On the topology of $\mathcal{M}_{0,n+1}/Σ_n$ and $\overline{\mathcal{M}}_{0,n+1}/Σ_n$
Tommaso Rossi


Designing, simulating, and performing the 100-AV field test for the CIRCLES consortiumMethodology and Implementation of the Largest mobile traffic control experiment to date
    MOSTAFA AMELI§, *,
    SEAN MCQUADE††,
    JONATHAN W. LEE*,
    MATT BUNTING‡,
    MATT NICE‡,
    HAN
    WANG**,
    WILL BARBOUR‡,
    RYAN WEIGHTMAN††
    CHRIS DENARO††
    RYAN DELORENZO‡‡,
    SHARON HORNSTEIN,
    JON F. DAVIS*,
    DAN TIMSIT†,
    RILEY WAGNER‡,
    RITA XU*,
    MALAIKA MAHMOOD‡‡,
    MIKAIL MAHMOOD‡‡,
    MARIA LAURA DELLE MONACHE**,
    BENJAMIN SEIBOLD,
    DAN WORK‡,
    JONATHAN SPRINKLE‡,
    BENEDETTO PICCOLI††, ‡‡,
    ALEXANDRE M. BAYEN*
    April 29, 2024
===================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================


This paper contains some results about the topology of _0,n+1/Σ_n and _0,n+1/Σ_n, where _0,n+1 is the moduli space of genus zero Riemann surfaces with marked points and _0,n+1 is its Deligne-Mumford compactification. We show that _0,n+1/Σ_n and _0,n+1/Σ_n are not topological manifolds for n≥ 4, and they are simply connected for any n∈. We also present some homology computations: for example we show that the homology of _0,n+1/Σ_n is all torsion and that _0,p+1/Σ_p has no p torsion, where p is a prime. Lastly we compute H_*(_0,n+1/Σ_n;) for small values of n, proving that _0,n+1/Σ_n is contractible for n≤ 5 while _0,7/Σ_6 is not. 





 

§ INTRODUCTION

Let _0,n+1 be the moduli space of genus zero Riemann surfaces with n+1 marked points and _0,n+1 be its Deligne-Mumford compactification. In this paper we study the topology of the quotients ℳ_0,n+1/Σ_n and ℳ_0,n+1/Σ_n. The author's interest in these spaces comes from operad theory: the collections of graded vector spaces (here s denotes the degree shift) 

    Grav{sH_*(_0,n+1;)}_n≥ 2     Hycom{H_*(_0,n+1;)}_n≥ 2

turns out to have a natural operad structure, coming from the geometry of _0,n+1 and _0,n+1. These operads are called respectively the Gravity and the Hypercommutative operad and were studied intesively in the nineties by Getzler <cit.>, <cit.>, Barannikov-Kontsevich <cit.> and Kontsevich-Manin <cit.>. Algebras over these operads can be thought as a generalization respectively of Lie algebras and commutative algebras (for a precise definition see Paragraph <ref> and Paragraph <ref>). The analogy do not stops here: as the Lie operad and the commutative operad are Koszul dual to each other, in the same way Grav and Hycomm are Koszul dual operads (see <cit.> and <cit.>). It turns out (Proposition <ref> and Proposition <ref>) that 

    ⊕_n= 2^∞sH_*(_0,n+1/Σ_n;)   and   ⊕_n= 2^∞H_*(_0,n+1/Σ_n;)

are respectively the free Gravity algebra and the free Hypercommutative algebra on a generator of degree zero. The main results are listed below:

    
  * ℳ_0,n+1/Σ_n and ℳ_0,n+1/Σ_n are not topological manifolds for any n≥ 4 (Theorem <ref>).
    
  * ℳ_0,n+1/Σ_n and ℳ_0,n+1/Σ_n are simply connected (Theorems <ref> and <ref>).
    
  * ℳ_0,n+1/Σ_n has the same rational homology of the point (Theorem <ref>).
    
  * ℳ_0,p+1/Σ_p and ℳ_0,p+2/Σ_p+1 have no p-torsion (Theorem <ref>), where p is a prime number.
    
  * ℳ_0,n+1/Σ_n is contractible for n≤ 5 (see Paragraph <ref>).

Here is the outline of the paper:

    
  Section <ref> presents an embedding of _0,n+1/Σ_n into the weighted projective space ℙ(n,n-1,…,2) as an open dense subset. This is crucial to prove that ℳ_0,n+1/Σ_n and ℳ_0,n+1/Σ_n are not topological manifolds for n≥ 4 (Theorem <ref>). We also describe _0,n+1/Σ_n as the complement of an explicit algebraic subvariety of ℙ(n,n-1,…,2).
    
  Section <ref> contains a cambinatorial model for _0,n+1/Σ_n based on cacti. This is useful to do computations when n is small. 
    
  Section <ref> deals with the computation of the fundamental group and the rational homology of ℳ_0,n+1/Σ_n and ℳ_0,n+1/Σ_n.
    
  Section <ref> contains some results about H_*(_0,n+1/Σ_n;_p), where p is a prime number. Firstly we give an upper bound for the order of a class in H_*(_0,n+1/Σ_n;) (Theorem <ref>). Then we construct a long exact sequence (Proposition <ref>) that express H^*(X/S^1;_p) in terms of H^*_S^1(X;_p), H^*_S^1(X^/p;_p) and H^*(X^/p/S^1;_p), where X is any suitably nice S^1-space. This sequence is an easy consequence of classical facts about transformation groups, but I was not able to find a reference for it.  So I decided to include a detailed construction. Since _0,n+1/Σ_n turns out to be homotopy equivalent to C_n()/S^1, where C_n() is the unordered configuration space of point in the plane, we will use the long exact sequence of Proposition <ref> to get some information about H_*(_0,n+1/Σ_n;_p)≅ H_*(C_n()/S^1;_p). In particular we compute H_*(_0,n+1/Σ_n;_p) when n≠ 0,1 mod p, and we prove that there is no p torsion in _0,p+1/Σ_p and _0,p+2/Σ_p+1 . We also compute H_*(_0,n+1/Σ_n;) for small values of n, and this proves that _0,n+1/Σ_n is contractible for any n≤ 5, while it has non trivial homology when n=6. 
    
  Section <ref> contains the computation of H_*^S^1(C_n(^*);_p) when n≠ 0 mod p. This computation turns out to be useful to prove that _0,n+1/Σ_n is contractible for any n≤ 5, even if it is an interesting topic on its own.



  
Notations and conventions: if X is a topological space we will indicate by F_n(X) (resp. C_n(X)) the ordered (resp. unordered) configuration space.


  
Acknowledgments: This project is part of the author's PhD thesis, funded by the PhD program of the University of Roma Tor Vergata and by the MIUR Excellence Project MatMod@TOV awarded to the
Department of Mathematics, University of Roma Tor Vergata,  CUP E83C18000100006. I can not omit the invaluable help that my advisor Paolo Salvatore gave to me in these years of PhD. I am grateful to him for the patience and kindness he showed every time we discussed about this project, and for sharing with me many good ideas about it. Without his guidance and support this paper would have not be possible.  It was a privilege to be introduced to research by him.





§ TOPOLOGY OF ℳ_0,N+1/Σ_N




 §.§ Orbifold structure


	In this paragraph we show that _0,n+1/Σ_n and _0,n+1/Σ_n are not topological manifolds for n≥ 4. For completeness we also discuss the case n=3.
	
		_0,4/Σ_3 is homeomorphic to S^2. _0,4/Σ_3 is homeomorphic to S^2-{0}.
	
	
		_0,4 is a compact Riemann surface, which is well known to be homeomorphic to S^2. Since Σ_3 acts faithfully and by biholomorphisms we get that the quotient _0,4/Σ_3 is again a Riemann surface. By the Riemann-Hurwitz formula we can compute the genus of _0,4/Σ_3, which turns out to be zero giving the first part of the statement. For _0,4/Σ_3 just observe that _0,4 is _0,4 minus three points (corresponding to the three stable curves). These points are all identified in _0,4/Σ_3, so _0,4/Σ_3 is _0,4/Σ_3 minus a point, as claimed.
	
	We now focus on _0,n+1/Σ_n and _0,n+1/Σ_n when n≥ 4.
 
      Observe that _0,n+1 is homeomorphic to F_n()/⋊^*: a point in _0,n+1 is just a configuration of points (p_0,p_1,…,p_n) in the Riemann sphere up to biholomorphisms. Un to rotations we can suppose that p_0 is the point at the infinity ∞∈∪{∞}. Deleting this point and using the stereographic projection we obtain a configuration of n points in the complex plane C up to translation, dilatations and rotations, as claimed. From now on we will identify freely _0,n+1 with F_n()/⋊^*. 
 

There is a homeomorphism 

    ϕ:ℳ_0,n+1/Σ_n   →C_n()/⋊^*
    
      [[z_1,…,z_n]]   ↦[{z_1,…,z_n}]

  
where [[z_1.…,z_n]] denotes the class associated to [(z_1,…,z_n)]∈ℳ_0,n+1≅ F_n()/⋊^*.


Σ_n acts on F_n() by permuting the coordinates, and this action commutes with that of ⋊^* by translations, rotations and dilations. Therefore the quotients (F_n()/⋊^*)/Σ_n=_0,n+1/Σ_n and (F_n()/Σ_n)/⋊^*=C_n()/Σ_n are homeomorphic.



The unordered configuration space is naturally a subspace of the symmetric power SP^n() which is homeomorphic to ^n. The homeomorphism maps an unordered n-uple {z_1,…,z_n} to the coefficients (a_0,…,a_n-1) of the unique monic polinomial a_0+a_1z+…+a_n-1z^n-1+z^n=(z-z_1)⋯(z-z_n) which have {z_1,…,z_n} as roots. 



If 𝐳(z_1,…,z_n) is a n-uple of complex numbers, its barycenter is 

    B(𝐳)z_1+…+z_n/n

Clearly, the barycenter does not depend on the order of the points z_1,…,z_n.

Before we get into the topology of _0,n+1/Σ_n we recall the definitions of weighted projective space and lens complex, just to fix the notation.

    Let (b_0,…,b_n) be a (n+1)-tuple of positive integers. The weighted projective space (of weights (b_0,…,b_n)) is defined as 
    
    ℙ(b_0,…,b_n)^n+1-{0}/∼

    where (z_0,…,z_n)∼ (t^b_0z_0,…,t^b_nz_n) for any t∈^*.


