Superposition of configurations and scanning
Andreas Stavrou



[
    [
    25 April 2024
=================


We endow the cohomology of configuration spaces of a manifold with a product arising from superposing configurations. We prove that, under the scanning isomorphism, this product corresponds to the cup-product of the section space of the standard scanning bundle of the manifold.




§ INTRODUCTION

Let M be the interior of a  compact connected d-manifold, possibly with boundary. We are interested in the (unordered) configuration spaces 

    C_n(M):={(m_1,...,m_n)∈M^n:m_i≠ m_j  if  i≠ j}/𝔖_n,

for n≥ 0,
where the symmetric group 𝔖_n acts by permuting the coordinates.

The homology of C_n(M) was classically studied using the scanning map, originally due to <cit.>, whose following incarnation is from <cit.>. 
Let T^+M be the fibrewise one-point compactification of the tangent bundle TM of M, and let T^+M∧_fS^2l be the fibrewise smash of T^+M with S^2l, so that T^+M∧_fS^2l is an S^d+2l-bundle over M. 
Denote by Γ_∂(M,S^2l) the space of its sections over M that map the boundary ∂ M constantly to the basepoint of the fibre.
[<cit.>]
For l≥ 1, there is an isomorphism of graded abelian groups

    s:H^*(Γ_∂(M,S^2l))⟶⊕_n≥ 0H^*(C_n(M))[2ln].



The domain of s is naturally a graded ring with the cup-product; our aim is to exhibit a meaningful interpretation of this ring structure on the target of s. Consider the coloured configuration space C_n,m(M) of n blue points and m red points, all distinct. There are maps 


    
        C_n(M)×C_m(M)     C_n,m(M)[hook,"i"']["p"]     C_n+m(M),
    


where i is the open inclusion, and p is the n+m  m-sheeted covering map which forgets the colours.
Combining Künneth, the cohomology map i^* and the transfer map p_!, we obtain the superposition product 
    sup_n,m:=p_!∘ i^*:H^*(C_n(M))⊗ H^*(C_m(M))⟶ H^*(C_n+m(M)),

for n,m≥ 0. In Section <ref>, we give an alternative but equivalent definition which justifies the name “superposition” and appears in <cit.>.

[Theorem <ref>]
The isomorphism s is multiplicative with the cup-product on H^*(Γ_∂(M,S^2l)) and the superposition product on ⊕_n≥ 0H^*(C_n(M))[2ln].



To prove Theorem <ref>, we analyse the topological definition of the map s following <cit.> which involves symmetric products. In Section <ref>, we set up the relevant preliminaries on symmetric products and multivalued functions.



 §.§ Acknowledgements
 The author would like to thank his PhD advisor Oscar Randal-Williams for the conversations which led to this paper. The paper was written while the author was affiliated with the University of Cambridge and funded by the Engineering and Physical Sciences Research Council (project reference: 2261124).



§ SYMMETRIC PRODUCTS AND MULTIVALUED FUNCTIONS



Given a based space (X,e), the symmetric product (X) is the free unital abelian monoid on X with identity e, topologised as the quotient

    (X):=(_n≥ 1 X^n/𝔖_n )/∼

where ∼ identifies (x_1,...,x_n,e)∼ (x_1,...,x_n). In other words, elements of (X) are formal sums ∑_i∈ I x_i with x_i∈ X and I a finite indexing set. If X is an unbased space, we set ^+(X)=(X⊔ *), with * the basepoint.


A multivalued function of based spaces is a based map (A,*)→ ((B),e). A multivalued function of unbased spaces is a map A→^+(B).




 §.§ Cohomology maps

We explain how multivalued functions induce cohomology maps[This trick was suggested by Oscar Randal-Williams.]. In literature, the Dold–Thom isomorphism <cit.> is usually used to define homology maps instead.

We fix the abelian monoid (S^n) as our model of K(,n) (it is indeed a model by the Dold–Thom theorem). Denoting by [X,Y]_* the set of based maps X→ Y up to based homotopy, we identify the functors of based spaces H^n(-)=[-,(S^n)]_*.
Every map α:Y→(S^n), representing a cohomology class [α]∈H^n(Y), factors uniquely through 


    
        Y[hook, "ι_Y"]    (Y)["α"]     (S^n), 
    


where ι_Y is the natural inclusion of Y into (Y) and α is a continuous homomorphism. Now, given a map f:X→(Y),
we define 
    f^*:H^*(Y)→H^*(X)
 by mapping [α]↦ [α∘ f]∈ [X,K(,n)]_*.

