α i − corresponds to merging two consecutive blocks. We start with α andadd restrictions on the coordinates ( p , . . . , p n ) one step at a time, so that on the i th step we will have fixed the differences between coordinates within each block of α i , but we think of the separate blocks sliding freely from side to side. After all the EPRESENTATION STABILITY FOR DISKS IN A STRIP 9 steps, we will have specified the configuration ( p , . . . , p n ) up to translating everycoordinate by the same real number.More precisely, at step 1 we require that if k appears before ℓ in α , then p k
i − .In the final step, step n means merging two blocks to get α n which has onlyone block, and at step n we set the difference between p s and p t to be 3 n , where s is the first (leftmost) number in α and t is the last (rightmost) number in α .At this stage we have specified the point p up to translation in R , and it is in U α ∩ · · · ∩ U α n . (cid:3) The lemma above gives the bulk of the proof that no k ( n, R ) is homotopy equiv-alent to desc( n, k − Proof of Theorem 2.2.
The barycentric subdivision of desc( n, w ) has one vertexfor every cell in desc( n, w ), and one simplex for every chain of incident cells indesc( n, w ). Taking w = k −
1, the nerve N has one vertex for each U α , and thusfor each cell in desc( n, k − U α withnonempty intersection—corresponding to a simplex in N —corresponds to a chainof incident cells in desc( n, k − N is equal to the barycentricsubdivision of desc( n, k − N is homotopy equivalentto the union of the various U α , which is no k ( n, R ). (cid:3) Corollary 2.4.
For any j ≥ , the homology group H j (no w +1 ( n, R )) is a directsummand of H j (config( n, w )) .Proof. Let p : cell( n, w ) → desc( n, w ) be the cellular map that sends the cell α tothe cell in which the numbers in each block of α are rearranged to be in descendingorder. If i : desc( n, w ) → cell( n, w ) is the inclusion map, then p ◦ i is the identity ondesc( n, w ). Thus, the induced maps on homology satisfy the relation that p ∗ ◦ i ∗ isthe identity on each H j (desc( n, w )). These maps give a way to write H j (desc( n, w ))as a direct summand of H j (cell( n, w )), and thus give a way to write H j (no w +1 ( n, R ))as a direct summand of H j (config( n, w )). (cid:3) Discrete gradient vector field
In the next two sections, we use discrete Morse theory to compute the homol-ogy of the cell complexes cell( n,
2) and desc( n, w ), which we have shown in the | | | | | | | | | | | | | | | |
21 31 | | ≃ | | | Figure 8. A discrete gradient vector field consists of a set of dis-joint pairs of cells, each pair incident and of consecutive dimen-sions. The complex is homotopy equivalent to one in which thepaired cells are collapsed, and only the critical (unpaired) cellsremain.previous section are homotopy equivalent to the configuration spaces config( n, w +1 ( n, R ). In any cell complex, the cellular homology comes from a chaincomplex generated by the cells; very broadly, discrete Morse theory gives a wayto decompose the chain complex as a direct sum of a chain complex that has nohomology (which we discard) and a chain complex generated by a smaller subsetof cells, the critical cells. This section concerns the reduction to the smaller chaincomplex, and the next section shows that in fact, in the smaller chain complex alldifferentials are zero, so the homology has a Z –basis in bijection with the set ofcritical cells.The basic definitions in discrete Morse theory are as follows. In any polyhedralcell complex, we say that cell f is a face of cell g if f is in the boundary of g anddim f = dim g −
1, and we say that g is a coface of f if f is a face of g . A discretevector field on a polyhedral cell complex is a set V of pairs of cells [ f, g ] such that f is a face of g and each cell can be in at most one pair; an example is shown inFigure 8. A discrete vector field V is gradient if there are no closed V –walks. A V –walk is a sequence of pairs [ f , g ] , . . . , [ f r , g r ] with [ f i , g i ] ∈ V , such that each f i +1 is a face of g i other than f i . The V –walk is closed if f r = f .A cell is critical with respect to a discrete gradient vector field V if the cell isnot in any pair in V . The fundamental theorem of discrete Morse theory [For02]states that there is a cell complex that is a strong deformation retraction of theoriginal cell complex, in which there is one cell per critical cell of V . Thus, we cancompute the homology groups H j (cell( n, H j (desc( n, w ) by defining discretegradient vector fields and computing the homology of the collapsed chain complexesgenerated by the critical cells.One way to define a discrete gradient vector field on a polyhedral cell complexis by defining a total ordering on all the cells. Given a total ordering, the resultingvector field contains a pair [ f, g ] if and only if both f is the greatest face of g and g is the least coface of f ; using the fact that the cell complex is polyhedral,one can prove that this vector field is gradient (see Lemma 3.7 of [Bau19]). Inwhat follows, we define a total ordering on all of cell( n ), the polyhedral complex EPRESENTATION STABILITY FOR DISKS IN A STRIP 11 that contains both cell( n,
2) and desc( n, w ) as subcomplexes. We use the resultingdiscrete gradient vector fields to compute the homology.To describe the ordering, let α = α | α | · · · | α r and β = β | β | · · · | β s be symbols in cell( n ). We say that a block α i or β i is a singleton if it has onlyone element. We say that a block α i is a follower if the preceding block α i − is asingleton less than every element of α i . Lemma 3.1.
