Asymptotic homology of graph braid groups
aa r X i v : . [ m a t h . A T ] M a y ASYMPTOTIC HOMOLOGY OF GRAPH BRAID GROUPS
BYUNG HEE AN, GABRIEL C. DRUMMOND-COLE, AND BEN KNUDSEN
Abstract.
We give explicit formulas for the asymptotic Betti numbers of theunordered configuration spaces of an arbitrary finite graph over an arbitraryfield. Introduction
The (co)homology of configuration spaces is a classical topic of perennial interest[Arn69, BCT89, Tot96, FT00, Chu12, DCK17]. This article is concerned with thehomology of configuration spaces of graphs [FS05, KP12, MS17, CL18, Ram18,ADCK19], which for our purposes are simply finite -dimensional cell complexes.For a graph Γ , we write B k ( Γ ) = { ( x , . . . , x k ) ∈ Γ k : x i = x j if i = j } / Σ k for the k th unordered configuration space of Γ . These spaces classify their fun-damental groups, the graph braid groups [Abr00], so this homology is also grouphomology.Experience has shown that Betti numbers of configurations spaces are profitablyviewed as functions of k . Fixing a homological degree, our main result asserts thatthe resulting function is asymptotic to a simple, explicit formula in the combina-torics of Γ .Recall that the degree or valence d ( w ) of a vertex w is the number of componentsof its complement in a small neighborhood, and w is called essential if d ( w ) ≥ .Given a set of essential vertices W , we write ∆ W Γ for the number of components ofthe complement of W in Γ . The i th Ramos number of Γ is defined as the maximum ∆ i Γ = max | W | = i ∆ W Γ of these numbers [Ram18]. Theorem 1.1.
Fix a field F and i ≥ . If Γ is a connected graph with an essentialvertex and ∆ i Γ > , then dim H i ( B k ( Γ ); F ) ∼ X W i Γ − Y w ∈ W ( d ( w ) − k ∆ i Γ − , where W ranges over sets of essential vertices of cardinality i such that ∆ W Γ = ∆ i Γ . As the authors have previously shown, the i th Betti number of B k ( Γ ) is even-tually a polynomial of degree ∆ i Γ − [ADCK20], so the theorem amounts to thecalculation of the leading coefficient of this polyomial.We comment briefly on our hypotheses. The inequality ∆ i Γ > always holdsexcept sometimes in the case i = 1 , where it is possible that ∆ Γ = 1 . This case iscompletely understood by work of Ko–Park [KP12]—in particular, the conclusionof the theorem is known not to hold when ∆ i Γ = 1 . The case of a disconnected graphfollows in light of the Künneth formula, and a connected graph without an essential ertex is homeomorphic to a point, an interval or a circle, whose configuration spacesare easily understood.We now contextualize Theorem 1.1 within three lines of research of independentinterest.1.1. Homological stability.
The idea of studying the homology of configurationspaces asymptotically is an old one, with roots in the scanning and group completiontechniques of McDuff [McD75] and Segal [Seg73] and present already in the veryearliest computations [Arn69]. In more recent years, this idea has flowered intothe study of homological stability and stability phenomena in general [CEF15].The analogue of classical homological stability [McD75, Chu12] in the context ofconfiguration spaces of graphs is the aforementioned fact that the Betti numbersare eventually equal to polynomials in the number of particles.Given homological stability, several questions arise. Theorem 1.1 is an answerto the analogue of one such question: what is the stable homology? We pose twomore, the first also purely computational.
Question . What is the stable range? That is, when does the i th Betti numberof B k ( Γ ) begin to equal a polynomial in k ?The second question is more conceptual, and we can only formulate it vaguely. Question . What does the stable homology represent? Does an analogue ofscanning or group completion apply?1.2.
Universal presentations.
The first interesting example in the homology ofconfiguration spaces of graphs is the star class , in which two particles orbit oneanother by passing successively through an essential vertex (see Section 2.3). Per-forming the same local move simultaneously at different essential vertices, perhapswith the addition of stationary particles on edges, produces a large family of toricclasses in homology.It was once thought that, at least rationally, all homology might be generatedby such tori, together with the homology of the graph itself. This idea was putto rest by the discovery that the configuration space of unordered particles inthe graph obtained by suspending points is a compact orientable surface of genus , up to homotopy [WG17, CL18]. Other exotic homology classes have since beendiscovered, and we refer to the resulting zoology problem as the problem of universalgeneration.Problem . Give a finite list of atomic graphs providing generators for the ho-mology of the configuration spaces of all graphs (perhaps within a certain class) infixed degree.For example, star classes together with cycles in the graph itself generate in de-gree [KP12]; toric classes generate in all degrees for trees and wheel graphs [MS18](see [CL18] for a related result in the ordered context); and, as the authors showin forthcoming work, the only “exotic” generator in degree is the aforementionedfundamental class of the surface of genus , at least for planar graphs [ADCK].In the course of proving Theorem 1.1, we show that the dimension of H i ( B k ( Γ ); F ) is asymptotic to the dimension of the submodule spanned by toric classes satisfyinga certain rigidity condition (see Section 2.4); indeed, it is the combinatorics of thissubmodule that account for the asymptotic formula. Thus, although tori are notuniversal generators, they are so asymptotically . he result also touches on the companion problem of universal relations , whereit asserts that the only non-obvious relation among these asymptotic generators isthe X - relation of Lemma 2.10.1.3. Torsion.
Elements of finite order in the homology of configuration spacesof graphs are exceedingly rare. At the level of ordered configuration spaces, noexamples are known, and it has been proven that none exist for certain limitedclasses of graphs [CL18]. In the unordered case of interest to us, Kim–Ko–Parkhave shown that -torsion in the first homology of the configuration space of twopoints detects planarity [KKP12]. In the non-planar case, the size of the -torsionsubgroup is computable in terms of invariants of the background graph [KP12], andthis -torsion propagates to higher degrees and larger configurations via disjointunion and the addition of stationary particles to edges. Apart from this singlesource of torsion, essentially nothing is known. Question . Is there nontrivial odd torsion in the homology of graph braid groups?Are there elements of even order greater than ?To see the connection between this question and Theorem 1.1, recall that, by theuniversal coefficient theorem, a reasonably finite space has torsion-free homologyif and only if its Betti numbers are independent of the coefficient field. For thisreason, Theorem 1.1 should be interpreted as implying that the homology of graphbraid groups is asymptotically torsion-free .1.4. Idea of proof.