    Let h:(Y_1,y_1)→ (Y_2,y_2) be a based map so that the diagram of based maps
    
    (X_1,x_1)["g"]["f_1"]    ((Y_1),y_1)["(h)"]
    
                (X_2,x_2)["f_2"]     ((Y_2),y_2)

    commutes. Then f_1^*∘ h^*=g^*∘ f_2^*.


    Let α:(Y_2,y_2)→ ((S^n),e) represent a class in H^*(Y_2). Then we have (f_1^*∘ h^*)([α])=[α∘ h∘ f_1] and 
    (g^*∘ f_2^*)([α])=[α∘ f_2∘ g]=[α∘(h)∘ f_1]
 where the last equality is by the commutativity of diagram (<ref>). Now it suffices to see that the maps α∘ h and α∘(h) are equal: indeed, both maps are homomorphisms restricting to the same map α∘ h on the generating set Y_1 of (Y_1).


It is easy to check that if f is really a single-valued map f:X→ Y, then this definition of f^* agrees with the standard definition.


By “inverting” a finite-sheeted covering map p:X→ X, we obtain a multivalued function p^-1:X→^+(X) mapping

    x⟼∑_x∈ p^-1(x)x.

Similarly, if p:(X,e)→ ((X),e) is a covering space away from the basepoint, we get a based multivalued function p^-1:(X,e)→ ((X),e). The cohomology map (p^-1)^* coincides with the transfer maps p_! in both the based and unbased case.




 §.§ Products of maps

Given two based multivalued functions f:A→(B) and g:C→(D), there is a multivalued function f g:A∧ C→(B∧ D) given by

    (f g)(a,c)=∑_(i,j)∈ I× J (a_i,c_j)

where f(a)=∑_I a_i and g(c)=∑ c_i. 
More precisely, f g is the composition of the tautological continuous map 
    ι_X,Y:(X)∧(Y)→(X∧ Y),

and  
the continuous maps f∧ g:A∧ C→(X)∧(Y).

Under the identification H^n(-)=[-,(S^n)]_*, the cross product × is the composition 


    
        [X,(S
^m)]_*×[Y,(S
^n)]_*["-∧-"] 

        [X∧Y,(S
^m)∧(S
^n)]_*["ι_m,n∘-"] 

        [X∧Y,(S
^m+n)]_*,
    


where ι_m,n:=ι_S^m,S^n:(S
^m)∧(S
^n)→(S
^m+n) represents a generator of H^m+n((S
^m)∧(S
^n))≅.
We thus obtain a diagram

    H^*(C)⊗H^*(D) ["f^*⊗ g^*"]["×"]   H^*(A)⊗H^*(B) ["×"]
    H^*(C∧ D)["(f g)^*"]    H^*(A∧ B).


    The diagram (<ref>) commutes.


    We evaluate the two directions of the diagram on a representative α⊗β for a class in H^*(C)⊗H^*(D). Going right-then-down, we have
    
    α⊗β   ↦ (α∘ f )⊗ (β∘ g)
       ↦ι_n,m∘((α∘ f)∧ (β∘ g))=ι_n,m∘ (α∧β)∘ (f∧ g),

    whereas going down-then-right,
        
    α⊗β   ↦ι_n,m∘ (α∧β)
       ↦ι_n,m∘ (α∧β)∘ (f g).

    These are equal because the diagram 
    

        
                 C∧D["α∧β"]["ι_C∧ι_D"]     (S^n)∧(S^m)["ι_n,m"]    (S^n∧S^m)

                (C)∧(D)[ur,"α∧β"]["ι_C,D"]        

             A∧B["fg"][ur,"f∧g"]    (C∧D)[rruu,"ι_n,m∘(α∧β)"']         
        
    

    commutes: every tilded map is the unique homomorphic extension of its untilded counterpart.




 §.§ Suspensions and cofibrations

Let Σ X=S^1∧ X be the reduced suspension of a based space X. A based multivalued function f:(X,x)→ ((Y),y) induces a suspension map Σ f:=𝕀_S^1 f:(Σ X,Σ x)→ ((Σ Y),Σ y).