There is a total ordering ≺ on cell( n ) with the following properties.Suppose that α and β first differ at block i . Then, (1) If α i and β i are both followers, if β i has more elements than α i then α ≺ β . (2) If neither α i nor β i is a follower, if β i has a lesser first element than α i then α ≺ β . (3) If neither α i nor β i is a follower, and α i and β i have the same first element,if β i has more elements than α i then α ≺ β . (4) If α i is a follower and β i is not, then α ≺ β .Proof. To define the ordering, we first define a “key” function that maps each cellto an element of L ∞ i =1 Z . Then we order the symbols lexicographically by key, andextend this partial order arbitrarily to a total order. For any cell α , each block α i of α contributes two entries to key( α ). The (2 i − α ) is n + 1 minusthe first element of the block α i if α i is not a follower, or 0 if α i is a follower. The(2 i )th entry of key( α ) is the number of elements in α i . Past twice the number ofblocks, all the entries of key( α ) are zero.One can verify that the lexicographical ordering of keys has the properties givenin the lemma statement. (cid:3) This total ordering gives rise to different discrete gradient vector fields on cell( n, n, w ). The next two lemmas describe the set of critical cells for each. Al-though each lemma only proves that every critical cell has the properties specifiedin the lemma, the theorems of the next section imply that the converse is also true. Lemma 3.2.
If a cell in cell( n, is critical with respect to the discrete gradientvector field that comes from the total ordering from Lemma 3.1, then the cell hasthe following properties: (1) Every two consecutive singletons are in decreasing order. (2)
If a given –element block has its elements in decreasing order, then theblock is a follower.Proof. We describe the pairing on the remaining cells, and then verify that it comesfrom the total ordering. An example of two paired cells is shown in Figure 9.Suppose that f is a cell such that there are two consecutive singletons in increasingorder, but in the string of blocks preceding those, the two conditions for beingcritical are met. Let g be the cell in which those two consecutive singletons arecombined such that the resulting 2–element block has its elements in decreasingorder. Then the discrete vector field contains [ f, g ]. From the reverse point ofview, suppose that g is a cell such that there is a 2–element block with elementsin decreasing order, not immediately preceded by a lesser singleton, such that inthe string of blocks preceding this 2–element block, the two conditions for beingcritical are met. Let f be the cell in which this 2–element block is split into twosingletons in increasing order. Then the discrete vector field contains [ f, g ]. | | | | | | ,
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1; 0 ,
2; 3 , Figure 9.
In cell(9 , f = 6 | | | | | | g = 6 | | | | | f is the greatestamong faces of g (shown in left column) and g is the least amongcofaces of f (shown in right column). In the picture, below eachsymbol appears the corresponding value of the key function.We need to show that for such a pair [ f, g ], f is the greatest face of g and g is theleast coface of f . To show the former, suppose that f comes from splitting the k thblock of g . Any face of g that comes from splitting an earlier block is less than f ,because if the block is ascending it gets shorter and its first entry cannot decrease(properties (2) and (3) of Lemma 3.1), and if the block is descending it is a followerand remains a follower while getting shorter (property (1) of Lemma 3.1). Becausethe k th block of g is descending but not a follower, any face of g that comes fromsplitting a later block, or the face that comes from splitting the same block in theother way (i.e., such that the resulting singletons remain in decreasing order), isless than f because the first element of the k th block is greater than that of f (property (2) of Lemma 3.1). Thus f is the greatest face of g .Similarly, suppose that [ f, g ] is a pair in the discrete vector field and g comesfrom combining the k th and ( k + 1)st blocks of f , which must then be ascend-ing singletons. Any coface of f that comes from combining two earlier blocks isgreater than g , because the blocks are decreasing singletons, so the first of the twoblocks gets longer and its first element cannot increase (properties (2) and (3) ofLemma 3.1). Any coface of f that comes from combining two later blocks, or fromcombining the k th and ( k + 1)st blocks in the other way (i.e., in ascending order),is greater than g because the k th block of that coface has a lesser first element than g and is not a follower (property (2) of Lemma 3.1). Thus g is the least coface of f . EPRESENTATION STABILITY FOR DISKS IN A STRIP 13 | | | | | g | | | | | g | | | | | g | | | | | g | | | | | g | | | | | g | | | | | g | | | | | g | | | | | g | | | | | g | | | | | | | | | | | | | | g | | | | g | | | | Figure 10.
To find the discrete gradient vector field on desc( n, w ),we use the ordering of cells from cell( n ), but the resultingpairing is different. Pictured are (left column) the faces of1 | | | | | | | | | , (cid:3) A similar analysis gives the set of critical cells for desc( n, w ). Lemma 3.3.
