title: Secondary representation stability and the ordered configuration space of the once-punctured torus
authors: Nicholas Wawrykow
text: 
  
  Mechanisms and dynamics of the NH_2^+ + H^+ and NH^+ + H^+ + H fragmentation channels upon single-photon double ionization of NH_3
      Thorsten Weber
      December 30, 2023
  ==================================================================================================================================
  
  
  In this paper we study stability patterns in the homology of the ordered configuration space of the once-punctured torus. In the last decade Church <cit.> and Church–Ellenberg–Farb <cit.> proved that the homology groups of the ordered configuration space of a connected noncompact orientable manifold stabilize in a representation theoretic sense as the number of points in the configuration grows, with respect to a map that adds each new point “at infinity." Miller and Wilson <cit.> proved that there is a secondary representation stability pattern among the unstable homology classes, with respect to adding a pair of orbiting points “near infinity." This pattern is formalized by considering sequences of homology classes as FIM^+-modules. We prove that, as FIM^+-modules, the sequence of “new" homology generators in the n^th homology of the ordered configuration space of 2n-2 points on the once-punctured torus is neither “free" nor “stably zero." We also show that this sequence is generated by homology classes on at most 4 points. Our proof uses Pagaria's work on the Betti numbers of the ordered configuration space of the torus <cit.> to calculate the growth rate of the Betti numbers of the ordered configuration space of the once-punctured torus. Our computations are the first to demonstrate that secondary representation stability is a non-trivial phenomenon in positive-genus surfaces.
  
  
  
  
  
  
  
  
  § INTRODUCTION
  
  
  For a topological space X, let 
  
      F_n(X):={(x_1, …, x_n)|x_i∈ X, x_i≠ x_j if i≠ j}⊆ X^n
  
  denote the ordered configuration space of n distinct points on X. When X=^d, the homology groups of F_n(^d) are isomorphic to the space of n-ary operations of the operad 𝒫ois^d(n); for more see, for example, Sinha's expository paper <cit.>. For most other manifolds explicit descriptions of the homology groups of their ordered configuration spaces are unknown. In his recent paper, Pagaria <cit.> proved that for k≥ 3, the k^th Betti number of the ordered configuration space of the torus is a polynomial of degree 2k-2 in the number of marked points, and that the 0^th, 1^st, and 2^nd Betti numbers are polynomials of degree 0, 1, 3, respectively, in the number of marked points. We use that result to prove
  
  
  Let T^∘ denote the once-punctured torus. Then, for k≥ 3, the k^th Betti number of F_n(T^∘) is a polynomial in n of degree 2k-2. For k=0,1,2, the k^th Betti number of F_n(T^∘) is a polynomial in n of degree 0,1,3, respectively.
  
  
  If X is an noncompact manifold such as the once-punctured torus, the ordered configuration spaces have additional structure: if (X)=d, there is an embedding
  
      e:X⊔^d↪ X.
  
  Figure <ref> provides an example of such an embedding (e.g., see Kupers and Miller <cit.>).
  
  
  
  The embedding induces an inclusion of ordered configuration spaces
  
      ι:F_n-1(X)↪ F_n(X)
  
  
      ι(x_1, …, x_n-1)↦ (e(x_1), …, e(x_n-1), e(0)),
  
  where 0 denotes the origin in ^d. See Figure <ref>. Thus, ι maps a configuration of n-1 points in X to its image under e, and it adds a new point corresponding to the image of the origin in ^d under e.
  
  
  
  The inclusion ι is an example of a more general result: the embedding e induces inclusions on the product of the ordered configuration spaces of X and ^d
  
      F_n(X)× F_m(^d)↪ F_n+m(X).
  
  
  In this paper we only consider one of these maps, which we will call ι' 
  
      ι':F_n-2(X)× F_2(^d)→ F_n(X)
  
  where
  
      ι'((x_1, …, x_n-2), (x'_n-1, x'_n)) = (e(x_1), …, e(x_n-2), e(x'_n-1), e(x'_n)).
  
  
  Figure <ref> provides an example of ι'.
  
  
  
  The inclusions ι and ι' induce maps on homology:
  
      ι_*:H_k(F_n-1(X))→ H_k(F_n(X))andι'_*:H_k-1(F_n-2(X))⊗ H_1(F_2(^d))→ H_k(F_n(X)).
  
  
  The symmetric group S_n acts on the configuration space F_n(X) by permuting the coordinates, and this induces an action of S_n on H_k(F_n(X);) for all k. Thus, we can view the homology groups of the ordered configuration space on X as symmetric group modules. When n is large with respect to k, the homology groups of the ordered configuration space of a noncompact manifold X are representation stable in the following sense.
  
  
  (Church–Ellenberg–Farb <cit.> in the orientable case and Miller–Wilson <cit.> in the general case) Let X be a connected noncompact d-manifold with d≥ 2. For k≤n-1/2,
  
      [S_n]·ι_*(H_k(F_n-1(X);))=H_k(F_n(X);).
  
  
  
  We define another map, which we also denote ι'_*, that leads to a notion of secondary representation stability
  
      ι'_*:H_k-1(F_n-2(X))→ H_k(F_n(X)).
  
  This map can viewed as a restriction of the other map ι'_*, and is obtained by pairing a homology class in H_k-1(F_n-2(X)) with the class in H_1(F_2(^d)) corresponding to the point n orbiting the point labeled n-1 counterclockwise. Miller and Wilson <cit.> were able to show that if n is large with respect to k, the homology groups of the ordered configuration space of a noncompact manifold were secondary representation stable in the sense that they satisfied the following theorem.
  
  
  (Miller–Wilson <cit.>) Let X be a connected noncompact finite type d-manifold, with d≥ 2. There is a function r:_≥ 0→_≥ 0 tending to infinity such that for k≤n-1/2+r(n),
  
      [S_n]·(ι_*(H_k(F_n-1(X);))+ι'_*(H_k-1(F_n-2(X), )))= H_k(F_n(X); ).
  
  
  
  Combining first and second order representation stability we see that there is a range where the k^th homology group of the ordered configuration space of n points on X is determined by the (k-1)^th homology group of n-2 points on x and the k^th homology group of n-1 points on X. See Figure <ref>. In Section <ref> we will see that secondary representation stability can be thought of as a condition on the generation degree of certain “FIM^+-modules." 
  