The proof of Theorem 1.1 proceeds in two steps. First, weshow that the toric classes mentioned above account for most of the homology ofthe configuration spaces of a graph (Theorem 3.13). Second, and easier, we counttori (Theorem 4.7).In pursuing the first task, an argument by induction on the first Betti number of Γ suggests itself, since tori account for everything in the case of a tree. This inductiveargument is facilitated by vertex explosion , a technique whereby the configurationspaces of Γ are related to those of a simpler graph by a long exact sequence (seeSection 2.2). In order to achieve the inductive step, we require more than knowledgeof the rates of growth of the modules in question; we must be assured that theserates of growth obtain for specific geometric reasons . This fine control is expressedin the concept of tameness , our principal technical innovation (see Section 3.1).1.5. Relation to previous work.
A stability phenomenon in k for H i ( B k ( Γ )) wasfirst observed by Ko–Park in the case i = 1 [KP12]. Ramos subsequently observeda generalization of this phenomenon for all i in the case where Γ is a tree, and hefurther identified the invariant ∆ i Γ controlling the degree of polynomial growth inthis case [Ram18]. The authors interpreted this stability phenomenon geometricallyin terms of edge stabilization (see Section 2.1), permitting its generalization to allgraphs, and computed the degree of growth in general [ADCK20]. The first authorformulated Theorem 1.1 as a conjecture in the summer of 2019 and presented it atthe AIM workshop “Configuration spaces of graphs” in early 2020. The extensivecomputer calculations of the second author provided evidence for the conjecture[DC]. .6. Acknowledgments.
The first author was supported by the National Re-search Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(No. 2020R1A2C1A01003201). The third author was supported by NSF grant DMS1906174. The authors benefited from the hospitality of the American Institute ofMathematics during the workshop “Configuration spaces of graphs.”1.7.
Conventions.
Given functions f, g : Z ≥ → Z ≥ , we say that f and g areasymptotic, written f ∼ g , provided both are eventually nonzero and lim k →∞ f ( k ) g ( k ) = 1 . We work over a fixed ground ring R , which we take to be a Noetherian integraldomain. At times, we require the ground ring to be a field, writing F instead.Homology is taken implicitly with coefficients in the ground ring.All gradings are non-negative. Given a degreewise finite dimensional graded F -vector space V , we write dim V for the function k dim V k , regarded as a functionfrom Z ≥ to itself. 2. Preliminaries
In this section, we collect the facts and tools used in the proof of Theorem 1.1.Although largely a review, it contains a few crucial observations not appearingelsewhere, namely Corollary 2.11, Lemma 2.20, and Corollary 2.21.2.1.
The Świątkowski complex.
This section is a brief summary of some neces-sary terminology and results from [ADCK20].A graph is a finite -dimensional CW complex, whose -cells and open -cellsare called vertices and edges, respectively. A contractible graph is called a tree. Ahalf-edge is a point in the preimage of a vertex under the attaching map of a -cell;thus, every edge determines two half-edges. In general, sets of vertices, edges, andhalf-edges are denoted V ( Γ ) , E ( Γ ) , and H ( Γ ) , respectively, but we omit Γ from thenotation wherever doing so causes no ambiguity.A half-edge h has an associated vertex v ( h ) and an associated edge e ( h ) , and wewrite H ( v ) = { h ∈ H : v = v ( h ) } for the set of half-edges incident on v ∈ V . Thedegree or valence of v is d ( v ) = | H ( v ) | . A vertex is essential if its valence is at least , and an edge is a tail if its closure contains a vertex of valence .A subgraph is a subcomplex of a graph. A graph morphism is a finite compositionof isomorphisms onto subcomplexes and inverse subdivisions—see [ADCK20, §2.1]for details. Definition 2.1.
Let Γ be a graph. For v ∈ V , write S ( v ) = Z h ∅ , v, h ∈ H ( v ) i .The Świątkowski complex (with coefficients in R ) is the R [ E ] -module S ( Γ ) = R [ E ] ⊗ Z O v ∈ V S ( v ) , equipped with the bigrading | ∅ | = (0 , , | v | = | e | = (0 , , and | h | = (1 , , togetherwith the differential determined by the equation ∂ ( h ) = e ( h ) − v ( h ) .When denoting elements of S ( Γ ) , we systematically omit all factors of ∅ andall tensor symbols, and we regard half-edge generators at different vertices as per-mutable up to sign. See [ADCK20, §2.2] for further discussion of S ( Γ ) and itselements. rite B ( Γ ) = F k ≥ B k ( Γ ) . The following result is [ADCK20, Thm. 2.10], butsee [Świ01] and [CL18] for precursors. Theorem 2.2.
There is a natural isomorphism of bigraded R [ E ] -modules H ∗ ( B ( Γ )) ∼ = H ∗ ( S ( Γ )) . Two comments are in order. First, the action of R [ E ] on the lefthand side arisesfrom an E -indexed family of edge stabilization maps . Stabilization at e replaces thesubconfiguration of particles lying in the closure of e with the collection of averagesof consecutive particles and endpoints—see Figure 1 and [ADCK20, §2.2]. Second,regarding the implied functoriality, we direct the reader to [ADCK20, §2.3]. ⇓ Figure 1.
Edge stabilizationThe reduced
Świątkowski complex is obtained by replacing S ( v ) in the definitionof S ( Γ ) with the submodule e S ( v ) ⊆ S ( v ) spanned by ∅ and the differences ofhalf-edges. The inclusion e S ( Γ ) ⊆ S ( Γ ) is a quasi-isomorphism as long as Γ has noisolated vertices [ADCK19, Prop. 4.9]. Note that, for any h ∈ H ( v ) , a basis for e S ( v ) is given by { ∅ } ∪ { h − h } h ∈ H ( v ) \{ h } . In this way, a (non-canonical) basisfor e S ( Γ ) may be obtained.2.2. Exploding vertices.
