The cohomology rings of the unordered configuration spaces of the torus
aa r X i v : . [ m a t h . A T ] J un THE COHOMOLOGY RINGS OF THE UNORDEREDCONFIGURATION SPACES OF THE TORUS
ROBERTO PAGARIA
Abstract.
We study the cohomology ring of the configuration space of un-ordered points in the two dimensional torus. In particular, we compute themixed Hodge structure on the cohomology, the action of the mapping classgroup, the structure of the cohomology ring and we prove the formality overthe rationals.
Introduction
We fully describe the cohomology with rational coefficients of the configurationspaces of unordered points in an elliptic curve (frequently called torus).Configuration spaces of points are related to physics (state spaces of non-collidingparticles on a manifold), robotics (motion planning), knot theory, and topology.Configuration spaces give invariants of the homeomorphism type of the base space.In the algebraic setting, configuration spaces are open in the moduli spaces ofpoints.Since the literature is very extensive, we compare our work only with the mainresults on the (co-)homology of configuration spaces. The first computation of thecohomology algebra of configuration spaces is due to Arnol’d [Arn69, Arn14] in thecase of R . This result has been generalized by Cohen, Lada, and May [CLM76]to the configuration space of R n and later by Goresky and Macpherson [GM88].Partially additive results have been obtained: by B¨odigheimer and Cohen [BC88]for once-punctured oriented surfaces, by the same authors and Taylor [BCT89]for odd dimensional manifolds, and by Drummond-Cole and Knudsen [DCK17]for surfaces in general. However there is no description of the ring structure; weprovide it in the case of elliptic curves. The Betti numbers C n ( X ) are described inthe following cases: for X = P ( R ) by Wang [Wan02], for X a sphere by Salvatore[Sal04], for X = P ( C ) by Felix and Tanr´e [FT05] and for elliptic curves by Maguireand Schiessl [Mag16, Sch16].In this paper we improve the previous results on configuration spaces in anelliptic curve in three ways. We describe: • the mixed Hodge structure on the cohomology (Theorem 3.3), • the action of the mapping class group (Theorem 3.3), • the ring structure (Theorem 4.1).The formality result over the rationals is proven in Corollary 4.3.We prove these results using the Kriˇz model [Kri94, Tot96, Bib16, Dup15] andthe representation theory on it [AAB14, Aza15].In Section 1 we recall the Kriˇz model, then in Section 2 we improve the re-sult on the decomposition of the Kriˇz model into irreducible representations, seeTheorem 2.7. Descriptions of the mixed Hodge structure and of the action of the mapping class group are obtained in Section 3 by computing the cohomology of themodel. Finally, the ring structure is presented in the last section.1. The Kriˇz model
Let E be an elliptic curve and consider the configuration space of n ordereddistinct points F n ( E ) def = { p ∈ E n | p i = p j } . The symmetric group S n acts on F n ( E ) by permuting the coordinates and thequotient is the configuration space of n unordered points C n ( E ) def = F n ( E ) / S n . We also consider the space M n ( E ), defined by M n ( E ) def = (cid:8) p ∈ F n ( E ) | X p i = 0 (cid:9) . Notice that there exists a non canonical isomorphism F n ( E ) ∼ = E × M n ( E ).In this section we recall a rational model for the cohomology algebra of F n ( E ).The model is a commutative differential bi-graded algebra (dga) that can be obtainin two different ways: as a specialization of the Kriˇz model