    Let (b_0,…,b_n) be a (n+1)-tuple of positive integers and ζ e^2π i/b_n. The lens complex is defined as  
    
    L(b_n;b_0,…,b_n-1) S^2n-1/∼

    where (z_0,…,z_n-1)∼ (ζ^b_0z_0,…,ζ^b_n-1z_n-1).


    The topology of ℙ(b_0,…,b_n) and of L(b_n;b_0,…,b_n-1) is well known. For example, the cohomology rings of these spaces were computed by Kawasaki in <cit.>.


There is an homeomorphism 

    ψ:SP^n()-Δ/⋊^*   →ℙ(n,n-1,…,2)  
    
      [{z_1,…,z_n}]   ↦ [a_0:…:a_n-2]

where (a_0,…,a_n-2) are the coefficients of the monic polynomial (z-z_1+B(𝐳))⋯(z-z_n+B(𝐳)) and Δ is the diagonal of SP^n(). 


 
The coefficient a_n-1 of the monic polynomial (z-z_1+B(𝐳))⋯(z-z_n+B(𝐳)) is 0, indeed
 
    a_n-1   =-z_1+B(𝐳)+…-z_n+B(𝐳)=-(z_1+…+z_n)+nB(𝐳)=0



Consider the map

    f:SP^n()   → SP^n()
    {z_1,…,z_n}   ↦{z_1-B(𝐳),…,z_n-B(𝐳)}

whose image consists of the unordered n-uples whose barycenter is the origin. By Remark <ref> there is an homeomorphism g:SP^n()→^n which sends a set of n complex numbers {z_1,…,z_n} to the coefficients (a_0,…,a_n-1) of the monic polynomial (z-z_1)⋯(z-z_n). The composite g∘ f maps {z_1,…,z_n} to the coefficients of the monic polynomial (z-z_1+B(𝐳))⋯(z-z_n+B(𝐳)). By Remark <ref> the image of g∘ f is {(a_0,…,a_n-2,0)| a_i∈}≅^n-1 therefore we get a map

    SP^n()-Δ^n-1-{0}→ℙ(n,n-1,…,2)

where the last map is the natural projection ^n-1-{0}→ℙ(n,n-1,…,2). It is easy to show that this map is constant on the ⋊^* orbits, so it induces a bijective map 

    ψ:SP^n()-Δ/⋊^*→ℙ(n,n-1,…,2)

which is an homeomorphism because the source is compact and the target is Hausdorff.


 The composition ψ∘ϕ:_0,n+1→ℙ(n,n-1,…,2) is an embedding.


Just combine Proposition <ref> and Proposition <ref>.



ℳ_0,n+1/Σ_n and _0,n+1/Σ_n are not a topological manifolds for any n≥ 4. 


Since ℳ_0,n+1/Σ_n is an open subset of ℳ_0,n+1/Σ_n, it suffices to show that there exist a point in ℳ_0,n+1/Σ_n which does not have an Euclidean neighbourhood. Let us consider the point p∈ℳ_0,n+1/Σ_n consisting of the roots of the polynomial z^n+1. By the embedding of Corollary <ref>, this corresponds to the point [1:0:…:0] of ℙ(n,n-1,…,2). Since ℳ_0,n+1/Σ_n is an open subset of ℙ(n,n-1,…,2) our claim is equivalent to show that p does not have an Euclidean neighbourhood contained in ℙ(n,n-1,…,2). Consider the open set 

    U_0{[1:a_1:…:a_n-2]∈ℙ(n,n-1…,2)}

If [1:a_1:…:a_n-2]=[1:b_1:…:b_n-2] then exists λ∈^* such that 

    λ^n=1
    λ^n-ka_k=b_k  for  k=1,…,n-2

Therefore U_0≅^n-2/∼
where (a_1,…,a_n-2)∼(b_1,…,b_n-2) if and only if there is a n-th root of unity λ such that λ^n-ka_k=b_k for all k=1,…,n-2. Therefore U_0 is homeomorphic to a cone on the lens complex L(n;n-1,…,2). The point p=[1,0,…,0] is precisely the vertex of this cone, therefore (by excission) we have 

    H_k(ℙ(n,n-1,…,2),ℙ(n,n-1,…,2)-{p})   =H_k(U_0,U_0-{p})
       =H_k-1(U_0-{p}) 
       =H_k-1(L(n;n-1,…,2))

If p has an euclidean neighbourhood U, the by excission

    H_k(ℙ(n,n-1,…,2),ℙ(n,n-1,…,2)-{p})   =H_k(U,U-{p})
       = if  k=0,2(n-2)-1
    
        0  otherwise

But this is a contradiction, because the lens complex L(n;n-1,…,2) is not a homology sphere if n≥ 4.




 §.§ An algebro-geometric description


Probably an algebraic-geometry minded reader already noticed that the embedding 

    ψ∘ϕ:_0,n+1/Σ_n↪ℙ(n,n-1,…,2)

we described in Paragraph <ref> exhibit _0,n+1/Σ_n as the complement of an algebraic subvariety of ℙ(n,n-1,…,2). In this short paragraph we simply make this description explicit. We first recall the definition and the main properties of the discriminant of a polynomial.

    Let p(z)=a_0+a_1z+…+a_nz^n be a polynomial with complex coefficients. The discriminant of p(z) is defined as
    
    Disc(p(z))(-1)^n(n-1)/2/a_n· Res(p(z),p'(z))

    where Res(p(z),p'(z)) is the resultant of p(z) and its derivative p'(z), i.e. it is the determinant of the Sylvester matrix of p(z) and p'(z). In particular, Disc(p(z)) is a homogeneus polynomial in a_0,…,a_n of degree 2n-2.

Two important properties of the discriminant are the following:

    
  * Disc(p(z))=0 if an only if p(z) has a multiple root.
    
  * If a_0^i_0a_1^i_1… a_n^i_n is a monomial of Disc(p(z)), then i_0,…,i_n satisfy the equation
    
    ni_0+(n-1)i_1+…+i_n-1=n(n-1)

    In particular, if a_n=1 we get that Disc(p(z))∈[a_0,…,a_n-1] is a weighted homogeneus polynomial (with a_i variable of weight n-i) and it makes sense to talk about the zero locus of Disc(p(z)) inside ℙ(n,n-1,…,1).


    Let p(z)=a_0+a_1z+…+a_n-2z^n-2+z^n be a generic polynomial. The embedding
    
    _0,n+1/Σ_n↪ℙ(n,n-1,…,2)

    maps the space _0,n+1/Σ_n to the complement of {Disc(p(z))=0}⊆ℙ(n,n-1,…,2).


    If we identify _0,n+1/Σ_n with C_n()/⋊^*, then the embedding maps a configuration [{z_1,…,z_n}] to the coefficients [a_0:…:a_n-2] of the polynomial p(z)=(z-z_1+B(𝐳))⋯(z-z_n+B(𝐳)). The complement of the image of this embedding consists exactly of the points [a_0:…:a_n-2]∈ℙ(n,n-1,…,2) such that the polynomial a_0+a_1z+… a_n-2z^n-2+z^n has a multiple root. But this is equivalent to say that the discriminant of such a polynomial is zero. Equation <ref> tell us that Disc(p(z)) is a weighted polynomial with variables a_i of weight n-i, thus its zero locus is well defined. 


	

§ A COMBINATORIAL MODEL USING CACTI

The main goal of this section is to provide a combinatorial model for _0,n+1/Σ_n. The key observation is that _0,n+1/Σ_n is homotopy equivalent to C_n()/S^1, the quotient of the unordered configuration space by the circle action. Starting from the homotopy equivalence between the ordered configuration space F_n() and the space of cacti (see the work by McClure-Smith <cit.>) we will build a CW-complex which is homotopy equivalent to C_n()/S^1.
 

 §.§ Cacti

In this paragraph we recall the definition and properties of the space of cacti. A 	combinatorial construction of this space was introduced by McClure and Smith <cit.>, while geometric constructions are due to Voronov <cit.> and Kaufmann <cit.>. Salvatore compared the two approaches in <cit.>. 
	
		Let 𝒞_n be the set of partitions x of S^1 into n closed 1-manifolds I_1(x),…,I_n(x) such that:
		
			
  * They have equal measure.
			
  * They have pairwise disjoint interiors.
			
  * There is no cyclically ordered sequence of points (z_1,z_2,z_3,z_4) in S^1 such that z_1,z_3∈ I_j(x), z_2,z_4∈ I_k(x) and j≠ k.
		
		We can equip this set with a topology by defining a metric on it: for any x,y∈𝒞_n we set
		
    d(x,y)=∑_j=1^n μ(I_j(x)-I_j(y))

		where μ denotes the measure. We will call 𝒞_n (with this topology) the space of based cacti with n-lobes. See Figure <ref> for an example. 
	
	
		𝒞_n is called the space of cacti for the following reason: given x∈𝒞_n, let us define a relation ∼ on S^1: two points z_1,z_2∈ S^1 are  equivalent if there is an index j∈{0,…,n} such that z_1 and z_2 are the boundary points of the same connected component of S^1-I_j. The quotient space c(x) S^1/∼ by this relation is a pointed space (the basepoint is just the image of 1∈ S^1) called the cactus associated to x: topologically it is a configuration of n-circles in the plane, called lobes, whose dual graph is a tree. The dual graph is a graph with two kind of vertices: a white vertex for any lobe and a black vertex for any intersection point between two lobes. An edges connects a white vertex w to a black vertex b if b represent the intersection point of the lobe corresponding to w with some other lobe. See Figure <ref> for some examples. Note that the lobes are the image of the 1-manifolds I_1(x),…,I_n(x) under the quotient map S^1→ S^1/∼. In what follows we will freely identify a partition x∈𝒞_n and its associated cactus c(x).
	
	
	

  
Cell decomposition: to any point x∈𝒞_n we can associate a sequence (X_1,…,X_l) of numbers in {1,…,n} by the following procedure: start from the point 1∈ S^1 and  move along the circle clockwise. The sequence (X_1,…,X_l) is obtained by writing one after the other the indices of all the 1-manifolds you encounter and has the following properties:
	
		
  * All values between 1 and n appear.
		
