Cohomology of generalised configuration spaces of points on R r
aa r X i v : . [ m a t h . A T ] A p r Cohomology of generalised configuration spaces ofpoints on R r Marcel B¨okstedt, Erica Minuz ∗ April 20, 2020
Abstract.
We compute the cohomology ring of a generalised type ofconfiguration space of points in R r . This configuration space is indexed bya graph. In the case the graph is complete the result is known and it is dueto Arnold and Cohen. However, our computations give a generalisation toany graph and an alternative proof of the classical result. Moreover, we showthat there are deletion-contraction short exact sequences for this cohomologyrings. Contents R r (Γ) R r (Γ) . . . . . 42.2 Proof of Theorem
Conf r (Γ) R r (Γ). . . . . . . . . . . . . . . . . . . . . . . 12 The configuration space of n points in R r , that we denote as Conf r ( n ), isdefined asConf r ( n ) = { ( x , . . . , x n ) ∈ R rn ; x i = x j for i = j, < i, j ≤ n } . Its cohomology ring has been computed by Arnold [1] in the case r = 2 andCohen [3] for r ≥ H ∗ (Conf r ( n )) = Z [ e α i,j ] / ∼ ∗ The authors are based in Aarhus University, Science and Thechnology, Department ofMathematics. The work is supported by
Det frie forskningsr˚ad, natur og univers − A . < i, j ≤ n , Z [ e α i ] is the free commutative graded algebra generatedby e α i,j of degree r − ∼ are the relations • e α i,j = ( − r e α j,i • e α i,j = 0 if r is odd • e α a,b e α b,c + e α b,c e α c,a + e α c,a e α a,b = 0In this paper we will study a generalisation of the definition of the con-figuration spaces Conf r ( n ) to configuration spaces depending on a graph.These were defined by Eastwood and
Huggett [4], and also described in [2].Given a graph Γ, we denote the configuration space of points in R r depend-ing on a graph by Conf r (Γ). Let α i,j denote an edge in Γ between the vertices v i and v j , the generalised configuration space of points in R r is is defined asConf r (Γ) = { ( x , . . . , x n ) ∈ R rn ; x i = x j if α i,j is an edge in Γ } . The main result in the article is provided by the computation of thecohomology of Conf r (Γ) for any graph Γ. The cohomology of Conf r (Γ) isgiven by a commutative graded ring that depends on the parity of the integer r . Let Z [ e α i ] be the free commutative graded algebra generated by e α i,j ofdegree r −
1, where α i,j is an edge in Γ between the vertices i and j orientedfrom i to j . Let w be a circuit in Γ, that is graph consisting of an orderedsets of edges w , w , . . . , w k and vertices v ( w ) , . . . , v k ( w ) , v k +1 ( w ) = v ( w )such that v i ( w ) , v i +1 ( w ) are the two vertices incident to w i . We denote by e w = e v , · e v , . . . · e v s, the product of the generators corresponding to theedges in the cycle w . We prove that the cohomology ring is given by thefollowing quotient of graded rings H ∗ (Conf r (Γ)) = Z [ e α i,j ] / ∼ where ∼ are the relations • e α i,j = ( − r e α j,i • e α i,j = 0 if r is odd • A ( w ) = P i ( − ( r − i e v , · · · ˆ e v i,j · · · e v sj, = 0 for every circuit w in Γ.We call the relations A ( w ) generalised Arnold relations . Moreover, let Γ r a denote the graph obtained from Γ by deleting the edge α , and Γ /α thegraph obtained by contracting the edge α . There is a deletion-contractionshort exact sequence in cohomology0 → H ∗ (Conf r (Γ r e )) → H ∗ (Conf r (Γ)) → H ∗− r +1 (Conf r (Γ /e )) → . Our approach is as follows. In section 2 we define a graded commutativering depending on a graph Γ. The definition is purely algebraic. There are2wo cases, depending on whether the generators are in even or odd degrees.We establish a deletion-contraction exact sequence for these rings. Such longexact deletion-contraction sequences are well known in graph cohomology,but our main algebraic result is that these long exact sequences actuallybreak up into short exact sequences.In section 3 we show that the cohomology rings of the graph indexedconfiguration space of points in an open discs are given by the rings definedin the previous section. The method is to first establish deletion-contractionlong exact sequences of cohomology of generalized configuration spaces, thento show that the algebraic exact sequences of the previous paragraph mapsto these short exact sequences by surjections, and conclude that these surjec-tions are actually isomorphisms by induction over the number of edges in thegraphs. This method also gives an alternative approach to the computationof the cohomology of not generalized configuration spaces.The results here presented are part of the second author’s Ph.D. thesis