for the configurationspaces or as the second page of the Leray spectral sequence (also known as theTotaro spectral sequence) for elliptic arrangements. Our main references for thefirst approach are [Kri94, AAB14, Aza15] and for the second one are [Tot96, Dup15,Bib16]. In the following we define the models for the cohomology of F n ( E ) and of M n ( E ).Let Λ be the exterior algebra over Q with generators { x i , y i , ω i,j } ≤ i 0) and the degree of ω i,j equal to(0 , → Λ of bi-degree (2 , − 1) on generators as follows:d x i = 0 and d y i = 0 for i = 1 , . . . , n andd ω i,j def = ( x i − x j )( y i − y j ) . For the sake of notation we set ω i,j := ω j,i for i > j .We define the dga A • , • as the quotient of Λ by the following relations:( x i − x j ) ω i,j = 0 and ( y i − y j ) ω i,j = 0 ,ω i,j ω j,k − ω i,j ω k,i + ω j,k ω k,i = 0 . Notice that the ideal is preserved by the differential map, thus the differentiald: A • , • → A • , • is well defined. Remark . The model A • , • coincides with the Kriˇz model E •• introduced in [Kri94]up to shifting the degrees, ie A p,q ∼ = E p + qq . The dga E •• is a rational model for X , as shown in [Kri94, Theorem 1.1].In order to study the cohomology of A • , • we need to introduce the elements u i,j = x i − x j , v i,j = y i − y j and γ = P ni =1 x i , γ = P ni =1 y i ∈ A , . HE COHOMOLOGY RINGS OF THE CONFIGURATION SPACES OF THE TORUS 3 We define the dga B • , • as the subalgebra of A • , • generated by u i,j , v i,j and ω i,j for 1 ≤ i < j ≤ n . Let D • , be the subalgebra of A • , • generated by γ and γ endowedwith the zero differential map. Notice that(1) A • , • ∼ = B • , • ⊗ Q D • , as differential algebras and that D • , is the cohomology ring of the elliptic curve E .The mixed Hodge structure on the cohomology of algebraic varieties definesa bigrading compatible with the algebra structure (see [Del75, p.81] or [Voi07,Theorem 8.35]). In our case the bigrading given by the mixed Hodge structurecoincides with the one given by the Leray spectral sequence as shown by Totaro[Tot96, Theorem 3] and by Gorinov [Gor17]. Explicitly, the subspace A p,q hasweight p + 2 q and degree p + q .The following result is a particular case of [Bib16, Theorem 3.3] and of [Dup15,Theorem 1.2]. Theorem 1.2. The cohomology algebra of F n ( E ) (or of M n ( E ) ) with rationalcoefficients is isomorphic to the cohomology of the dga A • , • (respectively of B • , • ).Moreover, the n -sheeted covering E × M n ( E ) → F n ( E )( q, p ) ( p i + q ) i =1 ,...,n induces the isomorphism of eq. (1) . Representation theory on the Kriˇz model Now we study the action of the symmetric group S n and of SL ( Q ) on thealgebras A • , • and B • , • . Those actions are given by a geometric action on F n ( E ).For general reference about the representation theory of the Lie group and of theLie algebra we refer to [Hal15] and to [FH91], respectively. The cases of SL ( C )and of sl ( C ) can be found in [GW09].2.1. Definition of the actions. Consider the action of S n on F n ( E ) defined by σ − · ( p , . . . , p n ) = ( p σ (1) , . . . , p σ ( n ) )for all σ ∈ S n . This induces an action on A • , • and on B • , • defined by σ − ( x i ) = x σ ( i ) ,σ − ( y i ) = y σ ( i ) ,σ − ( ω i,j ) = ω σ ( i ) ,σ ( j ) for all 1 ≤ i < j ≤ n and all σ ∈ S n .The mapping class group MCG( E ) acts naturally on F n ( E ) and on C n ( E ). Theorem 2.1 (Theorem 2.5 [FM12]) . The mapping class group MCG( E ) of thetorus is isomorphic to SL ( Z ) and the isomorphism is given by the natural actionof MCG( E ) on H ( E ; Z ) . Let f be an automorphism of E , the map induces the following vertical mor-phisms R. PAGARIA F n ( E ) E n F n ( E ) E nf n |F n ( E ) f n and by functoriality of the Leray spectral sequence it induces the action of SL ( Z )on A • , • . We explicitly describe this action on the generators ω i,j , x i , and y i : since f n : E n → E n fixes the divisor { p i = p j } , then f · ω i,j = ω i,j . The other generatorsbelongs to A , = H ( E n ) ∼ = H ( E ) ⊗ n . Therefore the action of MCG( E ) ∼ = SL ( Z )on A , is given by (cid:18) a bc d (cid:19) · x i = ax i + cy i (cid:18) a bc d (cid:19) · y i = bx i + dy i . This action extends to SL ( Q ) and since the actions of S n and of SL ( Q ) commute,then A • , • , B • , • and D • , become S n × SL ( Q )-modules.2.2. Decomposition into S n -representations. We recall a result of [AAB14,Theorem 3.15] on the decomposition of A • , • into S n -modules. The notations usedhere follow the ones in [AAB14].Let L ∗ = ( λ , . . . , λ t ) be a partition of the number n , ie λ i ∈ N + and P ti =1 λ i = n . We mark all blocks with labels in { , x, y, xy } , an ordered basis of H • ( E ). Theorder is 1 ≺ x ≺ y ≺ xy . Definition 2.2. A marked partition ( L ∗ , H ∗ ) is a partition L ∗ ⊢ n together withmarks H ∗ = ( h , . . . , h t ), h i ∈ { , x, y, xy } such that: if λ i = λ i +1 then h i (cid:23) h i +1 .Let C k be the cyclic group of order k . For any partition L ∗ ⊢ n define C L ∗ as theproduct of the cyclic groups C λ i for i = 1 , . . . , t . It acts on { , . . . , n } in the naturalway. Consider a marked partition ( L ∗ , H ∗ ) and define N L ∗ ,H ∗ as the group thatpermutes the blocks of L ∗ with the same labels. The group N L ∗ ,H ∗ is a product ofsymmetric groups. Call Z L ∗ ,H ∗ the semidirect product C L ∗ ⋊ N L ∗ ,H ∗ . Example 2.3. Let ( L ∗ , H ∗ ) be the marked partition L ∗ = (5 , , , , , , ⊢ H ∗ = ( xy, xy, xy, , x, x, x ). The group C L ∗ ∼ = ( Z / Z ) < S is generated by(1 , , , , , (6 , , , , , (11 , , , , , , , , N L ∗ ,H ∗ ∼ = S × S is generated by the permutations (1 , , , , , , , , , , , , Z L ∗ ,H ∗ is a groupisomorphic to ( Z / Z ≀ S ) × Z / Z × S .Given two representations V, W of two groups G and H respectively, define thetensor representation V ⊠ W of G × H by the vector space V ⊗ W with the action( g, h )( v ⊗ w ) = g ( v ) ⊗ h ( w ).We define the following one-dimensional representations. Let ϕ n be a faithfulcharacter of the cyclic group and ϕ L ∗ the character of C L ∗ ∼ = Z /λ Z × · · · × Z /λ t Z given by ϕ L ∗ def = sgn n | C L ∗ · ( ϕ λ ⊠ · · · ⊠ ϕ λ r ) . Recall that the degree deg of 1 , x, y, xy are respectively 0 , , , 2. Let α L ∗ ,H ∗ be the one dimensional representation of N L ∗ ,H ∗ ∼ = S µ × · · · × S µ l defined ongenerators by α L ∗ ,H ∗ ( σ ) def = ( − λ +deg( h )+1 , HE COHOMOLOGY RINGS OF THE CONFIGURATION SPACES OF THE TORUS 5 where σ is the permutation that exchange two blocks of size λ and label h . Set ξ L ∗ ,H ∗ to be the one dimensional representation of Z L ∗ ,H ∗ such that Res Z L ∗ ,H ∗ C L ∗ ξ L ∗ ,H ∗ = ϕ L ∗ and Res Z L ∗ ,H ∗ N L ∗ ,H ∗ α L ∗ ,H ∗ .We define | L ∗ | = n − t for a partition L ∗ = ( λ , . . . , λ t ) of n and for a mark H ∗ the numbers | H ∗ | = P ti =1 deg( h i ) and k H ∗ k = |{ i | h i = x }| − |{ i | h i = y }| . Theorem 2.4 ([AAB14, Theorem 3.15]) . There exist S n -representations A L ∗ ,H ∗ ⊂ A p,q such that A • , • = M | L ∗ | = q | H ∗ | = p A L ∗ ,H ∗ as S n -representation. Moreover: A L ∗ ,H ∗ ⊗ Q C ≃ Ind S n Z L ∗ ,H ∗ ξ L ∗ ,H ∗ . Example 2.5. Consider the marked partition ( L ∗ , H ∗ ) of Example 2.3, the char-acters are shown in the following table.