Deletion and contraction in configuration spaces of graphs
aa r X i v : . [ m a t h . A T ] M a y DELETION AND CONTRACTION IN CONFIGURATIONSPACES OF GRAPHS
SANJANA AGARWAL, MAYA BANKS, NIR GADISH , DANE MIYATA Abstract.
The aim of this article is to provide space level maps betweenconfiguration spaces of graphs that are predicted by algebraic manipulationsof cellular chains. More explicitly, we consider edge contraction and half-edgedeletion, and identify the homotopy cofibers in terms of configuration spaces ofsimpler graphs. The construction’s main benefit lies in making the operationsfunctorial - in particular, graph minors give rise to compatible maps at thelevel of fundamental groups as well as generalized (co)homology theories.As applications we provide a long exact sequence for half-edge deletion inany generalized cohomology theory, compatible with cohomology operationssuch as the Steenrod and Adams operations, allowing for inductive calcula-tions in this general context. We also show that the generalized homology ofunordered configuration spaces is finitely generated as a representation of theopposite graph minor category. Introduction
For every graph Γ (a finite 1-dimensional CW complex) denote the configurationspace of n distinct points on Γ by Conf n (Γ) := { ( x , . . . , x n ) ∈ Γ n | ∀ i < j x i = x j } . The symmetric group S n acts on this space by permuting the labels, and the quo-tient by this action is the unordered configuration space U Conf n (Γ) .This paper provides space level maps between such configuration spaces corre-sponding to deletion and contraction of edges in the graph. Such constructionshave been predicted by algebraic manipulations of cellular chains and applied infinite generation proofs. The immediate implication of our constructions is thatvarious deletion and contraction operations on graphs induce well-defined maps onthe fundamental groups of the configuration spaces as well as on any generalized(co)homology theory with its cohomology operations such as Steenrod and Adamsoperations.Specifically, we discuss the following two constructions: • It has been observed ([ADCK17, Lemma C.7]) that edge contraction on thegraph induces well defined maps on a chain model of the graph’s configura-tion spaces. This chain contraction was implicitly known to lift to a uniquehomotopy class of space level maps, as explained in [MPR20, Remark 1.10].We construct a (zigzag of) space level maps exhibiting this homotopy class.
Mathematics Subject Classification.
Primary 55R80; Secondary 05C10, 20F36.
Key words and phrases. graph configuration space, edge deletion and contraction. N.G. is supported by NSF Grant No. DMS-1902762. , DANE MIYATA In particular, this allows us to describe the homotopy cofiber of thecontraction map as a certain configuration space with constraints. • We identify the homotopy cofiber of the inclusion of a subgraph with somehalf-edges deleted with (suspensions of) the configuration space of a simplergraph. This gives an inductive tool for computing topological invariants viathe long exact sequence of a pair, with all three terms being configurationspaces of graphs.
Remark 1.0.1 (Functoriality) . All of our constructions will be obviously functorialin the data of a graph Γ along with some additional input such as a choice of vertex,a set of half-edges, or a topological sub-tree. We will not belabor the point of thisfunctoriality throughout the constructions.Let us discuss a number of applications of our constructions.1.1. Application 1: LES in K-theory and generalized cohomology.
Usingthe characterization of homotopy cofiber for half-edge deletion in §4 one gets thefollowing inductive machinery for computing generalized homology and cohomologyof configuration spaces. This sequence is already new for ordinary homology whendeleting only one half-edge.
Theorem 1.1.1.
Let Γ be a graph and fix a vertex v ∈ V (Γ) along with a set of halfedges H = { h , . . . , h r } incident on v . For every generalized cohomology theory E one has a long exact sequence, compatible with cohomology operations and naturalwith respect to graph embeddings and automorphisms . . . → E i ( U Conf n (Γ \ ∪ H )) ⊕ ( add ∗ v − add ∗ h ) −→ E i ( U Conf n − (Γ \ { v } )) ⊕| H | d −→−→ E i +1 ( U Conf n (Γ)) ι ∗ H −→ E i +1 ( U Conf n (Γ \ ∪ H )) → . . . This LES is a reflection of the Puppe cofiber sequence of the deletion given in §5.In particular a similar LES exists for E -homology, as well as the S n -equivariantversions for ordered configuration space. Remark 1.1.2.
The above LES in the special case in which E is ordinary homologyand with H the set of all half edges incident on v has been a central tool in the workof An–Drummond-Cole–Knudsen [ADCK20], where it was discovered and used asa chain-level algebraic manipulation.One of the advantages of our version above is that one can elect to remove onehalf-edge at a time, thereby always considering a single configuration space in everyterm.Another interesting special case is the LES in K-theory, respecting Adams op-erations, with all three terms being configuration spaces of graphs – a result thatcould be of interest in quantum physics in light of [Mac19].1.2. Application 2: contraction maps on graph braid groups.