Graph complexes and cohomology of configuration spaces , supervised by thefirst author. R r (Γ) In this section we will describe a graded commutative ring R r (Γ). If r iseven, Γ can be any not oriented graph. If r is odd, we demand that eachedge of Γ come with an orientation, that is an ordering of its two adjacentvertices. In both cases we will assume that Γ does not have loops, but wedo allow multiple edges.We introduce some notation. A circuit in a graph consists of ordered setsof edges w , w , . . . , w k and vertices v ( w ) , . . . , v k ( w ) , v k +1 ( w ) = v ( w ) suchthat v i ( w ) , v i +1 ( w ) are the two vertices incident to w i . In case r is odd, thecircuit w comes with an additional signs ǫ i ( w ). The given orientation of theedge w i determines an order of the pair of vertices v i ( w ) , v i +1 ( w ), If in thisorder v i < v i +1 , then ǫ i ( w ) = 1, else ǫ i ( w ) = − w of length l ( w ), we denote by e w = w · w . . . · w l ( w ) theproduct of the generators corresponding to the edges in the circuit w . If onechanges the order of w by a cyclic permutation to obtain a new circuit w ′ ,one changes e w by at most by a sign: e w = ± e w ′ .For r and Γ we will define a graded commutative algebra Λ r (Γ), and anideal I r (Γ) in this algebra. The precise definitions depend on whether r iseven or odd. Definition 2.1. If r is even, the algebra Λ r (Γ) is the free graded commuta-tive algebra over the integers with one generator e α in degree r − α ∈ E (Γ). If r is odd, Λ r (Γ) is the quotient of the graded commutativealgebra over the integers with one generator e α for each edge α ∈ E (Γ) bythe relations e α = 0. In this case, Λ r [Γ] is actually commutative. For each3ircuit we define its Arnold class: A ( w ) = ( A ( w ) = P i ( − i w · · · ˆ w i · · · w l ( w ) if r is even, A ( w ) = P i ǫ i ( w ) w · · · ˆ w i · · · w l ( w ) if r is odd.In case w consists of a single edge, which then has to be loop, this will beinterpreted as A ( w ) = 1. Let the generalized Arnold ideal I r (Γ) be the idealof Λ r (Γ) generated by the Arnold classes. Finally, let R r (Γ) be the quotientring Λ r (Γ) /I r (Γ).A map of graphs f : Γ → Γ ′ induces a map f R : Λ r (Γ) → Λ r (Γ) whichpreserves the Arnold classes, so it also induces a map of rings f R : R r (Γ) → R r (Γ ′ ).We can usually assume that Γ has no multiple edges, since the followinglemma holds. Lemma 2.2.
Let Γ be a graph and e an edge of Γ such that there exists adifferent edge e ′ incident to the same vertices as e . Let Γ ′ = Γ r e . Then,the rings R r (Γ) and R r (Γ ′ ) are isomorphic.Proof. There is an inclusion of graphs i : Γ ′ → Γ, and a left inverse p to i , such that p ( e ) = e ′ . In R r (Γ) we have the generalised Arnold relation e ′ − e = 0, so that e ′ = e ∈ R r (Γ). It follows easily that the induced maps i ∗ : R r (Γ ′ ) → R r (Γ) and p ∗ : R r (Γ) → R r (Γ ′ ) are inverse isomorphisms. R r (Γ) Let Γ be a graph and α ∈ E (Γ). We will mainly be interested in graphswithout loops and multiple edges, but it is convenient not to exclude thesecases, in order to be able to formulate certain induction arguments in asmooth way. We can delete the edge α from the graph Γ to obtain thegraph i α : Γ r α ⊂ Γ. We can also contract the edge α to obtain a graphΓ /α . The graph Γ /α might have multiple edges, but it does not have loops.There is a map p α : Γ → Γ /α which identifies the two vertices incident to α . There are induced maps of graded algebras i Λ α : Λ r [Γ r α ] → Λ r (Γ) and p Λ α : Λ r [Γ] → Λ r [Γ /α ].If β ∈ E (Γ r α ) we alternatively denote its image i α ( β ) ∈ E (Γ) by β . If γ ∈ E (Γ), we alternatively denote its image in p α ( γ ) ∈ E (Γ /α ) by [ γ ].Let Λ[ e α ] be the exterior algebra on the generator e α of degree r − Definition 2.3.
We consider the following two ring homomorphisms: ι α : Λ r [Γ /α ] → Λ r (Γ /α ) ⊗ Λ r [ e α ] ψ α : Λ r (Γ) → Λ r (Γ /α ) ⊗ Λ r [ e α ] ι α ( e [ η ] ) = e [ η ] ⊗ ψ α ( e [ e η ] ) = ( e [ η ] ⊗ η = α ,1 ⊗ e [ α ] if η = α .4onsider the generalized Arnold classes in Λ r (Γ /α ) ⊗ Λ r [ e α ]. We definethe Arnold ideal in Λ r (Γ /α ) ⊗ Λ r [ e α ] to be the ideal generated by the gen-eralized Arnold classes in Λ r (Γ /α ) ⊂ Λ r (Γ /α ) ⊗ Λ r [ e α ]. If Γ ′ is the graphobtained from Γ /α by adding a single vertex and a single edge connectingthe new vertex to α , there is an obvious isomorphism preserving Arnoldideals between this ring and Λ(Γ ′ ). Lemma 2.4.