  * X _i≠ X_i+1 for every i=1,…,l-1.
		
  * There is no subsequence of the form (i,j,i,j) with i≠ j.
	
	This sequence is just a combinatorial way to encode the shape of the cactus c(x). For any such a sequence (X_1,…,X_l), the subspace of 𝒞_n consisting of partitions whose associated sequence is (X_1,…,X_l) turns out to be a product of simplices. More precisely, if m_i is the cardinality of {j∈{1,…,l}| X_j=i}, i=1,…,n, then 
	
    (X_1,…,X_l)≅∏_i=1^nΔ^m_i-1

	For an example look at the cactus of Figure <ref>: it belongs to the cell (3,4,2,1,2,3)≅Δ^0×Δ^1×Δ^1×Δ^0.
	Intuitively all the cacti c(x) associated to partitions x∈(X_1,…,X_l) have the same shape. So we will represent pictorially a cell by drawing a cactus, meaning that the cell contains all the partitions x∈𝒞_n whose associated cactus c(x) has that shape. From this point of view, the parameters
	of a cell (X_1,…,X_l) can be thought as the lengths of the arcs between two consecutive intersection points of lobes. The boundary of a cell is obtained by collapsing some of these arcs. See Figure <ref> for some examples. This gives a regular CW-decomposition of   𝒞_n.
	
	
	
	
		There are two relevant groups acting on 𝒞_n: S^1 acts by rotating the basepoint of a cactus, Σ_n acts by relabelling the lobes.
	
	The important thing about cacti is that they are a very small cellular model for the configuration space F_n():
	
	
		The space of cacti 𝒞_n is (S^1×Σ_n)-equivariantly homotopy equivalent to F_n().
	
 
     The moduli space  _0,n+1 is homotopy equivalent to 𝒞_n/S^1. 
 
 
      Recall that _0,n+1 is homeomorphic to F_n()/⋊^*. By Theorem <ref> 𝒞_n/S^1 is homotopy equivalent to F_n()/S^1. Therefore we get that 
     
    _0,n+1≅F_n()/⋊^*≃F_n()/S^1≃𝒞_n/S^1

     where the first homotopy equivalence holds because the group ⋊^* is homotopy equivalent to S^1.
 

    The quotient 𝒞_n/S^1 is still a regular CW-complex, and its cells are described by the same combinatorics of 𝒞_n except that we do not have a basepoint on our cacti (see Figure <ref> for an example). We call 𝒞_n/S^1 the space of unbased cacti. 
	
		
	

 

 §.§ A combinatorial model for ℳ_0,n+1/Σ_n

 In this paragraph  we describe a small combinatorial model for ℳ_0,n+1/Σ_n. We will use this model in Paragraph <ref> to compute H_*(_0,n+1/Σ_n;) for small values of n.
	
		_0,n+1/Σ_n is homotopy equivalent to 𝒞_n/(S^1×Σ_n). 
	
	
		By Proposition <ref> F_n() is (S^1×Σ_n)-equivariantly homotopy equivalent to 𝒞_n, so we have a homotopy equivalence between the quotients
		
    F_n()/(S^1×Σ_n)≃𝒞_n/(S^1×Σ_n)

		The left hand space is homotopy equivalent to _0,n+1/Σ_n, and this proves the statement.
	
 
     As we know from Theorem <ref> the space of cacti 𝒞_n is (S^1×Σ_n)-equivariantly homotopy equivalent to the ordered configuration space F_n(). Therefore we have a homotopy equivalence between 𝒞_n/(S^1×Σ_n)  and C_n()/S^1. Combining this with Proposition <ref> we conclude that _0,n+1/Σ_n is homotopy equivalent to C_n()/S^1. We will freely switch between _0,n+1/Σ_n and  C_n()/S^1 depending on the situation.
 
	Now let us describe in details the space 𝒞_n/(S^1×Σ_n): recall that the space of cacti 𝒞_n is a regular CW-complex whose cells are described by cacti (with numbered lobes and a basepoint). The boundary of a cell is described by collapsing arcs. The (S^1×Σ_n)-action is encoded as follows:
	
		
  * S^1 acts by rotating the basepoint. 
		
  * Σ_n acts by relabelling the lobes.
	
	To better understand the quotient 𝒞_n/(S^1×Σ_n) it is useful to first consider the space of unbased cactus 𝒞_n/S^1≃_0,n+1 and then quotient by the Σ_n-action: 𝒞_n/S^1 is a CW-complex where a cell is given by an unbased cactus, and the boundary of each cell is obtained by collapsing arcs as in 𝒞_k. Now we quotient by the Σ_n-action: this corresponds to relabelling the lobes. This time we have some cells with non-trivial stabilizer and we need to take this into account (some examples of these cells are depicted in Figure <ref>). 
	
	To sum up, 𝒞_n/(S^1×Σ_n) is built up by two types of cells, those with trivial stabilizer ("not symmetric cells") and those with non-trivial stabilizer ("symmetric cells"):
	
		
  * Non symmetric cells: take a cactus c (without basepoint and labelling of the lobes) and fixes an arbitrary labelling of the lobes. Being a non symmetric cell means that the only permutation of Σ_n which fix c as a labelled cactus is the identity. The cell σ(c) associated to c is given by: 
		
    σ(c)∏_i=1^nΔ^n_i-1

		where n_i is the number of intersection points of the i-th lobe with the other lobes.  
		As for the space of cacti 𝒞_n the parameters of Δ^n_i-1 represent the length of the arcs between two intersection points. The boundary of σ(c) is given by sending to zero some parameter, i.e. collapsing some arc.
		
  * Symmetric cells: let c be a cactus (without basepoint and labelling of the lobes) and fix an arbitrary labelling of the lobes. Being a symmetric cell means that the isotropy group of c as a labelled cactus is a non trivial subroup G_c≤Σ_n.  This cactus gives us a cell
		
    σ(c)∏_i=1^nΔ^n_i-1/G_c

		In other words, σ(c) is the quotient of the cell associated to c as a labelled cactus (with an arbitrary fixed labelling) by the action of its isotropy group.
		The boundary of σ(c) is given by obtained by sending to zero some parameter, i.e. collapsing some arc.
	
	We conclude this paragraph by discussing some explicit examples.
	[n=3] There are only two cactus (unlabelled, without basepoint) with three lobes, let us call them c_0 and c_1 (see Figure <ref>). c_0 is fixed by a rotation of 2π/3, so its stabilizer is /3. The corresponding cell is a point. c_1 is fixed by a rotation of π, so its stabilizer is /2. The corresponding cell σ(c_1) is obtained from Δ^1={(t_0,t_1)∈[0,1]^2| t_0+t_1=1} quotienting by the relation (t_0,t_1)∼(t_1,t_0). Therefore 𝒞_3/(S^1×Σ_3) is contractible.     
	  
	
	[n=4] There are four cactus (unlabelled, without basepoint) with four lobes, let us call them c_0, c_1, c_2 and c_3 (see Figure <ref> where such cacti are depicted with an arbitrary labelling of the lobes). c_0 is fixed by a rotation of 2π/4, so its stabilizer is /4. The corresponding cell is a point. c_1 is not fixed by any rotation, so its corresponding cell σ(c_1) is a copy of Δ^1.  c_2 is fixed by a rotation of π, so its stabilizer is /2. The corresponding cell σ(c_2) is obtained from Δ^1×Δ^1 by imposing the relation
		
    (t_0,t_1)× (s_0,s_1)∼ (s_1,s_0)× (t_1,t_0)

		Finally, c_3 is fixed by a rotation of 2π/3, so its stabilizer is /3. The corresponding cell σ(c_3) is obtained from Δ^2 by quotienting the /3-action. To be explicit, the generator of /3 acts on Δ^2 by permuting cyclically the coordinates (t_0,t_1,t_2). 
	  
	
	


§ FUNDAMENTAL GROUP AND RATIONAL HOMOLOGY



 §.§ Fundamental group


In this paragraph we prove that ℳ_0,n+1/Σ_n and _0,n+1/Σ_n are simply connected. First we recall some facts on the fundamental group of orbit spaces.
	

  §.§.§ On the fundamental group of orbit spaces

	
	The following results can be found in the paper of B. Noohi <cit.>, which extend a previous work by M.A. Armstrong <cit.>. The goal of this paragraph is to explain the relation between π_1(X) and π_1(X/G), when G is a topological group acting on a nice topological space X. By nice topological space we mean that X is connected, locally path-connected, semilocally simply-connected. We also assume that the orbit space X/G is semilocally simply-connected. From now on we will always assume that we are in this situation.

		The statement of the next Theorem contain the notion of action with slice property, which is discussed in <cit.> and  <cit.>. Instead of giving the precise definition we give some important examples where this property is satisfied:
		
			
  * If G is a finite group then any action of G on X has the slice property.
			
  * G Lie group (not necessarily compact), X locally compact. If the map 
			
    G× X    → X× X 
    
    				(g,x)   ↦ (gx,x)

			is proper, then the action has the slice property.
			
  * If G is compact Lie group acting on a completely regular space, then the action of G on X has the slice property.
		
	
	
	
		Let G be a topological group acting on a topological space X which is connected, locally path-connected and semilocally simply-connected. We also assume that X/G is semilocally simply connected. Fix a base point x_0∈ X and let [x_0] be its image in X/G. Suppose that the action has the slice property. If all stabilizer groups G_x are locally path-connected then we have an exact sequence
		
    π_1(X,x_0)[r]    π_1(X/G,[x_0])[r]   π_0(G)/I[r]     1

		where I⊂π_0(G) is the subgroup generated by the path components of G containing an element g which fix some point of X.
	
	
		The first map in the exact sequence above is the one induced by the projection p:X→ X/G. The second map is defined as follows: given a loop α in X/G we can lift it to a path α̃ in X beginning at the basepont x_0 and ending at α̃(1). Since α is a loop in X/G we have that α̃(1)=g_αx_0 for some g_α∈ G. Let us denote by [g_α] the element in π_0(G)/I which corresponds to the path component of G containing g_α. It is easy to verify that the assignment 
		
    π_1(X/G,[x_0])   →π_0(G)/I
    
    			[α]↦ [g_α]

		is a well defined function.
	