As mentionedin [MPR20, Remark 1.10], the edge contraction maps between configuration spacesare associated with well-defined homomorphisms between the respective fundamen-tal groups – the so called graph braid groups . Our explicit space level constructionof these maps provides a way to study the induced maps on π directly. Problem 1.2.1.
Describe the edge contraction homomorphism between graph braidgroups, e.g. with respect to the Farley-Sabalka presentation of tree braid groups [FS05] . ELETION AND CONTRACTION IN CONFIGURATION SPACES OF GRAPHS 3
For our next couple of items we observe that the space level lift of edge contrac-tion turns the assignments Γ U Conf n (Γ) and Conf n (Γ) into functors from the Miyata-Ramos-Proudfoot opposite graph-minor category[MPR20] to the homotopy category of spaces. In particular, the application of anyhomotopy invariant functor, such as π and generalized homology theories givesrepresentations of this category.We consider fundamental groups first. The action of graph minors on Γ π ( U Conf n (Γ)) allows us to make the following conjecture: Let γ i π ( − ) by theterms in the lower central series of π ( U Conf n ( − )) , considered as representationsof the opposite graph-minor category. Conjecture 1.2.2 ( Finite generation of LCS quotients ) . Every successivequotient of the LCS, γ i π ( − ) /γ i +1 π ( − ) , forms a finitely generated representation ofthe opposite graph-minor category. Remark 1.2.3.
The above conjecture is known to hold in the case i = 1 , equiv-alently for H ( U Conf n ( − )) (see [MPR20]). Then a possible path to proving theconjecture would be to show that the Lie ring ⊕ ∞ i =1 γ i π/γ i +1 π is generated by itsdegree elements.1.3. Application 3: finite generation for generalized homology theories.
As above, consider
U Conf n ( − ) as a functor from the Miyata-Ramos-Proudfootopposite graph-minor category [MPR20] to the homotopy category of spaces. Thenthe application of a generalized homology theory gives a linear representation ofthis category. For these representations we prove, Theorem 1.3.1.
Let E be any connective multiplicative generalized homology the-ory such that its coefficient ring E ∗ is Noetherian. Then for every i ∈ Z , thefunctor Γ E i ( U Conf n (Γ)) is a finitely generated representation of the opposite graph-minor category.Explicitly, this implies that for every fixed n and i there exist finitely manygraphs Γ , . . . , Γ r and E -homology classes α j ∈ E i ( U Conf n (Γ j )) whose imagesunder deletion and contraction of graphs span E i ( U Conf n (Γ)) for every graph Γ . Remark 1.3.2 ( Generalized cohomology theories ) . The theory of graph-anyons in [Mac19] expresses the interest of quantum physicists in vector bundlesover
U Conf n ( G ) , and thus in the K -group K ( U Conf n ( G )) . Following this, weask whether the above theorem can be extended in some way to multiplicative coho-mology theories with Noetherian coefficient ring E ∗ . That is, whether the functors Γ E i ( U Conf n (Γ)) are in some sense finitely generated representations of the graph-minor category.An approach to finite generation of cohomology is to consider its linear duals.Explicitly, one could try and apply the Noetherian property of the opposite graph-minor category to prove finite generation for the functors Hom E ∗ ( E i ( U Conf n ( − )) , E ) or related constructions. This idea appears e.g. in [KM18], where Kupers and Millerprove that duals of homotopy groups of configuration spaces are finitely generatedFI-modules. Thus we propose, SANJANA AGARWAL, MAYA BANKS, NIR GADISH , DANE MIYATA Conjecture 1.3.3.
For every fixed n , the dual of the Grothendieck group of vectorbundles over configuration spaces of graphs Γ Hom Z ( K ( U Conf n (Γ)) , Z ) is generated by finitely many functions on vector bundles under deletion and con-traction. Acknowledgements.
We are deeply grateful to AIM for facilitating the work-shop on Configuration Spaces of Graphs, Feb 2020, at which this project emerged.We also thank John Wiltshire-Gordon for suggesting this problem, and Safia Chet-tih, John Wiltshire-Gordon, and Ben Knudsen for helping us develop the ideaspresented here. Special thanks to Gabriel Drummond-Cole for being a key part ofthis project throughout the workshop and for providing various important sugges-tions and advice. 2.
