Secondary representation stability and the ordered configuration space of the once-punctured torus
TThe Betti Numbers of the Ordered Configuration Space of theOnce-Punctured Torus are Polynomial in the Number of Points
Nicholas WawrykowAugust 26, 2020
Abstract
We prove that for k at least 3 the k -th Betti number of the ordered configuration space of the once-punctured torus is a polynomial in the number of points of degree 2 k −
2. We do this by buildingon the work of Pagaria [Pag20] who proved the same growth relation for Betti numbers of the orderedconfiguration space of the torus. Since the once-punctured torus is an open manifold, the homology groupsof its ordered configuration space are secondary representation stable in the sense of Miller and Wilson[MW19]. We use the growth rate of the Betti numbers of the ordered configuration space of the once-punctured torus to show that this space exhibits a secondary representation stability pattern not yet seenin other surfaces. The homology groups of the ordered configuration space of Euclidean space are well-understood; in contrast, much less is known of the homology groups of the ordered configuration spacesof positive-genus surfaces. Our computations are the first to demonstrate that secondary representationstability is a non-trivial phenomenon in positive-genus surfaces.
For a topological space X , let F n ( X ) := { ( x , . . . , x n ) | x i ∈ X, x i (cid:54) = x j if i (cid:54) = j } ⊆ X n denote the ordered configuration space of n distinct points on X . When X = R d , the homology groupsof F n ( R d ) are isomorphic to the operad P ois d ( n ); for more see [Sin06]. For most other manifolds explicitdescriptions of the homology groups of their ordered configuration spaces are unknown. In his recent paper,Pagaria [Pag20, Corollary 2.9] proved that for k ≥
3, the k -th Betti number of the ordered configurationspace of the torus was polynomial in the number of marked points and of degree 2 k −
2. We build on this toprove
Theorem 1.1.
Let T ◦ denote the once-punctured torus. Then, for k ≥ , the k -th Betti number of F n ( T ◦ ) is a polynomial in n of degree k − . For k = 0 , , , the k -th Betti number of F n ( T ◦ ) is a polynomial in n of degree , , , respectively. The symmetric group S n acts on the configuration space F n ( X ) by permuting the coordinates. When X is an open manifold like the once-punctured torus, the ordered configuration spaces have additional structure.If X is an open manifold of dimension d , then there is an embedding e : X (cid:116) R d (cid:44) → X. Such an embedding exists, for example, by Kupers and Miller [KM15, Lemma 2.4]. R T ◦ R T ◦ Figure 1: The embedding e : T ◦ (cid:116) R (cid:44) → T ◦ a r X i v : . [ m a t h . A T ] A ug he embedding induces an inclusion of ordered configuration spaces: ι : F n − ( X ) → F n ( X )by setting ι ( x , . . . , x n − ) (cid:55)→ ( e ( x ) , . . . , e ( x n − ) , e (0)) , swhere 0 denotes 0 ∈ R d . Thus, ι maps a configuration of n − X to its image under e and adds anew point corresponding to the image under e of the origin in R d . Figure 2: The inclusion ι : F ( T ◦ ) (cid:44) → F ( T ◦ )The embedding also induces a map on the product of two configuration spaces: ι (cid:48) : F n − ( X ) × F ( R d ) → F n ( X )given by ι (cid:48) (( x , . . . , x n − ) , ( x (cid:48) , x (cid:48) )) = ( e ( x ) , . . . , e ( x n − ) , e ( x (cid:48) ) , e ( x (cid:48) )) . Figure 3: The inclusion ι (cid:48) : F ( T ◦ ) (cid:44) → F ( T ◦ )The inclusions ι and ι (cid:48) induce maps on homology: ι ∗ : H k ( F n − ( X )) → H k ( F n ( X )) and ι (cid:48)∗ : H k − ( F n − ( X )) ⊗ H ( F ( R d )) → H k ( F n ( X )) . The symmetric group action on ordered configuration space induces an action of Q [ S n ] on H k ( F n ( X ); Q )for all k . Theorem 1.2. (Church–Ellenberg–Farb [CEF15, Theorem 6.4.3] in the orientable case and Miller–Wilson[MW19, Theorem 3.12] in the general case) Let X be a connected, noncompact d-manifold with d ≥ . For k ≤ n − , Q [ S n ] · ι ∗ ( H k ( F n − ( X ); Q )) = H k ( F n ( X ); Q ) . We define another stabilization map, also denoted ι (cid:48)∗ , that leads to a notion of secondary representationstability: ι (cid:48)∗ : H k − ( F n − ( X )) → H k ( F n ( X )) , by pairing a class in H k − ( F n − ( X )) with the class in H ( F ( R d )) corresponding to the point n orbitingthe point labeled ( n −