  §.§.§ Computations

We now apply the results of the previous paragraph to compute the fundamental group of ℳ_0,n+1/Σ_n and ℳ_0,n+1/Σ_n.
	
		The quotient ℳ_0,n+1/Σ_n is simply connected.
	
	
		
		Since ℳ_0,n+1 is simply connected, Theorem <ref> gives us an isomorphism between π_1(ℳ_0,n+1/Σ_n) and Σ_n/I. Let (i,j)∈Σ_n be a transposition. It has at least one fixed point since it fixes a configuration whose clustering is of the form ((i,j),1…,î,…,ĵ,…,n). Therefore the subgroup I contains all the transpositions, so it must be Σ_n itself.  This concludes the proof.
	

    The quotient ℳ_0,n+1/Σ_n is simply connected.


     The space ℳ_0,n+1/Σ_n is homotopy equivalent to C_n()/S^1, therefore it suffice to prove statement for this last space. Let us denote by p:C_n()→ C_n()/S^1 the quotient map. The proof follows several steps:
     
         
  * Since S^1 is connected we get that π_0(G)/I=1 in the exact sequence of Theorem <ref>. This implies that p_*: B_n→π_1(C_n()/S^1) is surjective. Therefore we have a short exact sequence
         
    1[r]    Ker(p_*) [r,hook]    B_n [r, "p_*"]    π_1(C_n()/S^1) [r]     1

        
         
  * π_1(C_n()/S^1) is abelian: the previous step tell us that π_1(C_n()/S^1) is a quotient of the braid group, in particular it is generated by elements σ_1,…,σ_n-1  and in addition to the relations of B_n there are some extra relations coming from Ker(p_*). Let us make explicit some of these relations: the braid Δσ_1(σ_2σ_1)…(σ_n-1σ_n-2…σ_1) belongs to Ker(p_*) since it is the given by a rotation of π. Therefore we have the relation 
         
    σ_i=σ_iΔ=Δσ_n-i=σ_n-i

         Now suppose n is odd. The previous relation enable us to prove that σ_i and σ_i+1 commute in π_1(C_n()/S^1):
         
    σ_iσ_i+1= σ_n-iσ_i+1=σ_i+1σ_n-i=σ_i+1σ_i

         where the middle equality holds because n is odd and therefore n-i≠ i for each i=1,…,n-1. To get the statement in the case n=2k we can do the same procedure to show that σ_i and σ_i+1 commute for each i≠ k. So it remains to prove the equality σ_kσ_k+1=σ_k+1σ_k: combining σ_k+1σ_k+2σ_k+1=σ_k+2σ_k+1σ_k+2 and  σ_k+2σ_k+1=σ_k+1σ_k+2  we get that σ_k+2=σ_k+1. Therefore σ_kσ_k+1=σ_kσ_k+2=σ_k+2σ_k=σ_k+1σ_k.
         
  * π_1(C_n()/S^1) is generated by σ_1: by the previous point we know that π_1(C_n()/S^1) is abelian. Combining this fact with the braid relation σ_iσ_i+1σ_i=σ_i+1σ_iσ_i+1 we get σ_i=σ_i+1 for all i=1,…,n-2. Thus π_1(C_n()/S^1) is generated by σ_1.
         
  * π_1(C_n()/S^1) is the trivial group: by the previous discussion we know that π_1(C_n()/S^1) is an abelian group, so it is isomorphic to H_1(C_n()/S^1;) by Hurewicz. Moreover π_1(C_n()/S^1) it is generated by σ_1. So it suffice to show that σ_1=0. We proceed  as follows: consider the n-th roots of unity {ζ_1,…,ζ_n}∈ C_n(). Now take the loop 
         
    α:[0,1]   → C_n()
    
                 t   ↦{e^2tπ i/nζ_1,…,e^2tπ i/nζ_n}

In plain words α rotates the n-agon {ζ_1,…,ζ_n} counter-clockwise between 0 and 2π/n degrees. It is easy to see that this loop represent the class (n-1)σ_1∈ H_1(C_n();) (see figure <ref> for a picture). Therefore p_*(α) is (n-1) times the generator of H_1(C_n()/S^1;). But p_*(α) is a constant loop, so it is the zero class in homology. Therefore we get the equation (n-1)σ_1=0 in H_1(C_n()/S^1;). Similarly, let {ζ_1,…,ζ_n-1} be the set of (n-1)-th roots of unity. Consider the loop
  
    β:[0,1]   → C_n()
    
                 t   ↦{e^2tπ i/(n-1)ζ_1,…,e^2tπ i/(n-1)ζ_n-1,0}

In plain words α rotates the configuration {ζ_1,…,ζ_n-1,0} counter-clockwise between 0 and 2π/(n-1) degrees. It is easy to see that this loop represent the class nσ_1∈ H_1(C_n();) (see Figure <ref>). Therefore p_*(α) is n times the generator of H_1(C_n()/S^1;). But p_*(α) is a constant loop, so it is the zero class in homology. Therefore we get the equation nσ_1=0 in H_1(C_n()/S^1;). Finally we can conclude: we know that π_1(C_n()/S^1)≅ H_1(C_n()/S^1;) is a cyclic group with generator σ_1. However nσ_1=0=(n-1)σ_1, therefore σ_1=0.




     




 §.§ Rational homology

This paragraph contains some results on the rational homology of ℳ_0,n+1/Σ_n and _0,n+1/Σ_n. To do such computations we will use some operad theory, which will be reviewed in the next paragraphs.


  §.§.§ The Gravity operad

Let X be a S^1-space. The action θ:S^1× X→ X induces an operator Δ:H_*(X;)→ H_*+1(X;) by the composition

    H_*(X)[r]    H_*(S^1)⊗ H_*(X)[r,"×"]   H_*(S^1× X)[r,"θ_*"]    H_*(X)

where the first map take a class x∈ H_*(X) and send it to [S^1]⊗ x. We will call Δ the BV-operator (see <cit.>). In what follows all the homology groups are taken with integer coefficients, unless otherwise stated. To easy the notation we sometimes write H_*(X) instead of H_*(X;).

Let 𝒟_2 be the little two disk operad. S^1 acts on 𝒟_2(n) by rotations, so we get a BV-operator Δ:H_*(𝒟_2(n))→ H_*+1(𝒟_2(n)). This map is compatible with the operadic structure and induces a morphism of operads Δ:H_*(𝒟_2)→ H_*+1(𝒟_2). The kernel of this map is a sub-operad of H_*(𝒟_2), called the Gravity operad. We will denote it by Grav.


    S^1 acts freely on 𝒟_2(n) so we can identify the kernel of Δ:H_*(𝒟_2(n))→ H_*+1(𝒟_2(n)) with H_*^S^1(𝒟_2(n))≅ s H_*(_0,n+1), where this last isomorphism holds because _0,n+1 and 𝒟_2(n)/S^1 are homotopy equivalent (Corollary <ref>). To sum up we have the following identification:
    
    Grav(n)= sH_*(_0,n+1)



    The action of Σ_n+1 on _0,n+1 by relabelling the points induces an action in homology, making Grav a cyclic operad.

Unlike many familiar operads, the Gravity operad is not generated by a finite number of operations. However, it has a nice presentation with infinitely many generators:

     As an operad Grav is generated by (graded) symmetric operations of degree one
     
    {a_1,…,a_n}∈ Grav(n)    for  n≥ 2

      Geometrically, {a_1,…,a_n} corresponds to the generator of H_0(ℳ_0,n+1,). These operations (called brackets) satisfy the so called generalized Jacobi relations: for any k≥ 2 and l∈
      
    ∑_1≤ i<j≤ k(-1)^ϵ(i,j){{a_i,a_j},a_1,…,â_i,…,â_j,…,a_k,b_1,…,b_l}=
        {{a_1,…,a_k},b_1,…,b_l}

where the right hand term is interpreted as zero if l=0 and ϵ(i,j)=(a_1+…+ a_i-1)a_i+(a_1+…+a_j-1)a_j+ a_ia_j.


A Gravity algebra (in the category of chain complexes over ) is an algebra over the Gravity operad. To be explicit, it is a chain complex (A,d_A) together with graded symmetric chain maps {-,…,-}:A^⊗ k→ A of degree one such that for k≥ 3, l≥ 0 and a_1,…,a_k,b_1,…,b_l∈ A Equation <ref> is satisfied.


    Let (A,d_A)∈ Ch() be a chain complex. The free Gravity algebra on (A,d_A) is the following chain complex:
    
    Grav(A)⊕_n=2^∞Grav(n)⊗_Σ_nA^⊗ n

    


    (Free) Gravity algebras can be defined in the same way in the category of chain complexes over some commutative ring R. In this paper we will focus in the case R=.



  §.§.§ The Hypercommutative operad

Let _0,n+1 be the Deligne-Mumford compactification of the moduli space of genus zero Riemann surfaces with marked points. For any m,n∈ and i=1,…,n we can define gluing maps
	
    ∘_i:_0,n+1×_0,m+1   →_0,n+m
    
    		(C_1,p_0,…,p_n)×(C_2,q_0,…,q_m)   ↦ (C,p_0,…,p_i-1,q_1,…,q_m,p_i+1,…,p_n)

	where C is the curve obtained from C_1⊔ C_2 identifying p_i and q_0 and introducing a nodal singularity.
	
	
		Let us denote by (n)_0,n+1. Σ_n acts on (n) by permuting the marked points p_1,…,p_n of a stable curve (C,p_0,…,p_n)∈_0,n+1. The gluing maps defined above define an operad structure on the collection {(n)}_n≥ 2. We call  the Deligne-Mumford operad.
	
Since homology is a lax monoidal functor, the operadic structure of  induce an operadic structure on H_*(;), giving an operad in graded abelian groups:
	
		The Hypercommutative operad Hycom is a (cyclic) operad in graded abelian groups whose arity n operations are given by
		
    Hycom(n) H_*(_0,n+1;)

		As an operad, it is generated by (graded) symmetric operations of degree 2(n-2)
		
    (a_1,…,a_n)∈ H_*(_0,n+1;)    n≥ 2

		which satisfy the following generalized associativity relations:
		
    ∑_S_1⊔ S_2={1,…,n}± ((a,b,x_S_1),c,x_S_2)=  ∑_S_1⊔ S_2={1,…,n}± (a,(b,c,x_S_1),x_S_2)

		
		
		where if S={s_1,…,s_k} is a subset of {1,…,n}, x_S is an abbreviation for x_s_1,…,x_s_k. The sign in front of each summand is determined by the Koszul sign rule. We report below some explicit examples of these relations:
		
			
  * n=0: we get ((a,b),c)=(a,(b,c)), i.e. the binary operation is associative (and commutative).
			