If a cell in desc( n, w ) is critical with respect to the discrete gradientvector field that comes from the total ordering from Lemma 3.1, then the cell hasthe following properties: (1) Every two consecutive singletons are in decreasing order. (2)
Every non-singleton block has w elements and is a follower.Proof. The pairing on the remaining cells is defined as follows. Suppose that f isa cell such that there is a singleton immediately followed by a block of size lessthan w , for which every element is greater than the singleton, and in the string ofblocks preceding those, the two conditions for being critical are met. Let g be thecell in which the singleton is combined with the following block. Then the discretevector field contains [ f, g ]. From the reverse point of view, suppose that g is a cellsuch that there is a non-singleton block that is not preceded by a lesser singleton, and in the string of preceding blocks, the two conditions for being critical are met.Let f be the cell in which this non-singleton block is split into two blocks, the firstof which is the least element as a singleton block. Then the discrete vector fieldcontains [ f, g ].Suppose that [ f, g ] is in the discrete vector field and f comes from splitting the k th block of g . To show that f is the greatest face of g , consider the result ofsplitting any earlier block of g . Because g looks like a critical cell at that stage,that block has size w and is a follower; after splitting, it is shorter and is still afollower, so the key is less than that of g (property (1) of Lemma 3.1); in contrast,among ways to split the k th block another way, or to split a later block, f is thegreatest because it is the only one for which the k th block begins with that leastelement of the k th block of g (property (2) of Lemma 3.1).To show that g is the least coface of f , consider the result of combining anyearlier blocks of f . Because f looks like a critical cell at that stage, the two blockswould be non-follower singletons in decreasing order, so the combined block wouldbe larger, not be a follower, and have the same first element as the first of thetwo singletons, giving a greater key than that of f (property (3) of Lemma 3.1);in contrast, among ways to combine later blocks of f , g is the least because itis the only one that increases the first element of the k th block (property (2) ofLemma 3.1). (cid:3) Basis for homology
In the previous section we constructed discrete gradient vector fields on cell( n, n, w ) and described their critical cells. In this section we construct Z –bases for H ∗ (cell( n, H ∗ (desc( n, w )) with one basis cycle per critical cell.The following general lemma shows that it suffices to construct, for each criticalcell e , a cycle z ( e ) such that e is its maximum cell and has coefficient ±
1. Thenthe rest of the section is devoted to the construction.
Lemma 4.1.
Let X be any finite polyhedral cell complex with a total ordering onthe cells, giving a discrete gradient vector field. Suppose that for each critical cell e , there is a cycle z ( e ) such that e has coefficient ± in z ( e ) and is the greatest cellappearing with nonzero coefficient in z ( e ) . Then every homology class in H ∗ ( X ) can be written uniquely as a Z –linear combination of the homology classes of thecycles z ( e ) .Proof. For any pair [ f, g ] in the discrete vector field, we refer to f as a “match-up cell” and refer to g as a “match-down cell”. We also define z ′ ( f ) to be theboundary of g ; we know that f is the greatest cell appearing in z ′ ( f ), and that ithas coefficient ± X is polyhedral.First, we show that every j –cycle z is a Z –linear combination of cycles z ( e ) and z ′ ( f ), where e ranges over the critical j –cells and f ranges over the match-up j –cells. This follows from the following observation: if a match-down cell g is thegreatest cell in a j –chain, then in the boundary of that chain, the correspondingmatch-up cell f appears with nonzero coefficient, because g is the least coface of f ,so no other cell in the chain has f as a face. Thus, for any j –cycle z , the greatestcell of z cannot be a match-down cell. It is either a critical cell e or a match-upcell f , so we subtract the appropriate multiple of z ( e ) or z ′ ( f ) to get a new cyclewith lesser maximum. Repeating this process gives us z as a linear combination of EPRESENTATION STABILITY FOR DISKS IN A STRIP 15 cycles z ( e ) and z ′ ( f ), so because each z ′ ( f ) is a boundary, this implies that z ishomologous to a linear combination of the cycles z ( e ) only.To show the uniqueness, we need to show that no nontrivial linear combinationof cycles z ( e ) is null-homologous. Because the cycles z ( e ) and z ′ ( f ) have distinctmaxima, they are linearly independent. Thus, it suffices to show that if a j –cycle z is a boundary, it is a linear combination of the boundaries z ′ ( f ). To see this, welook at the set of all j + 1–chains. The chains z ( e ), z ′ ( f ), and g (as e ranges over allcritical ( j + 1)–cells, f ranges over all match-up ( j + 1)–cells, and g ranges over allmatch-up ( j + 1)–cells) form a Z –basis for the set of all ( j + 1)–chains, because theyhave distinct maxima equal to the set of all j –cells. When we apply the boundarymap to this basis, the cycles z ( e ) and z ′ ( f ) map to zero, and the match-down cells g map to the j –dimensional boundaries z ′ ( f ). Thus indeed every j –dimensionalboundary is a linear combination of these boundaries z ′ ( f ).Thus, every homology class in H ∗ ( X ) can be written as a Z –linear combinationof the homology classes of the cycles z ( e ), and the combination is unique. (cid:3) In both cell( n,
2) and desc( n, w ), each critical cell e consists of a sequence oflarger blocks, with descending sequences of singletons in between, and each largerblock has a rigidly specified form. The construction of the cycle z ( e ) is based onthis structure. To make this precise, we define a bilinear concatenation product ofchains in the following way.In Section 2 we have defined injected cell and concatenation product of injectedcells: given two cells with disjoint sets of labels, we can write the two cells witha vertical bar between them. Just as we can apply an injection on [ n ] to a cell incell( n ) (that is, we relabel the disks), we can also apply an injection to a Z –linearcombination of cells. Applying injections commutes with taking boundary maps,and applying an injection to a cycle gives an injected cycle . If two injected cycles z = P i α ,i f ,i and z = P j α ,j f ,j have disjoint labels, then we define theirconcatenation product to be z | z = X i,j α ,i α ,j · f ,i | f ,j , which the Leibniz rule from Section 2 implies is also an injected cycle.We typically restrict our attention to order-preserving injections on [ n ]; the totalordering on cell( n ) does not respect arbitrary injections, but it does respect order-preserving injections. Every critical cell e in cell( n,
2) can be written uniquely asa concatenation product of some number of images of the cells 1, 1 2, and 1 | n, w ) can be writtenuniquely as a concatenation product of images of cells 1 and 1 | ( w + 1) w · · · irreducible criticalinjected cells. We can associate a cycle z ( e ) to each critical cell e by first doing soon the irreducibles. We set z (1) = 1 and z (1 2) = 1 2 + 2 1. We set z (1 | z (1 | | | | − | − | − |
3. We set z (1 | ( w + 1) w · · · w + 1) of the cell ( w + 1) w · · · z to commute with order-preserving injections on the labels andwith taking concatenation products—i.e., z ( f | f ) = z ( f ) | z ( f ), as in Figure 11—we obtain a definition of z for all critical cells in cell( n,
2) and in desc( n, w ).Next we check the hypothesis of Lemma 4.1.