Given a graph Γ and v ∈ V , we write Γ v for the graphobtained by exploding the vertex v —see Figure 2 and [ADCK20, Def. 2.12]—whichwe regard as a subgraph of a subdivision of Γ , uniquely up to isotopy. In general,there is a long exact sequence relating the homology of B ( Γ ) with that of B ( Γ v ) [ADCK20, Prop. 2.3]. We state only the special case we require. Γ Γ v Figure 2.
A local picture of vertex explosion
Proposition 2.3. If v ∈ V is a bivalent vertex with corresponding edges e, e ′ ∈ E ,then the sequence · · · → H i ( B k ( Γ v )) ι −→ H i ( B k ( Γ )) ψ −→ H i − ( B k − ( Γ v )) e − e ′ −−−→ H i − ( B k ( Γ v )) → · · · is exact. Here,(1) the map ι is induced by the inclusion of Γ v ,(2) the map ψ is induced by the chain map on reduced Świątkowski complexessending β + ( h − h ′ ) α to α , where β involves no half-edge generators at v ,and(3) the map e − e ′ is multiplication by the ring element e − e ′ ∈ R [ E ] .Moreover, all maps shown are compatible with edge stabilization. ore generally, given a subset W ⊆ V , we write Γ W for the graph obtainedby exploding each of the vertices in W . The analogue of the exact sequence ofProposition 2.3 is a spectral sequence interpolating between the homology of B ( Γ W ) and that of B ( Γ ) , which arises by filtering the reduced Świątkowski complex by thenumber of vertices of W needed to write an element [ADCK20, §3.2].In general, this spectral sequence is rather mysterious. Fortunately, we needbut little knowledge of it. Denoting the r th page by E r ∗ , ∗ = E r ∗ , ∗ ( W ) , we have thefollowing result (see the proof of [ADCK20, Lem. 3.15]. Proposition 2.4.
There is a canonical R [ E ] -linear inclusion and an isomorphism E ∞| W | , ⊆ E | W | , ∼ = R [ π ( Γ W )] h X i , where X is the set of generators of e S ( Γ ) of the form N w ∈ W h w , with h w a differenceof half-edges at w . Thus, in degree | W | , the top graded piece of the filtration on homology associatedto W is a submodule of an R [ E ] -module that is free over a specific quotient ringtied to the combinatorics of W in Γ .2.3. Loops, stars, and relations.
In this section, we explore two basic sourcesof nontrivial elements of H ∗ ( B ( Γ )) and the relations between them. The reader isdirected to [ADCK19, §5.1] for further details. Example 2.5.
Since Γ = B ( Γ ) is a subspace of B ( Γ ) , an oriented cycle in Γ determines an element of H ∗ ( B ( Γ )) , called a loop class .A standard chain level representative of a loop class is obtained by summingthe differences of half-edges involved in the cycle in question. For example, thestandard representative of the unique loop class in the graph L depicted in Figure3, oriented clockwise, is h − h ′ ∈ e S ( L ) . h ′ he ′ e Figure 3.
The lollipop graph L Example 2.6.
There is a canonical homotopy equivalence S ≃ B ( S ) , where S is the cone on three points—see Figure 4 for a depiction of the cycle witnessing thishomotopy equivalence. By functoriality, then, the choice of half-edges h , h , and h sharing a common vertex determines a star class in H ( B ( Γ )) , which dependson the ordering only up to sign. We will denote star classes by α or, e.g., α ifwe wish to emphasize the particular choice of half-edges.Writing e j for the edge associated to h j , a standard chain level representativefor a star class is given by the sum a = e ( h − h ) + e ( h − h ) + e ( h − h ) . his expression is symmetric and exhibits well the geometry of the class (see Figure4), but the alternative expression a = ( e − e )( h − h ) − ( e − e )( h − h ) in the basis for e S ( Γ ) privileging h is also useful. − + − + − e h e h e h e h e h e h Figure 4.
The basic star class and its standard representativeStar classes and loop classes interact according to a relation called the
Q-relation .Our notation will refer to the graph L of Figure 3, but functoriality propagates therelation to any graph with a subgraph isomorphic to a subdivision of L . Writing γ for the clockwise oriented loop class and α for the counterclockwise oriented starclass in L , we have the following. Lemma 2.7 (Q-relation) . In the homology of the configuration spaces of the graph L , there is the relation ( e − e ′ ) γ = α . The Q-relation implies that, modulo star classes, a loop class is annihilatedby the appropriate difference of edges. Consequently, loop classes generate R [ E ] -submodules exhibiting strictly slower growth (again modulo star classes), hencecontributing nothing asymptotically. Thus, despite its simplicity, the Q-relationalready carries the germ of Theorem 1.1. Remark . Roughly speaking, Theorem 3.13 below asserts that every generatorin H ∗ ( B ( Γ )) satisfies a relation analogous to the Q-relation forcing slow growthmodulo tori, and the difficulty of the argument lies in having to establish this factwithout having access to the generators or relations themselves. As discussed inSection 1.2, it would be extremely interesting to have such access.Different star classes at a vertex of valence at least are also related to oneanother. The first relation is essentially obvious. Lemma 2.9.
In the homology of the configuration spaces of a graph containing avertex with distinct half-edges h , h , h , and h , there is the relation of star classes α − α + α − α = 0 . The second relation, called the
X-relation , has almost the same form.