  
  
  
  Miller and Wilson also calculated which parts of H_k(F_n(X); ) came from just the [S_n]-span of the image of ι'_* in several cases.
  
  
  
  (Miller–Wilson <cit.>) 
  
      ([S_2n] ·ι'_*(H_n-1(F_2n-2(^2);)))/ ([S_2n]·ι_*(H_n(F_2n-1(^2);))) ≅⊕_λ∈ D_2nV_λ
  
  as S_2n representations, where V_λ is the irreducible S_2n-representation associated to the partition λ, and λ is in D_2n, if and only if when the associated Young diagram is cut in two along the upper staircase, the resultant two skew subdiagrams are symmetric under reflection in the line of slope -1. 
  
  
  
  
  (Miller–Wilson <cit.>) Let X be a connected noncompact surface. If X is not orientable or of genus greater than zero, then
  
      H_0(F_0(X);) ≅,
  
  and for n>0
  
      ([S_2n] ·ι'_*(H_n-1(F_2n-2(X);)))/ ([S_2n]·ι_*(H_n(F_2n-1(X);))) ≅ 0.
  
  
  
  In the first example, the structure of the homology groups had previously been determined, and, as we will see, this quotient is a “free FIM^+-module"; in the second case the classes of interest stabilize to zero. This raises the question: are there spaces such that the homology arising solely from the [S_n]-span of ι_*' of their ordered configuration space are secondary representation stable, while being neither free as FIM^+-modules nor eventually zero?
  
  We answer this in the affirmative. The sequence of n^th homology group generators of the the ordered configuration space of 2n-2 points on the once-punctured torus T^∘ is secondary representation stable, while being neither “free" nor eventually zero. 
  
  
  
  Let T^∘ denote the once-punctured torus. For n≥ 2,t
  
      [S_2n-2]·ι_*(H_n(F_2n-3(T^∘);))≠ H_n(F_2n-2(T^∘);).
  
  For n≥4,
  
      [S_2n-2]·(ι_*(H_n(F_2n-3(T^∘);))+ι'_*(H_n-1(F_2n-4(T^∘), ))) = H_n(F_2n-2(T^∘);).
  
  Moreover, the sequence
  
      {([S_2n-2]ι'_*(H_n-1(F_2n-4(T^∘), )))/([S_2n-2]ι_*(H_n(F_2n-3(T^∘), )))}_n∈_≥ 1
  
  is not a free FIM^+-module.
  
  
  
  
   §.§ Acknowledgements
  
  Jennifer Wilson was of utmost help in understanding <cit.> and writing this paper. I would like to thank John Wiltshire-Gordon for sharing his computations of small-degree Betti numbers of F_n(T^∘) with Miller and Wilson; these computations were an indirect inspiration for this paper. I would also like to thank Jeremy Miller, Ben Knudsen, Zachary Himes, and Bradley Zykoski for their insightful comments on this paper.
  
  
  
  § FB-MOD, FI-MOD, AND FIM^+-MOD
  
  
  
  We introduce the categories FB-mod, FI-mod, and FIM^+-mod. We use these categories to formalize the notion of second order representation stability. One could rephrase the definitions and results for FI-mod and FIM^+-mod in the language of (skew)-twisted commutative algebras; for some examples of this see <cit.> and <cit.>.
  
  
  Let FB be the category whose objects are all finite (possibly empty) sets and whose morphisms are bijective maps.
  
  
  Every finite set is isomorphic to [n] := {1, …, n} for some n; a choice of such isomorphisms for all finite sets provides an equivalence between FB and its full subcategory that has one set [n] for each n∈_0^+.
  
  
  The category of FB-modules over a ring R has covariant functors from FB to the category of R-modules as objects, and natural transformations between these functors as morphisms.
  
  
  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≃End_FI([n]); therefore, we can view {W_n} as a sequence of symmetric group representations.
  
  
  Let FI be the category whose objects are all finite (possibly empty) sets and whose morphisms are injective maps.
  
  
  Just as for FB, there is an equivalence between FI and its full subcategory that has one set [n] for each n∈_≥0^+.
  
  
  The category of FI-modules over the ring R has covariant functors from FI to the category of R-modules as objects, and natural transformations between these functors as morphisms. Similarly, objects of the the category of FI-(homotopy)-space are covariant functors from FI to the (homotopy)-category of topological spaces, and the morphisms are natural transformations between these functors.
  
  
  Much like an FB-module, an FI-module is a sequence of symmetric group representations; however, there are nontrivial maps between these representations. Let V be an FI-module, if ι_n,m, n<m, denotes the standard inclusion of [n] into [m], then (ι_n, m)_*:V_n→ V_m must be S_n-equivariant; moreover, (ι_n, m)_*(V_n) must be invariant under the action of S_[m]-[n].
  
  
  An FI-submodule V' of an FI-module V, is a sequence of symmetric group representations {V'_n}⊆{V_n} that is closed under the action of the FI-morphism.
  
  
  We want to build an FI-module out of an FB-module. Let  
  
      M(-)_n :=R[Hom_FI([-],[n]);
  
  letting n vary over the nonnegative integers makes this an FI-module. If W_d is an S_d-representation, we can define M(W_d) by setting
  
      M(W_d)_n:=W_d⊗_R[S_d]M(d)_n.
  
  Letting n vary over the nonnegative integers one can check 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 degree of an FB-module W; this is a functor from FB-mod to FI-mod.
  
  
  Given an FB-module W, let M(W) denote the associated free FI-module
  
      M(W):=⊕_d≥ 0M(W_d).
  
  
  
  
  An FI-module V is generated by a set S⊆∐_n≥ 0 V_n if V is the smallest FI-submodule containing S. If there is some finite set S that generates V, then V is finitely generated. If V is generated by ∐_0≤ n≤ dV_n, then V is generated in degree ≤ d. 
  
  
  We want to recover a generating set for an FI-module V, preferably a minimal one. Such a generating set consists of subrepresentations of V_n, for all n, not arising from the FI-structure in smaller degrees. We use the language of FI-homology to formalize this.
  
  
  The zeroth FI-homology group of an FI-module V in degree n, denoted H_0^FI(V)_n, is the set of S_n-representations in V_n not arising from the image of V_n-1 under the maps induced by all inclusions f:[n-1]↪ [n]:
  
      H_0^FI(V)_n:= V_n\⊕_f:[n-1]↪ [n]f_*V_n-1.
  