1) counterclockwise. Miller and Wilson [MW19] were able to show that for n largewith respect to k , the homology groups of ordered configuration space of an open manifold were secondaryrepresentation stable in that they satisfied the following theorem.2 heorem 1.3. (Miller–Wilson [MW19, Theorem 1.2]) Let X be a connected noncompact finite type -manifold. There is a function r : Z ≥ → Z ≥ tending to infinity such that for k ≤ n − + r ( n ) , Q [ S n ] · ( ι ∗ ( H k ( F n − ( X ); Q )) + ι (cid:48)∗ ( H k − ( F n − ( X ) , Q ))) = H k ( F n ( X ); Q ) . In their paper, Miller and Wilson [MW19, Propositions 3.33 and 3.35] calculated Q [ S n ]-span of the imageof ι (cid:48)∗ for R and k = n and for general surfaces for k = 0 and k = n . In the two first cases the homologygroups were already known, and in the last case the image of ι (cid:48)∗ is homologically trivial, i.e., the image of ι (cid:48)∗ is 0. No other examples have been computed to the author’s knowledge.Since T ◦ is a connected non-compact finite type 2-manifold, the homology groups of its ordered configu-ration spaces are secondary representation stable. Moreover, theorem 1.1 implies Corollary 1.4.
Let T ◦ denote the once-punctured torus. Then, for sufficiently large n , Q [ S n ] · ( ι ∗ ( H n ( F n − ( T ◦ ); Q )) + ι (cid:48)∗ ( H n − ( F n − ( T ◦ ) , Q ))) = H n ( F n − ( T ◦ ); Q ); moreover, this is the first example of secondary representation stability in the homology groups of a configu-ration space where the homology groups are unknown or the image of ι (cid:48)∗ is homologically non-trivial. This paper was inspired by Jeremy Miller and Jenny Wilson’s paper on secondary representation stabilityfor ordered configuration spaces of manifolds [MW19]. I would like to thanks John Wiltshire-Gordon forsharing his computations of small-degree Betti numbers of F n ( T ◦ ) with Miller and Wilson; these computationswere an indirect inspiration for this paper. Jenny was incredibly helpful to me in both understanding herpaper and writing this one. I would also like to thank Andrew Snowden for insightful conversations on TCAsand Karen Butt for her comments on this paper. We introduce the language of FI-mod and FIM + -mod. These category-theoretic constructions allow us toformalize the concepts of first and second order representation stability. Definition 1.
Let FB be the category whose objects are all f inite (possibly empty) sets and whose morphismsare b ijective maps.Every finite set is isomorphic to [ n ] := { , . . . , n } for some n ; this provides an equivalence between FBand its full subcategory that has one set [ n ] for each n ∈ Z +0 . Definition 2. A FB-module over the ring R is a covariant functor from FB to the category of R -modules.For an FB-module W and a finite set S , let W S denote the corresponding R -module. When S is the set[ n ], we write W n for W [ n ] . Each W n carries an action of S n arising from the equivalence S n (cid:39) End FI ([ n ]), so { W n } is a sequence of symmetric group representations. Definition 3.
Let FI be the category whose objects are all f inite (possibly empty) sets and whose morphismsare i njective maps.Just as for FB, there is an equivalence between FI and its full subcategory that has one set [ n ] for each n ∈ Z +0 . Definition 4. An FI-module over the ring R is a covariant functor from FI to the category of R -modules.Similarly, an FI-(homotopy)-space is a covariant functor from FI to the (homotopy)-category of topologicalspaces.Much like an FB-module, an FI-module is a sequence of symmetric group representations; however, thereare relations between representations of different degrees. Let V be an FI-module, if ι n,m , n < m , denotesthe standard inclusion of [ n ] into [ m ], ( ι n,m ) ∗ : V n → V m must be S n -equivariant; moreover, ( ι n,m ) ∗ ( V n ) mustbe invariant under the action of S m − n . 3e want to build up an FI-module from an FB-module. For an S d -representation W d , let M ( W d ) n := (cid:77) A ⊆ [ n ] , | A | = d W A . Letting n vary over the nonnegative integers we see that M ( W d ) is an FI-module.Since an FB-module is a sequence of symmetric group representations, we can apply M ( − ) to every degreeof an FB-module W ; this gives a functor that induces an FI-module structure on W : M ( W ) := (cid:77) d ≥ M ( W d ) . Definition 5.