Figure 11.
The cycle z (2 | | z (2 | | z (1 4), the result of putting the cycles z (2 | z (1 4) side by side in the strip. Lemma 4.2.
Let e be any critical cell of cell( n, or desc( n, w ) . Then in the cycle z ( e ) , the cell e has coefficient ± , and it is the greatest cell that appears in z ( e ) with nonzero coefficient.Proof. First we show the statement where e is an irreducible critical injected cell.In our ordering, 1 2 is greater than 2 1, so 1 2 is the greatest cell of z (1 2). Inany cell( n ), every coefficient in the boundary of any cell is ± w + 1) w · · · | ( w + 1) w · · · w = 2this applies to z (1 | e is anirreducible critical injected cell.Next we consider concatenation products and apply induction on the number ofirreducibles. Suppose that e = e | e , where e and e are critical (that is, theyare the images of critical cells under order-preserving injections), and suppose thatthe lemma statement is true for both e and e . By definition, the coefficient of e in z ( e | e ) = z ( e ) | z ( e ) is the product of the coefficient of e in z ( e ) and thecoefficient of e in z ( e ), so the coefficient is ±
1. Let f be any cell appearing in z ( e ), and let f be any cell appearing in z ( e ). We need to show that e = e | e is at least as great as f | f , knowing that e (cid:23) f and e (cid:23) f . Indeed, if e ≻ f ,then by the properties in Lemma 3.1 we know that e | e ≻ f | f no matter what f is. And, if f = e , then the fact that e ≻ f implies that e | e ≻ e | f . (Notethat the characterization of critical cells implies that the first block of e cannotbe a follower in e . The first block of f may become a follower in e | f , in whichcase we use property (4) from Lemma 3.1.)By induction on the number of irreducibles, e is the greatest cell in z ( e ) whetheror not it is irreducible. (cid:3) Putting together Lemmas 3.2, 3.3, 4.1, and 4.2, we have proved the followingtheorem.
Theorem 4.3.
A basis for H ∗ (cell( n, is given by the classes of the cycles z ( e ) ,where e ranges over all cells with the following properties: (1) Every two consecutive singletons are in decreasing order. (2)
If a given –element block has its elements in decreasing order, then theblock is a follower.A basis for H ∗ (desc( n, w )) is given by the classes of the cycles z ( e ) , where e rangesover all cells with the following properties: (1) Every two consecutive singletons are in decreasing order. (2)
Every non-singleton block has w elements and is a follower. EPRESENTATION STABILITY FOR DISKS IN A STRIP 17
Figure 12.
To compose two morphisms in FI d , we have ( ϕ ′ , c ′ ) ◦ ( ϕ, c ) = ( ϕ ′ ◦ ϕ, c ′′ ), where c ′′ ( i ) is equal to c ′ ( i ) if i is not in theimage of ϕ ′ (for instance, i = 3 has color 1 in the example shown)and is equal to c ( ϕ ′− ( i )) if i is in the image of ϕ ′ (for instance, i = 2 has color 2 in the composition because c (1) = 2 and ϕ ′ (1) =2). 5. Generating an FI d –module In this section we prove a general lemma about FI d –modules. If we want to provethat a given sequence of abelian groups is an FI d –module, many verifications areneeded: we need to specify a group homomorphism for each morphism in FI d , andwe need to prove that compositions that are equal in FI d give equal group homomor-phisms. To streamline such a proof, we can write every morphism as a compositionof permutations and what we call high-insertion maps, which correspond to thevarious inclusions [ n ] ֒ → [ n + 1]. Lemma 5.1 below states which compatibility prop-erties we need to check, in order for the permutations and high-insertion maps tospecify an FI d –module.First we review the precise definition of FI d –module, from [Ram17]. The categoryFI d has one object [ n ] = { , . . . , n } for each natural number n . The morphismsare pairs ( ϕ, c ), where ϕ is an injection, say, from [ n ] to [ m ], and c is a d –coloringon the complement of the image of ϕ ; that is, c is a map from [ m ] \ ϕ ([ n ]) to aset of size d , which in this paper we choose to be { , , . . . , d − } . The morphismscompose as illustrated in Figure 12: for each element colored by the first morphism,in the composition, the image of that element under the second morphism is theone that gets that color. (In the picture, the color of a given element is shownin a diamond just above the element.) More formally, if ( ϕ, c ) : [ n ] → [ n ] and( ϕ ′ , c ′ ) : [ n ] → [ n ] are two morphisms, then we have( ϕ ′ , c ′ ) ◦ ( ϕ, c ) = ( ϕ ′ ◦ ϕ, c ′′ ) , where c ′′ ( i ) is equal to c ′ ( i ) if i ϕ ′ ([ n ]), and is equal to c ( ϕ ′− ( i )) if i ∈ ϕ ′ ([ n ]).An FI d –module M over a commutative ring k is defined to be a functor from FI d to k –modules; that is, we have a k –module M n for each n , and for each ( ϕ, c ) : [ n ] → [ m ], we have a corresponding k –module map ( ϕ, c ) ∗ : M n → M m . In the presentpaper we use k = Z . An FI d –module is finitely generated if there exists a finiteset of elements x , . . . , x r ∈ F ∞ n =1 M n such that the only FI d –submodule of M containing x , . . . , x r is M itself.Any FI d –module is determined by the permutation action on each M n , alongwith the d different maps from M n to M n +1 that correspond to taking the inclusion Figure 13.