Lemma 2.10 ( X -relation) . In the homology of the configuration spaces of a graphcontaining a vertex with distinct half-edges h , h , h , and h , we have the relationof stabilized star classes e α − e α + e α − e α = 0 . These relations combine to give the following simple but important relation. Itis this relation that is responsible for the factor of d ( w ) − in the statement ofTheorem 1.1 (see the proof of Lemma 4.3). orollary 2.11. In the homology of the configuration spaces of a graph containinga vertex with distinct half-edges h , h , h , and h , there is the relation of stabilizedstar classes ( e − e ) α − ( e − e ) α + ( e − e ) α = 0 . In situations with a certain amount of degeneracy, star classes at distinct verticescan also be related. Referring to the graph Θ pictured in Figure 5, and writing α and α ′ for the clockwise oriented star classes at the top vertex and bottom vertices,respectively, we have the following relation. Figure 5.
The theta graph Θ Lemma 2.12 ( Θ -relation) . In the homology of the configuration spaces of the graph Θ , there is the relation of star classes α − α ′ = 0 . This relation, which is an easy consequence of the Q-relation, gives rise to twophenomena of interest: first, the dichotomy of rigidity and non-rigidity and its im-plications for rates of growth (Proposition 2.16); second, -torsion in the homologyof non-planar graph braid groups. As discussed in Section 1.3, Theorem 1.1 impliesa more general causal relationship between growth and torsion, the precise natureof which remains mysterious.2.4. Tori.
Since B ( Γ ⊔ Γ ) ∼ = B ( Γ ) × B ( Γ ) , one can form the external product ofclasses supported in disjoint subgraphs of a larger graph [ADCK19, Def. 5.10]. Definition 2.13.
Let W be a set of essential vertices. A class α ∈ H | W | ( B | W | ( Γ )) is called a W -torus if it is the external product of star classes at each w ∈ W .These classes have the following fundamental property. Observation 2.14. If e , e ∈ E can be connected by a path of edges avoiding W ,then ( e − e ) α = 0 for any W -torus α .Therefore, the action of R [ E ] on α factors through the quotient shown in thecommuting diagram R [ E ] h α i R [ E ] · α ⊆ H | W | ( B ( Γ )) R [ π ( Γ W )] h α i It turns out that W -tori for which this map is an isomorphism can be partiallycharacterized [ADCK20, Lem. 3.15]. Definition 2.15 ([ADCK20, Def. 3.10]) . The set W of essential vertices is well-separating if the open star of each w ∈ W intersects more than one component of Γ W .Recall that E r ∗ , ∗ denotes the r th page of the spectral sequence arising from thefiltration induced by W (see Section 2.2). roposition 2.16. Let W be a well-separating set of essential vertices and α a W -torus. The following conditions are equivalent.(1) The natural map R [ π ( Γ W )] h α i → R [ E ] · α is an isomorphism.(2) The image of α in E ∞| W | , is nonzero.(3) Any cycle in e S ( Γ ) representing a star factor of α involves half-edges inmultiple components of Γ W . When the equivalent conditions of Proposition 2.16 hold, we say that α is rigid .Concretely, the second condition means that α does not arise from Γ w for any w ∈ W . These notions relate to the Ramos numbers as follows [ADCK20, Cor.3.16, Lem. 3.18]. Proposition 2.17.
Let W be a set of essential vertices of cardinality i .(1) If W is well-separating, then a rigid W -torus exists.(2) If ∆ W Γ = ∆ i Γ , then either W is well-separating or i = ∆ i Γ = 1 . Together, Propositions 2.16 and 2.17 imply the lower bound in the followingresult [ADCK20, Thm. 1.2], which was the authors’ original motivation for consid-ering rigid tori.
Theorem 2.18.
Under the conditions of Theorem 1.1, the dimension of H i ( B k ( Γ ); F ) coincides with a polynomial of exact degree ∆ i Γ − for k sufficiently large. In particular, dim H i ( B k ( Γ ); F ) is asymptotic to a positive constant multiple of k ∆ i Γ − , so Theorem 1.1 amounts to the identification of this constant. Building onthe fact expressed in Proposition 2.16 that rigid tori achieve the highest possiblerate of growth, we will show that all other classes exhibit strictly slower growth, atleast modulo tori (Theorem 3.13).The remainder of this section is devoted to producing a convenient—in particular,independent—set of rigid tori. Construction 2.19.
Let W be a well-separating set of essential vertices. For each w ∈ W , we fix two edges incident on w and lying in distinct components of Γ W .Abusively, we write A w for the set of d ( w ) − star classes at w (with signs chosenarbitrarily) involving these two edges. We write A W ∼ = Q w ∈ W A w for the resultingset of W -tori and A W ( Γ ) ⊆ H | W | ( B ( Γ )) for the R [ E ] -submodule generated by A W .Note that each α ∈ A W is rigid by construction. Lemma 2.20. If W is a well-separating set of essential vertices, then the naturalmap R [ π ( Γ W )] h A W i → A W ( Γ ) is an isomorphism.Proof. We will show that the kernel of the composite R [ E ] h A i π W −−→ R [ π ( Γ W )] h A i → A W ( Γ ) ⊆ H | W | ( B ( Γ )) is the kernel of π W . We proceed by considering a ( | W | + 1) -chain b whose boundarywitnesses an element in the kernel, writing ∂b = X I p I ( E ) O w ∈ W a wi w . ere the summation runs over multi-indices I = ( i w ) w ∈ W with ≤ i w ≤ d ( w ) − , p I ( E ) is a polynomial in the edges, and a wi is the standard representative of α wi ,i.e., a wi = ( e w − e ′ w ) h wi − ( e w − e ′ w,i ) h w with e w and e ′ w lying in distinct components of π ( Γ W ) (note that three of the edgesand one of the half edge differences in the righthand term are independent of i ).Substituting this expression, we see that, for each I , the coefficient of N w ∈ W h wi w is equal to p I ( E ) Y w ∈ W ( e w − e ′ w ) . We will argue that this coefficient lies in ker( π W ) , which will imply that p I ( E ) ∈ ker( π W ) , as desired. Indeed, the kernel is a prime ideal, as R [ π ( Γ W )] is an integraldomain, and e w − e ′ w / ∈ ker( π W ) by construction.We examine the bounding chain b . Necessarily, b is of the form b ≡ X s q s ( E ) h s ⊗ O w ∈ W h ws (mod F | W |− ) , where h ws is a half-edge generator at w , h s is a half-edge generator at a vertexnot in W , and F p = F p e S ( Γ ) denotes the filtration induced by W . Then, writing ∂h s = e s − e ′ s , we obtain the equivalence X I p I ( E ) O w ∈ W a wi w ≡ X s q s ( E )( e s − e ′ s ) ⊗ O w ∈ W h ws (mod F | W |− ) The filtration splits at the level of graded modules, so we actually have the equality X I p I ( E ) O w ∈ W a wi w = X s q s ( E )( e s − e ′ s ) ⊗ O w ∈ W h ws , since each side is a sum of terms involving every vertex of W . Since e s and e ′ s liein a common component of Γ W for each s , we conclude that all coefficients in bothsums lie in ker( π W ) . (cid:3) We record a consequence of this result for later use.