The following diagram of ring maps is commutative and themaps preserve the generalized Arnold ideals. Λ r (Γ r α ) Λ r (Γ /α )Λ r (Γ) Λ r (Γ /α ) ⊗ Λ r [ e α ] p Λ α ◦ i Λ α i Λ α ι Λ α ψ Λ α Proof.
The diagram commutes since if e η ∈ E [Γ r α ] is a generator bothpaths to the lower right square takes it to [ β ] ⊗ i Λ α and p Λ α takes an Arnoldclass to an Arnold class, since they are induced by maps of graphs. Themap ι Λ α obviuously preserves the Arnold elements, so we only need to checkthat ψ Λ α does. Let w be a circuit in Γ. We have to prove that ψ Λ α ( A ( w )) iscontained in the ideal generated by A ( u ) ⊗ u a circuit in Γ /α . Thereare three cases. If w i = α for all i , then u = p Λ α ( w ) is a circuit in Γ /α suchthat p Λ α ( A ( w )) = A ( u ), and we are done. If α occurs more than once in w ,then A ( w ) = 0, and we are done again. If finally e i = α for a unique i , then u = [ w ][ w ] . . . d [ w i ] . . . [ w l ( w ) ] is a circuit in Γ /α such that ψ Λ α ( A ( w )) = A ( u ),and all our work is done.It follows from lemma 2.4 that we get an induced commutative diagramof ring maps R r (Γ r α ) R r (Γ /α ) R r (Γ) R r (Γ /α ) ⊗ Λ r [ e α ] p Rα ◦ i Rα i Rα ι Rα ψ Λ α Theorem 2.5.
The above diagram is a pullback diagram. The map i Rα : R r (Γ r α ) → R r (Γ) is injective, and the map ψ Rα : R r (Γ) → R r (Γ /α ) ⊗ Λ r [ e α ] is surjective. We will prove this theorem in the nex subsection. As consequences wehave
Corollary 2.6.
Suppose that Γ ′ is a subgraph of Γ such that V (Γ ′ ) = V (Γ) .The map induced by inclusion i ∗ : R r (Γ ′ ) → R r (Γ) is injective. orollary 2.7. For every α ∈ E (Γ) there is a short exact sequence ofAbelian groups → R r (Γ r α ) k → R r (Γ) k → R r (Γ /α ) k − r +1 → where the indices k and k − r + 1 denote the grading in the ring. Theorem
Let Λ r [Γ /α ] ⊗ e α denote the ideal in Λ r [Γ /α ] ⊗ Λ r [ e α ] generated by e α . As anAbelian group, Λ r [Γ /α ] ⊗ Λ r [ e α ] is the direct sum of the image of the injectivemap ι Λ α and the ideal Λ r [Γ /α ] ⊗ e α . Let π : Λ r [Γ /α ] ⊗ Λ[ e α ] → Λ r [Γ /α ] ⊗ e α be the projection, and define g Λ α = π ◦ ψ Λ α : Λ[Γ] → Λ r [Γ /α ] ⊗ e α . Since ψ Λ α and π preserve the ideal generated by the Arnold classes, so does g Λ α . We obtain a restricted map g Iα : I (Γ) → I (Γ /α ) ⊗ e α and a quotient map g Rα : R r (Γ) → R r (Γ /α ) ⊗ e α . We also immediately obtain a commutativediagram: 0 0 00 I r (Γ r α ) Λ r [Γ r α ] R r (Γ r α ) 00 I r (Γ) Λ r [Γ] R r (Γ) 00 I r (Γ /α ) ⊗ e α Λ r [Γ /α ] ⊗ e α R r (Γ /α ) ⊗ e α
00 0 0 i Iα i Λ α i Rα g Iα g Λ α g Rα (1)The rows of this diagram are short exact by definition. Most of this subsec-tion will go into proving that the three columns are short exact. Lemma 2.8.
The middle column in diagram 1 is exact.Proof.