(1 , , , , 5) (16 , , , , 20) (1 , , , , , 10) (21 , ϕ ζ ζ α − ξ ζ ζ − Decomposition into S n × SL ( Q ) -representations. Let T = { H t } ∼ = Q ∗ be the maximal torus in SL ( Q ) generated by the diagonal matrices H t = (cid:0) t t − (cid:1) .Let V be the irreducible representation Q with the standard action of matrix-vector multiplication and let V k = S k V be the irreducible representation given bythe symmetric power of V . The representation V k has dimension k + 1 and can beview as Q [ x, y ] k , ie the vector space of homogeneous polynomials in two variables.The action of T on the monomials is given by H t · x a y k − a = t a − k x a y k − a , thus V k decomposes, as representations of T (2) V k = k M a =0 V (2 a − k ) , where V (2 a − k ) is the subspace where H t acts with character t a − k , ie the subspacegenerated by x a y k − a . Since the group SL ( Q ) is dense in SL ( C ), each irreducibleregular representation of SL ( Q ) is isomorphic to V k for some k ∈ N . For a proofsee [GW09, Proposition 2.3.5] and use a density reasoning.As a consequence we can decompose a representation V of SL ( Q ) using itsdecomposition V = ⊕ a ∈ Z V ( a ) ⊕ n a as representation of T : indeed V ∼ = ⊕ k ∈ N V ⊕ m k k as representation of SL ( Q ), where m k = n k − n k +2 . By setting V = V m ⊗ V n , weobtain the following formula for n ≤ m : V m ⊗ V n ∼ = V m + n ⊕ V m + n − ⊕ · · · ⊕ V m − n . As observed in Section 2.1, the group SL ( Q ) acts trivially on ω i,j for all 1 ≤ i < j ≤ n and the two dimensional subspace generated by x i and y i is isomorphicto V as representation of SL ( Q ). R. PAGARIA We will use the decomposition of Theorem 2.4 to obtain a decomposition of A • , • into S n × SL ( Q )-modules. Let A p,qa = M | L ∗ | = q | H ∗ | = p, k H ∗ k = a A L ∗ ,H ∗ be a S n × T -stable subspace of A p,q and hence we have A p,q = ⊕ pa = − p A p,qa . Let p a : A p,q → A p,qa be the S n × T -equivariant projection and define Y = ( ) ∈ SL ( Q ) and, for a ≥ 0, define π a : A p,qa +2 → A p,qa by v p a ( Y · v ).Notice that, π a is a morphism of S n -representations and that A p,qa is zero if a p mod 2, or if a > p , or if a > n − q . Lemma 2.6. The map π a is injective.Proof. If V ⊆ A p,q is a SL ( Q )-representation, then p a ( V ) ⊆ V , thus it is enoughto prove that π a : V ( a + 2) → V ( a ) is injective for all irreducible representation V k ⊆ A p,q . If k a mod 2 or if k < a + 2 then V ( a + 2) = 0. Otherwise k = a + 2 b + 2 for some b ≥ V ( a + 2) is a one dimensional vector spacegenerated by the homogeneous monomial x a + b +2 y b . The projection p a has kernelequal to ⊕ l = a V ( l ), thus p a ( Y · x a + b +2 y b ) = p a (( x + y ) a + b +2 y b ) = ( a + b + 2) x a + b +1 y b +1 . This last term is non-zero since a, b ≥ π a is injective. (cid:3) Theorem 2.7. The algebra A • , • decomposes as S n × SL ( Q ) -representation in thefollowing way: (3) A p,q ∼ = p M a =0 coker π a ⊠ V a . Proof. Observe that the maximal torus T of SL ( Q ) acts on A p,qa with character t a , thus by Theorem 2.4 we have A p,q = p M a = − p A p,qa as S n × T representations. By using eq (2) we obtain that L pa =0 coker π a ⊠ V a is isomorphic to L pa = − p A p,qa as representation of S n × T . The representationtheory of SL ( Q ) ensure that the representations A p,q and L pa =0 coker π a ⊠ V a areisomorphic as S n × SL ( Q )-representations. (cid:3) Define the S n -invariant subalgebra of A • , • by UA • , • and of B • , • by UB • , • . Ob-viously we have UA • , • = UB • , • ⊗ Q D • . We use the previous calculation to compute UA • , • Corollary 2.8. For q > p + 1 we have UA p,q = 0 .Proof. Let n be the trivial representation of S n . We use Theorem 2.4 to showthat h n , A p,q i S n = 0for q > p + 1. Indeed, it is enough to prove that h n , Ind S n Z L ∗ ,H ∗ ξ L ∗ ,H ∗ i S n = 0 HE COHOMOLOGY RINGS OF THE CONFIGURATION SPACES OF THE TORUS 7 for all ( L ∗ , H ∗ ) with | L ∗ | = q and | H ∗ | = p . By Frobenius reciprocity we have h n , Ind S n Z L ∗ ,H ∗ ξ L ∗ ,H ∗ i S n = h Res S n Z L ∗ ,H ∗ n , ξ L ∗ ,H ∗ i Z L ∗ ,H ∗ Since the representations in the right hand side are one-dimensional the value of h n , Ind S n Z L ∗ ,H ∗ ξ L ∗ ,H ∗ i S n is non zero if and only if ξ L ∗ ,H ∗ = .By definition ξ L ∗ ,H ∗ = is equivalent to ϕ L ∗ = and α L ∗ ,H ∗ = . From the factthat ϕ k = sgn k only for k = 1 , ψ L ∗ = if and only if λ i = 1 , i = 1 , . . . , t .The condition α L ∗ ,H ∗ = implies that the only marked blocks of ( L ∗ , H ∗ ) thatappear more than once are the ones with λ i = 2 and deg( h i ) = 1 or the ones with λ i = 1 and deg( h i ) = 1.Consequently, h n , Ind S n Z L ∗ ,H ∗ ξ L ∗ ,H ∗ i S n = 0 only if L ∗ = (2 q , n − q ) and thedegree of h i is 1 for i < q , this implies p ≥ q − (cid:3) Corollary 2.9. For q > p + 1 we have UB p,q = 0 . (cid:3) The additive structure of the cohomology We compute the cohomology with rational coefficients of the unordered config-uration spaces of n points, taking care of the mixed Hodge structure and of theaction of SL ( Q ). The integral cohomology groups are known only for small n in[Nap03, Table 2], where a cellular decomposition of ordered configuration spacesis given. In this section, we use the calculation of the Betti numbers of C n ( E ) todetermine the Hodge polynomial in the Grothendieck ring of SL ( Q ).Observe that H • ( C n ( E )) = H • ( F n ( E )) S n by the Transfer Theorem. Define theseries T ( u, v ) = 1 + u v (1 − u v ) = 1 + 2 u v + u v + 3 u v + 2 u v + . . . and let T n ( u, v ) be its truncation at degree n in the variable u .The computation of the Betti numbers of unordered configuration space of n points in an elliptic curve was done simultaneously by [DCK17], [Mag16], and[Sch16] in different generality. We point to the last reference because [Sch16, The-orem] fits exactly our generality. Theorem 3.1. The Poincar´e polynomial of C n ( E ) is (1 + t ) T n − ( t, . We use the notation V u k v h to denote a vector space V in degree k with a Hodgestructure of weight h . The Grothendieck ring of SL ( Q ) is the free Z -module withbasis given by [ V ] for all finite-dimensional irreducible representations V of SL ( Q )and product defined by the tensor product of representations. Definition 3.2. The Hodge polynomial of C n ( E ) with coefficients in the Grothendieckring of SL ( Q ) is n X i =0 2 i X k = i " W k H i ( C n ( E ); Q ) (cid:30) W k − H i ( C n ( E ); Q ) u i v k , where W k H i ( C n ( E ); Q ) is the weight filtration on H i ( C n ( E ); Q ). The ordinaryHodge polynomial is n X i =0 2 i X k = i dim Q W k H i ( C n ( E ); Q ) (cid:30) W k − H i ( C n ( E ); Q ) ! u i v k . R. PAGARIA q V q q − V q − V q − ... . . . . . . V V V V V · · · q (a) Case n = 2 q + 1 odd. q V q V q − ... . . . . . . V V V V V V V · · · q q + 1 (b) Case n = 2 q + 2 even. Figure 1. The algebra H • , • ( UB ) as representation of SL ( Q ).We prove a stronger version of Theorem 3.1. Theorem 3.3. The Hodge polynomial of C n ( E ) with coefficients in the Grothen-dieck ring of SL ( Q ) is (4) ([ V ] + [ V ] uv + [ V ] u v ) ⌊ n − ⌋ X i =0 [ V i ] u i v i + ⌊ n ⌋− X i =1 [ V i − ] u i +1 v i +1 and the ordinary Hodge polynomial is (1 + uv ) T n − ( u, v ) . Figure 1 represents the module H • , • ( UB ) that corresponds to the right factor ofeq. (4).3.1. Some elements in cohomology.Definition 3.4. Let α, α ∈ A , , β ∈ A , be the elements α def = X i,k The element α belongs to UB , , is non-zero, and d α = 0 .Proof. First observe that α = P i,k 1. This proves that α = 0. Finally, we compute d α :d α = X i,k The element β belongs to UB , , is non-zero, and d β = 0 .Proof. Observe that β = X i,j,k For n > q the element α q is non-zero.Proof. Let us rewrite α as α = P i,k For n > q + 1 the element α q − β is non-zero.Proof. Let us rewrite β as β = X i,j,k Proof of Theorem 3.3. It is enough to prove that the Hodge polynomial of UB inthe Grothendieck ring of SL ( Q ) is ⌊ n − ⌋ X i =0 [ V i ] u i v i + ⌊ n ⌋− X i =1 [ V i − ] u i +1 v i +1 Observe that Im d q,p = 0 for q > p + 1 by Corollary 2.9. From Lemma 3.5 andLemma 3.7 we have that the elements α k for 2 k < n generate as SL ( Q )-module asubspace of dimension at least k + 1 in H k,k ( UB, d). Analogously, from Lemma 3.6and Lemma 3.8 the elements α k − β for 2 k + 1 < n generate as SL ( Q )-modulea subspace of dimension at least k in H k,k +1 ( UB, d). Since the Betti numbers of UB • , • (Theorem 3.1) coincides with the above dimensions, we have that H k ( UB ) ∼ = V k u k v k and H k +1 ( UB ) ∼ = V k − u k +1 v k +1 . (cid:3) The cohomology ring In this section we determine the cup product structure in the cohomology of C n ( E ) and we prove the formality result.In the following we consider graded algebras with an action of SL ( Q ). Wewill write ( x i | i ∈ I ) SL ( Q ) for the ideal generated by the elements M x i for all M ∈ SL ( Q ) and all i ∈ I . Theorem 4.1. The cohomology ring of C n ( E ) is isomorphic to Λ • V ⊗ S • V [ b ] (cid:30) ( a ⌊ n +12 ⌋ , a ⌊ n ⌋ b, b ) SL ( Q ) , where a is a non-zero degree-one element in V (1) ⊂ V and b an SL ( Q ) -invariantindeterminate of degree .Proof. It is enough to prove that H • ( UB ) ∼ = S • V [ b ] / ( a ⌊ n +12 ⌋ , a ⌊ n ⌋ b, b ) SL ( Q ) . De-fine the morphism ϕ : S • V [ b ] / ( a ⌊ n +12 ⌋ , a ⌊ n ⌋ b, b ) SL ( Q ) → H • ( UB ) that sends a, b to α, β respectively. It is well defined because H k ( UB ) = 0 for k ≥ n and β = 0since it has odd degree. The map ϕ is surjective since H • ( UB ) is generated by α i and α i β as SL ( Q )-module by Theorem 3.3. A dimensional reasoning shows theinjectivity of the map ϕ . (cid:3) Corollary 4.2. The cohomology H • ( C n ( E )) is generated as an algebra in degreesone, two and three.Proof. A minimal set of generators is given by α, α, β, γ, γ . (cid:3) Corollary 4.3. The space C n ( E ) is formal over the rationals.Proof. We prove that UB is formal. Consider the subalgebra K • , • of UB • , • generatedby α, α, β endowed with the zero differential. It is concentrated in degrees ( i, i ) and( i, i + 1) because β = 0. Since K ∩ Im d = 0 (Corollary 2.9), K ֒ → U B is a quasi-isomorphism. The fact that K ∼ = H ( U B ) implies that the algebra UB is formal.As a consequence UA is formal. The space C n ( E ) is formal since our model UA isequivalent to the Sullivan model. (cid:3) Acknowledgements. I would like to thank the referee for several useful sugges-tions. References [AAB14] Samia Ashraf, Haniya Azam, and Barbu Berceanu, Representation theory for the Kriˇzmodel , Algebr. Geom. Topol. (2014), no. 1, 57–90. MR 3158753[Arn69] Vladimir I. 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