  * n=1: ((a,b),c,d)+(-1)^cd((a,b,d),c)=(a,(b,c),d)+(a,(b,c,d)).
		
		
		Further details can be found in <cit.> and <cit.>.
	
	
		Geometrically, the n-ary operation (a_1,…,a_n) corresponds to the fundamental class [_0,n+1].
	
	
		An hypercommutative algebra  (in the category of chain complexes over ) is just an algebra over the Hypercommutative operad. Explicitly, it is a chain complex (A,d_A) equipped by (graded) symmetric products (-,⋯,-):A^⊗ n→ A of degree 2(n-2), such that Equation <ref> is satisfied for any choice of variables a,b,c,x_1,…,x_n∈ A.
	
 
    Let (A,d_A)∈ Ch() be a chain complex. The free Gravity algebra on (A,d_A) is the following chain complex:
    
    Hycom(A)⊕_n=2^∞Hycom(n)⊗_Σ_nA^⊗ n

    


    (Free) Hypercommutative algebras can be defined in the same way in the category of chain complexes over some commutative ring R. In this paper we will focus in the case R=.
	


  §.§.§ Computations

We start with the computation of the rational homology of _0,n+1:

H_k(ℳ_0,n+1/Σ_n;)= if  k=0

0  otherwise

To proof of this Theorem is elementary but a bit long, so we will subdivide it in several steps. The key observation is that up to a shift in degrees the vector space

    ⊕_n=2^∞H_*(ℳ_0,n+1/Σ_n;)

carries a very nice algebraic structure:

Let us denote by Grav(x) the free gravity algebra on a generator x of degree zero (over ). Then

    Grav(x)=⊕_n=2^∞sH_*(ℳ_0,n+1/Σ_n;)



Let x be the graded vector space of dimension 1 spanned by x. By definition

    Grav(x)⊕_n=2^∞Grav(n)⊗_Σ_n( x)^⊗ n

where Grav(n)=sH_*(ℳ_0,n+1,). Explicitly, if p∈ H_*(ℳ_0,n+1,) and σ∈Σ_n we have the identification (σ· p)⊗ x⊗⋯⊗ x= p⊗σ·(x⊗⋯⊗ x). Since x is a variable of degree zero Σ_n acts trivially on ( x)^⊗ n, so we get the identification 

    (σ p)⊗ x⊗⋯⊗ x∼ p⊗ x⊗⋯⊗ x

Therefore Grav(x)=⊕_n=2^∞sH_*(ℳ_0,n+1;)_Σ_n=⊕_n=2^∞sH_*(ℳ_0,n+1/Σ_n;).



In Grav(x), we have {{x,x},x,…,x}=0.


Let us consider generalized Jacobi relation (Equation <ref>) with l=0 and n≥ 3:

    ∑_1≤ i<j≤ n±{{a_i,a_j},a_1,…,â_̂î,…,â_̂ĵ,…,a_n}=0

The relevant case for us is when x_k=x for all k=1,…,n and x is a variable of degree zero. In this case all the signs are positive and we get

    n2{{x,x},x,…,x}=0

and the statement follows. 




The only non trivial elements in Grav(x) are of the form {x,…,x}.


Let us consider the generalized Jacobi relation associated to any k∈ and l>0 :

    ∑_≤ i<j≤ k±{{a_i,a_j},a_1,…,â_̂î,…,â_̂ĵ,…,a_k,b_1,…,b_l}={{a_1,…,a_k},b_1,…,b_l}

If all the variables a_i, b_i are equal to a variable x of degree zero, the left hand side of the equation written above is 0 because of the previous Lemma. It follows that for all k≥ 3, l>0 we have

    {{x,…,x}_k,x,…,x}=0

Therefore all the iterated brackets are 0 and the only non trivial elements of Grav(x) are of the form {x,…,x}.

Now we are ready to prove Theorem <ref>:
(of Theorem <ref>)
By the previous Corollary, in Grav(x) the only non trivial elements are of the arity k brackets {x,…,x}. Since x has degree 0 and the arity k bracket {-,…,-} has degree one, the element {x,…,x} has degree one. By Proposition <ref> we conclude that the arity k bracket {x,…,x} is the suspension of the generator of H_0(ℳ_0,n+1/Σ_n,). Since these are the only non zero elements of Grav(x), the statement of Theorem <ref> follows.


	A similar approach can be used to compute H_*(ℳ_0,n+1/Σ_n;), as the following Proposition suggests:
	
		
		
		Let us denote by Hycom(x) the free hypercommutative algebra on one generator x of degree zero (over ). Then we have
		
    Hycom(x)=⊕_n=2^∞H_*(ℳ_0,n+1/Σ_n;)

	
	
		Let x be the graded vector space of dimension 1 spanned by x. By definition
		
    Hycom(x)⊕_n=2^∞Hycom(n)⊗_Σ_n( x)^⊗ n

		where Hycom(n)= H_*(ℳ_0,n+1;). Explicitly, if p∈ H_*(ℳ_0,n+1;) and σ∈Σ_n we have the identification (σ· p)⊗ x⊗⋯⊗ x= p⊗σ·(x⊗⋯⊗ x). The right hand side is equal to p⊗ x⊗⋯⊗ x since x is a degree zero variable. It follows that 
		
    (σ p)⊗ x⊗⋯⊗ x∼ p⊗ x⊗⋯⊗ x

		Finally, Hycom(x)=⊕_n=2^∞H_*(ℳ_0,n+1;)_Σ_n=⊕_n=2^∞H_*(ℳ_0,n+1/Σ_n;).
		
	
	
		This Proposition is the Koszul dual (in the sense of Ginzburg-Kapranov <cit.>, see also <cit.>) of Proposition <ref>.  While ℳ_0,n+1/Σ_n has the rational homology of a point, the case of ℳ_0,n+1/Σ_n is highly non trivial. I started thinking about this problem in the Fall 2022 and some computations I did at that time are reported below. However, in March 2023 the problem was solved in the paper <cit.>:  ℳ_0,n+1 can be constructed by iterated blow-ups from P^n-2. Since the centers of these blow-ups are Σ_n-equivariant, the blow-up formula for cohomology allows the authors to compute H^*(ℳ_0,n+1;) as Σ_n-representation. By taking the Σ_n-invariant part they get a formula for the Poincarè polynomial of ℳ_0,n+1/Σ_n, even if the combinatorics involved looks difficult. So, if you want to know the ranks of the homology of ℳ_0,n+1/Σ_n you have to count some combinatorial objects, but there is not any closed formula/recursion that may help you in this task. It would be interesting to see if our operadic approach can bring to a better formula for these ranks.
		I thank Professor Vladimir Dotsenko for pointing out to me the paper <cit.>.
	
	
	
		In principle Proposition <ref> gives us a complete computation of H_*(ℳ_0,n+1/Σ_n;): indeed the Hypercommutative operad  has a nice presentation in terms of generators and relations (see Paragraph <ref>); however there are infinitely many of them, making explicit computations a bit involved. 
	
	We end this paragraph by stating some results about the ranks of H_*(_0,n+1/Σ_n;).
	
		The vector spaces H_i(_0,n+1/Σ_n;) and H_2(n-2)-i(_0,n+1/Σ_n;) have the same dimension. 
	
	
		By Poincarè duality we have an isomorphism
		
    φ: H^i(_0,n+1;)   → H_2(n-2)-i(_0,n+1;)
    ψ   ↦ [_0,n+1]∩ψ

		If g∈Σ_n, then
		
    φ(g·ψ)   =[_0,n+1]∩ g^*ψ
       =(g_*)^-1g_*([_0,n+1]∩ g^*ψ)
       =(g_*)^-1(g_*([_0,n+1])∩ψ)
       =(g_*)^-1([_0,n+1]∩ψ)=g^-1·φ(ψ)

		where the fourth equality holds because Σ_n acts on _0,n+1 by complex automorphism, so each g∈Σ_n  preserves the orientation and, as a consequence, the fundamental class. Therefore the Poincarè duality isomorphism induce an isomorphism between the coinvariants
		
    H^i(_0,n+1;)_Σ_n H_2(n-2)-i(_0,n+1;)_Σ_n

		The right hand term is isomorphic to H_2(n-2)-i(_0,n+1/Σ_n;) Since we are working rationally. The left hand term is isomorphic (by the norm map) to the invariants H^i(_0,n+1;)^Σ_n≅ H^i(_0,n+1/Σ_n;).
		Now the statement follows by the Universal Coefficient Theorem which allow us to identify H^i(_0,n+1/Σ_n;) with H_i(_0,n+1/Σ_n;).
	
	A simple application of Proposition <ref> and Proposition <ref> is the following computation:
	
		H_2(_0,n+1/Σ_n;) has dimension n-2.    
		
	
	
	
		By Proposition <ref> H_2(_0,n+1/Σ_n;) is isomorphic to H_2(n-2)-2(_0,n+1/Σ_n;), so it suffices to compute the dimension of this last vector space. By the presentation of the Hypercommutative operad, H_2(n-2)-2(_0,n+1/Σ_n;) is generated by classes of the form
		
    ((x,…,x)_k  times ,x,…,x_n-k  times )      k=2,…,n-1

		Now we claim the these classes are actually a basis for H_2(n-2)-2(_0,n+1/Σ_n;): indeed in this case the generalized associativity relations are of the form
		
    ∑_k=2^n-1n-3k-2((x,…,x)_k  times ,x,…,x_n-k  times )=∑_k=2^n-1n-3k-2(x,(x,…,x)_k  times ,x,…,x_n-k-1  times )

		But x and the symmetric products (x,…,x) are all variables of even degree, so we can switch them without producing signs. Therefore the right hand term of the equality above is really the same as the left hand term, so the generalized associativity is actually the trivial relation 0=0. The claim is now proved since there are no other relations.
	