The inclusion i Λ α is clearly injective by its definition.An element in Λ[Γ] can be uniquely written as x + ye α where x, y areproducts of edges different from α , that is x, y are both in the image of i Iα .The image of i Iα are the classes for which y = 0. Since g Λ α ( x + ye α ) = π ( x ⊗ y ⊗ e α ) = y ⊗ e α , g Λ α also consists of the classes for which y = 0. This provesexactness at Λ r (Γ). Finally, the map p Λ α ◦ i Λ α : Λ r (Γ r α ) → Λ r (Γ /α ) isan isomorphism, so that for any class x ⊗ e α ∈ Λ(Γ /α ) ⊗ e α we can find¯ x ∈ Λ r (Γ r α ) such that p Λ α i Λ α ¯ x = x and g Λ α ( i Λ α (¯ x ) e α ) = π ( ψ Λ α ( i Λ α (¯ x ) e α )) = π ( p Λ α i Λ α ¯ x ⊗ e α ) = π ( x ⊗ e α ) = x ⊗ e α . It follows that g Λ α is surjective. Lemma 2.9.
The map g Iα in diagram 1 is surjective.Proof. The map g Iα is a map of Λ r (Γ r α )-modules since g Iα (( x )( x + y e α )) = g Iα ( x x + ( x y ) e α ) = x y ⊗ e α = x g Iα ( x + y α )This means that it is sufficient to prove that each element of a set of gen-erators for I (Γ /α ) ⊗ e α as Λ(Γ r α )-module is in the image of g Iα . Notethat the map x x ⊗ e α defines an isomorphism of Λ(Γ r α ) modules I (Γ /α ) → I (Γ /α ) ⊗ e α . It follows that if we define¯ A ( w ) = A ( w ) ⊗ e α for circuits w in Γ /α , then the classes ¯ A ( w ) form a set of generators for I (Γ /α ) ⊗ e α . We conclude that it suffices to show that for each circuit w inΓ /α , the class ¯ A ( w ) is in the image of g Iα .Let v , v ∈ V (Γ) = V (Γ r α ) be the vertices incident to α , and v ∈ V (Γ /α ) the vertex given be collapsing v and v . The vertex v might beincident to some of the edges [ w i ] ∈ E (Γ /α ). Since we are assuming that Γhas no multiple edges or loops, Γ /α also has no loops, although it might havedouble edges. We can decompose the circuit w as a composition of circuits w ( i ) , each starting and ending with the vertex v , such that w (1)1 , w (1)2 , · · · w (1) l ( w (1) ) , w (2)1 · · · w (2) l ( w (2) ) . . . w ( k ) l ( w ( k ) ) is a cyclic reordering of w . . . w l ( w ) . A cyclic reordering will at most flip thesign of A ( w ) so we get that A ( w ) = X i ± e w (1) e w (2) · · · e w ( i − A ( w ( i ) ) e w ( i +1) · · · e w ( k ) This reduces the lemma further to the case when at most two edges of w are incident to v .Let the circuit be { w , w , . . . w l ( w ) } . Since the map p ◦ i : E (Γ r α ) → E (Γ /α ) is a bijection, each edge w i ∈ E (Γ /α ) is the image of some unique w ′ i ∈ E (Γ r α ). If the edges w ′ , . . . w ′ l ( w ) form a circuit w ′ in Γ r α , then¯ A ( w ) = g Iα ( w ′ ), and we are finished here.7f the the edges w ′ , . . . , w ′ l ( w ) do not form a circuit, this is because thereis an i so that w i and w i +1 are adjacent to v and v (in either order). Since α is an edge incident to the vertices v , v , we can form the circuit w ′′ to becircuit w ′ , . . . w ′ i , α, w ′ i +1 . . . w ′ l ( w ) . Then g Iα ( w ′′ ) = ± ¯ A ( w ), and the proof iscomplete. Corollary 2.10.
The sequence R r (Γ r α ) i Rα −→ R r (Γ) g Rα −→ R r (Γ /α ) → is exact.Proof. The map g Rα is surjective since g Λ α is surjective. That the composite g Rα i Rα is trivial follows from a simple diagram chase, using that the compositein the middle column is trivial, and that the quotient map Λ[Γ r α ] → R r (Γ r α ) is surjective. The only thing left to check is that im( i Rα ) = ker( g Rα ).The columns of the diagram 1 are chain complexes, so that the diagramdefines a short exact sequence of chain complexes. By lemma 2.8, the homol-ogy of the middle column vanishes. Using the long exact sequence of a shortexact sequence of chain complexes, we see that the quotient ker( g Rα ) / im( i Rα )is isomorphic to the cokernel of the map g Iα . According to lemma 2.9, thiscokernel is trivial.In preparation for the proof of theorem 2.5, we need a lemma. Lemma 2.11.
Let α, β be two different edges of Γ . The following diagramis commutative R r (Γ r α ) R r (Γ r α/β ) ⊗ e β R r (Γ) R r (Γ /β ) ⊗ e βg Rβ i Rα i Rα ⊗ Id g Rβ (2) Proof.