    The suspended map Σ f commutes with the suspension isomorphism, that is, if s_Z:H^*(Z)→H^*+1(Σ Z) is the suspension of space Z, then s_X∘ (Σ f)^*=f^*∘ s_Y. 


    Apply Proposition <ref> to C=A=S^1 and f=𝕀_S^1. The cross-product maps are the suspension isomorphisms in cohomology.



    Let 
        A_i["f_i"]    B_i["g_i"]     C_i
     be based cofibration sequences for i=1,2, and suppose the diagram of based maps 
    
    A_1["f_1"]["a"]    B_1["g_1"]["b"]     C_1["c"]
    (A_2)["(f_2)"]   (B_2)["(g_2)"]   (C_2)

    commutes. Then the induced cohomology maps a^*,b^*,c^* commute with the long exact sequences of the two cofibration sequences, that is the diagram
    
    ...["∂^*_2"]   H^*(C_2)["g_2^*"]["c^*"]    H^*(B_2)["f_2^*"]["b^*"]     H^*(A_2)["a^*"] ["∂_2^*"]    H^*+1(C_2)["g_2^*"]["c^*"]    ... 
    
            ...["∂^*_1"]   H^*(C_1)["g_1^*"]    H^*(B_1)["f_1^*"]    H^*(A_1)["∂_1^*"]    H^*+1(C_1)["g_1^*"]   ...

    commutes.


    The left two squares commute by Proposition <ref>, so we focus on the rightmost square. Let us recall the construction of the connecting homomorphism ∂_i^* using the reduced mapping cone CA_i of A_i. Since f_i are cofibrations, we have C_i=B_i/A_i and we have the sequence of maps  B_i/A_i   CA_i∪_A_iB_i [hook,"∼"', "ι"][two heads, "k"]    ΣA_i
    where ι collapses the cone and is a homotopy equivalence, whereas k collapses the subspace B. The connecting homomorphism ∂_i^* is the composition (ι^*)^-1∘ k^*∘ s_A_i. But all three composed maps commute with the induced cohomology maps by the naturality of this topological construction, by Proposition <ref>, and by Corollary <ref>.




§ SCANNING


We recall the set-up from <cit.> for the space-level model of the scanning isomorphism s. Let (M,M_0) be a manifold submanifold pair, and (X,x_0) a based space. The labelled configuration space C(M,M_0;X) is defined as 

    C(M,M_0;X)={{(m_i,x_i)}_i∈ I: I  finite set , m_i∈ M,x_i∈ X, m_i≠ m_j  if  i≠ j}/∼,

where ∼ is the relation generated by ξ_I∼ξ_I∪{(m,x)} if m∈ M_0 or x=x_0. We will denote a generic element of C(M,M_0;X) by ξ_I, and given a subset J⊂ I, we will write ξ_J:={(m_i,x_i)}_i∈ J. 

The space C:=C(M,M_0;X) has a filtration C_1⊂ ... ⊂ C_k⊂ C_k+1⊂ ...⊂ C where C_k contains labelled configurations of at most k points. The filtration quotients D_k=C_k/C_k-1 are naturally based at ∞, corresponding to when (m,x)∈ξ_I with m∈ M_0 or x=x_0. Define the wedge sum V=⋁_k≥ 1D_k. The scanning map is

    σ: C   ⟶(V)
    ξ_I   ⟼∑_J⊆ Iξ_J.

We define three further maps
    
        
  * Δ:C→ C∧ C, the diagonal map ξ↦ (ξ, ξ);
        
  * Φ: V →(V∧ V) mapping ξ_I↦∑_I=A⊔ B(ξ_A,ξ_B);
        
  * Ψ: V →(V∧ V) mapping ξ_I↦∑_I=A∪ B(ξ_A,ξ_B).
    
By abuse of notation, Φ and Ψ will also denote their natural extensions of the form (V)→(V∧ V). We thus obtain a square of maps 
    C["σ"] ["Δ"]    (V)["Φ", shift left =2.5pt]["Ψ"', shift right=2.5pt]
    
                C∧ C["σσ"]    (V∧ V),

which should be thought of as two diagrams, one with Φ and one with Ψ.


The cohomology map σ^*:H^*(V)→H^*(C) is an isomorphism.    