  
  
  Note that H_0^FI(V)_n is an S_n-representation, so {H_0^FI(V)}_n is a sequence of symmetric group representations, i.e., an FB-module. Thus, we can think of H_0^FI(-) as a functor from FI-mod to FB-mod.
  
  
  A based set S_* is a set with a distinguished element *∈ S_*, the basepoint. A map of based sets f:S_*→ T_* takes *∈ S_* to *∈ T_*. We define FI# to be the category whose objects are finite  based sets and whose morphisms are maps of based sets that are injective away from the basepoint, i.e., if f:S_*→ T_* is an FI#-morphism, then |f^-1(t)|≤ 1 for all t∈ T_*, t ≠ *.
  
  
  
  The category of FI#-modules over the commutative ring R has covariant functors from FI# to the category of R-modules as objects, and natural transformations as morphisms. We can similarly define the category of FI#-homotopy-spaces.
  
  
  An FI#-module can be viewed as an FI-module by forgetting the morphisms in FI# that are not injections. Similarly, an FI#-module can also be seen to be an FI^op-module by considering only surjective morphisms in FI#. Moreover, FI# is equivalent to FI#^op.
  
  
  (Church–Ellenberg–Farb <cit.>)  The category of FI#-modules is equivalent to the category of FB-modules via the equivalence of categories
  
      M(-):FB-Mod⇆FI#-Mod :H_0^FI(-).
  
  Thus, every FI#-module V is of the form ⊕^∞_n=0M(H_0^FI(V)_n).
  
  
  Recall the definition of the ordered configuration space of n points on a open manifold X and the inclusion map ι:F_n-1(X)↪ F_n(X) given in the introduction. The inclusion ι is well-behaved up to homotopy with respect to the symmetric group action, making F_*(X) an FI-homotopy-space. Taking the homology groups of these ordered configuration spaces gives us a sequence of FI-modules: for fixed k≥ 0, H_k(F_*(X)) is an FI-module. We can say even more, namely that the forgetful map 
  
      π:F_n(X)→ F_n-1(X),
  
  given by forgetting the last coordinate
  
      π(x_1, …, x_n)=(x_1, …, x_n-1),
  
  is well behaved with respect to the symmetric group action, and F_*(X) is an FI#-homotopy-space. Fixing k, this makes H_k(F_*(X)) an FI#-module. By Theorem <ref>, the zeroth FI-homology of these homology groups, H^FI_0(H_k(F_*(X))), is an FB-module. 
  
  For a thorough overview of FI and FI-mod see <cit.> or <cit.>.
  
  
  A matching of a set A is a set of disjoint 2-element subsets of A, and a matching is said to be 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 perfect matching on the complement of the image we get the category FIM.
  
  
  Let FIM denote the category whose objects are finite sets and whose morphisms are injective maps f:A↪ B along with a perfect matching on B\ f(A).
  
  
  Morphisms between two objects A, B of FIM exist only when |A| and |B| have the same parity. If there are morphisms from A to B and we insist on an ordering of the perfect matching, then the symmetric group S_m, m=|B|-|A|/2, acts on an ordered perfect matching B_1, …, B_m on the complement of the image of A in B by permuting the ordering:
  
      σ·(B_1, …, B_m)= (B_σ(1), …, B_σ(m)).
  
  
  This inspires the definition of FIM^+, a category enriched over R-mod.
  
  
  Let FIM^+ be the category whose objects are finite sets and whose module of morphisms f, consist of injective maps with an ordered perfect matching on the complement quotiented by a signed symmetric group action:
  
      R⟨ (f:A→ B, B_1, …, B_m)|f  is injective,  |B_i|=2, B=Im(f)⊔ B_1⋯⊔ B_m⟩/⟨ (f, B_1, …, B_m)=sign(σ)(f, B_σ(1), …, A_σ(m))  for all σ∈ S_m⟩.
  
  
  
  
  The category of FIM^+-modules over the ring R has covariant functors from FIM^+ to the category of R-modules as objects, and natural transformations between these functors as morphisms.
  
  
  
  An FIM^+-submodule W' of an FIM^+-module W is a sequence of symmetric group representations W'_n⊆ W_n that is closed under the action of the FIM^+-morphism.
  
  
  
  An FIM^+-module W is generated by a set S⊆∐_n≥ 0 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 ∐_0≤ n≤ dV_n, then W is generated in degree ≤ d. 
  
  
  Let
  
      M^FIM^+(-)_n:= R[Hom_FIM^+([-], [n])].
  
  By letting n vary we see that this takes in an integer and produces an FIM^+-module. We can extend this to a functor from the category of S_d-representations by setting
  
      M^FIM^+(W_d)_n:=W_d⊗_R[S_d]M^FIM^+(d)_n.
  
  
  By taking a sequence of symmetric group representations {W_d} we can define free FIM^+-modules.
  
  
  Given an FB-module W, let M^FIM^+(W) denote the associated free FIM^+-module
  
      M^FIM^+(W):=⊕_d≥ 0M^FIM^+(W_d).
  
  
  
  Note that
  
      (M^FIM^+(0)_2n)=(2n)!/n!2^n,
  
  and
  
      (M^FIM^+(W_d)_2n)=2nd(W_d)(M^FIM^+(0)_2n-d).
  
  
  When X is a noncompact manifold, we let 𝒲_i^X(n) denote the sequence of minimal generators of the homology groups of its ordered configuration space
  
      𝒲_i^X(n):=H^FI_0(H_n+i/2(F(X);))_n.
  
  
  By setting non-integer homology to be 0, it follows that there is an S_n-action on 𝒲_i^X(n) for all n≥ 0. Thus, 𝒲_i^X is an FB-module. Miller and Wilson proved that it also has an FIM^+-module structure <cit.>. Additionally, they proved that as an FIM^+-module it is secondary representation stable in the sense that it satisfies the following theorem, a reformulation of Theorem <ref>.
  
  
  (Miller–Wilson <cit.>) If K is a field of characteristic zero and X is a connected noncompact d-manifold of finite type and d≥ 2, then, for each i≥ 0, the sequence of minimal generators
  
      𝒲_i^X(n)=H^FI_0(H_n+i/2(F(X);K))_n
  
  is finitely generated as an FIM^+-module.
  