Key Lemma
In this section we present and prove a lemma that will be central to the geometricconstructions of this paper. Let U ⊆ Γ be an open set and denote Conf n,k (Γ , U ) for the subspace of configurations of n points in Γ (either ordered or unordered) nomore than k of which lie in U . Lemma 2.0.1.
Fix a vertex v ∈ Γ and let U be the ball of radius / around v .Then the inclusion Conf n, (Γ , U ) ֒ → Conf n (Γ) is a homotopy equivalence.Proof. For r > let f r : [0 , → [0 , be the monotonic homeomorphism x x r .This defines a homeomorphism h r of the graph Γ treated as a CW-complex, so thatevery edge comes equipped with an identification with [0 , as follows: map everyedge to itself via the identity map, except for edges incident on v – orient theseedges so that v is identified with ∈ [0 , and map them to themselves via f r .As the parameter r varies, the functions h r assemble to a continuous isotopy h : (0 , × Γ → Γ , and therefore we have an induced isotopy H : (0 , × Conf n (Γ) → Conf n (Γ) on the configuration spaces (either ordered or unordered), where at map at time r is denoted by H r .Now let d : Conf n (Γ) → [0 , ∞ ] be distance of the 2nd closest point to v . To beclear, if there are two equidistant points closest to v then d will take on this minimaldistance, and if there is no more than one point in the connected component of v then d will take the value ∞ . Clearly this is a continuous function, similarly tohow min( x, y ) is continuous on R .With these at hand, consider the continuous map H d ∧ : Conf n (Γ) → Conf n (Γ) , ¯ x H d (¯ x ) ∧ (¯ x ) . First, observe that this map is well defined: d > , as there can not be two distinctpoints at distance to v . The effect of this map on configurations is to push pointsaway from v : if a point x i ∈ Γ of the configuration is at distance < s ≤ from v ,then it is mapped to the point on the same edge but at distance s d ∧ ≥ s from v .After application of the above map, the second closest point to v will be atdistance d (¯ x ) d (¯ x ) ∧ . But this function always takes value greater than / (recall ELETION AND CONTRACTION IN CONFIGURATION SPACES OF GRAPHS 5 that the minimum of x x is e − /e ≥ . ). Thus there is at most one point at distance ≤ / to v .Lastly, the isotopy H (1 − t )+ t ( d ∧ connects the identity on Conf n (Γ) at H withthe above map that pushes all but the closest point away from v . (cid:3) Contraction
We wish to realize geometrically the An–Drummond-Cole–Knudsen homologicaledge contraction map from [ADCK17, Appendix C]. For this purpose we use thefollowing,
Lemma 3.0.1. If U i ⊆ Γ i are open sets such that Γ \ U = Γ \ U and ¯ U ≃ ¯ U are homotopy equivalent relative to ∂ ¯ U = ∂ ¯ U , then Conf n, (Γ , U ) ≃ Conf n, (Γ , U ) . Proof.
For any continuous function f : ¯ U i → ¯ U j that fixes the boundary, one getsa map on configuration spaces M ( f ) : Conf n, (Γ i , U i ) → Conf n, (Γ j , U j ) by therule that every point not in U i is mapped to itself, and the (at most one) point in U i is mapped to U j via f . Note that this definition patches to a continuous map.Now for { i, j } = { , } let f ij : ¯ U i → ¯ U j be maps such that f ji ◦ f ij is homotopicrel ∂ to Id U i , say via the homotopy h ( i ) t . Then the induced maps on configurations M ( f ij ) are homotopy equivalences, where the homotopies are given by M ( h ( i ) t ) . (cid:3) Corollary 3.0.2.
The “contraction of a subtree" map on homology is realized geo-metrically as follows.Let T ⊆ Γ be a tree and let U ⊇ T be an ǫ -neighborhood with ǫ ≤ / . Then thecontraction is realized by the span Conf n (Γ /T ) ˜ ← Conf n, (Γ /T, U/T ) ˜ → Conf n, (Γ , U ) ֒ → Conf n (Γ) Now, the cone of the contraction can be described explicitly: it is the space ofconfigurations that have at least two points in U , where all other configurations arecollapsed to a point. 4. Deletion
Let
Γ = (
V, E ) be a graph. A half-edge h in Γ is formally an incident pair ( v, e ) ∈ V × E . Denote e ( h ) := e and v ( h ) := v . Geometrically, consider the half-edge h to be the subspace of the edge e ( h ) identified with the interval (0 , / ⊂ [0 , ,parametrized so that corresponds to the vertex v ( h ) . We will abuse the notationand freely treat h as an open set in the topological space Γ . Theorem 4.0.1.