	More generally, one can combine Proposition <ref> and Proposition <ref> to compute H_*(_0,n+1/Σ_n;) for small values of n. We summarize the results in the following table:
	
    n    Poincarè polynomial 
    2   1
    3   1+t^2
    4   1+2t^2+t^4
    5   1+3t^2+3t^4+t^6
    6   1+4t^2+7t^4+4t^6+t^8

	



§ ON THE TORSION OF H_*(ℳ_0,N+1/Σ_N;)
 



 §.§ An upper bound on the order of the elements

Theorem <ref> tell us that rationally ℳ_0,n+1/Σ_n has the same homology of a point. In particular each class in the integral homology is a torsion class. This paragraph contains some partial answers to the question "what kind of torsion appears in H_*(ℳ_0,n+1/Σ_n;)?". 


    Let x be a class in H_*(ℳ_0,n+1/Σ_n;). Then the order of x divides n!.


    The quotient map p:ℳ_0,n+1→ℳ_0,n+1/Σ_n is a n! fold ramified covering in the sense of <cit.>. Therefore we have a transfer map τ:H_*(ℳ_0,n+1/Σ_n;)→ H_*(ℳ_0,n+1;Z) such that the diagram 
    
    H_*(ℳ_0,n+1/Σ_n;) [rr,"· n!"] [dr,"τ"]        H_*(ℳ_0,n+1/Σ_n;) 
           H_*(ℳ_0,n+1;) [ur,"p_*"]

    commutes. Given x∈ H_*(ℳ_0,n+1/Σ_n;) we know that it is a torsion class; however H_*(ℳ_0,n+1;) is torsion free, so τ is the zero map and we get the statement.

In order to get more information on the torsion of H_*(ℳ_0,n+1/Σ_n;) we will follow this approach:

    
  * First of all we focus on the calculation of H_*(ℳ_0,n+1/Σ_n;_p), with p be any prime number. The advantage is that now we have coefficients in a field and we can use linear algebra. 
    
  * Secondly we note that _0,n+1/Σ_n is homotopy equivalent to C_n()/S^1, which is a bit easier to handle (Remark <ref>).
    
  * To compute H_*(ℳ_0,n+1/Σ_n;_p)≅ H_*(C_n()/S^1;_p) the main idea is that we can compare it to H^S^1_*(C_n();_p), which is known by the computation of <cit.>. I thank my advisor Paolo Salvatore to suggesting this approach to me.

	We will get a complete calculation of H_*(ℳ_0,n+1/Σ_n;_p) when n≠ 0,1 mod p and n=p,p+1. The other cases seems more complicated. Moreover we will prove that ℳ_0,n+1/Σ_n is contractible for any n≤ 5. Before going into the details of these computations we discuss a bit the relation between homotopy quotients and strict quotients.


 §.§ Homotopy quotients vs strict quotients
 

Let X be an Hausdorff space equipped with an S^1-action. In particular, being Hausdorff ensures that the isotropy groups G_x are closed subgroups, i.e. G_x=S^1 or it is a finite group. In this paragraph we discuss the relation between the homotopy quotient X_S^1 and the strict quotient X/S^1. Most of the results presented are easy consequences of the theory of transformation groups, which is  developed in the books <cit.> and <cit.>. I believe that what follows is well known to the experts in the field, but I was not able to find it in the classical references cited above.


    Let X be a S^1-space, A⊆ X be a closed invariant subspace and F be an abelian group of coefficients. Suppose that for any x∈ X-(A∪ X^S^1) we have H^i(BG_x;F)=0 for all i>0. Then for any i∈ the map f:X_S^1→ X/S^1 induces an isomorphism
    
    f^*:H^i(X/S^1,A/S^1∪ X^S^1;F)→ H^i_S^1(X,A∪ X^S^1;F)


If we fix _p as coefficients and A=X^/p we get:

    Let X be an S^1-space. Then for any i∈ the map f:X_S^1→ X/S^1 induces an isomorphism
     
    f^*:H^i(X/S^1,X^/p/S^1;_p)→ H^i_S^1(X,X^/p;_p)

   


    If there are not /p-fixed points, then we have
    
    H^i(X/S^1;_p)≅ H^i_S^1(X;_p)

    Intuitively we can justify this statement like this: if there are no /p-fixed points the S^1-action on X behave as it would be free, so there is no difference (in (co)homology mod p) between the homotopy quotient and the strict quotient.

We can use this Corollary to get a Mayer-Vietoris sequence as follows: the map f:X_S^1→ X/S^1 sends (X^/p)_S^1 to X^/p/S^1 therefore we get a map of long exact sequences 

    [column sep=small]
           ⋯[r]     H^i(X/S^1,X^/p/S^1)[r][d,red]    H^i(X/S^1)[r][d]     H^i(X^/p/S^1)[r][d]   H^i+1(X/S^1,X^/p/S^1)[r][d,red]   ⋯
       ⋯[r]     H^i_S^1(X,X^/p)[r]    H^i_S^1(X)[r]     H^i_S^1(X^/p)[r]    H^i_S^1(X,X^/p)[r]    ⋯

where the red vertical arrows are isomorphisms by the previous Corollary. Therefore we get:

    Fix p a prime and use _p as field of coefficients for cohomology. Then we have a long exact sequence
    
    [column sep=small]
                ⋯[r]     H^i(X/S^1)[r]     H^i(X^/p/S^1)⊕ H^i_S^1(X)[r]   H^i_S^1(X^/p)[r]     H^i+1(X/S^1)[r]   ⋯



    Proposition <ref> continues to hold if we replace X^/p with X^S^1 and use rational coefficients for cohomology.

We are going to use Proposition <ref> for X=C_n() in order to do some computations of H_*(C_n()/S^1;_p). Before going on let us observe that the space of fixed points C_n()^/p is well understood:
	
		Let n=pq or n=pq+1. Then the fixed points C_n()^/p are homeomorphic to C_q(^*).
	
	
		Let us prove the statement when n=pq, the other case in similar. Let us denote by ζ e^i2π/p the generator of /p. Consider the quotient space
		
    H{z∈^*| arg(z)∈[0,2π/p]}/∼

		where ∼ identifies a point z∈{z∈^*| arg(z)=0} with ζ z. So H is homeomorphic to ^*. Now observe that any configuration in C_n()^/p is of the form {z_1,ζ z_1,…ζ^p-1z_1,…, z_q,ζ z_q,…ζ^p-1z_q}, where z_1,…,z_q are distinct points in {z∈^*| arg(z)∈[0,2π/p)}. The association {z_1,ζ z_1,…ζ^p-1z_1,…, z_q,ζ z_q,…ζ^p-1z_q}↦{z_1,…,z_q} defines a continuous map 
		
    f: C_n()^/p→ C_q(H)

		Conversely, if we have a configuration {z_1,…,z_q}∈  C_q(H), we can produce a configuration of C_n()^/p by taking the /p-orbits of every point. More precisely, the association {z_1,…,z_q}↦{z_1,ζ z_1,…ζ^p-1z_1,…, z_q,ζ z_q,…ζ^p-1z_q} defines a continuous function C_q(H)→ C_n()^/p, which is the inverse of f. See Figure <ref> for a pictorial description of f.
	
	


 §.§ Recollection on H_*^S^1(C_n();_p) 

In this paragraph we recall the computation of H_*^S^1(C_n();_p) for any n∈ and any prime p (see <cit.> for details).
Combining this with the Mayer-Vietoris sequence of Proposition <ref> we will be able to compute H_*(C_n()/S^1;_p)≅ H_*(ℳ_0,n+1/Σ_n;_p) is some cases. 

The computation of H_*^S^1(C_n();_p) was done using the Serre spectral sequence associated to the fibration

    C_n()↪ C_n()_S^1→ BS^1

The homology of the fiber is known, thanks to the work of F. Cohen (see <cit.>):
	
		Consider the disjoint union C()_n∈C_n() and let p be any prime. Then H_*(C();_p) has the following form:
		
			
  p=2: H_*(C();_2) is the free graded commutative algebra on classes {Q^i(a)}_i∈, where Q^i(a) is a class of degree 2^i-1 in H_*(C_2^i();_2).
			
  p≠ 2: H_*(C();_p) is the free graded commutative algebra on classes 
   
    {a,[a,a], Q^i[a,a], β Q^i[a,a]}_i≥ 1

   where a is the generator of H_0(C_1();_p), [a,a] is the generator of H_1(C_2();_p)  and Q^i[a,a] (resp. β Q^i[a,a]) is a class of degree 2p^i-1 (resp. 2p^i-2) in H_*(C_2p^i();_p).
		
	
The differential of the second page of the homological spectral sequence can be determined by the following Proposition:
	
		Let X be a topological space of finite type equipped with an S^1 action. Fix  a field of coefficients for (co)homology. Then the differential d^2 of the second page of the homological spectral sequence associated to X↪ X_S^1→ BS^1 is given by 
		
    d^2(x⊗ y_2i)= 0  if  i=0
    Δ(x)⊗ y_2i-2 otherwise

		where y_2i is the generator of H_2i(BS^1;) and Δ:H_*(X;)→ H_*+1(X;) is the BV-operator.
	
Consider a class a^k[a,a]^lx∈ H_*(C_n();_p), where k∈, l=0,1  and x is a monomial which contains only the letters {Q^i[a,a],β Q^i[a,a]}_i≥ 1. It turns out that the operator Δ acts as follows:
		
    Δ(a^kx)=k(k-1)a^k-2[a,a]x     Δ(a^k[a,a]x)=0


Therefore the second page of the homological spectral sequence associated to 

    C_n()↪ C_n()_S^1→ BS^1

is quite explicit and we are ready to state the main Theorems of <cit.>:

		Let p be a prime, n∈ such that n=0,1 mod p. Then the (co)homological spectral sequence (with _p-coefficients) associated to the fibration
  
    C_n()↪ C_n()_S^1→ BS^1

  degenerates at the second page. In particular
		
    H_*^S^1(C_n();_p)≅ H_*(C_n();_p)⊗ H_*(BS^1;_p)

	

		Let p be an odd prime, n∈ such that n≠ 0,1 mod p. The homological spectral sequence (with _p-coefficients) associated to the fibration
  
    C_n()↪ C_n()_S^1→ BS^1

  degenerates at the third page. In particular
		
    H_*^S^1(C_n();_p)≅ coker(Δ)

		where Δ:H_*(C_n();_p)→ H_*+1(C_n();_p) is the BV-operator.
	