  
  To prove this theorem, Miller and Wilson used the complex of injective words, the arc resolution spectral sequence, and a Noetherianity result of Nagpal, Sam, and Snowden <cit.>. In addition to proving this theorem, they computed some explicit examples of secondary representation stability.
  
  
  (Miller–Wilson <cit.>) 
  
      𝒲_0^^2(2n)≅ M^FIM^+(0)_2n≅⊕_λ∈ D_2nV_λ
  
  where V_λ is the irreducible S_2n representation corresponding to the partition λ, and λ is in D_2n if and only if when the associated Young diagram is cut in two along the upper staircase, the resultant two skew subdiagrams are symmetric under reflection in the line of slope -1.
  
  
  
  (Miller–Wilson <cit.>) Let X be a connected, noncompact surface. If X is not orientable or of genus greater than zero, then
  
      𝒲_0^X(0) ≅ and 𝒲_0^X(2n)≅ 0 for n>0.
  
  
  
  These explicit examples of secondary representation stability are edge cases. When X=^2 the FIM^+-module 𝒲_i^^2(2n) is a free FIM^+-module for all i≥ 0 and all the homology groups of F_*(^2) are known. For a general connected surface X, it was known that 𝒲_0^X(0) = H^FI_0(H_0(F(X)))_0=H_0(F_0(X))≅. For n>0, 𝒲_0^X(2n)=0, i.e., stably zero. This is the opposite extreme of 𝒲_0^^2(2n) being free. 
  
  All examples of secondary representation stability where the FIM^+-module structure is known, including the ones above, are either free FIM^+-modules or stably zero. We seek an example of secondary representation stability that lies between these extremes; namely, a finitely generated FIM^+-module arising from the homology groups of the ordered configuration space of a surface that is neither free nor stably zero.
  
  
  
  § ARC RESOLUTION SPECTRAL SEQUENCE
  
  
  
  In this section we describe the arc resolution spectral sequence that Miller and Wilson used to prove their secondary representation stability results. We state a few important results on the differentials of this spectral sequence, which we will later use to bound the generation degree of two FIM^+-modules of the form 𝒲_i^T^∘(n).
  
  The arc resolution spectral sequence arises from the augmented semi-simplical space Arc_*(F_*(M)), see <cit.> for a definition and <cit.> for more details about this spectral sequence. For more about spectral sequences arising from augmented semi-simplical spaces see, for example, Randal-Williams <cit.>.
  
  
  
  (Miller–Wilson <cit.>) Let M be a noncompact connected smooth d-manifold d≥ 2. There is a spectral sequence called the arc resolution spectral sequence that satisfies:
  
      E^2_p, q[M](n)≅Ind^S_n_S_p+1× S_n-p-1𝒯_p+1⊠ H_0^FI(H_q(F(M)))_n-p-1,
   
  where 𝒯_p is a free abelian group of rank ∑^p_i=0(-1)^ip!/i!. Moreover, 
  
      E^∞_p,q[M](n)=0  for p+q+2 ≤ n.
  
  See Figure <ref>.
  
  
  
  
  The differentials of the arc resolution spectral sequence satisfy a Leibniz rule.
  
  
  
  (Miller–Wilson <cit.>) Let M be a noncompact connected smooth d-manifold d≥ 2. There exist maps
  
      t^r:E^r_p,q[^d](S)⊗ E^r_p',q'[M](T)→ E^r_p+p'+1, q+q'[M](S⊔ T)
  
  arising from the embedding of Arc_p(F_S(^d))×Arc_p'(F_T(M))→Arc_p+p'(F_S⊔ T(M)) given in <cit.>. The maps t^r satisfy a Leibniz rule with respect to the differentials: if a∈ E^r_p,q[^d](S) and b∈ E^r_p',q'[M](T), then
  
      d^r(t^r(a⊗ b))=t^r(d^r(a)⊗ b)+(-1)^p+qt^r(a⊗ d^r(b)).
  
  
  
  When n = 2 or 3, the group 𝒯_n is isomorphic a free group of rank n-1, with -basis [1,2] and [[1,2],3], [[1,3],2], respectively, by <cit.> and <cit.>. These are Reutenauer's basis elements for ℒ_n, a subgroup of 𝒯_n. For more information on 𝒯_n and ℒ_n see <cit.>. There are explicit formulae for differentials in the arc resolution spectral sequence when applied to Reutenauer's basis elements.
  
  
  
  (Miller–Wilson <cit.>) Let E^r_p,q[M](n) be the arc resolution spectral sequence, and let
  
      L=[[[⋯[m_1, m_2], m_3],…],m_n]∈ E^1_k-1,0[M](n).
  
  Then d^r(L)=0 for r<k and d^n(L) is the homology class of ψ(⋯ψ(ψ(m_1, m_2), m_3), …, m_n) in E^n_-1, n-1[M](n), where the m_i are distinct points in M and ψ(x,y) denotes y orbiting x. See Figure <ref> for a representative of ψ(ψ(m_1, m_2), m_3)) in F_3(T^∘).
  
  
  We can use the Leibniz rule and our knowledge of the differentials from ℒ_n to calculate many of the differentials of the arc resolution spectral sequence and relate them to FIM^+-morphisms.
  
  
  
  Let M be a noncompact 2-manifold. The E^2[M]-page differential from the 1^st column to the -1^st column of the arc resolution spectral sequence,
  
      d^2:E^2_1,q[M]→ E^2_-1,q+1[M],
  
  is the FIM^+-morphism arising from the standard inclusion [n-2]↪[n].
  
  
  
  Note that
  
      E^2_1,q[M](n) = Ind^S_n_S_2× S_n-2𝒯_2⊠ H_0^FI(H_q(F(M)))_n-2.
  
  Since M is 2-dimensional,
  
      E^2_1,q[M](n) = t^2(E^2_1, 0[^2](2)⊗ E^2_-1,q[M](n-2)).
  
  For p<-1, all the elements E_p,q[M] of the arc resolution spectral sequence are 0, so
  
      d^2(E^2_-1,q[M](n-2))=0.
  