Let H = { h , . . . , h k } be a subset of the half-edges incident to v . Then the cone of the inclusion ι H : Conf n (Γ \ ∪ H ) ֒ → Conf n (Γ) is homotopyequivalent to the reduced suspensions _ h ∈ H ˜Σ Conf n − (Γ \ { v } ) + . In the ordered case the cone is equivalent to the induction of this wedge with its S n − -action to S n . That is, S n ∧ S n − W h ∈ H ˜Σ( Conf n − (Γ \ { v } ) + . SANJANA AGARWAL, MAYA BANKS, NIR GADISH , DANE MIYATA Proof.
We start with the unordered case. Let U be the ball of radius / around v . Lemma 2.0.1 gives a homotopy equivalence of pairs ( Conf n, (Γ , U ) , Conf n, (Γ \ ∪ H, U \ ∪ H )) ≃ ( Conf n (Γ) , Conf n (Γ \ ∪ H )) and the former is a cofibration. Thus, since cones are homotopy invariant, it issufficient to construct a homeomorphism Conf n, (Γ , U ) /Conf n, (Γ \ ∪ H, U \ ∪ H ) → _ h ∈ H ˜Σ Conf n − (Γ \ { v } ) + . Suppose a configuration i
Conf n, (Γ , U ) has a point x on h ∈ H . Then map itto the suspension labelled by h , where the configuration is obtained by forgetting x and setting the suspension parameter equal to d ( x, v ) . If no such x exists, mapthe configuration to the basepoint ∗ . Note that this function is well-defined, asthere can be at most one point on our set of half-edges. Note also that if x = v then the suspension parameter is set to which lands on the basepoint. Lastly, as x leaves the half-edge, the cone parameter goes to and its image under our mapwill approach the basepoint.Since configurations in Γ \ ∪ H are all sent to the basepoint, the map abovefactors through the quotient by Conf n, (Γ \ ∪ H, U \ ∪ H ) .The inverse map is defined by sending a point (¯ x, t ) ∈ ˜Σ Conf n − (Γ \ { v } ) + onthe wedge-summand labeled by h to the configuration that has an additional pointon h at distance t/ to v . Of course, the basepoint has to map to the basepoint.These maps clearly patch to a continuous map, and are inverses to the above mapsthat would forget the newly added point.To adapt the above argument for the ordered case, index the wedge sum by [ n ] × H , accounting for the label of the point x i ∈ h . The rest of the construction works injust the same way. Clearly this construction is compatible with the S n -action, andthe stabilizer of n acts by the ordinary permutation action on Conf n − (Γ \{ v } ) . (cid:3) Lastly, we wish to describe the "boundary map"
Cone ( ι H ) → ˜Σ Conf n (Γ \ ∪ H ) + obtained by crushing the base of the cone ∼ = Conf n (Γ) to a point. Proposition 4.0.2.
Under the identification
Cone ( ι H ) ≃ _ h ∈ H ˜Σ Conf n − (Γ \ { v } ) + the boundary map to ˜Σ Conf n (Γ \ ∪ H ) + on the wedge summand labeled by h hasthe form ˜Σ add v − ˜Σ add h where the map add v adds the vertex v to a configuration, while add h adds a newclosest point to v on the edge e ( h ) .Proof. An explicit homotopy equivalence from the collapse
Conf n, (Γ , U ) /Conf n, (Γ \ ∪ H, U \ ∪ H ) to the mapping cone on the inclusion can be constructed by the following recipe. Fora configuration with a point x i on a half-edge h ∈ H , say at distance d = d ( x i , v ) , • when d ≤ / , i.e. x i is on the third of h nearest to v , move x i to v whilefixing all other points and set the cone parameter to d ; ELETION AND CONTRACTION IN CONFIGURATION SPACES OF GRAPHS 7 • when / − d ≤ / , i.e. x i is on the third of h farthest away from v , scalethe edge e ( h ) down away from v , moving all points on it until x i appearsat distance / and set the cone parameter to / − d ) ; • lastly, when x i is in the middle third of h , i.e. / ≤ d ≤ / , scale thismiddle third up to encompass the whole of h . This amounts to moving x i to be at distance d − /
4) + 1 / to v . All other points remain fixed.These three maps patch together to give the desired homotopy equivalence (e.g. ahomotopy H t can be constructed by replacing the terms / in the above construc-tion with t/ and adjusting the formulas accordingly).Now, the boundary map is constructed by collapsing the base of the mappingcone to a point. This is the set of points in Conf n, (Γ , U ) , which in the aboveconstruction is the range of configurations with a point in the middle third ofsome h ∈ H . But under the homeomorphism with W h ∈ H ˜Σ Conf n − (Γ \ { v } ) + ,such configurations correspond to the middle thirds in every suspension. Aftercollapsing the base of the cone to a point, the boundary map factors through mapsof the form Σ X → Σ bottom X ∨ Σ top X where one collapses the middle third to apoint. Now observe that on the bottom third of the suspensions, the boundarymap is precisely Σ add v , while on the top third we find maps that add a new closestpoint to v on the edge e ( h ) but with the suspension parameter going backwards(this is the − d term appearing in the cone parameter). Thus this map is homotopicto − Σ add h .The same arguments apply in the case of ordered configurations. (cid:3) Applications: Generalized homology theories