    This is analogous to the proof of Theorem 4.1 from <cit.>. Write V_k=∨_i=1^kD_i and observe that σ restricts to a multivalued function of cofibrations from C_k→ C_k+1→ C_k+1/C_k to V_k→ V_k+1→ V_k+1/V_k, in both of which the third space is D_k+1, of the form
    
    C_k[hook]["σ_k"]   
        C_k+1[two heads]["σ_k+1"]     D_k+1[hook,"ι_D_k+1"]
    (V_k)[hook]   (V_k+1)[two heads]   (D_k+1)
 
    The map σ_1^* is the identity map. We induct using the map of long exact sequences
    
    ...["∂^*"]   H^*(D_k)[equal]    H^*(V_k+1)["σ_k+1^*"]    H^*(V_k)["∂^*"]["σ_k^*"]     ...
    
                ...["∂^*"]   H^*(D_k)   H^*(C_k+1)   H^*(C_k)["∂^*"]     ...

     from Lemma <ref>, and the 5-lemma, to conclude that σ_k^* is an isomorphism for all k≥ 1. As C and V are the colimits of the filtrations {C_k} and {V_k}, σ^* is an isomorphism by the naturality of cohomology maps.




 §.§ Commutativity of diagram <ref>

 The diagram <ref> with Ψ commutes, that is Ψ∘σ=(σσ) ∘Δ.


    By direct computations 
    
    (Ψ∘σ)(ξ_I)=Ψ(∑_J⊆ Iξ_J)=∑_A,B,J⊆ I:A∪ B=J (ξ_A,ξ_B)

    and 
    
    (σσ) ∘Δ(ξ_I)=(σσ)(ξ_I,ξ_I)=∑_A,B⊆ I(ξ_A,ξ_B).

    The two summations agree: in the first summation J is fully determined by A and B and can be ignored.


The remaining of this section deals with the technicalities of proving the following.

    If M_0=∅ and X=S^2l is an even sphere, the maps Φ and Ψ are based homotopic.


For an integer k≥ 1, write [k]={1,...,k}. Let  ℐ={I_i: i∈ [k]} be a sequence of finite sets. The coloured configuration space on M of type ℐ is the space 
    C_ℐ(M):=F_⊔_i∈ [k]I_i(M)/∏_i∈ [k](I_i),

where F_S(M) is the ordered configuration on M of points labeled by S, and the product of symmetric groups acts ∏_i∈ [k](I_i) acts by permuting the corresponding co-ordinates. We may make a similar definition 
    D_ℐ(M,X)=F_⊔_i∈ [k]I_i(M)×_∏_i∈ [k](I_i) X^⊔_i∈ [k]I_i/∼,
 where ∼ collapses to one point, called ∞, all configurations with some x_i=x_0.  

    Let l>0, and A,B,C be three disjoint sets. If A is non-empty, then the map 
    
    μ:D_A, B, C(M,S^2l)   ⟶ D_A∪ B(M,S^2l)∧ D_A∪ C(M,S^2l),
    
            (ξ_A,ξ_B, ξ_C)   ⟼ (ξ_A∪ξ_B, ξ_A∪ξ_C)
 is based-null-homotopic.


   For brevity, we write D_ℐ=D_ℐ(M,S^2l). As observed in <cit.>, D_ℐ given any type ℐ,   is homeomorphic to the Thom space of the vector bundle
    
    η_ℐ:E_ℐ:= F_ℐ×_∏_i∈ [k](I_i)^2l⊗(⟨⊔_i∈ [k]I_i⟩) ⟶
                C_ℐ.

    Under these homeomorphisms, the map μ is identified with the Thomification, (f), of the bundle morphism
    

        
            E_A, B, C["η_A,B,C"]["f"]    
            E_A⊔B ×E_A ⊔C["η_A⊔B×η_A⊔C"] 
            

            C_A,B,C["f"]     C_A⊔B×C_A⊔C,
        
    

    which on F_ℐ×^2l⊗⟨⊔_i∈ [k]I_i⟩ is defined as the product of the map
    
    (x_A,x_B,x_C)⟼ (x_A ∪ x_B,x_A ∪ x_C),

    the identity on ^2l and the map
    
    ⟨ A⊔ B ⊔ C⟩   ⟶⟨ A⊔ B⟩⊕⟨ A⊔ C⟩
    
            (a⊕ b⊕ c)   ⟼((a⊕ b)⊕ (a⊕ c)),

    and descends after quotienting by the diagonal action. 
    Observe that f is an injective linear map on the fibres, which is a necessary condition for a morphism of vector bundles to give a map of Thom spaces. 