  Thus Lemma <ref>, shows
  
      d^2(E^2_1,q[M](n))=d^2(t^2(E^2_1, 0[^2](2)⊗ E^2_-1,q[M](n-2)))=t^2(d^2(E^2_1, 0[^2](2))⊗ E^2_-1,q[M](n-2) )
  
  By Lemma, <ref> the image of E^2_1, 0[^2](2) under d^2 corresponds to two points orbiting each other in H_1(F_2(^2))_2. Therefore, the image of
  
      t^2(d^2(E^2_1, 0[^2](2))⊗ E^2_-1,q[M](n-2) )⊂ E^2_-1, q+1[M](n)
  
  corresponds to the cup product of the homology of two points orbiting each other in H_1(F_2(^2))_2 and the homology classes of E^2_1,q[M](n-2)=H_0^FI(H_q.(F(M)))_n-2 in H_0^FI(H_q+1.(F(M)))_n. This is the FIM^+-morphism arising from the inclusion [n-2]↪[n].
  
  
  
  
  § ORDERED CONFIGURATION SPACE OF THE ONCE-PUNCTURED TORUS
  
  
  
  We begin this section by using Pagaria's calculation of the growth rate of the Betti numbers of the ordered configuration space of the torus <cit.> to calculate the growth rate of the Betti numbers of the ordered configuration space of the once-punctured torus. We use this result and the arc resolution spectral sequence to prove an FIM^+-module version of Theorem <ref>.
  
  
  (Pagaria <cit.>) For k≥ 3 the k^th Betti number of F_n(T) is of the form
  
      b_k(F_n(T))=c_kn2k-2+o(n^2k-2),
  
  where c_k is a constant greater than or equal to 2k-3k-3.
  
  For k≤ 5 the Betti numbers are
  
      b_0(F_n(T)) = 1,
  
  
      b_1(F_n(T))  = 2n,
  
  
      b_2(F_n(T))  = 2n3+3n2+n,
  
  
      b_3(F_n(T)) = 14n4+8n3+2n2,
  
  
      b_4(F_n(T)) = 32n6+74n5+33n4+5n3,
  
  
      b_5(F_n(T)) = 63n8+427n7+490n6+154n5+18n4.
  
  
  
  We prove a similar result for the ordered configuration space of the once-punctured torus by noting that the torus is additive group.
  
  
  Let T^∘ denote the once-punctured torus. For k≥ 3, the k^th Betti number of F_n(T^∘) is a polynomial in n of degree 2k-2.  For k≤ 5 the formulae for the k^th Betti numbers are
  
      b_0(F_n(T^∘)) = 1,
  
  
      b_1(F_n(T^∘)) = 2n,
  
  
      b_2(F_n(T^∘))
      =2n3+5n2,
  
  
      b_3(F_n(T^∘))
      = 14n4+18n3,
  
  
      b_4(F_n(T^∘)) = 32n6+106n5+79n4,
  
  
      b_5(F_n(T^∘)) = 63n8+490n7+853n6+432n5.
  
  
  
  
  Since T≃^2/ ^2 is an additive group, we can decompose the ordered configuration space of the torus as a product:
  
      F_n(T)≃ T× F_n-1(T^∘)
      
      (x_1, …, x_n)↦ x_1× (x_2-x_1, …, x_n-x_1),
  
  where coordinates in F_n-1(T^∘) are taken modulo ^2, and x_1 is the location of the puncture in T^∘. See, for example, Cohen <cit.>.
  
  Since Poincare polynomials respect product decompositions, we can write
  
      P(F_n(T))=P(T)× P(F_n-1(T^∘)) = (1+2t+t^2)P(F_n-1(T^∘)).
  
  
  These equations can be rearranged to give the Poincare polynomial for F_n-1(T^∘) in terms of the Poincare polynomial for F_n(T), 
  
      P(F_n-1(T^∘))=P(F_n(T))/1+2t+t^2=∑^∞_i=0(-1)^i(i+1)t^iP(F_n(T)),
  
  where the second equality arises by expanding (1+2t+t^2)^-1 as a Taylor series in t. 
  
  
  We have
  
      ∑_k=0^∞b_k(F_n-1(T^∘))t^k=(∑^∞_i=0(-1)^i(i+1)t^i)(∑_j=0^∞b_j(F_n(T))t^j) =∑^∞_k=0∑^k_m= 0(-1)^k-m(k+1-m)b_m(F_n(T))t^k.
  
  
  This gives us a formula for the Betti numbers of F_n-1(T^∘) in terms of the Betti numbers for F_n(T),
  
      b_k(F_n-1(T^∘))=∑^k_m= 0(-1)^k-m(k+1-m)b_m(F_n(T)).
  
  
  By replacing n-1 with n we see that
  
      b_k(F_n(T^∘))=∑^k_m= 0(-1)^k-m(k+1-m)b_m(F_n+1(T)),
  
  e.g., 
  
      b_0(F_n(T^∘)) = b_0(F_n+1(T)),
   
  
      b_1(F_n(T^∘))=b_1(F_n+1(T))-2b_0(F_n+1(T)).
  
  
      b_2(F_n(T^∘))= b_2(F_n+1(T))-2b_1(F_n+1(T))+3b_0(F_n+1(T)),
  
   etc.
  
  Now apply Theorem <ref>, which states that b_k(F_n(T)) is a polynomial in n of degree 2k-2, and for k=0,1,2, it is a polynomial of degree 0, 1, 3. Reindexing from n to n+1 does not change this since there cannot be any cancelation in the top degree. For k≥ 3, the above calculations prove that b_k(F_n(T^∘)) can be written as a linear combination of k+1 polynomials in n of degree ≤2k-2, and that only one of these polynomials is of degree 2k-2. When k=0, 1, or 2, b_k(F_n(T^∘)) can be expressed as linear combination of k+1 polynomials in n of degree ≤ 0,1, or  3, respectively, with only one polynomial in each of these sums having top degree. Therefore, the top degree term in the sum has nonzero coefficient. Thus, for k≥ 3, b_k(F_n(T^∘)) is polynomial in n of degree 2k-2, and, when k=0, 1, 2, it is polynomial in n of degree 0, 1, 3, respectively.
  
  To get the formula for b_k(F_n(T^∘)) for k≤5 apply the formula
  
      b_k(F_n(T^∘))=∑^k_m= 0(-1)^k-m(k+1-m)b_m(F_n+1(T))
  
  to the formulae for the Betti numbers of the ordered configuration space of the torus from Theorem <ref>.
  
  
  
  Since T^∘ is a noncompact manifold, we can apply the results of Church, Ellenberg, and Farb <cit.> and Miller and Wilson <cit.> to study sequences of minimal generators of the homology groups of its ordered configuration space.
  