Corollary 2.21.
The composite map A W ( Γ ) ⊆ H | W | ( B ( Γ )) → E ∞| W | , is injective.Proof. It suffices to show injectivity after further composing with the inclusion E ∞| W | , ⊆ E | W | , of Proposition 2.4. The full composite sends α ∈ A W to theimage of its standard representative in the associated graded module. The proofof Lemma 2.20 shows that the resulting classes in E | W | , are linearly independentover R [ π ( Γ W )] , implying the claim. (cid:3) Tori dominate
In pursuing Theorem 1.1, our most difficult task is to estimate the size of thequotient H i ( B ( Γ )) /T i ( Γ ) . This quotient vanishes in the case of a tree, so we inducton the first Betti number of Γ . In order to carry out this induction, it will benecessary to retain a certain degree of module-theoretic control over the quotient.We begin by specifying the nature of this control. .1. Tame partitions.
Recall that a partition of a set X is an unordered collection P of pairwise disjoint nonempty subsets of X (called blocks) whose union is X . Wewrite x ∼ P y when x and y lie in the same block of P . The set of partitionsof X forms a partially ordered set by declaring that P ≤ P ′ if and only if everyblock of P ′ is contained in some block of P —thus, the maximal partition is thediscrete partition consisting of the singletons, and the minimal partition is thetrivial partition with a single block. Example 3.1.
Given a graph Γ and a set W of essential vertices, we write π ( Γ W ) for the partition of E such that e ∼ π ( Γ W ) e if and only if e and e lie in the inthe same connected component of Γ W .We now formulate our main definition. Definition 3.2.
Let Γ be a graph. A nontrivial partition P of E is called i - tame (with respect to Γ ) if there is a set of essential vertices W of cardinality at most i with P ≤ π ( Γ W ) such that, if W is well-separating, then(1) P = π ( Γ W ) , and(2) if e and e are tails and e ∼ P e , then e ∼ π ( Γ W ) e .By convention, the trivial partition is i -tame for every i > .There are no -tame partitions. The convention regarding the trivial partitionis necessary only when i = ∆ i Γ = 1 . AB C DE F GH
Figure 6.
Example 3.3.
Consider the graph depicted in Figure 6.(1) The partition with blocks { AC } , { BC } , { CD } , { DE } , { DF, F G, F H } is π ( Γ { C,D } ) . This partition is not i -tame for any i (but see Example 3.12).(2) The partition with blocks { AC } , { BC, CD } , { DE } , { DF, F G, F H } is -tame, since it is obtained from π ( Γ { C,D } ) by merging the blocks containing BC and CD , which are not both leaves.We record a simple fact about tame partitions, which is immediate from thedefinition. Lemma 3.4. If P is i -tame, then P is ( i + 1) -tame. Given a graph Γ , we work in the ambient Abelian category of finitely generated R [ E ] -modules. This category receives a contravariant functor from the partiallyordered set of partitions of E , which sends P to R [ P ] and P ≤ P ′ to the quotientmap R [ P ′ ] → R [ P ] . emma 3.5. Let F be a field. For any partition P of E , the function dim F [ P ] : k dim( F [ P ]) k is a polynomial of degree | P | − . Thus, if P is i -tame, then the growth of F [ P ] is slower than that of the submodulegenerated by a W -torus with | W | = i and ∆ W Γ = ∆ i Γ , hence slower than that of H i ( B ( Γ )) . In practice, it is this rate of growth that is of interest, but, as will beborn out in the arguments to come, the utility of tameness lies in guaranteeing that this rate of growth obtains for specific geometric reasons. We will be concerned with certain subcategories of the ambient Abelian category.Recall that a subcategory of an Abelian category is called a
Serre subcategory if itis nonempty, full, and closed under subobjects, quotients, and extensions.
Definition 3.6.
Let Γ be a graph. An R [ E ] -module M is called i - tame (withrespect to Γ ) if it belongs to the Serre subcategory C R ( Γ , i ) generated by gradedshifts of the modules R [ P ] , where P ranges over i -tame partitions of E .From our point of view, the main interest in tameness lies in controlling growth. Definition 3.7.
We say that a graded vector space M is n - small if lim k →∞ k n dim M = 0 . Proposition 3.8.
Let F be a field. If M is an i -tame F [ E ] -module and either i > or ∆ i Γ > , then M is (∆ i Γ − -small.Proof. By rank-nullity, the category of (∆ i Γ − -small F [ E ] -modules is itself a Serresubcategory; therefore, we may assume that M = F [ P ] for some i -tame partition P of E . Since ∆ W Γ ≤ ∆ i Γ whenever | W | ≤ i , it follows from Proposition 2.17 and thedefinition of tameness that | P | < ∆ i Γ , and the claim follows from Lemma 3.5. (cid:3) We close with three simple criteria for tameness.
Lemma 3.9. If P ′ is an i -tame partition and P ≤ P ′ , then R [ P ] ∈ C R ( Γ , i ) .Proof. The module R [ P ] is a quotient of R [ P ′ ] ∈ C R ( Γ , i ) , and C R ( Γ , i ) is closedunder quotients. (cid:3) Lemma 3.10.