An FI-module V is generated by a set S ⊆ (cid:96) n ≥ V n if V is the smallest FI-submodulecontaining S . If there is some finite set S that generates V , then V is finitely generated . If V is generated by (cid:96) ≤ n ≤ d V n , then V it generated in degree ≤ d .We want to recover a generating set for an FI-module V , preferably a minimal one. One such generatingset consists of subrepresentations of V n , for all n , not arising from the FI-structure in smaller degrees. Weuse the language of FI-homology to formalize this. Definition 6.
The zeroth FI-homology group of an FI-module V in degree n , denoted H FI0 ( V ) n , are the S n representations not arising from V A , for all A ⊂ [ n ], | A | = n − H FI0 ( V ) n := V n \ (cid:77) A ⊂ [ n ] , | A | = n − V A . We have special notation when V is the homology of the ordered configuration space of an open manifold X : Definition 7.
Given i, n ≥
0, let W Xi ( n ) denote the sequence of minimal generators W Xi ( n ) := H FI0 (cid:16) H n + i ( F ( X ); R ) (cid:17) n . Note that H FI0 ( V ) n is an S n representation, so { H FI0 ( V ) } n is a sequence of symmetric group representa-tions, i.e., an FB-module. Thus, we can think of H ( − ) as a functor from FI-mod to FB-mod. Definition 8. A based set S ∗ is a set with a distinguished element ∗ ∈ S ∗ , the basepoint . A map of basedsets f : S ∗ → T ∗ takes ∗ ∈ S ∗ to ∗ ∈ T ∗ . Then, FI is the category whose objects are based f inite sets andwhose morphisms are maps of based sets that are i njective away from the basepoint, i.e., if f : S ∗ → T ∗ is isan FI | f − ( t ) | ≤ t ∈ T ∗ , t (cid:54) = ∗ . Definition 9. An FI over the commutative ring R is a covariant functor from FI R -modules. Similarly, an FI is a functor from FI op -module by only considering surjective morphismsin FI op . Theorem 2.1. (Church–Ellenberg–Farb [CEF15, Theorem 4.1.5]) The category of FI M ( − ) : FB-Mod (cid:28)
FI : H F I ( − ) . Thus, every FI V is of the form ⊕ ∞ n =0 M ( H F I ( V ) n ) . n points on a open manifold X and the inclusionmap ι : F n − ( X ) (cid:44) → F n ( X ) defined in the introduction. The inclusion ι is well-behaved up to homotopywith respect to the symmetric group action on the indices making F ∗ ( X ) an FI-homotopy-space. Taking thehomology groups of these ordered configuration spaces gives us a sequence of FI-modules: for fixed k ≥ H k ( F ∗ ( X )) is an FI-module. We can say even more, namely that forgetful map π : F n ( X ) → F n − ( X ) , given by forgetting the last coordinate π ( x , . . . , x n ) = ( x , . . . , x n − ) , is well behaved with respect to the symmetric group action, and F ∗ ( X ) is an FI k , this makes H k ( F ∗ ( X )) an FI S m correspond to partitions of m . If λ = ( λ , λ , . . . , λ l ) with λ ≥ λ ≥· · · ≥ λ l > m , then we write V λ for the irreducible representation of S m corresponding to λ . Let λ [ n ] denote the partition of n of the form ( n − m, λ , λ , . . . , λ l ). Every partition of n can be uniquelywritten in this form for some λ . We let V ( λ ) n denote the S n -representation V ( λ ) n := (cid:40) n < m + λ V λ [ n ] , n ≥ m + λ . Definition 10.
Let V n be a sequence of rational S n -representations with decomposition into irreducibleconstituents V n = (cid:77) λ c nλ V ( λ ) n . Then V n is (uniformly) multiplicity stable if there exists some N ≥ λ and for all n ≥ N ,the multiplicities c nλ = c Nλ are independent of n . Theorem 2.2. (Church–Ellenberg–Farb [CEF15, Theorem 6.4.3] in the orientable case and Miller–Wilson[MW19, Theorem 3.12] in the general case) Let X be a connected, non-compact d -manifold with d ≥ . For n ≥ k + 1 , H k ( F n ( X ); Z ) is multiplicity stable. For a thorough overview of FI and FI-mod see [CEF15] or [Wil18].