In the formulation of the lemma we have tacitely and legitimatelyidentified the graph (Γ r α ) /β with the graph (Γ /β ) r α . We first note thecommutativity of the diagramΛ r (Γ r α ) Λ r (Γ r α/β ) ⊗ Λ r [ e β ]Λ r (Γ) Λ r (Γ /β ) ⊗ Λ r [ e β ] ψ Λ β i Rα i Λ α ⊗ Id ψ Λ β (3)Since ψ Λ β and i Λ α are ring maps, it suffices to check this on generators e η ,which is trivial to do. Applying the projection π , we obtain that the following8iagram is commutative:Λ r (Γ r α ) Λ r (Γ r α/β ) ⊗ e β Λ r (Γ) Λ r (Γ /β ) ⊗ e βg Λ β i Rα i Λ α ⊗ Id g Λ β (4)The statement of the lemma follows from that there is a surjective mapfrom diagram 4 to diagram 2,Note that if Γ does not have any loops, the ring map c : Λ[Γ] → Z , c (1) = 1 and c ( e α ) = 0 for all edges α in Γ factors over R r (Γ). This is notthe case if Γ has a loop α because the Arnold relation corresponding to thecircuit consisting of he single edge α is not mapped to 0 by c . It follows thatif Γ has no loops, the canonical map Z → R r (Γ) is a split inclusion, with leftinverse the map η that maps each e α to 0. We say that a graph Γ satisfies( ∗ ) if both of the following two statement are true. • If Γ has no loops, for each α ∈ E (Γ) the map i α : R r (Γ r α ) → R r (Γ)is injective. • If Γ does not have loops or multiple edges and if x ∈ R r (Γ) and x = Z ,there exists a β ∈ E (Γ) such that g Rβ ( x ) = 0. Lemma 2.12.
Every graph Γ satisfies ( ∗ ) .Proof. We will argue by induction on the number of edges of Γ. The graphwith one vertex and no edges satisfies ( ∗ ) for trivial reasons.The induction hypothesis is that every graph with at most n − ∗ ). Let Γ be a graph with n edges. We need to show that Γ satisfies( ∗ ).We first show that i Rα : R r (Γ r α ) → R r (Γ) is injective. Using lemma 2.2we easily reduce to the case that Γ has no multiple edges. The map i α preserves the direct sum decomposition R r (Γ) ∼ = Z ⊕ ker( η ), so it suffices toshow that if α ∈ E (Γ), x ∈ R r (Γ) \ Z , then i Rα ( x ) = 0.Since Γ r α has no multiple edges and satisfies ( ∗ ) by assumption, thereis an edge β ∈ E (Γ r α ) such that g Rβ ( x ) = 0 ∈ R r (Γ r α/β ). Since Γ /β hasno loops, it satisfies ( ∗ ), i Rα g Rβ ( x ) = 0. Now apply lemma 2.11 to prove that i Rα ( x ) = 0 ∈ R r (Γ) as required.We finally need to prove that if Γ has no multiple edges, x ∈ R r (Γ) \ Z and g Rη ( x ) = 0 for all η ∈ E (Γ), then x = 0. Pick any α ∈ E (Γ). Since g Rα ( x ) = 0, by lemma 2.10 there is an y ∈ R r (Γ r α ) such that i Rα ( y ) = x .Using lemma 2.11 again, we see that for any β ∈ E ( γ r α ):( i Rα ⊗ Id) g Rβ ( y ) = g Rβ i Rα ( y ) = g Rβ ( x ) = 09ecause Γ /β satisfies ( ∗ ), and because (Γ r α ) /β either equals Γ /β or (Γ /β ) r α the map i Rα ⊗ Id : R r (Γ r α/β ) → R r ( γ/β ) is injective, so that g Rβ ( y ) = 0.Because this is true for every β ∈ E (Γ r α ), and Γ r α also satisfies ( ∗ ), itfollows that y = 0 so that x = 0.We sum up in Theorem 2.13.
The columns of diagram 1 are short exact.Proof.
The middle column is exact by lemma 2.8. The right hand column isexact by lemma 2.10 and lemma 2.12. The exactness of the left hand columnfollows from this by the nine-lemma (or by simple diagram chase).
Proof of theorem 2.5.
The injectivity follows from lemma 2.12. The map ψ Rα is surjective and the map ι Rα is injective, so it suffices to show that if ψ Rα ( x ) ∈ im ι Rα , then x ∈ im( i Rα ). But ψ Rα ( x ) ∈ im ι Rα if and only if g Rα ( x ) = 0, so thetheorem follows from the exactness of the right column in diagram 1. Conf r (Γ) In this section we will prove that there is an isomorphism between the ring R r (Γ) defined in the previous section and the cohomology ring of Conf r (Γ).Moreover, Lemma → H ∗ (Conf r (Γ r e )) → H ∗ (Conf r (Γ)) → H ∗− r +1 (Conf r (Γ /e )) → . The first step is to describe the deletion-contraction long exact sequencethat occurs for configuration spaces.