Any morphism in FI d can be decomposed as a se-quence of high-insertion maps, followed by a permutation that pre-serves the order of the newly inserted elements. The morphism inthe figure is equal to (1 4 2)(3 5) ◦ i ◦ i .from [ n ] into [ n + 1] and coloring the element n + 1 each of the d different colors. Werefer to these latter maps as the high-insertion maps . Notationally, we denoteby [ i k ] : M n → M n +1 the k th high-insertion map, which colors element n + 1 withthe color k . We denote by [ σ ] : M n → M n the permutation map corresponding toa permutation σ ∈ S n .In order for a choice of permutation action and high-insertion maps to correspondto an FI d –module, we need to check some compatibility properties. We say that“high-insertion maps commute with permutations” if for every color k , every n ,and every σ ∈ S n , we have [ i k ] ◦ [ σ ] = [ e σ ] ◦ [ i k ] , where e σ ∈ S n +1 fixes the element n + 1 and permutes the other elements accordingto σ . We say that “insertions are unordered” if for every pair of colors k, ℓ andevery n , we have the following relation of maps from M n to M n +2 :[( n + 1 n + 2)] ◦ [ i k ] ◦ [ i ℓ ] = [ i ℓ ] ◦ [ i k ] . Here ( n +1 n +2) denotes the permutation in S n +2 that transposes the greatest twoelements. The following lemma says that checking these two properties is enoughto define an FI d –module. Lemma 5.1.
Suppose we have modules M n , with S n –actions on the various M n and d high-insertion maps from each M n to M n +1 . If “high-insertion maps commutewith permutations” and “insertions are unordered”, then the compositions of thesemaps form an FI d –module.Proof. For each morphism ( ϕ, c ) in FI d , we need to define a map ( ϕ, c ) ∗ : M n → M m . We already have a definition when ϕ is a permutation, that is, when ( ϕ, c ) =( σ, · ), where σ ∈ S n and · denotes an empty coloring. In this case ( σ, · ) ∗ = [ σ ].The high-insertion maps describe what happens when ϕ is the inclusion map from[ n ] to [ n + 1], that is, when ( ϕ, c ) = ( i, n + 1 k ), where i : [ n ] ֒ → [ n + 1] is theinclusion. In this case we set ( i, n + 1 k ) ∗ = [ i k ].Given an arbitrary morphism ( ϕ, c ) in FI d , as in Figure 13 we can write theinjection ϕ : [ n ] → [ m ] uniquely as σ ϕ ◦ i m − n , such that σ ϕ ∈ S m is order-preservingon the set [ m ] \ [ n ], and i m − n denotes the composition of inclusions i : [ n ] ֒ → [ n + 1], i : [ n + 1] ֒ → [ n + 2], and so on. In other words, σ ϕ takes the same values on [ n ]as ϕ , and maps [ m ] \ [ n ] to the complement of the image of ϕ in order. Looking atthe coloring c on [ m ] \ ϕ ([ n ]), we let c , . . . , c m − n denote the values of c in order; EPRESENTATION STABILITY FOR DISKS IN A STRIP 19 to be precise, we have c i = c ( σ ϕ ( i )) for i = n + 1 , . . . , m . Then we have( ϕ, c ) = ( σ ϕ , · ) ◦ ( i, m c m − n ) ◦ · · · ◦ ( i, n + 1 c ) , so we should define( ϕ, c ) ∗ = ( σ ϕ , · ) ∗ ◦ ( i, m c m − n ) ∗ ◦ · · · ◦ ( i, n + 1 c ) ∗ = [ σ ϕ ] ◦ [ i c m − n ] ◦ · · · ◦ [ i c ] . To check functoriality, we need to check that if we have another sequence of per-mutations and high-insertions that composes to ( ϕ, c ) in FI d , then the correspondingmaps on the various modules M n compose to ( ϕ, c ) ∗ . Given an arbitrary sequenceof permutations and high-insertion maps, the property that “high-insertion mapscommute with permutations” implies that we can push all the permutations to theleft past the high-insertion maps, without changing the composition map, to geta composition of permutations followed by a composition of high-insertion maps.Using the fact that the permutations in each S n form a group action on M n , wecan replace the composition of permutations by a single permutation.Thus, to prove that we have an FI d –module, it suffices to show that if( ϕ, c ) = ( σ ′ , · ) ◦ ( i, m c ′ m − n ) ◦ · · · ◦ ( i, n + 1 c ′ ) , then we have [ σ ′ ] ◦ [ i c ′ m − n ] ◦ · · · ◦ [ i c ′ ] = [ σ ϕ ] ◦ [ i c m − n ] ◦ · · · ◦ [ i c ] . Because σ ′ and σ ϕ both take the same values on [ n ] as ϕ , we can write σ ′ = σ ϕ ◦ σ ′′ ,where σ ′′ only permutes [ m ] \ [ n ]. Thus, canceling [ σ ϕ ] from both sides it sufficesto show that we have[ σ ′′ ] ◦ [ i c ′ m − n ] ◦ · · · ◦ [ i c ′ ] = [ i c m − n ] ◦ · · · ◦ [ i c ] . This identity comes from the property that “insertions are unordered”. Specif-ically, we can use induction on m − n . If m − n = 1, there is nothing to