   We now factor (f,f) as the composition of two bundle maps both injective on the fibres. First, we include η_A,B,C into its stabilisation η_A,B,C⊕. Secondly, observe that for any finite set I, the vector space ⟨ I⟩ contains the (I)-invariant vector v_I:=∑_i∈ Ii. By also picking an arbitrary non-zero vector v∈^2l, we obtain the nowhere vanishing section s of η_A∪ B×η_A∪ C given by p⟼ (p,v⊗ (v_A∪ B⊕ (-v_A∪ C))). Furthermore, on every fibre, s does not lie in the image of f; this can be checked on the direct summand ⟨ A⟩ of each of ⟨ A⊔ B⟩ and ⟨ A⊔ C⟩: f takes the same values in both copies ⟨ A⟩, whereas s takes the same values with opposite signs. In summary, we have obtained the diagram of bundle maps
    

        
            E_A,B,C[hook]     E_A,B,C⊕["f⊕s"]     E_A∪B×E_A∪C

            C_A,B,C[equal]     C_A,B,C["f"]     C_A∪B×C_A∪C[bend left, "s"]
        
    

    where both squares are injective on fibres. We conclude that the map μ factors through
    
    (E_A,B,C)→(E_A,B,C⊕)≅(E_A,B,C)∧ S^1→( E_A⊔ B )∧(E_A ⊔ C).

    The canonical inclusion of (E_A,B,C) in its reduced suspension is based-null-homotopic, and thus so is μ.




    By the definition V=∨_p≥ 1D_p, we may write the smash V∧ V as the wedge ∨_q,r≥ 1D_q∧ D_r, so that (V∧ V)=∧_q,r≥ 1(D_q∧ D_r). As a result, the maps Φ,Ψ:V→(V∧ V) decompose to (p,q,r)-components  
    Φ^(p,q,r),Ψ^(p,q,r): D_p⟶(D_q∧ D_r).

    We will prove that these component maps are pairwise base-homotopic.
    
    To begin with, the component Φ^(p,q,r) is given explicitly by  
    Φ^(p,q,r)(ξ_I)=∑_A,B:|A|=q,
    |B|=r,
     A⊔ B=I(ξ_A,ξ_B),
 where |I|=p. The summation clearly vanishes if p≠ q+r. The component Ψ^(p,q,r) has the analogous formula 
    
    Ψ^(p,q,r)(ξ_I)=∑_A,B:|A|=q,
    |B|=r, 
    A∪ B=I(ξ_A,ξ_B),
 which  only differs from Φ^(p,q,r) in that the union A∪ B is no longer disjoint. This too clearly vanishes if q+r<p and, for p=q+r, we have Φ^(p,q,r)=Ψ^(p,q,r). Thus for all p≤ q+r, the components Φ^(p,q,r) and Ψ^(p,q,r) are in fact equal on the nose. 
    
    For p>q+r, Ψ^(p,q,r) factors as the composition
    
    D_p⟶(D_q+r-p, p-r,p-q)⟶(D_q∧ D_r)

    by first partitioning the set I of size p into three disjoint sets C,A',B' of sizes q+r-p, p-r, p-q, respectively, and then mapping ξ_C,A',B' to ξ_C∪ A'∧ξ_C∪ B'. This latter map is μ from Lemma <ref> and is based-null homotopic. Thus Ψ^(p,q,r) is based-homotopic to the trivial map which is equal to Φ^(p,q,r).



The diagram
    

        
            
            H^*(V∧V) ["(σσ)^*"]["Δ^*"]     H^*(C∧C)["Φ^*"]

            H^*(V)["σ^*"]     H^*(C).
        
    

    commutes.

    


§ THE SUPERPOSITION PRODUCT AND ITS RELATION WITH Φ

Let C_n,m(M) be the coloured configuration space of type {[n],[m]}, 
and D_n,m:=D_n,m(M,S^2d) its labelled version. 
The space C_n,m(M) is both an open subset of the product C_n× C_m, and a degree
n+m n 
covering map of C_n+m, giving the maps


    
        C_n×C_m     C_n,m[hook,"i"']["p"]     C_n+m
    


Using the cohomology transfer map p_!, we obtain the superposition product 
    sup_n,m=p_!∘ i^*: H^*(C_n× C_m)→ H^*(C_n+m).

We will suppress the index of sup.