	
		We can explicitly describe coker(Δ): a basis of this vector space is given by (the image of) classes in H_*(C_n();_p) which do not contain the bracket [a,a] (Equation <ref>).
	


 §.§ Computations in some special cases


From now on we fix _p as field of coefficients for homology, where p is any prime number. We are interested in the quotient _0,n+1/Σ_n, which is homotopy equivalent to C_n()/S^1. Corollary <ref> allow us to do a very nice computation:


	If n≠ 0,1 mod p the map f:C_n()_S^1→ C_n()/S^1 induces an isomorphism
	
    f^*:H^i(C_n()/S^1;_p)→ H^i_S^1(C_n();_p)



	Just observe that if n≠ 0,1 mod p there are no /p-fixed points.

Now suppose that p divides n (or n-1). We would like to apply the Mayer-Vietoris sequence of Proposition <ref> 

    [column sep=small]
    		   ⋯[r]     H^i(C_n()/S^1)[r]     H^i(C_n()^/p/S^1)⊕ H^i_S^1(C_n())[r]   H^i_S^1(C_n()^/p)[r]    ⋯

to compute of H^i(C_n()/S^1;_p), but in this generality the problem is hard. Indeed we do not know H^*(C_n()^/p/S^1;_p), which could be as difficult to compute as H^*(C_n()/S^1;_p). The only advantage is that C_n()^/p/S^1 is a much smaller space with respect to C_n()/S^1. Moreover, even if we would be able to do such computation it remains to understand the maps that fit into the Mayer-Vietoris sequence, which can be an even harder problem. 

	In the Mayer-Vietoris sequence above the terms H^i_S^1(C_n()^/p;_p) can be computed easily: consider the map of fibrations
	
    C_n()^/p[r] [d,hook]     C_n()^/p[d,hook]
       (C_n()^/p)_/p[r] [d]    (C_n()^/p)_S^1[d]
        B/p[r]    BS^1

	and observe that the mod p Serre spectral sequence of the left fibration degenerates at the second page. The map of cohomological spectral sequences induced by the map of fibrations above is surjective at the E_2 page so the Serre spectral sequence of the right fibration degenerates at the second page as well. Therefore H^*_S^1(C_n()^/p;_p)=H^*(BS^1;_p)⊗ H^*(C_n()^/p;_p) as _p-vector space. Finally we note that H^*_S^1(C_n()^/p;_p) is known: C_n()^/p is homeomorphic to the space of unordered configurations of points in ^* ( Lemma <ref>) and the homology of this space is known thanks to the work of F. Cohen <cit.> (see also Proposition <ref> for a computation of these homology groups).


	In some cases we can compute all the terms in the Mayer-Vietoris sequence <ref>. For example take n=pq (or n=pq+1) with (p,q)=1. The terms H^*(C_n()^/p/S^1;_p) can be computed as follows: first recall that C_n()^/p is homeomorphic to C_q(^*) (Lemma <ref>), therefore C_n()^/p/S^1≅ C_q(^*)/S^1. Since p does not divide q the subspace C_q(^*)^/p is empty and by Remark <ref> we deduce that
	
    H^*(C_q(^*)/S^1;_p)≅ H^*_S^1(C_q(^*);_p)

	The advantage is that H^*_S^1(C_q(^*);_p) can be computed by the Serre spectral sequence associated to the fibration
	
    C_q(^*)↪ C_q(^*)_S^1→ BS^1

	We will postpone this computation to Section <ref> (Theorem <ref>).


We end this paragraph with an acyclicity result:

	Let p be a prime. Then 
	
    H^*(C_p()/S^1;_p)≅ H^*(C_p+1()/S^1;_p)≅_p  if  *=0
    
    		0  otherwise


In order to prove this Theorem we need a preliminary Lemma:

	Let  n∈ and p be a prime. Suppose n=0,1 mod p. If i:C_n()^/p↪ C_n() is the inclusion then 
    i^*:H^k_S^1(C_n();_p)→ H^k_S^1(C_n()^/p;_p)

	is a monomorphism.


	Consider the commutative diagram
	
    H^k_S^1(C_n())[r,"i^*"][d]     H^k_S^1(C_n()^/p)[d]
       ( S^-1H^*_S^1(C_n()))^k[r,"i^*"]   ( S^-1H^*_S^1(C_n()^/p))^k

	where S is the multiplicatively closed subset {1,c,c^2,c^3,…}⊆ H^*(BS^1;_p)=_p[c], with c be a variable of degree two. By the Localization Theorem (see <cit.>) the bottom horizontal arrow is an isomorphism. The cohomological version of Theorem <ref> tells us that the mod p spectral sequence associated to the fibration C_n()↪ C_n()_S^1→ BS^1 degenerates at the E_2 page, so H^*_S^1(C_n();_p) is a free H^*(BS^1)-module. Therefore the left vertical arrow is a monomorphism, and this proves the statement.

(of Theorem <ref>)
	We do the case of C_p(), the other one is completely analogous. First of all note that C_p()^/p≅ C_1(^*)=^*. Therefore
	
    C_p()^/p/S^1≅^*/S^1

	and we conclude that C_p()^/p/S^1 is contractible. If i≥ 1 the Mayer-Vietoris sequence <ref> becomes
	
    [column sep=small]
    		   ⋯[r]     H^i(C_p()/S^1)[r]    H^i_S^1(C_p())[r]   H^i_S^1(C_p()^/p)[r]    ⋯

	By Lemma <ref> the map H^k_S^1(C_p();_p)→ H^k_S^1(C_p()^/p;_p)
	is a monomorphism, therefore the Mayer-Vietoris sequence splits (for i≥ 1) as:
	
    [column sep=small]
    		    0[r]     H^i_S^1(C_p())[r]   H^i_S^1(C_p()^/p)[r]     H^i+1(C_p()/S^1)[r]     0

	To conclude the proof we show that H^i_S^1(C_p();_p) and H^i_S^1(C_p()^/p;_p) are vector spaces of the same dimension: by Theorem <ref> we have H^*_S^1(C_p();_p)≅ H^*(C_p();_p)⊗ H^*(BS^1;_p) as H^*(BS^1;_p)-module. Similarly H^*_S^1(C_p()^/p;_p)≅ H^*(C_p()^/p;_p)⊗ H^*(BS^1;_p) by Remark <ref>. Finally we observe that H_*(C_p();_p) is generated by a class of degree zero and one of degree one (i.e. a^p and [a,a]a^p-2) therefore H^i(C_p();_p) and H^i(C_p()^/p;_p)≅ H^i(S^1;_p) have the same dimension for each i∈. As a consequence we get that H^i_S^1(C_p();_p) and H^i_S^1(C_p()^/p;_p) have the same dimension, as claimed.





 §.§ Examples

In what follows we use the results of the previous paragraph to do some explicit computations of H_*(C_n()/S^1;_p). In particular, we will see that the quotients C_n()/S^1 are contractible until n=6, which is the first non contractible space.


  
n=1: C_1()/S^1 is homeomorphic to a half line, so it is contractible.


  
n=2: by the combinatorial model explained in Section <ref> C_2()/S^1 is homotopy equivalent to a CW-complex with only one cell of dimension zero, so it is contractible.


  
n=3: by the combinatorial model explained in Section <ref> C_3()/S^1 is homotopy equivalent to a CW-complex of dimension one (see also Figure <ref>), so there are no homology classes of degrees strictly greater than 1. But C_3()/S^1 is simply connected, so H_1(C_3()/S^1;)=0 and this implies that C_3()/S^1 is contractible. 


  
n=4: We prove that C_4()/S^1 is contractible by showing that H_i(C_4()/S^1;)=0 for any i≥ 1. This is enough to prove the claim since C_4()/S^1 is homotopic to a CW complex of dimension two, and it is simply connected. By the cellular model we know that there are no homology classes of degrees strictly greater than 2.  By Theorem <ref> we know that the Euler characteristic must be one, so

    1=b_0-b_1+b_2

where b_i is i-th Betti number. But C_4()/S^1 is simply connected, so b_1=0 and combining this fact with the equation above we get that b_2=0. Therefore C_4()/S^1 is contractible.



  
n=5: We will prove that C_5()/S^1 is contractible by showing that for any prime p and any i≥ 1 H_i(C_5()/S^1;_p)=0. The cellular model tells us that C_5()/S^1 has no homology classes of degrees strictly greater than three. By Theorem <ref> the order of any class in H_*(C_5();) divides 5!. So the only interesting cases are when we take _2, _3 and _5 as coefficients for homology.

	
  * _2-coefficients: by Proposition <ref> we have a Mayer-Vietoris sequence
	
    [column sep=small]
    		   ⋯[r]    H^i(C_5()/S^1)[r]      H^i(C_5()^/2/S^1)⊕ H^i_S^1(C_5())[r]   H^i_S^1(C_5()^/2)[r]    ⋯

	Now observe that C_5()^/2 is homeomorphic C_2(^*), whose homology with _2-coefficients is generated by the classes listed below:
	
    Homology class      Degree
    a^2b   0
    [b,a]a   1
    b· Qa   1
    [[b,a],a]   2

	For the notation and more details about H_*(C_n(^*);_p) see Proposition <ref>. Moreover, C_2(^*)/S^1 is homotopy equivalent to a circle (see Figure <ref> and its caption for an explanation). The second pages of the Serre spectral sequences that compute H^*_S^1(C_5();_2) and H^*_S^1(C_5()^/2;_2) are displayed below:
	
	
	[classes = draw = none , xscale = 0.6, yscale=0.8, no ticks]
		
		
		["1"](0,0)
		["1"](0,1)
		["1"](0,2)
		["1"](0,3)
		["0"](1,0)
		["0"](1,1)
		["0"](1,2)
		["0"](1,3)
		["1"](2,0)
		["1"](2,1)
		["1"](2,2)
		["1"](2,3)
		["0"](3,0)
		["0"](3,1)
		["0"](3,2)
		["0"](3,3)
		["1"](4,0)
		["1"](4,1)
		["1"](4,2)
		["1"](4,3)
		["0"](5,0)
		["0"](5,1)
		["0"](5,2)
		["0"](5,3)
		["1"](6,0)
		["1"](6,1)
		["1"](6,2)
		["1"](6,3)
		["⋯"](7,0)
		["⋯"](7,1)
		["⋯"](7,2)
		["⋯"](7,3)
		[background] at (3.5,-1.5) H^*_S^1(C_5();_2);
		