prove.Otherwise, let n + k = ( σ ′′ ) − ( n + 1); that is, in the alternative composition, el-ement n + k gets inserted with color c ′ k = c and then σ ′′ changes its number to n + 1. By the “insertions are unordered” property we can write[ i c ′ k ] ◦ [ i c ′ k − ] = [( n + k n + k − ◦ [ i c ′ k − ] ◦ [ i c ′ k ] , then [ i c ′ k ] ◦ [ i c ′ k − ] = [( n + k − n + k − ◦ [ i c ′ k − ] ◦ [ i c ′ k ] , and so on, until the composition ends with [ i c ′ k ] on the right. Then, applying the“high-insertion maps commute with permutations” property, we can move all thetranspositions to the left to make the composition σ ′′ ◦ ( n + k n + k − ◦ ( n + k − n + k − ◦ · · · ◦ ( n + 2 n + 1) , which is equal to σ ′′ ◦ ( n + k n + k − · · · n + 2 n + 1) , a permutation that fixes n + 1. Denoting this new permutation by σ ′′′ , we have[ σ ′′ ] ◦ [ i c ′ m − n ] ◦ · · · ◦ [ i c ′ ] = [ σ ′′′ ] ◦ [ i c ′ m − n ] ◦ · · · ◦ [ i c ′ k +1 ] ◦ [ i c ′ k − ] ◦ · · · ◦ [ i c ′ ] ◦ [ i c ′ k ] , and we know that c ′ k = c by how we have selected k . Applying the inductivehypothesis, we have[ σ ′′′ ] ◦ [ i c ′ m − n ] ◦ · · · ◦ [ i c ′ k +1 ] ◦ [ i c ′ k − ] ◦ · · · ◦ [ i c ′ ] = [ i c m − n ] ◦ · · · ◦ [ i c ] , and so composing with [ i c ] on the right, we obtain the desired equality. (cid:3)
6. FI d –module for disks in a strip The goal of this section is to prove the following two theorems, which are themain theorems of this paper.
Theorem 6.1.
For any j , the homology groups H j (cell( n, H j (config( n, form a finitely generated FI j +1 –module over Z . Theorem 6.2.
For any j ≥ and w ≥ , the homology groups H j (desc( n, w )) are zero unless j is a multiple of w − . If j = b ( w − for some integer b , thenthe homology groups H j (desc( n, w )) form a finitely generated FI b +1 –module over Z ,and thus for k = w + 1 , the no– k –equal homology groups H j (no k ( n, R )) also forma finitely generated FI b +1 –module. Theorem 4.3 implies that H j (desc( n, w )) = 0 if j is not a multiple of w − n, w ) has no critical j –cells. And, we know from Sec-tion 2 that the cell complexes cell( n,
2) and desc( n, w ) are homotopy equivalent tothe configuration space config( n,
2) and the no- k -equal space no w +1 ( n, R ), respec-tively. Thus, in both cases it remains to specify the permutation action and thehigh-insertion maps, to check the compatibility properties from the hypothesis ofLemma 5.1, and to verify that the resulting FI d –module is finitely generated.The permutation actions on H ∗ (cell( n, H ∗ (desc( n, w )) come from thepermutation actions on cell( n,
2) and desc( n, w ), which correspond to the permuta-tion actions on config( n,
2) and no w +1 ( n, R ) by permuting the labels. Specifically,for each cell in cell( n, n, n, w ), we apply the permuta-tion to the numbers in that symbol, and then rearrange the numbers within eachblock so that they are in descending order. For each permutation σ ∈ S n , we denotethe corresponding maps on homology by [ σ ]. We note that the permutations donot respect the basis for homology given in Theorem 4.3; applying a permutationto a basis cycle z ( e ) may give a cycle that is homologous to a linear combinationof several basis cycles.We define the high-insertion maps in terms of barriers , which roughly are thenon-singleton blocks. Specifically, as before we write each critical cell e as the(unique) concatenation product of images of the cells 1, 1 2, 1 | | ( w +1) w · · · | | ( w + 1) w · · · j –cell in cell( n, j , and for any critical j –cell in desc( n, w ), the number ofbarriers is b = jw − .For 0 ≤ k ≤ j in the case of cell( n, ≤ k ≤ b in the case of desc( n, w ),the k th high-insertion map is defined as follows and is depicted in Figure 14. Givena critical cell e of cell( n, i k ( e ) we insert a block containing only thenumber n + 1, right after the k th barrier of e (or as the first block, if k = 0). Weobserve that the result is also a critical cell. Thus, these maps give rise to mapson homology, [ i k ] : H j (cell( n, → H j (cell( n + 1 , i k ] : H j (desc( n, w )) → H j (desc( n + 1 , w )): Given a homology class, we write it in terms of the basis cycles z ( e ), and then replace each z ( e ) by z ( i k ( e )).The proof of compatibility between the permutation action and the high-insertionmaps is based on the following three useful properties of barriers: EPRESENTATION STABILITY FOR DISKS IN A STRIP 21 i i i Figure 14.