Definition 11. A matching of a set A is a set of disjoint 2-element subsets of A , and a matching is perfect if the union of these subsets is A .The category FI ignores the data of the complement of the image of a morphism; by insisting on a perfectmatching on the complement of the image we get the category FIM. Definition 12.
Let
FIM denote the category whose objects are f inite sets and whose morphisms are i njectivemaps f : A (cid:44) → B along with a perfect m atching on B \ f ( A ).Morphisms between two objects A, B of FIM exist only when | A | and | B | have the same parity. If thereare morphisms from A to B , then the symmetric group S m , m = | B |−| A | , acts on the perfect matching B , . . . , B m on the complement of the image of A in B by permuting the ordering: σ · ( B , . . . , B m ) = ( B σ (1) , . . . , B σ ( m ) ) . This inspires the definition of FIM + , a category enriched over R -mod. Definition 13.
Let
FIM + be the category whose objects are f inite sets and whose module of morphisms f , consist of i njective maps with a perfect m atching on the complement quotiented by a signed symmetricgroup action: R (cid:10) ( f : A → B, B , . . . , B m ) (cid:12)(cid:12) f is injective, | B i | = 2 , B = im ( f ) (cid:116) B · · · (cid:116) B m (cid:11) (cid:104) ( f, B , . . . , B m ) = sign( σ )( f, B σ (1) , . . . , A σ ( m ) ) for all σ ∈ S m . efinition 14. An FIM + -module over the ring R is a covariant functor from FIM + to the category of R -modules.Many of the definitions for FI-modules can be adapted to FIM + -modules including that of finite genera-tion. Definition 15.
An FIM + -module W is generated by a set S ⊆ (cid:96) n ≥ W n if W is the smallest FIM + -submodule containing S . If there is some finite set S that generates W , then W is finitely generated . If W is generated by (cid:96) ≤ n ≤ d V n , then W is generated in degree ≤ d .One could rephrase the definitions and results for FI-mod and FIM + -mod in the language of (skew)-twistedcommutative algebras, see [SS12] and [NSS19] for example.Miller and Wilson [MW19] showed that the sequence of minimal generators of the homology groups of theordered configuration space of an open manifold formed an FIM + -module, i.e., W Xi ( n ) is an FIM + -module.Moreover, they proved that for an open manifold X , the homology groups of its ordered configuration spacesare secondary representation stable in the sense that they satisfy the following theorem. Theorem 2.3. (Miller–Wilson [MW19, Theorem 1.4]) If K is a field of characteristic zero and X is aconnected non-compact manifold of finite type and dimension at least two, then, for each i ≥ , the sequenceof minimal generators W Xi ( n ) = H FI0 (cid:16) H n + i ( F ( X ); K ) (cid:17) n is finitely generated as an FIM + -module. k = n k = n homological degree k FI degree n homology vanishes First order representation stabilityFIM + -module W Xi ( n )Stable range of W Xi ( n )Figure 4: First and second order representation stability for surfacesTo prove this theorem, Miller and Wilson used the complex of injective words and a Noetherianity result ofNagpal, Sam, and Snowden [NSS19]. Furthermore, they computed some explicit examples of this secondaryrepresentation stability for the homology groups of the ordered configuration space on an open manifold in[MW19]. Proposition 2.4. (Miller–Wilson [MW19, Proposition 3.33]) W R (2 n ) ∼ = (cid:77) λ ∈ D n V λ where a partition λ of n is in D n if and only if when the associated Young diagram is cut in two along theupper staircase, then the resultant two skew subdiagrams are symmetric under reflection in the line of slope − . They made some calculations for generalized surfaces:
Proposition 2.5. (Miller–Wilson [MW19, Proposition 3.35]) Let X be a connected non-compact surface. If X is not orientable or of genus greater than zero, then W X (0) ∼ = Z and W X (2 n ) ∼ = 0 for n > . X = R the FIM + -module W R (2 n ) is a“free” FIM + -module and all the homology groups of F ∗ ( R ) are known. For the secondset of cases explicit calculation gives W X (0) = H FI0 ( H ( F ( X ))) = H ( F ( X )) ∼ = Z , a previously knownresult. The case of W X (2 n ) for n > W R (2 n ), as it is always the zeroFIM + -module.We seek an example of secondary representation stability that lies between these extremes; namely, afinitely generated FIM + -module arising from the homology groups of the ordered configuration space of asurface that is neither free nor zero and where the homology groups are unknown. In a recent paper [Pag20], Pagaria used a filtration on the Kriz model to show that Betti numbers of theordered configuration space of the torus were polynomial in the number of points, and that for large k , the k -th Betti number b k was of degree 2 k − Theorem 3.1. (Pagaria [Pag20, Corollary 2.9]) For k ≥ the Betti numbers of F n ( T ) are of the form b k = c k (cid:18) n k − (cid:19) + o ( n k − ) , where c k ≥ (cid:0) k − k − (cid:1) .For k ≤ the Betti numbers are b = 1 ,b = 2 n,b = 2 (cid:18) n (cid:19) + 3 (cid:18) n (cid:19) + n,b = 14 (cid:18) n (cid:19) + 8 (cid:18) n (cid:19) + 2 (cid:18) n (cid:19) ,b = 32 (cid:18) n (cid:19) + 74 (cid:18) n (cid:19) + 33 (cid:18) n (cid:19) + 5 (cid:18) n (cid:19) ,b = 63 (cid:18) n (cid:19) + 427 (cid:18) n (cid:19) + 490 (cid:18) n (cid:19) + 154 (cid:18) n (cid:19) + 18 (cid:18) n (cid:19) . We can use Pagaria’s theorem to calculate the Betti numbers of the ordered configuration space of theonce-punctured torus.