We recall that D_n is a Thom space over C_n and D_n∧ D_m is also a Thom space over C_n× C_m. Further recall that the map Φ restricts to a multivalued function Φ^(n+m,n,m):D_n+m→(D_n∧ D_m). We will henceforth suppress the superscript of Φ.

    The diagram 
    

        
            H^*(C_n×C_m)["sup"]["Th","≅"']    H^*(C_n+m)["Th","≅"'] 

            H^*(D_n∧D_m)["Φ^*"]     H^*(D_n+m),
        
    

    commutes. (The degree shifts of the Thom isomorphisms are kept implicit.)


     Observe that D_n,m is also the Thom space of an oriented vector bundle over C_n,m. Defining the maps i and p in the obvious way, we get two pullback squares of bundles
     

    
    D_n∧D_m-{∞}    D_n,m-{∞} [hook,"i"]["p"]    D_n+m-{∞}

        C_n×C_m     C_n,m[hook,"i"]["p"]     C_n+m.
    


Therefore, the Thom isomorphisms commute with the compactified maps i^*_∞
and p^*_∞. Moreover, from Lemma <ref> proven below, as p is a covering space, there are transfer maps p_! and (p_∞)_!:H^*(D_n,m)→H^*(D_n+m), which also commute with the Thom isomorphisms.

Now p_!∘i^* is by definition the superposition product. But also (p_∞)_!∘i^*_∞ equals Φ^*, because Φ^(n+m,n,m) factors as the composition D_n+m→(D_n,m)→( D_n∧ D_m), where the first map is the inverse of the covering map p_∞ and the second map is (i_∞). In cohomology, we obtain the equality (p_∞)_!∘i^*_∞=Φ^*, implying the desired diagram commutes.




    Let π:E→ B be an oriented rank k vector bundle, and p:B→ B a degree n covering space. Then the multivalued function (E)→((p^*E)) inverse to the covering map p̅:p^*E→ E induces a transfer map p̅_!:H^*((p^*E))→H^*((E)) which commutes with the transfer map p_!, so that the diagram
    

        
            H^*+k((p^*E))["p̅_!"]     H^*+k((E))

            H^*(B)["","≅"'] ["p_!"]     H^*(B)[""]
        
    

    commutes.


    For both bundles F=E, p^*E, we replace the pair ((F),∞) by the pair ((F),(F)) which is isomorphic by a radial homeomorphism, so that p̅:((p^*E),(p^*E))→ ((E),(E)) is a degree n covering space of pairs. Under the excision isomorphisms p̅_!:H^*((p^*E),(p^*E))→ H^*((E),(E)) is equal to the transfer map defined in the classical way, namely on singular cochains, it is the dual of the chain map: 
    p̅^!:C_*((E),(E))   ⟶ C_*((p^*E),(p^*E))
    σ   ⟼∑_n σ

    where the summation is taken over all n p̅-preimages of the simplex σ. The transfer map p_! has a similar definition. We pick a cochain u_E∈ C^k((E),(E)) representing the Thom class of E→ B, so that p^*(u_E) represents the Thom class of p^*E→B. It suffices to check that the diagram
    

        
            C^i+k((E),(E))["p̅_!"]       C^i+k((E),(E))

            C^i(B)["π^*(-)⌣p̅^*u_E"]["p_!"]     C^i(B)["π^*(-)⌣u_E"]
        
    

    commutes, i.e. that for any class α, we have 
    π^*(p_!(α))⌣ u_E=p̅_!(π^*(α)⌣p̅^*u_E).

    But from the definition of the cup product we have p̅_!(π^*(α)⌣p̅^*u_E)=p̅_!(π^*(α))⌣ u_E, and then by evaluating on a simplices, p̅_!(π^*(α))=π^*(p_!(α)).    




§ PROOF OF THEOREM <REF>


For any l≥ 1, the multivalued function σ induces a multiplicative isomorphism 
    σ^*:H^*(C(M;S^2l))⟶⊕_n≥ 1H^*(C_n(M))[2ln]

with the cup-product in the domain and sup in the target.


For positive l, and a fixed i, the abelian group H^i-2ln(C_n(M))≅H^i(D_n) is non-zero only for finitely many n, which justifies the identification

    H^i(∨_n≥ 1D_n)≅⊕_n≥ 1H^i( D_n),

with a direct sum as opposed to a direct product. That σ^* is an isomorphism was proved in Proposition <ref>.