  
  For all k≥ 3 there are FB-modules W dependent on k such that,
  
      H_k(F(T^∘);)=⊕_d=0^2k-2M(W_d),
  
  and
  
      H_0(F(T^∘);)=M(W_0),  H_1(F(T^∘);)=⊕_d=0^1M(W_d),and H_2(F(T^∘);)=⊕_d=0^3M(W_d);
  
  moreover, in each case the top W_d is nonzero.
  
  
  
  Since T^∘ is noncompact, H_k(F(T^∘); ) is an FI#-module. Theorem <ref> proves H_k(F(T^∘); ) is of the form M(W)=⊕_d≥ 0M(W_d), for some FB-module W. Recall 
  
      M(W_d)_n=W_d⊗_[S_d]·Hom_FI([d],[n]).
  
  From this we see that the dimension of M(W_d)_n is a polynomial in n of degree d:
  
      (M(W_d)_n)=(W_d)·nd.
  
  Therefore, the dimension of M(W)_n is a polynomial in n of degree d, with d the largest integer such that W_d≠ 0. By Proposition <ref>, for k≥ 3, b_k(F_n(T^∘) is a polynomial in n of degree 2k-2, and for k=0,1,2, b_k(F_n(T^∘) is a polynomial in n of degree 0, 1, 3 respectively. Therefore, for k≥ 3,
  
      H_k(F(T^∘);)=⊕_d=0^2k-2M(W_d),
  
  and
  
      H_0(F(T^∘);)=M(W_0),  H_1(F(T^∘);)=⊕_d=0^1M(W_d),and H_2(F(T^∘);)=⊕_d=0^3M(W_d),
  
  where the top degree term in each sum is nonzero.
  
  
  
  The FIM^+-module 𝒲_1^T^∘ is generated in degree 1. 
  
  
  
  By Proposition <ref> if k≥ 3,
  
      H_k(F(T^∘);)=⊕_d=0^2k-2M(W_d),
  
  and
  
      H_0(F(T^∘);)=M(W_0),  H_1(F(T^∘);)=⊕_d=0^1M(W_d),and H_2(F(T^∘);)=⊕_d=0^3M(W_d);
  
  moreover, in each case the top W_d is nonzero. By Theorem <ref> there is an equivalence H_0^FI(M(W))≅ W. Setting k=n+1/2, if n≥ 5, then k≥ 3 and
  
      H_n+1/2(F(T^∘); )=⊕^n-1_d=0M(W_d),and H^FI_0(H_n+1/2(F(T^∘); ))_n=0.
  
  If n=3, then
  
      H_2(F(T^∘); )=⊕^2_d=0M(W_d), and H^FI_0(H_2(F(T^∘); ))_2=W_2≠0.
  
  Similarly, if n=1, then 
  
      H_1(F(T^∘); )=⊕^1_d=0M(W_d), and H^FI_0(H_1(F(T^∘); ))_1=W_1≠0.
  
  
  
  Thus, 𝒲_1^T^∘(n)=H^FI_0(H_n+1/2(F(T^∘)))_n is nonzero only when n=1,3. Therefore, it suffices to check that 𝒲_1^T^∘(3) is generated by 𝒲_1^T^∘(1) as an FIM^+-module. 
  
  Note that 𝒲_1^T^∘(3) is a term in the arc resolution spectral sequence
  
      𝒲_1^T^∘(3)=H_0^FI(H_2(F(M)))_3=E^2_-1,2[T^∘](3).
  
  
  Since 𝒲_1^T^∘(1)=H_0^FI(H_1(F(M)))_1 and E^2_1,1[T^∘](3)=Ind^S_3_S_2× S_1𝒯_2⊠ H_0^FI(H_1(F(M)))_1, Proposition <ref> implies 𝒲_1^T^∘(3) is generated by 𝒲_1^T^∘(1) as an FIM^+-module if
  
      d^2(E^2_1,1[T^∘](3))=E^2_-1,2[T^∘](3).
  
  Since 
  
      p+q+2 = -1+2+2 =3 ≤ 3 = n,
  
  Proposition <ref> implies E^∞_-1,2[T^∘](3) =0. As the arc resolution spectral sequence is zero in the lower half plane, it follows that if
  
      d^3:E^3_2,0[T^∘](3)→ E^3_-1,2[T^∘](3)
  
  is the zero map, then 
  
      d^2(E^2_1,1[T^∘](3))=E^2_-1,2[T^∘](3)
  
  since for r≥ 2 the only entry of the spectral sequence with nonzero image under d^r mapping into E_-1,2^r[T^∘](3) would be E^2_1,1[T^∘](3), and this map must be surjective as E_-1,2^∞[T^∘](3)=0. See Figure <ref>. Since E^3_2,0[T^∘](3)= 𝒯_3=ℒ_3, Lemma <ref> implies the d^3 map from E^3_2,0[T^∘](3) being the zero map is equivalent to the image of two particles orbiting a third clockwise in T^∘ like a bullseye, as in Figure <ref>, being homologically trivial in H_2(F_3(T^∘)). 
  
  
  
  
  
  To see that two particles orbiting a third in F_3(T^∘) is homologically trivial, we show that there is an embedding of a 3-manifold with boundary in F_3(T^∘), whose boundary corresponds to two particles orbiting a third in T^∘. We start by embedding (S^1)^3 in (T^∘)^3. Identify T^∘ with (/)^2 where the puncture is located at (0,0), and let S^1 be identified with /. Consider the map
  
      f:(/)^3≅(S^1)^3→ (T^∘)^3≅((/)^2)^3
  
  
      f(a,b,c)=(1/2, a, 1/2+1/8cos(2π b), a+1/8sin(2π b), c, 1/2).
  
  
  The map f does not extend to an embedding of (S^1)^3 in F_3(T^∘). This can be seen by overlaying the three copies of T^∘ onto a single T^∘, as in the left side of Figure <ref>. To extend f to a map to F_3(T^∘), excise the region defined by the parameters 1/4<a<3/4, 0≤ b< 1, and 1/4<c<3/4, as in the right side of Figure <ref>. This is a solid torus S^1× D^2, where the S^1 arises from the parameter 0≤ b< 1, and D^2 is the region defined by the parameters 1/4<a<3/4, 1/4<c<3/4. The boundary of the resulting manifold is a 2-torus defined by the product of the boundary of the square given by restricting the parameters 1/4<a<3/4, 1/4<c<3/4 and the circle 0≤ b< 1. We show that this boundary corresponds to two particles orbiting a third particle counterclockwise, as in Figure <ref>.
  