Theorem 3.2.
Let T ◦ denote the once-punctured torus. For k ≥ , the k -th Betti number of the orderedconfiguration space of n points on T ◦ , b k ( F n ( T ◦ )) is polynomial in n of degree k − . When k = 0 , , ,it is polynomial in n of degree , , , respectively. For k ≤ the Betti numbers are given by the followingformulae: b ( F n ( T ◦ )) = 1 b ( F n ( T ◦ )) = 2 nb ( F n ( T ◦ )) = 2 (cid:18) n (cid:19) + 5 (cid:18) n (cid:19) b ( F n ( T ◦ )) = 14 (cid:18) n (cid:19) + 18 (cid:18) n (cid:19) b ( F n ( T ◦ )) = 32 (cid:18) n (cid:19) + 106 (cid:18) n (cid:19) + 79 (cid:18) n (cid:19) b ( F n ( T ◦ )) = 63 (cid:18) n (cid:19) + 490 (cid:18) n (cid:19) + 853 (cid:18) n (cid:19) + 432 (cid:18) n (cid:19) . roof. The torus T can be thought of as an additive group, namely T (cid:39) R \ Z . This allows us to decomposethe ordered configuration space of the torus as a product: F n ( T ) (cid:39) T × F n − ( T ◦ )( x , . . . , x n ) (cid:55)→ x × ( x − x , . . . , x n − x ) , where coordinates in F n − ( T ◦ ) are taken modulo Z . Here x is the location of the puncture in T ◦ .Since Poincare polynomials respect product decompositions, we can write P ( F n ( T )) = P ( T ) × P ( F n − ( T ◦ ))= (1 + 2 t + t ) P ( F n − ( T ◦ )) . These equations can be rearranged to give the Poincare polynomial for F n − ( T ◦ ) in terms of the Poincarepolynomial for F n ( T ): P ( F n − ( T ◦ )) = P ( F n ( T ))1 + 2 t + t = ∞ (cid:88) i =0 ( − i ( i + 1) t i P ( F n ( T )) , where the second equality arises by expanding (1 + 2 t + t ) − as a Taylor series in t .By explicitly expressing the Poincare polynomials in terms of Betti numbers, i.e., noting P = (cid:80) ∞ i =0 b i t i ,we have ∞ (cid:88) k =0 b k ( F n − ( T ◦ )) t k = (cid:32) ∞ (cid:88) i =0 ( − i ( i + 1) t i (cid:33) ∞ (cid:88) j =0 b j ( F n ( T )) t j = ∞ (cid:88) k =0 k (cid:88) m =0 ( − k − m ( k + 1 − m ) b m ( F n ( T )) t k . This gives us a formula for the Betti numbers of F n − ( T ◦ ) in terms of the Betti numbers for F n ( T ): b k ( F n − ( T ◦ )) = k (cid:88) m =0 ( − k − m ( k + 1 − m ) b m ( F n ( T )) . By replacing n − n we see that b k ( F n ( T ◦ )) = k (cid:88) m =0 ( − k − m ( k + 1 − m ) b m ( F n +1 ( T )) , e.g., b ( F n ( T ◦ )) = b ( F n +1 ( T )) ,b ( F n ( T ◦ )) = b ( F n +1 ( T )) − b ( F n +1 ( T )) .b ( F n ( T ◦ )) = b ( F n +1 ( T )) − b ( F n +1 ( T )) + 3 b ( F n +1 ( T )) , etc.Now apply theorem 3.1, which states that b k ( F n ( T )) is a polynomial in n of degree 2 k −
2, and for k = 0 , ,
2, it is a polynomial of degree 0 , ,
3. Reindexing from n to n + 1 does not change this. For k ≥ b k ( F n ( T ◦ )) can be written as a linear combination of k + 1 polynomials in n of degree ≤ k −
2, and that the degree 2 k − k − . When k = 0 , , or 2, b k ( F n ( T ◦ )) can be expressed as linear combination of k +1 polynomials in n of degree ≤ , , or 3, respectively,with the highest degree term having leading coefficient 1 , , or , respectively. Thus, for k ≥ b k ( F n ( T ◦ ))is polynomial in n of degree 2 k −