Let W be a proper subset of the essential vertices of the connectedgraph Γ . Then R [ π ( Γ W )] ∈ C R ( Γ , | W | + 1) .Proof. If W = ∅ , then π ( Γ W ) is the trivial partition, which is -tame. Otherwise,define W ′ by adding any vertex w to W that is adjacent to one of its vertices. Theedge connecting these two vertices is a singleton in π ( Γ W ′ ) and we define a newpartition P by identifying this block with the block of any other edge at w . Then P is ( | W | + 1) -tame, and π ( Γ W ) ≤ P , so the claim follows from Lemma 3.9. (cid:3) Lemma 3.11. If α is a non-rigid W -torus, then R [ E ] · α ∈ C R ( Γ , | W | ) .Proof. By Observation 2.14, R [ E ] · α receives a surjection from R [ π ( Γ W )] . If W is not well-separating, the partition π ( Γ W ) is | W | -tame by definition, so assumeotherwise. By Proposition 2.16, α has a star factor—at the vertex w ∈ W , say—whose half-edges all lie in a single component of Γ W . The Θ -relation implies that R [ E ] · α admits a surjection from R [ π ( Γ W \{ w } )] , which is | W | -tame by Lemma3.10. (cid:3) Example 3.12.
Although the partition π ( Γ { C,D } ) of Example 3.3 is not i -tamefor any i , Lemma 3.11 shows that the module R [ π ( Γ { C,D } )] is -tame. .2. Tameness modulo tori.
Our present goal is to prove that the inclusion of thesubmodule generated by tori is an isomorphism after localizing at the subcategoryof tame modules. Fixing W , we write T W ( Γ ) ⊆ H i ( B ( Γ )) for the R [ E ] -submodulegenerated by all W -tori. We write T i ( Γ ) for the R [ E ] -span of the T W ( Γ ) , where W ranges over all subsets of essential vertices of cardinality i . Theorem 3.13. If Γ is a connected graph with an essential vertex, then the quotient H i ( B ( Γ )) /T i ( Γ ) is i -tame. The strategy of the proof of Theorem 3.13 is induction on b ( Γ ) . For the inductionstep, we will choose an edge e of Γ , both vertices of which are essential, and subdivideit by adding a bivalent vertex v . We denote the resulting edges of Γ v by e and e ′ .Given a partition P of E ( Γ v ) , write P e ∼ e ′ for the partition of E ( Γ ) obtained byidentifying the respective blocks of P containing e and e ′ . Lemma 3.14. If P is i -tame with respect to Γ v , then P e ∼ e ′ is i -tame with respectto Γ .Proof. Let W be a set of essential vertices witnessing P as i -tame with respect to Γ v . We will show that W also witnesses P e ∼ e ′ as i -tame with respect to Γ . Since P e ∼ e ′ ≤ π (( Γ v ) W ) e ∼ e ′ = π ( Γ W ) , we may assume that W is well-separating in Γ , hence in Γ v . There are now twocases.If e ∼ P e ′ , then e and e ′ lie in the same component of ( Γ v ) W , since e and e ′ aretails of Γ v . Thus, the inclusion of ( Γ v ) W into Γ W induces a bijection on connectedcomponents. Given tails e and e of Γ lying in the same block of P e ∼ e ′ , it followsthat e ∼ P e , hence e ∼ π (( Γ v ) W ) e by tameness, and the claim follows.If e P e ′ , then e and e ′ lie in distinct components of ( Γ v ) W . Given tails e and e of Γ lying in the same block of P e ∼ e ′ , one of the following situations obtains,up to relabeling: either e ∼ P e , or else e ∼ P e and e ∼ P e ′ . Since all fourare tails of Γ v , it follows in either case that e ∼ π ( Γ W ) e . The verification that P e ∼ e ′ < π ( Γ W ) is left to the reader. (cid:3) This result has an important consequence for modules, which is the motivationfor the condition on tails in the definition of tameness.
Lemma 3.15. If M is i -tame with respect to Γ v , then M/ ( e − e ′ ) is i -tame withrespect to Γ .Proof. Since R [ P ] / ( e − e ′ ) ∼ = R [ P e ∼ e ′ ] , Lemma 3.14 implies the claim for M = R [ P ] with P an i -tame partition of E ( Γ v ) . Therefore, it suffices to show that thecollection of modules M for which M/ ( e − e ′ ) ∈ C ( Γ , i ) forms a Serre subcategory.From an exact sequence → M → N → P → of R [ E ( Γ v )] -modules, there arises the exact sequence Tor( P ) → M/ ( e − e ′ ) → N/ ( e − e ′ ) → P/ ( e − e ′ ) → , where Tor indicates the ( e − e ′ ) -torsion submodule. Assuming that M/ ( e − e ′ ) and P/ ( e − e ′ ) lie in C ( Γ , i ) , it is immediate that N/ ( e − e ′ ) does as well. On the otherhand, if N/ ( e − e ′ ) ∈ C ( Γ , i ) , then P/ ( e − e ′ ) ∈ C ( Γ , i ) ; therefore, since Tor( P ) injects into P/ ( e − e ′ ) , it follows that M/ ( e − e ′ ) ∈ C ( Γ , i ) , as desired. (cid:3) orollary 3.16. Suppose that M ∈ C R ( Γ v , i ) is ( e − e ′ ) -torsion. With the induced R [ E ( Γ )] -module structure, M ∈ C R ( Γ , i ) . The main technical input is the following result, whose proof we defer to Section4.1.
Lemma 3.17.
The intersection T i − ( Γ v ) ∩ ker( e − e ′ ) is i -tame with respect to Γ . Taking this result for granted, we complete the argument.
Proof of Theorem 3.13.