For the multiplicativity, we look at the following diagram (where C_n is shorthand for C_n(M) for economy of space)
   
    H^*(C)⊗H^*(C)["×"]["σ^*⊗σ^*"]
                   H^*(∨_n≥ 1D_n)⊗H^*(∨_n≥ 1D_n)   ["×"]["⊗"]
                   ⊕_n,m≥ 1H^*(C_n)⊗H^*(C_n)["×"]
                
    H^*(C∧ C)["(σσ)^*"]["Δ^*"]
                   H^*(∨_n≥ 1D_n∧∨_n≥ 1D_n)[""]["Φ^*"]
                   ⊕_n,m≥ 1H^*(C_n× C_m)["sup"]
                
    H^*(C)["σ^*"]
                   H^*(∨_n≥ 1D_n)[""]
                    ⊕_n≥ 1H^* (C_n).

    Here every square commutes: the top-left from Proposition <ref>, the bottom-left from Corollary <ref>, the top-right because the Thom isomorphism commutes with the cross-product, and the bottom-right by Proposition <ref>. Looking only at the corners of the large square, the leftmost vertical composition is the cup product and the rightmost column is the superposition product, whereas the top-most and bottom maps are the scanning maps. This commutativity proves exactly that σ^* respects the products.




There is an isomorphism of graded rings 
    s:(H^*(Γ_∂(M,S^2l)),⌣)⟶(⊕_n≥ 0H^*(C_n(M))[2nl],sup).



    The restricted map s:H^*(Γ_∂(M,S^2l))→⊕_n≥ 1H^*(C_n(M))[2nl] is an isomorphism of non-unital rings by combining Proposition 3.6 and Corollary 4.2 of <cit.>, with our Proposition <ref>. (That the target is a ring, i.e. that sup is associative and graded commutative, follows a posteriori since ⌣ has these properties.) Passing to the unreduced cohomology H^*(Γ_∂(M,S^2l)) adds back the unit to the cohomology ring. Similarly, the summand H^*(C_0(M))[0]≅[0] can be checked to be generated by the unit for sup, making the extended isomorphism s an isomorphism of unital rings. 





§ AN ALTERNATIVE DEFINITION OF THE SUPERPOSITION PRODUCT

We give an alternative definition of the superposition product appearing in <cit.>. Let C_n(M)^∞ be the one point compactification of C_n(M), where the point ∞ corresponds to configurations where two points collide or one point goes to the boundary. It is straightforward to define a superposition map 

    μ_n,m:C_n(M)^∞∧ C_m(M)^∞   ⟶  C_n+m(M)^∞
    
             (s,t)   ⟼∞ if  s∩ t≠∅ or  s=∞ or  t=∞, 
    
            s∪ t  otherwise.


From now on assume that M is orientable. We observe that C_n(M)^∞∧ C_m(M)^∞=(C_n(M)× C_n(M))^∞. Using the Poincare–Lefschetz duality isomorphism  and the homology map (μ_n,m)_*, we define sup_n,m so that the diagram

    H^*(C_n(M)× C_m(M))[dashed, "sup_n,m"]["≅",""']     H^*(C_n+m(M))["≅",""']
    H_d(n+m)-*(C_n(M)^∞∧ C_m(M)^∞)["(μ_n,m)_*"]   H_d(n+m)-*(C_n+m(M)^∞)

commutes. 


    The map sup_n,m coincides with sup_n,m from Section <ref>.


    Open inclusions give opposite maps after compactifying, thus μ_n,m factors as


    
        C_n(m)^∞∧C_m(m)^∞[hook,"i^∞"]     C_n,m(M)^∞["p^∞"]     C_n+m(M)^∞.
    


Consequently, we may factor 
    sup_n,m=(^-1∘ p^∞_* ∘)∘(^-1∘ i^∞_* ∘),

using the  isomoprhism for C_n,m(M). By standard properties of Poincaré–Lefschetz duality (^-1∘ i^∞_* ∘) is equal to the cohomology map i^*, whereas the Gysin map (^-1∘ p^∞_* ∘) is equal to the transfer map p_!. As a result sup_n,m=p_!∘ i^*=sup_n,m.

    Proposition <ref> can also be proved for non-orientable manifolds, but care must be taken in using the orientation sheaf for Poincaré–Lefschetz duality.
amsalpha