  
  
  
  
  Fix a=3/4. The first particle is fixed at (1/2, 3/4), the second particle freely orbits it in a circle given by (1/2+1/8cos(2π b), 3/4+1/8sin(2π b)), and the third particle goes in a straight line from (1/4, 1/2) to (3/4, 1/2), i.e., underneath the orbiting pair from left to right. See the top left image in Figure <ref>.
  
  Now fix c=3/4, and let a vary from 3/4 to 1/4. This fixes the third particle at (3/4, 1/2), and sends the first particle from (1/2, 3/4) to (1/2, 1/4) along a straight line, and the second particle freely orbits the first while following it. This corresponds to the orbiting pair going going to the left of the third particle from top to bottom, as in the top right of Figure <ref>.
  
  Fix the first particle at (1/2, 1/4), while the second freely orbits it along (1/2+1/8cos(2π b), 1/4+1/8sin(2π b)). Let the third particle moves from (3/4, 1/2) to (1/4, 1/2) in a straight line. This corresponds to the third particle going above the orbiting pair from right to left, as in the bottom right of Figure <ref>.
  
  Finally we finish the square, by fixing c=1/4, i.e., holding the third particle at (1/4, 1/2). The first particle moves up by letting a go from 1/4 to 3/4, while the second particle freely orbits it. This corresponds to sending the orbiting pair from bottom to top to the right of the third particle. See the bottom left of Figure <ref>.
  
  This boundary corresponds to the third particle orbiting the orbiting pair, as in Figure <ref>. Since this is the boundary of a 3-manifold in F_3(T^∘), it is homologically trivial in H_2(F_3(T^∘)). It corresponds to the image of 𝒯_3=ℒ_3 under d^3, so 
      d^3(E^3_2,0[T^∘](3))=0.
  
  Therefore,
  
      d^2(E^2_1,1[T^∘](3))=E^2_-1,2[T^∘](3),
   
  proving that 𝒲_1^T^∘(3) is generated by 𝒲_1^T^∘(1) as an FIM^+-module, and 𝒲_1^T^∘(n) is generated in degree 1.
  
  
  Now we prove a partial FIM^+-module version of Theorem <ref>.
  
  
  The sequence of minimal generators
  
      𝒲^T^∘_2(n)=H^FI_0(H_n+2/2(F(T^∘); ))_n
  
  is finitely generated as an FIM^+-module in degree ≤ 4; moreover, it is stably nonzero.
  
  
  
  If k≥ 3, then Proposition <ref> proves
  
      H_k(F(T^∘);)=⊕_d=0^2k-2M(W_d),
  
  and W_2k-2 is nonzero. By Theorem <ref> there is an equivalence H_0^FI(M(W))≅ W. Thus, if n is an even number bigger than 3 and we set k=n+2/2, then k≥ 3, 2k-2=2(n+2/2)-2=n, and
  
      𝒲^T^∘_2(n)=H^FI_0(H_n+2/2(F(T^∘); ))_n≅ W_n≠ 0.
  
  Therefore, for even n≥ 3, 𝒲^T^∘_2(n) is nonzero, and the FIM^+-module 𝒲^T^∘_2(n) is stably nonzero.
  
  Next, we show that 𝒲_2^T^∘ is generated in degree at most 6. The E^2-page of the arc resolution spectral sequence for the once-punctured torus is of the form
  
      E^2_p,q[T^∘](n)=Ind^S_n_S_p+1× S_n-p-1𝒯_p+1⊠ H^FI_0(H_q(F(T^∘)))_n-p-1,
  
  
  so 𝒲_2^T^∘(n) is an element of the -1^st-column of the E^2-page:
  
      𝒲_2^T^∘(n) = H^FI_0(H_n+2/2(F(T^∘)))_n = Ind^S_n_S_0× S_n𝒯_0⊠ H^FI_0(H_n+2/2(F(T^∘)))_n =  E^2_-1,n+2/2[T^∘](n).
  
  
  We will show that for r≥ 2 the only differential d^r with nonzero image in E^r_-1,n+2/2[T^∘](n) is d^2, and that this differential is surjective; see Figure <ref>. For n≥ 8, and 0≤ q≤n-2/2, consider E^2_p,q[T^∘](n) such that p+q=n+2/2, i.e., the slope -1 diagonal starting at, but not including, E^2_1,n/2[T^∘](n). The differentials of the spectral sequence will map the subquotients of these elements of the spectral sequence into E_-1,n+2/2[T^∘](n), and elements on this diagonal are of the form
  
      E^2_n+2/2-q, q[T^∘](n)=Ind^S_n_S_n+2/2-q+1× S_n-2/2+q-1𝒯_n+2/2-q+1⊠ H^FI_0(H_q(F(T^∘)))_n-2/2+q-1.
  
  For q≥ 3,
  
      H_q(F(T^∘))=⊕_d = 0^2q-2M(W_d).
  
  Since n≥ 8 and 3≤ q≤n-2/2, it follows that 2q-2 < n-2/2+q-1, so the W_n-2/2+q-1-term in the decomposition is 0. Additionally, recall that
  
      H_0(F(T^∘))=M(W_0), H_1(F(T^∘))=⊕_d = 0^1M(W_d),and H_2(F(T^∘))=⊕_d = 0^3M(W_d);
  
  since n≥ 8, we have 3<n-2/2+2-1, 1< n-2/2+1-1, and 0<n-2/2+0-1, so the W_n-2/2+q-1-term in the decomposition is 0 in these cases too. This proves that for n≥ 8, and 0≤ q≤n-2/2
  
      H^FI_0(H_q(F(T^∘)))_n-2/2+q-1=0.
  
  Thus, for n≥ 8,  0≤ q≤n-2/2, and r≥ 2, E^r_n+2/2-q, q[T^∘](k)=0.
  
  
  
  Thus, for r≥ 2, the only differential, d^r, that can have non-zero image in E^r_-1,n+2/2[T^∘](n) is d^2, as this is the only differential with nonzero domain. Since n≥ 8, Proposition <ref> proves E_-1, n+2/2^∞[T^∘](n) = 0 as p+q+2=-1+n+2/2+2≤ n. Therefore,
  
      d^2(E^2_1,n-2/2[T^∘](n))= E^2_-1,n+2/2[T^∘](n).
  