The k th high-insertion map i k inserts a singleton justafter the k th barrier, with label greater than all the existing labels.214 2 3 14 23 Figure 15.
The cycles z (1 4 | |
3) and z (1 4 | |
2) are ho-mologous, because their difference is the boundary of the chain z (1 4) | f is a critical cell and σ is a permutation, and we write σ ( z ( f )) in terms of the basis as σ ( z ( f )) = X i α i · z ( e i ) , then each critical cell e i has the same number of barriers as f has.(2) “Only barriers obstruct singletons”: Let e be a critical cell without anybarrier, and let i be a single number. Then the cycles z ( e ) | i and i | z ( e )are homologous.(3) “Critical cells can concatenate at a barrier”: If e and e are critical cellssuch that e ends with a barrier, then e | e is a critical cell.Property (1) is true because in cell( n,
2) or in desc( n, w ), every critical cell of agiven dimension has the same number of barriers. Property (3) is true accordingto our characterization of critical cells in Theorem 4.3. To prove Property (2), weobserve that the only critical cells without barriers have only singleton blocks, soit suffices to show that we can permute consecutive singletons, as in the followinglemma and in Figure 15.
Lemma 6.3.
Let z and z be injected cycles with disjoint sets of labels, and let p and q be numbers not appearing in z and z . Then the concatenation products z | p | q | z and z | q | p | z are homologous.Proof. The proof follows from the more general fact that the homology class of anyconcatenation product of cycles is preserved by replacing a factor by a homologousfactor. This is because of the Leibniz rule from Section 2: for any injected cells f
12 HANNAH ALPERT − z (1 | |
4) + z (1 | | − z (23 | | ∼ − z (1 | |
4) + z (1 | | − z (23 | | z (1 | | − z (1 |
32) + z (1 | − z (23 | z (1 | i (2 3)(2 3) i Figure 16.
The property that “high-insertion maps commutewith permutations” is not quite true for cycles, but it is true forhomology classes.and f with disjoint sets of labels, we have ∂ ( f | f ) = ∂f | f + ( − b ( f ) f | ∂f , where b ( f ) denotes the number of blocks in f . The two concatenation products z | p | q | z and z | q | p | z differ by the boundary of the chain z | q p | z , sothey are homologous. (cid:3) Using these three properties of barriers, we can verify the two compatibilityproperties between the permutations and the high-insertion maps.
Lemma 6.4.
For each j , the S n –actions and high-insertion maps on the homol-ogy groups H j (cell( n, and H j (desc( n, w )) have the property that “high-insertionmaps commute with permutations” and the property that “insertions are unordered”.Proof. First we verify the first property, that if σ ∈ S n , then for all k we have[ i k ] ◦ [ σ ] = [ e σ ] ◦ [ i k ] , where e σ is the corresponding permutation in S n +1 . Figure 16 lays out what weneed to prove in a specific example. It suffices to check the desired relation on thebasis cycles z ( e ), where e is a critical cell. We can write e as e | e , where e ends with the k th barrier of e , so that i k ( e ) = e | n + 1 | e . Let z and z bethe injected cycles resulting from applying σ to z ( e ) and z ( e ), so that we have[ σ ] z ( e ) = z | z . By definition we have([ e σ ] ◦ [ i k ]) z ( e ) = [ e σ ] z ( e | n + 1 | e ) = z | n + 1 | z , and we want to show that this cycle is homologous to ([ i k ] ◦ [ σ ]) z ( e ) = [ i k ]( z | z ).Applying a high-insertion map to z | z requires writing that cycle in termsof the basis cycles. Suppose that z is homologous to P i α i z ( e i ). Then z | z ishomologous to P i α i z ( e i ) | z , and z | n + 1 | z is homologous to P i α i z ( e i ) | n +1 | z , so it suffices to show, for any i , that we have (using the ∼ symbol forhomologous cycles) [ i k ]( z ( e i ) | z ) ∼ z ( e i ) | n + 1 | z . Using the “number of barriers is preserved by permutation” property, we know that e i has exactly k barriers. We write e i as c | t , where c (“core”) ends with the k thbarrier of e i , and t (“tail”) consists of the remaining blocks, containing no barrier. EPRESENTATION STABILITY FOR DISKS IN A STRIP 23
123 412124 3 i ◦ i (3 4) i ◦ i
12 3 41212 4 3 i ◦ i (3 4) ≀ Figure 17.
The “insertions are unordered” property says thatwhen disks are inserted, the homology class of the result does notdepend on the ordering of the insertions, as long as each disk goesbetween the right pair of barriers.We can write z ( t ) | z in terms of the basis as P i β i z ( e i ). Then, using the “criticalcells can concatenate at a barrier” property, each c | e i is a critical cell, so we have[ i k ]( z ( e i ) | z ) = [ i k ]( z ( c ) | z ( t ) | z ) = X i β i · [ i k ]( z ( c ) | z ( e i )) = X i β i · z ( c ) | n +1 | z ( e ) . To show this is homologous to z ( e i ) | n + 1 | z , we use the “only barriers obstructsingletons” property. This implies that z ( t ) | n + 1 is homologous to n + 1 | z ( t ),so we have z ( e i ) | n +1 | z = z ( c ) | z ( t ) | n +1 | z ∼ z ( c ) | n +1 | z ( t ) | z ∼ X i β i · z ( c ) | n +1 | z ( e ) . Thus for each i we have[ i k ]( z ( e i ) | z ) ∼ z ( e i ) | n + 1 | z , and so we have [ i k ]( z | z ) ∼ z | n + 1 | z , as desired. This completes the proof that “high-insertion maps commute withpermutations”.Next we check the property that “insertions are unordered”, which is depictedin Figure 17 and says that for all colors k, ℓ we have[( n + 1 n + 2)] ◦ [ i k ] ◦ [ i ℓ ] = [ i ℓ ] ◦ [ i k ] . If k = ℓ , then the equality holds on the level of cells; both sides result in a cellwith the number n + 1 inserted right after the k th barrier and n + 2 inserted rightafter the ℓ th barrier. If k = ℓ , then i k ◦ i k puts n + 2 | n + 1 right after the k thbarrier, and ( n + 1 n + 2) ◦ i k ◦ i k puts n + 1 | n + 2 right after the k th barrier.By the “only barriers obstruct singletons” property, these two resulting cells givehomologous cycles. (cid:3) We now have the necessary pieces to prove our two main theorems.