The quotient vanishes when i = 0 , so assume i > . Weproceed by induction on b ( Γ ) , the base case of a tree being trivial, since the quotientagain vanishes. Suppose that the claim holds for Γ v in every degree, and considerthe short exact sequence → im( ι ) → H i ( B ( Γ )) → im( ψ ) → arising from Proposition 2.3.We claim first that im( ψ ) ∈ C R ( Γ , i ) . The induction hypothesis implies that H i − ( B ( Γ v )) /T i − ( Γ v ) ∈ C R ( Γ v , i − . Since this category is closed under subob-jects, the third entry in the exact sequence → T i − ( Γ v ) ∩ im( ψ ) → im( ψ ) → im( ψ ) /T i − ( Γ v ) ∩ im( ψ ) → is ( i − -tame with respect to Γ v , hence ( i − -tame with respect to Γ by Corollary3.16—we use that im( ψ ) = ker( e − e ′ ) by Proposition 2.3. Lemmas 3.4 and 3.17now show that the first and third entries are i -tame with respect to Γ , and C R ( Γ , i ) is closed under extensions.It now suffices to show that im( ι ) /T i ( Γ ) ∈ C R ( Γ , i ) . Write Q = H i ( B ( Γ v )) /T i ( Γ v ) ,and consider the commuting diagram T i ( Γ v ) / ( e − e ′ ) H i ( B ( Γ v )) / ( e − e ′ ) Q/ ( e − e ′ ) 00 T i ( Γ ) im( ι ) Q/ ( e − e ′ ) 0 , ⊆ where the vertical arrows are induced by ι . By Proposition 2.3, the middle arrowis an isomorphism, so the dashed arrow is determined. The top sequence is rightexact, since it is obtained by tensoring a short exact sequence down to R [ E ( Γ )] .Since the lefthand vertical arrow is surjective, it follows that the bottom sequenceis right exact, hence short exact. Since Q/ ( e − e ′ ) ∈ C R ( Γ , i ) by induction andLemma 3.15, the claim follows. (cid:3) Tameness and tori
Proof of Lemma 3.17.
The main input is the following calculation.
Proposition 4.1.
For any connected graph Γ and well-separating proper subset W of essential vertices, the kernel of the map T W ( Γ ) → E ∞| W | , is ( | W | + 1) -tame withrespect to Γ . If R is a field, then this kernel is also (∆ W Γ − -small. This result also has the following useful consequence.
Corollary 4.2.
The kernel and cokernel of the natural map M W T W ( Γ ) → T i ( Γ ) re ( i + 1) -tame and i -tame, respectively, where W ranges over well-separating setsof essential vertices of cardinality i . If R is a field and ∆ i Γ > , then both kerneland cokernel are (∆ i Γ − -small.Proof. If ∆ i Γ = 1 , then i = 1 , and the source of the map in question vanishes. Thus,the claim regarding the kernel is vacuous in this case, and the claim regarding thecokernel follows from Lemma 3.11, since every star class in Γ is non-rigid.Assume that ∆ i Γ > , so that a well-separating set W exists. The kernel inquestion is a submodule of the kernel of the composite M W T W ( Γ ) → T i ( Γ ) → M W E ∞| W | , ( W ) , which is ( i + 1) -tame and (∆ i Γ − -small by Proposition 4.1. The claim regardingthe cokernel follows from Lemma 3.11, since the cokernel is generated by the imagesof tori supported at non-well-separating vertex sets, which are in particular non-rigid. (cid:3) Before turning to the proof of Proposition 4.1, we make a few first reductionsand establish notation.At each vertex w ∈ W , choose two half-edges in distinct components of Γ W .We may make this choice so that the edges associated to each pair of half-edgesare not both tails. If | W | > , this claim follows from the assumption that Γ isconnected, while assuming otherwise in the case | W | = 1 leads to the conclusionthat Γ has only one essential vertex, in which case H ( B ( Γ )) = E ∞ , , and theconclusion of Proposition 4.1 is vacuous. Let A W ⊆ T W ( Γ ) denote the set of W -tori corresponding to these choices, as exhibited in Construction 2.19. Lemma 4.3.
For any connected graph Γ and well-separating set W of essentialvertices, T W ( Γ ) /A W ( Γ ) is p -tame with respect to Γ , where p = ( | W | + 1 W proper | W | otherwise . Regardless, if R is a field, then T W ( Γ ) /A W ( Γ ) is (∆ W Γ − -small. There is reason to find this result surprising. For example, if Γ is a tree and W contains an essential vertex of high valence, then there are many rigid W -tori notin A , and the naive expectation is that the images of such tori in T W ( Γ ) /A W ( Γ ) should exhibit the fastest possible growth. As the proof will show, it is the relationof Corollary 2.11 that dampens this growth. Remark . Given the smallness estimate of Lemma 4.3, it is natural to wonderwhether T W ( Γ ) /A W ( Γ ) is always | W | -tame. We do not know the answer to thisquestion. Proof of Proposition 4.1.
By Lemma 2.20 and Corollary 2.21, the composite map A W ( Γ ) ⊆ T W ( Γ ) → E ∞| W | , is injective, so the kernel in question is isomorphic to a submodule of T W ( Γ ) /A W ( Γ ) ,and the claim follows from Lemma 4.3. (cid:3) Order the half-edges at each w ∈ W subject to the requirement that, for each w ,the first two half-edges in the ordering are the two privileged in the construction f A W ( Γ ) . Given tuples I = ( i w ) w ∈ W , J = ( j w ) w ∈ W , and K = ( k w ) w ∈ W suchthat ≤ i w , j w , k w ≤ d ( w ) and i w < j w < k w , we have the W -torus α IJK = N w ∈ W α i w j w k w , and every W -torus is uniquely of the form ± α IJK for some suchchoice.There are two important observations to be made about these indices. First,Lemma 2.9 implies that every W -torus is a linear combination of W -tori α IJK satisfying i w ≡ . Second, if i w ≡ , then α IJK ∈ A W if and only if j w ≡ . Weintroduce a filtration A W ( Γ ) ⊆ M ⊆ M ⊆ · · · ⊆ M | W | ⊆ T W ( Γ ) by declaring M r to be generated over R [ E ] by those W -tori α IJK such that i w ≡ and { w : j w = 2 } ≤ r . Rephrasing the observations above, we have M | W | = T W ( Γ ) and M = A W ( Γ ) . Lemma 4.5.