We will prove the following theorem.
Theorem 3.1.
There is a long exact sequence in cohomology · · · −→ H ∗ (Conf r (Γ r e )) −→ H ∗ (Conf r (Γ)) −→ H ∗− r +1 (Conf r (Γ /e )) −→ H ∗ +1 (Conf r (Γ r e )) −→ · · · Before we turn to the proof, we make a few preliminary observations.Let e be an edge in Γ between the vertices a and b . The space Conf r (Γ)is an open subspace of Conf r (Γ r e ). The complement A e (Γ) = Conf r (Γ r e ) − Conf r (Γ)is a closed subspace in Conf r (Γ r e ) and A e (Γ) = { ( x , . . . , x n ) ∈ R rn ; x i = x j if α i,j ∈ E (Γ) r { e } while x a = x b } . A e (Γ) and Conf r (Γ /e ) send-ing ( x , . . . , x n ) to ( x , . . . , x a , . . . , b x b , . . . , x n ). Let m a,b ( x ) > | x a − x c | such that c = b , but c is connected byan edge to a . This number will be independent of x b . We define an openneighborhood V e (Γ) of A e (Γ) in Conf r (Γ r e ) in the following way V e (Γ) = { x = ( x , . . . , x n ) ∈ Conf r (Γ r e ); | x a − x b | < m a,b ( x ) } Lemma 3.2.
Conf r (Γ) ∩ V e is homotopy equivalent to S r − × Conf r (Γ /e ) .Proof. Conf r (Γ) ∩ V e is the space { ( x , . . . , x n ) ∈ Conf r (Γ) : 0 < | x a − x b | < m a,b ( x ) } We define the maps f : Conf r (Γ) ∩ V e → S r − × Conf r (Γ /e )by f (( x , . . . , x n )) = (cid:18) x a − x b | x a − x b | , ( x , . . . , x a , . . . , b x b , . . . , x n ) (cid:19) and g : S r − × Conf r (Γ /e ) → Conf r (Γ) ∩ V e by g ( y, ( x , . . . , x n )) = ( x , . . . , x a , . . . , x a + m a,b ( x ) y, . . . , x n )Now gf is clearly homotopic to the identity and f g equals the identity. Proof of theorem 3.1.
We have two open subspaces Conf r (Γ) and V e (Γ) ofConf r (Γ r e ) such that Conf r (Γ) ∪ V e (Γ) = Conf r (Γ r e ). There is a pushoutdiagram V e (Γ) ∩ Conf r (Γ) V e (Γ)Conf r (Γ) Conf r (Γ r e )We obtain a Mayer-Vietoris long exact sequence in cohomology · · · −−−→ H ∗ (Conf r (Γ r e )) φ ∗ −−−→ H ∗ (Conf r (Γ)) ⊕ H ∗ ( V e (Γ)) ψ ∗ −−−→ H ∗ (Conf r (Γ) ∩ V e (Γ)) δ ∗ −−→ H ∗ +1 (Conf r (Γ r e )) −−−→ · · · where φ is the map assigning to each cohomology class x its restrictions( x | Conf r (Γ) , x | V e (Γ) ) and ψ ( x, y ) = x − y .11e notice that V e (Γ) is homotopy equivalent to Conf r (Γ /e ) and by Lemma r (Γ) ∩ V e is homotopy equivalent to S r − × Conf r (Γ /e ).Let [ µ ] denote the fundamental class of S r − . By the Kunneth formula, wecan rewrite the long exact sequence as · · · −→ H ∗ (Conf r (Γ r e )) −→ H ∗ (Conf r (Γ)) ⊕ H ∗ (Conf r (Γ /e )) −→ M k + l = ∗ H k ( S r − ) ⊗ H l (Conf r (Γ /e )) −→ H ∗ +1 (Conf r (Γ r e ) −→ · · · This implies the existence of the long exact sequence · · · −→ H ∗ (Conf r (Γ r e )) −→ H ∗ (Conf r (Γ)) −→ [ µ ] H ∗ (Conf r (Γ /e )) −→ H ∗ +1 (Conf r (Γ r e )) −→ · · · Finally, using the isomorphism [ µ ] H ∗− r +1 (Conf r (Γ /e )) ∼ = H ∗− r +1 (Conf r (Γ /e ))we have the deletion-contraction long exact sequence for generalised config-uration spaces: · · · −→ H ∗ (Conf r (Γ r e )) −→ H ∗ (Conf r (Γ)) −→ H ∗− r +1 (Conf r (Γ /e )) −→ H ∗ +1 (Conf r (Γ r e )) −→ · · · R r (Γ) . Let Γ be a graph and r a natural number. For any edge e = e ( v , v ) ∈ E (Γ),ordered by that v < v , there is a map p e : Conf r (Γ) → S r − defined by p e ( x ) x v − x v | x v − x v | ∈ S r − ⊂ R r \ { } . If all edges in Γ have an orientation, we can combine these maps to a map p (Γ) : Conf r (Γ) → ( S r − ) E (Γ) . We choose a standard generator [ S r − ] ∈ H r − ( S r − ). After choosing a totalorder of the edges, we can identify H ∗ (( S r − ) E (Γ) ) with the ring Λ[ E (Γ)]. If r is even, this identification depends of the order of the edges, but not onthe orientation of the edges. If r is odd, the identification depends on theorientation of the edges, but not of the order of the edges. In both cases,two different choices differ by an isomorphism. Definition 3.3.