Proof of Theorem 6.1.
Lemma 6.4 verifies the two hypotheses of Lemma 5.1, whichare that “high-insertion maps commute with permutations” and “insertions areunordered”. Then Lemma 5.1 implies that the compositions of these maps give anFI d –module for d = j + 1.We claim that a finite generating set for this FI j +1 –module consists of the basiscycles z ( e ) where e is a critical cell for n ≤ j . Given any basis cycle z ( e ) with n > j , we can write e as the concatenation product of irreducible critical injectedcells. There are j order-preserving images of the irreducibles 1 2 and 1 | e ′ be the result of deletingthese additional singleton blocks and shifting the numbers down so that they remainconsecutive. Then z ( e ) is the result of applying some high-insertion maps and apermutation to z ( e ′ )—the permutation preserves the order of the numbers in e ′ —and z ( e ′ ) is in the proposed generating set. Thus the finitely many basis cycleswith n ≤ j do generate the FI j +1 –module. (cid:3) Proof of Theorem 6.2. If j is not a multiple of w −
1, there are no critical cells indimension j and thus no homology. If j = b ( w − H j (desc( n, w )) is an FI b +1 –module. The critical cells with n equal to b ( w + 1)—that is, those consisting only of barriers—form a finite generating set forthe FI b +1 –module. (cid:3) Conclusion
To generalize the results of this paper to config( n, w ) for w >
Conjecture 7.1.
For any j and w , the homology groups H j (config( n, w )) form afinitely generated FI d –module for d = 1 + j jw − k . For various reasons this conjecture seems trickier to prove than the results ofthis paper. The first difficulty is in finding a Z –basis for H j (config( n, w )). Thestrategy that produces the bases given in this paper goes roughly as follows. Thereis a total ordering on the cells of cell( n ) such that the critical cells of the discretegradient vector field are in bijection with a Z –basis of H ∗ (cell( n )). (To order, wemodify the “key” function from Lemma 3.1 so that there is no special case forfollower blocks. The critical cells are those where the first element of the block isthe least element, and where furthermore the blocks appear in descending orderof first element.) When a cell in cell( n,
2) matches up to a higher-dimensional cellin cell( n ) \ cell( n, w = 2 we construct basis cycles asconcatenation products of two kinds of cycles: those that generate the homologyof cell( n ), such as z (1 2) = 1 2 + 2 1, and those that are boundaries in cell( n,
2) ofcells of cell( n ) \ cell( n, z (1 | ∂ (3 2 1).However, this strategy makes less sense for larger w . For instance, in cell(6), thecell 4 5 6 | | | , ∂ (4 5 6 1 2 3), which is in cell(6 ,
5) but is not in cell(6 , w . EPRESENTATION STABILITY FOR DISKS IN A STRIP 25
Question 7.2.
For w >
2, is H ∗ (config( n, w )) a free abelian group? If so, is therea discrete gradient vector field on cell( n, w ) that has the same number of criticalcells as the rank of H ∗ (config( n, w ))?The second difficulty is in counting barriers in the various cycles. In the theoremsof this paper, for each space config( n,
2) or no k ( n, R ), every cycle of a given dimen-sion has the same number of barriers. However, this is not true for config( n, w )with w >
2. For instance, in config(8 ,
3) we can construct a 4–cycle with no barriersby using four disjoint circling pairs, and we can also construct a 4–cycle with twobarriers by using two clusters of three disks and two fixed singleton disks. TheFI d –module structure depends completely on being able to recognize barriers in aconsistent way. How can we make the notion of barrier precise? Roughly, we cansay that a cycle z on n disks has at least one barrier if ( n + 1) | z and z | ( n + 1) arenot homologous. But, if a cycle is not a concatenation product, how can we countthe barriers? Question 7.3.
Is there a collection of single-barrier cycles in H ∗ (config( n, w )),such that there is an S n –equivariant way of breaking arbitrary cycles (or homologyclasses) into sums of concatenation products of these single-barrier cycles?Counting barriers is related to estimating the growth of the ranks of the homol-ogy groups. The proofs in this paper imply that not only are H j (config( n, H j (no k ( n, R )) finitely generated FI d –modules for d = j + 1 and d = jk − respec-tively, but in fact the rank of each of these free abelian groups is equal to d n timesa polynomial function of n . In the case of config( n, w ) for w > d − n times a polynomial of n , ( d − n times a polynomial, and so on, but in the setting of this paper we do not have theseadditional terms. Question 7.4.
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