2. When k = 0 , ,
2, it is polynomial in n of degree 0 , ,
3, respectively.To get the formulae for b k ( F n ( T ◦ )) for k ≤ b k ( F n ( T ◦ )) = k (cid:88) m =0 ( − k − m ( k + 1 − m ) b m ( F n +1 ( T ))to the calculations for Betti numbers of the configuration space of the torus computed in Pagaria’s paper.8ince T ◦ is a non-compact manifold, we can apply the results of Church, Ellenberg, and Farb [CEF15]and Miller and Wilson [MW19] to study the sequence of homology group generators. Corollary 3.3.
The sequence of minimal generators W T ◦ ( n ) = H FI0 (cid:16) H n +22 ( F ( T ◦ ); R ) (cid:17) n is finitely generated as an FIM + -module. Moreover, it is not stably zero and it arises from a set of spaceswhose homology groups are unknown.Proof. Since T ◦ is an open manifold, Theorem 2.3 applies, proving that this sequence is finitely generated asan FIM + -module. Since H k ( F ( T ◦ ); R ) is FI M ( W ) = ⊕ d ≥ M ( W d ),for some FB-module W . Recall that M ( W d ) n = (cid:76) A ⊆ [ n ] , | A | = d W A , so M ( W ) n = (cid:77) d ≥ M ( W d ) n = (cid:77) d ≥ (cid:77) A ⊆ [ n ] , | A | = d W A . The dimension of M ( W d ) n is a polynomial in n of degree d :dim( M ( W d ) n ) = dim( W d ) · (cid:18) nd (cid:19) Since H k ( F ( T ◦ ); R ) is of the form M ( W ) = (cid:76) d ≥ M ( W d ) and b k ( F n ( T ◦ )) is a polynomial in n of degree2 k − k ≥
3, we see that H k ( F ( T ◦ ); R ) = k − (cid:77) d =0 M ( W d )by theorem 3.2 and that W k − is not the zero module. Setting k = n +22 , we see that H n +22 ( F ( T ◦ ); R ) = n (cid:77) d =0 M ( W n ) . By theorem 2.1 H FI0 ( M ( W )) ∼ = W , so W T ◦ ( n ) = H FI0 (cid:16) H n +22 ( F ( T ◦ ); R ) (cid:17) n ∼ = W n is not zero for n > Corollary 3.4.
The FIM + -module W T ◦ ( n ) is stably zero for all n > , and the FIM + -module W T ◦ ( n ) (cid:54) = 0 for n = 1 , , but is stably zero for all n ≥ .Proof. Our proofs of corollary 3.3 and theorem 3.2 show that for k ≥ k term in H k ( F ( T ◦ ); R ) = k − (cid:77) d =0 M ( W d ) . So W T ◦ ( n ) is stably zero for all n ≥
3. When k = 1 ,
2, the computations of the k -th Betti numbers intheorem 3.2 show that H ( F ( T ◦ ); R ) = (cid:77) d =0 M ( W d )and H ( F ( T ◦ ); R ) = (cid:77) d =0 M ( W d );moreover, the computations shows that the top terms are not zero. This proves that W T ◦ ( n ) (cid:54) = 0 for n = 1 , W T ◦ ( n ) is stably zero for all n >
0, providing a new proof of 2.5 in thecase X = T ◦ . 9 eferences [CEF15] Thomas Church, Jordan S Ellenberg, and Benson Farb, FI-modules and stability for representationsof symmetric groups , Duke Mathematical Journal (2015), no. 9, 1833–1910. 2, 4, 5, 9[KM15] Alexander Kupers and Jeremy Miller,
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