The quotient M r /M r − is p -tame for every ≤ r ≤ | W | , where p isas in Lemma 4.3. If R is a field, the quotient is (∆ W Γ − -small.Proof. Since this quotient is generated by the images of those α IJK such that i w ≡ and { w : j w = 2 } = r , it suffices to show that R [ E ] · [ α IJK ] ∈ C R ( Γ , p ) for such I , J , and K , where [ α IJK ] = α IJK + M r − . Suppose that j w = 2 , anddefine J ′ and K ′ by j ′ w = ( w = w j w w = w and k ′ w = ( j w w = w k w w = w . Corollary 2.11 implies that ( e − e ) α IJK = ( e j w − e ) α IJ ′ K − ( e k w − e ) α IJ ′ K ′ ∈ M r − , where each edge shown is incident on w with associated half-edge given by itssubscript. Thus, we have the extra relation ( e − e )[ α IJK ] = 0 in the quotient.By assumption, e and e lie in distinct blocks of π ( Γ W ) , and we obtain a newpartition P < π ( Γ W ) by identifying these blocks. In the diagram of surjectionsamong R [ E ] -modules R [ E ] h α IJK i R [ E ] · α IJK R [ E ] · [ α IJK ] R [ π ( Γ W )] h α IJK i R [ P ] h α IJK i . the inner dashed filler is supplied by Observation 2.14, and the outer dashed filleris supplied by the extra relation. The smallness claim follows, and it now sufficesto show that R [ P ] is p -tame.Without loss of generality, e is not a tail, and its other endpoint u = w isessential. If u ∈ W , then it follows easily that P is | W | -tame, hence p -tame. If u / ∈ W , then π ( Γ W ) is ( | W | + 1) -tame by Lemma 3.10, so Lemma 3.9 implies theclaim. (cid:3) roof of Lemma 4.3. The module in question is M | W | /M , which admits the finitefiltration M /M ⊆ · · · ⊆ M | W | /M with p -tame and (∆ W Γ − -small associatedgraded by Lemma 4.5. (cid:3) Lemma 4.6.
Let W be a proper set of essential vertices and α a W -torus in Γ v .If e and e ′ lie in the same component of ( Γ v ) W , then R [ E ( Γ v )] · α is ( | W | + 1) -tamewith respect to Γ .Proof. The assumption implies that α = ψ (˜ α ) , where ˜ α is the external product of α with the loop formed by e , e ′ , and any path joining e and e ′ in ( Γ v ) W . Therefore,we have the isomorphism R [ E ( Γ v )] · α ∼ = R [ E ( Γ )] · ˜ α im( ι ) ∩ R [ E ( Γ )] · ˜ α . Now, e is incident on the essential vertex w , so the Q-relation gives the equation ( e − e )˜ α = α ′ , where e is an edge internal to the loop, e = e is an edge incidenton w , and α ′ is the external product of α and a star class at w . Since α ′ ∈ im( ι ) , wemay apply this observation repeatedly to conclude that the quotient shown aboveadmits a surjection from R [ π ( Γ W )] , and the claim follows from Lemma 3.10. (cid:3) Proof of Lemma 3.17.
We claim first that T W ( Γ v ) ∩ ker( e − e ′ ) ∈ C ( Γ , i ) , where W is a well-separating subset of i − essential vertices. If e and e ′ lie in the samecomponent of ( Γ v ) W , the claim follows from Lemma 4.6, so assume otherwise. Inthis case, the intersection in question is a submodule of the kernel of the map T W ( Γ v ) → F Wi − /F Wi − , since the target is ( e − e ′ ) -torsion-free. The claim in thiscase now follows from Proposition 4.1, which asserts that this kernel is i -tame withrespect to Γ v , hence also with respect to Γ by Lemma 3.15.Now, summing over well-separating subsets of cardinality i − and writing Tor for ( e − e ′ ) -torsion submodules, we have the exact sequences Tor(ker( f )) → M W Tor( T W ( Γ v )) → Tor M W T W ( Γ v ) ! / ker( f ) ! → ker( f ) / ( e − e ′ )Tor M W T W ( Γ v ) ! / ker( f ) ! → Tor( T i − ( Γ v )) → Tor(coker( f )) , where f is the map of Corollary 4.2. It follows from that result that the first andfourth entries of the first sequence are i -tame, and we have already shown that thesecond is so. Therefore, the first entry of the second sequence is i -tame. Since thethird is so by Corollary 4.2, the conclusion follows. (cid:3) Counting tori.
The goal of this section is to calculate the asymptotic dimen-sion of the module T i ( Γ ) of tori. Theorem 4.7.
Fix a field F and i ≥ . If Γ is a connected graph with an essentialvertex and ∆ i Γ > , then dim T i ( Γ ) ∼ X W i Γ − Y w ∈ W ( d ( w ) − k ∆ i Γ − , where W ranges over sets of vertices of cardinality i such that ∆ W Γ = ∆ i Γ . heorem 1.1 now follows by combining this result with Theorem 3.13, Proposi-tion 3.8, and Theorem 2.18.The key ingredient in the proof of Theorem 4.7 is the following local version,which is interesting in its own right. Proposition 4.8.
For any well-separating set W of essential vertices, dim T W ( Γ ) ∼ W Γ − Y w ∈ W ( d ( w ) − k ∆ W Γ − . Proof.
We have dim A W ( Γ ) ∼ dim T W ( Γ ) by Observation 2.14 and Lemma 4.3. Onthe other hand, Lemma 2.20 shows that A W ( Γ ) is freely generated over F [ π ( Γ W )] by the set A W . Therefore, dim T W ( Γ ) ∼ dim A W ( Γ )= (cid:18) k + ∆ W Γ − W Γ − (cid:19) | A W | = ( k + ∆ W Γ − · · · ( k + 1)(∆ W Γ − Y w ∈ W ( d ( w ) − ∼ W Γ − Y w ∈ W ( d ( w ) − k ∆ W Γ − , as claimed. (cid:3) Proof of Theorem 4.7.
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