By the previous section, the Puppe cofiber sequence for the inclusion of half-edgedeletion takes the form
U Conf n (Γ \ ∪ H ) ֒ → U Conf n (Γ) → _ h ∈ H ˜Σ U Conf n − (Γ \ { v } ) + →→ ˜Σ U Conf n (Γ \ ∪ H ) + ֒ → ˜Σ U Conf n (Γ) + → . . . where the connecting map between the rows is given on the wedge-summand withlabel h ∈ H by ˜Σ add v − ˜Σ add h . Also, as mentioned in the previous section, asimilar sequence exists in the S n -equivariant context for the ordered configurationsspaces, but with an appropriate induction on configurations of n − points.Applying any generalized homology or cohomology theory to this Puppe sequencenow yields the long exact sequence claimed in Theorem 1.1.1.We next turn to the proof of Theorem 1.3.1, regarding the generalized homologyof unordered configuration spaces as a representation of the Miyata-Proudfoot-Ramos opposite graph-minor category defined in [MPR20]. Proof of Theorem 1.3.1.
The basic inputs to the proof are the Noetherianity of thecoefficient ring E ∗ , the fact that the singular chains C i ( U Conf n ( − )) are equivalentto a finitely generated representation of the opposite graph minor category, and theAtiyah-Hirzebruch spectral sequence – the AHSS for short.Atiyah-Hirzebruch provide a spectral sequence for every space X with first page E p,q = C p ( X ; E q ) := C p ( X ) ⊗ E q = ⇒ E p + q ( X ) . More explicitly, E -page of the spectral sequence looks as follows- SANJANA AGARWAL, MAYA BANKS, NIR GADISH , DANE MIYATA C ( X ) ⊗ E C ( X ) ⊗ E C ( X ) ⊗ E C ( X ) ⊗ E C ( X ) ⊗ E C ( X ) ⊗ E C ( X ) ⊗ E C ( X ) ⊗ E C ( X ) ⊗ E C ( X ) ⊗ E C ( X ) ⊗ E C ( X ) ⊗ E · · ···· ··· [ADCK17] shows that the singular chains C ∗ ( U Conf n (Γ)) are quasi-isomorphicto a smaller chain complex, the reduced Świątkowski complex e S ∗ ,n (Γ) , for everygraph Γ . We may thus replace replace every occurrence of C ∗ ( X ) in the AHSSabove with these Świątkowski complexes.[ADCK17] further shows that e S ∗ ,n ( − ) is a representation of the opposite graph-minor category. But our geometric construction of edge contraction in Corollary3.0.2 shows that the quasi-isomorphism with singular chains is compatible witha homotopy action of the opposite graph-minor category. Furthermore [MPR20,Proof of 1.15] shows that e S i,n ( − ) is finitely generated as a representation of theopposite graph-minor category for every i ≥ .Note that all terms e S i,n ( − ) ⊗ E j are E -modules, and recall the fact that E ∗ being Noetherian implies E is a Noetherian ring and every E j is a finitely generated E -module. Thus all terms in the E -page above, with C ∗ replaced by e S ∗ ,n , formfinitely generated representations of the opposite graph minor category over thering E .Now, all terms in the later pages of the spectral sequence are subquotients ofthose appearing in the first page. But [MPR20, Theorem 1.2] shows that finitelygenerated representations of the opposite graph-minor category over a Noetherianring are again Noetherian, and thus finite generation passes to subquotients. Thesame can be said of the terms at the E ∞ -page, which gives us the graded factorsof a filtration on the functor E i ( U Conf n ( − )) . Thus the latter functor is filteredwith quotients finitely generated as representations of the opposite graph minorcategory. Hence, it is itself a finitely generated representation. (cid:3) References [ADCK17] Byung Hee An, Gabriel C Drummond-Cole, and Ben Knudsen. Subdivisional spacesand graph braid groups. arXiv preprint arXiv:1708.02351 , 2017.[ADCK20] Byung Hee An, Gabriel Drummond-Cole, and Ben Knudsen. Edge stabilization in thehomology of graph braid groups.
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