Let r be an even number, the maps p r (Γ) induce ringhomeomorphisms p r (Γ) ∗ : Λ[Γ] → H ∗ (Conf r (Γ)); p r (Γ)( e ) = p ∗ e ([ S r − ]) . emma 3.4. The map p r (Γ) ∗ is surjective. There is a short exact sequence → H ∗ (Conf( r Γ r e )) → H ∗ (Conf r (Γ)) → H ∗− r +1 (Conf r (Γ /e )) → . Proof.
We prove it by induction on the number of edges in Γ. The lemmais true if Γ has one edge. Now we suppose the result true for graphs with n − α ∈ E (Γ).If we have made choices of orientation of edges and order of edges forΓ, we can make compatible choices for Γ r α respectively Γ /α , so thatthe maps i : E (Γ r α ) → E (Γ) respectively p : E (Γ) → E (Γ /α ) preservethe orientations and orders of the edges. Assume that we have made suchcompatible choices.We have a commutative diagram0 / / Λ[Γ r α ] p r (Γ r α ) ∗ (cid:15) (cid:15) i Λ α / / Λ[Γ] p r (Γ) ∗ (cid:15) (cid:15) p Λ α / / Λ[Γ /α ] p r (Γ /α ) ∗ (cid:15) (cid:15) / / · · · / / H ∗ (Conf r (Γ r α )) φ ∗ / / H ∗ (Conf r (Γ)) ψ ∗ / / H ∗− r +1 (Conf r (Γ /α )) / / · · · The first and last vertical maps are surjective by the induction hypothesisand p Λ α is also surjective by lemma 2.8. By the commutativity of the diagram p r (Γ /e ) ◦ p Λ α = ψ ∗ ◦ p r (Γ). Moreover p r (Γ /e ) ◦ p Λ α is surjective since it is thecomposition of surjective maps. It follows that ψ ∗ is surjective, so that thelong exact sequence at the bottom row breaks up into short exact sequences.Therefore the diagram above is a map of short exact sequences, and it followsby the five lemma that the middle vertical map is surjective. Lemma 3.5.
The map p r (Γ) ∗ maps elements in the ideal generated by thegeneralised Arnold relations to .Proof. It suffices to show that if w is a circuit in Γ, and if A ( w ) ∈ Λ[Γ]is the corresponding Arnold element, then p r (Γ) ∗ ( A ( w )) = 0. Let C n be acyclic graph with vertices v , v , . . . v n and edges c i = e ( v i , v i +1 ) for i ≤ n − c n = e ( v n , v ). The edges of C n form a circuit c n . There is amap of graphs f : C l ( w ) → Γ which maps the edge c ni ∈ E ( C n ) to w i ∈ E (Γ).This map induces f Λ ∗ : Λ[ C n ] → Λ[Γ] and ( f conf ) ∗ : H ∗ (Conf r ( C r )) → H ∗ (Conf r (Γ)). By naturality, there is a commutative diagram:Λ r ( C n ) Λ r (Γ) H ∗ (Conf r ( C r )) H ∗ (Conf r (Γ)) f ∗ Λ p r ( C n ) ∗ p r (Γ) ∗ ( f conf ) ∗ Using that A ( w ) = f Λ ∗ ( A ( c n )), it follows from this diagram that it sufficesto show that for every n , the Arnold class A ( c n ) is in the kernel of the map p r ( C n ) ∗ . 13n order to prove the lemma, we investigate the kernel of the map ofclasses in degree ( n − r − p r ( C n ) ∗ : (Λ[ C r ]) ( n − r − → H ( n − r − (Conf r ( C r )) is generated bythe Arnold element A ( c n ). Note that the group (Λ( C n )) ( n − r − ∼ = Z n isgenerated by the classes c n c n · · · b c ni · · · c nn .We make a preliminary remark. Let I n be the linear graph C n r c n . ThenConf r ( I n ) = { ( x , . . . , x n ) ∈ R kr : x = x , . . . , x n − = x n } The map p ( I n ) is a homotopy equivalence, and p r ( I n ) ∗ an isomorphism. Inparticular H ( n − r − (Conf r ( I n )) ∼ = Z , generated by the class p r ( I r ) ∗ ( c n c n · c nn − ).Let n = 2. In this case C has a double edge, so that Conf r ( I ) → Conf r ( C ) is a homeomorphism, A ( c ) = ± c ± c and p r ( C ) ∗ : (Λ[ C ] / { A ( c ) = 0 } ) ( r − ∼ = H ( r − (Conf r ( C )) , so that we have an induction start.Let n ≥