  Proposition <ref> proves that d^2 from the 1^st column to the -1^st column corresponds to the FIM^+-module morphism induced from [n-2]↪[n]. Since d^2 is surjective for all n≥ 8, 𝒲_2^T^∘(n) is generated as a FIM^+-module by 𝒲_2^T^∘(n-2).
  
  Now we prove that 𝒲_2^T^∘(6) is generated as an FIM^+-module from 𝒲_2^T^∘(4). We first show that E^2_-1, 4[T^∘](6) is nonzero.
  
  Since 
  
      E^2_-1, 4[T^∘](6)=H^FI_0(H_4(F(T^∘)))_6=𝒲_2^T^∘(6),
  
  and by Proposition <ref>,
  
      H_4(F(T^∘))=⊕_d = 0^6M(W_d),with W_6≠0
  
  it follows that E^2_-1, 4[T^∘](6) is nonzero. Moreover, Proposition <ref> proves that
  
      H_1(F(T^∘))=⊕_d=0^1M(W_d)and H_0(F(T^∘))=M(W_0).
  
  Therefore,
  
      H^FI_0(H_1(F(T^∘)))_2=0and H^FI_0(H_0(F(T^∘)))_1=0,
  
  so
  
      E^2_3, 1[T^∘](6)=Ind^S_6_S_4× S_2𝒯_4⊠ H^FI_0(H_1(F(T^∘)))_2=0,
  
  and 
  
      E^2_4, 0[T^∘](6)=Ind^S_6_S_5× S_1𝒯_5⊠ H^FI_0(H_0(F(T^∘)))_1=0.
  
  Thus for r≥ 2,
  
      E^r_3, 1[T^∘](6)=0,
  
  and
  
      E^r_4, 0[T^∘](6)=0.
  
  If E^3_2,2[T^∘](6)=0, then, for r≥ 3, all differentials d^r into E^r_-1, 4[T^∘](6) have trivial image. Then, as p+q+2=-1+4+2≤6=n, Proposition <ref> would show that
  
      d^2(E^2_1,3[T^∘](6))=E^2_-1, 4[T^∘](6).
  
  This, along with Proposition <ref>, which states that the differential d^2 from the 1^st column to the -1^st column corresponds to FIM^+-morphism induced from [4]↪ [6], would prove that 𝒲_2^T^∘(6) is generated as an FIM^+-module from 𝒲_2^T^∘(4).
  
  We will show that the subset of E^2_4,1[T^∘](6), given by t^2(E^2_2,0[^2](3)⊗ E^2_1,1[T^∘](3)) surjects onto E^2_2,2[T^∘](6) under d^2, proving that E^3_2,2[T^∘](6)=0.
  
  The 0^th-column of the E^2-page of the arc resolutions spectral sequence is zero, so the d^2-differential to the 0^th-column of the arc resolution spectral sequence, 
  
      d^2:E^2_2,0[M](3)→ E^2_0,1[M](3)=0,
  
  must be the zero map for all manifolds M. In particular
  
      d^2:E^2_2,0[^2](3)→ E^2_0,1[^2](3)
  
  is the zero map. Proposition <ref> proves that 
  
      d^2:E^2_1,1[T^∘](3)→ E^2_-1,2[T^∘](3)
  
  is surjective. By Lemma <ref> the map t^2 satisfies a Leibniz rule, so
  
      d^2(t^2(E^2_2,0[^2](3)⊗ E^2_1,1[T^∘](3)))=t^2(E^2_2,0[^2](3)⊗ E^2_-1,2[T^∘](3)).
  
  Since E^2_2,2[T^∘](6)=Ind^S_6_S_3× S_3𝒯_3⊠ H_0^FI(H_2(F(T^∘)))_3, E^2_2,0[^2](3)= 𝒯_3, and E^2_-1,2[T^∘](3) = H_0^FI(H_2(F(T^∘)))_3, it follows from the definition of t that
  
      t^2(E^2_2,0[^2](3)⊗ E^2_-1,2[T^∘](3))= E^2_2,2[T^∘](6).
  
  Therefore,
  
      d^2(E^2_4,1[T^∘](6))= E^2_2,2[T^∘](6),
  
  and E^3_2,2[T^∘](6) =0. Thus, d^2 is the only nontrivial differential for r≥ 2 into E^r_-1, 4[T^∘](6), and 𝒲_2^T^∘(6) is generated as an FIM^+-module from 𝒲_2^T^∘(4). It follows that 𝒲_2^T^∘(n) is generated in degree at most 4 as an FIM^+-module.
  
  
  
  The sequence of minimal generators 𝒲_2^T^∘ is not a free FIM^+-module.
  
  
  
  
  Theorem <ref> proves 𝒲_2^T^∘ is generated in degree at most 4 and that 𝒲_2^T^∘(4)≠ 0. It follows from the definition of ordered configuration space that
  
      H_1(F_0(T^∘),)=0,
  
  so
  
      𝒲_2^T^∘(0)=H_0^FI(H_1(F(T^∘),))_0=0.
  
  Proposition <ref> proves that
  
      H_2(F(T^∘),)_1=0and H_2(F(T^∘),)_2≠0,
  
  so
  
      𝒲_2^T^∘(2)=H_0^FI(H_2(F(T^∘),))_2≠0.
  
  Therefore, to show that 𝒲_2^T^∘ is not free, it suffices to prove that M^FIM^+(𝒲_2^T^∘(2))⊈𝒲_2^T^∘.
  
  Proposition <ref> proves that (𝒲_2^T^∘(2))=5, so 
  
      (M^FIM^+(W_2)_4) = 42×(M^FIM^+(0))_4-2×(W_2)= 6× 1× 5=30.
  
  Since, (𝒲_2^T^∘(2)) = 14 <30 it follows that M^FIM^+(W_2)_4⊈𝒲_2^T^∘(4), and 𝒲_2^T^∘ is not a free FIM^+-module.
  
  
  This corollary, along with Theorem <ref>, provide an FIM^+-module version of Theorem <ref>. Moreover, they demonstrate the first example of a non-free, stably nonzero secondary representation stability sequence in positive genus, This is the first stably nonzero example for any manifold for which the homology groups of the ordered configuration spaces were not already explicitly known.
  
  amsalpha
  
  
  