3, and assume the induction hypothesis for n −
1. Let c i be anyedge of C n , so that C n r c i is isomorphic to I n . Let m = ( n − r − C n r c i ] ( m ) Λ[ C n ] ( m ) Λ[ C n /c i ] ( m ) H ( m ) (Conf r ( C n r c i )) H ( m ) (Conf r ( C n )) H ( m ) (Conf r ( C n /c i )) i Λ ci p r ( C n r c i ) ∗ p Λ ci p r ( C n ) ∗ p r ( C n /c i ) ∗ φ ∗ ψ ∗ Because the rows are short exact, we have an induced exact sequence ofkernels and cokernels:ker( p r ( C n r c i ) ∗ ) i Λ ci −→ ker( p r ( C n ) ∗ ) p Λ ci −−→ ker( p r ( C n /c i ) ∗ ) ∂ −→ coker( p r ( C n r c i ) ∗ )The map p r ( C n r c i ) ∗ is an isomorphism by the preliminary remark. There-fore p Λ c i restricts to an isomorphism ker( p r ( C n ) ∗ ) → ker( p r ( C n /c ni ) ∗ ) ∼ = Z .Notice also that p Λ c i ( A ( c n )) = A ( c n − ), where we in the notation have iden-tified C n /c i with C n − . In order to complete the proof, we only need to showthat A ( c n ) ∈ ker( p r ( C n ) ∗ ).By the diagram and the inductive assumption, p r ( C n ) ∗ ( A ( c n )) is in theimage of the map φ ∗ , so there is an x ∈ Λ[ C n r c i ] such that p r ( C n ) ∗ ( A ( c n ) − i Λ c i ( x )) ∈ ker p r ( C n ) ∗ . We conlude: For every i , 1 ≤ i ≤ n there is a number n i such that p r ( C n ) ∗ ( A ( c n ) − i Λ c i ( n i c n · · · b c ni · · · c nn )) = 0 (5)14e need to show that n i = 0. To prove this, we pick j = i . p r ( C n /c ni ) ∗ p Λ c ni ( c n )( A ( c n ) − i Λ c i ( n i c · · · b c i · · · c n ))= p r ( C n /c ni ) ∗ (( A ( c n − ) − n i c c · · · b c i · · · b c j . . . c n = − p r ( C n /c ) ∗ ( n i c n · · · b c i · · · b c j . . . c n ) . Since by the inductive assumption the kernel of p r ( C n /c i ) ∗ is the subgroupgenerated by A ( c n − ), and since n i c n · · · b c i · · · b c j . . . c n is only in this subgroupif n i = 0, we have proved p Λ c j ( i Λ c i ( n i c i · · · b c i · · · c n )) = c c · · · b c i · · · b c j · · · c n is not in the subgroup of Λ[ C n /c j ] generated by A ( c n − ), since n − >
1. Itfollows from the induction hypothesis that n i = 0, so that p r ( C n ) ∗ ( A ( c n )) =0. This finishes the proof. Corollary 3.6.
The map p r (Γ) ∗ : Λ[Γ] → H ∗ (Conf r (Γ)) factors uniquelyover the map λ r : R r (Γ) → H ∗ (Conf r (Γ)) . Theorem 3.7.
There is a isomorphism of graded commutative rings λ r : R r (Γ) → H ∗ (Conf r (Γ)) . Proof.
We prove the theorem by induction on the number of edges in thegraph. Assume that the lemma is true for all graphs with n − n edges. If Γ has multiple edges, the lemma follows fromthe induction hypothesis and lemma 2.2. Consider the following map of shortexact sequences:0 / / R r (Γ r e ) ∼ = (cid:15) (cid:15) / / R r (Γ) (cid:15) (cid:15) / / R r (Γ /e ) ∼ = (cid:15) (cid:15) / / / / H ∗ (Conf r (Γ r e )) φ ∗ / / H ∗ (Conf r (Γ)) ψ ∗ / / H ∗ (Conf r (Γ /e )) / / References [1] V. I. Arnol’d. The cohomology ring of the colored braid group.
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