	
	    [classes = draw = none , xscale = 0.6, yscale=0.8, no ticks]
		["1"](0,0)
		["2"](0,1)
		["1"](0,2)
		["0"](1,0)
		["0"](1,1)
		["0"](1,2)
		
		["1"](2,0)
		["2"](2,1)
		["1"](2,2)
		
		["0"](3,0)
		["0"](3,1)
		["0"](3,2)
		
		["1"](4,0)
		["2"](4,1)
		["1"](4,2)
		
		["0"](5,0)
		["0"](5,1)
		["0"](5,2)
		
		["1"](6,0)
		["2"](6,1)
		["1"](6,2)
		
		["⋯"](7,0)
		["⋯"](7,1)
		["⋯"](7,2)
		[background] at (3.5,-1.5) H^*_S^1(C_5()^/2;_2);
	
	
	As we know, both the spectral sequences degenerate at the second page (Theorem <ref> and Remark <ref>) and summing over the diagonals we get the ranks of H^i_S^1(C_5()) and H^i_S^1(C_5()^/2). By Lemma <ref> the inclusion i: C_5()^/2↪ C_5() induces a monomorphism i^*:H^i_S^1(C_5();_2)→ H^i_S^1(C_5()^/2;_2) in each degree, so we can conclude that it is an isomorphism for any i≥ 2 by looking at the ranks. If we put all these information in the Mayer-Vietoris sequence we conclude immediately that
	
    H^i(C_5()/S^1;_2)=0    for all  i≥ 3

	Now C_5()/S^1 is simply connected, so the first (co)homology group is zero. By the same argument as before using the Euler characteristic we can conclude that H^2(C_5()/S^1;_2)=0 as well. So there is no 2-torsion in H_*(C_5()/S^1;).
	
	
  * _3-coefficients: by Theorem <ref> H_*(C_5()/S^1;_3)≅ H_*^S^1(C_5();_3). Theorem <ref> tells us that H_*^S^1(C_5();_3) is isomorphic to the subspace of H_*(C_5();_3) spanned by the classes which do not contain the bracket [a,a]. In this case H_*(C_5();_3) has only two classes: a^5 of degree zero and a^3[a,a] of degree one, so we can conclude that H_*^S^1 (C_5();_3) is trivial. Hence there is no 3-torsion in H_*(C_5()/S^1;).
	
	
  * _5-coefficients: By Theorem <ref> H^i(C_5()/S^1;_5)=0 for any i≥ 1. Thus there is no 5-torsion in H_*(C_5()/S^1;). 





  
n=6: as we said at the beginning of this paragraph, C_6()/S^1 is the first non contractible space. In particular its homology with _3-coefficients will be not trivial. By Theorem <ref> the order of any class in H_*(C_6()/S^1;) divides 6!. So the only interesting coefficients for homology are _2, _3 and _5. From the cellular model described in Section <ref> we know that C_6()/S^1 is homotopy equivalent to a CW-complex of dimension 4, therefore H^i(C_6()/S^1;)=0 for any i≥ 5. Moreover C_6()/S^1 is simply connected and its Euler characteristic is equal to one, so we get 0=b_2-b_3+b_4. 

	
  * _5-coefficients: by Theorem <ref> H^i(C_6()/S^1;_5)=0 for any i≥ 1, so there is no 5-torsion in H_*(C_5()/S^1;). 
	
  * _3-coefficients: consider the Mayer-Vietoris sequence
	
    [column sep=small]
    		   ⋯[r]     H^i(C_6()/S^1)[r]     H^i(C_6()^/3/S^1)⊕ H^i_S^1(C_6())[r]   H^i_S^1(C_6()^/3)[r]    ⋯

	Now observe that C_6()^/3 is homeomorphic C_2(^*), whose homology with _3-coefficients is generated by the classes listed below:
	
    Homology class      Degree
    ba^2   0
    b[a,a]   1
    [b,a]a   1
    [[b,a],a]   2

	As we already know C_2(^*)/S^1 is homotopy equivalent to a circle, so it has no classes of degrees greater that two. The second pages of the Serre spectral sequences that compute H^*_S^1(C_6();_3) and H^*_S^1(C_6()^/3;_3) are displayed below:
	
	
	[classes = draw = none , xscale = 0.6, yscale=0.8, no ticks]
		
		
		["1"](0,0)
		["1"](0,1)
		["0"](0,2)
		["0"](0,3)
		["1"](0,4)
		["1"](0,5)
		
		
		["0"](1,0)
		["0"](1,1)
		["0"](1,2)
		["0"](1,3)
		["0"](1,4)
		["0"](1,5)
		
		["1"](2,0)
		["1"](2,1)
		["0"](2,2)
		["0"](2,3)
		["1"](2,4)
		["1"](2,5)
		
		["0"](3,0)
		["0"](3,1)
		["0"](3,2)
		["0"](3,3)
		["0"](3,4)
		["0"](3,5)
		
		["1"](4,0)
		["1"](4,1)
		["0"](4,2)
		["0"](4,3)
		["1"](4,4)
		["1"](4,5)
		
		["0"](5,0)
		["0"](5,1)
		["0"](5,2)
		["0"](5,3)
		["0"](5,4)
		["0"](5,5)
		
		["1"](6,0)
		["1"](6,1)
		["0"](6,2)
		["0"](6,3)
		["1"](6,4)
		["1"](6,5)
		
		["⋯"](7,0)
		["⋯"](7,1)
		["⋯"](7,2)
		["⋯"](7,3)
		["⋯"](7,4)
		["…"](7,5)
		
		[background] at (3.5,-1.5) H^*_S^1(C_6();_3);
		
	
	    [classes = draw = none , xscale = 0.6, yscale=0.8, no ticks]
		["1"](0,0)
		["2"](0,1)
		["1"](0,2)
		
		
		["0"](1,0)
		["0"](1,1)
		["0"](1,2)
		
		["1"](2,0)
		["2"](2,1)
		["1"](2,2)
		
		["0"](3,0)
		["0"](3,1)
		["0"](3,2)
		
		["1"](4,0)
		["2"](4,1)
		["1"](4,2)
		
		["0"](5,0)
		["0"](5,1)
		["0"](5,2)
		
		["1"](6,0)
		["2"](6,1)
		["1"](6,2)
		
		
		["⋯"](7,0)
		["⋯"](7,1)
		["⋯"](7,2)
		
		[background] at (3.5,-1.5) H^*_S^1(C_6()^/3;_3);
	
	
	As we know, both the spectral sequences degenerate at the second page and summing over the diagonals we get the ranks of H^i_S^1(C_6()) and H^i_S^1(C_6()^/3). By Lemma <ref> the inclusion i: C_6()^/3↪ C_6() induces a monomorphism i^*:H^i_S^1(C_6();_3)→ H^i_S^1(C_6()^/3;_3) in each degree, so we can conclude that it is an isomorphism for any i≥ 4 by looking at the ranks. If we put all these information in the Mayer-Vietoris sequence we conclude immediately that
	
    H^i(C_6()/S^1;_3)=
    		0    for all  i≥ 5  and  i=1,2 
    _3    for  i=0,3,4

	Thus C_6()/S^1 is not contractible, since there are 3-torsion classes in H_*(C_6()/S^1;).







§ EXTRA: COMPUTATION OF H_*^S^1(C_N(^*);_P) WHEN P∤ N

In this section we compute H^*_S^1(C_n(^*);_p) in the case n≠ 0 mod p. We will do this by analyzing the Serre spectral sequence associated to the fibration

    C_n(^*)↪ C_n(^*)_S^1→ BS^1

We start by reviewing the homology of the fiber.
	

 §.§ Preliminaries on H_*(C_n(^*);_p)
 
 
	
		Let M be a manifold and (X,∗) be a based CW-complex, not necessarily connected. The space of configurations in M with labels in X is defined as 
		
    C(M;X)_n∈F_n(M)×_Σ_n X^n/∼

		where (p_1,…,p_n;x_1,…,x_n)∼ (p_1,…,p̂_i,…,p_n;x_1,…,x̂_i,…,x_n) if x_i=∗.
	

	When M=^n the homology of C(^n;X) is known, thanks to the work of Cohen <cit.>. The idea is the following: C(^n;X) is homotopy equivalent to the free 𝒟_n-algebra on X, where 𝒟_n is the little n-disk operad. Therefore H_*(C(^n;X);_p) can be described as a functor of H_*(X;_p). For the purpose of this work it is enough to recall the results in the case n=2. 
	
	
		Let p be a prime number. Fix a basis ℬ of H_*(X;_p)/[∗], where [∗]∈ H_0(X;_p) is the class of the basepoint. We define a basic bracket of weight k inductively as follows:
		
			
  * A basic bracket of weight 1 is just an element a∈ℬ. Its degree is by definition the homological degree of a. Observe that any class of H_*(X;_p) can be seen as a class in H_*(C(;X);_p).
			
  * By induction assume that the basic brackets of weight j have been defined and equipped with a total ordering compatible with weight for j<k. Then a basic bracket of weight k is a homology class [a,b]∈ H_*(C(;X);_p), where [-,-] is the Browder bracket and a,b are basic brackets such that:
			
				
  * weight(a)+weight(b)=k.
				
  * a<b and if b=[c,d] then c≤ a.
			
			The degree of [a,b] is by definition deg(a)+deg(b)+1.
			
		
		In the case p≠2 we also include as basic brackets classes of the form [a,a], where a is a basic bracket of even degree. 
	
	
		Let p be any prime, and Q:H_q(C(;X);_p)→ H_pq+p-1(C(;X);_p) be the first Dyer-Lashof operation (when p is odd it acts only on classes of odd degree q). Then H_*(C(;X);_p) has the following form:
		
			
  p=2: H_*(C(;X);_2) is the free graded commutative algebra on classes Q^i(x), where Q^i denotes the i-th iteration of Q and x is a basic bracket.
			
  p≠ 2: H_*(C(;X);_p) is the free graded commutative algebra on classes Q^i(x) and β Q^i(x), where β is the Bockstein operator, Q^i denotes the i-th iteration of Q and x is a basic bracket of odd degree.
		
	
 Now consider the discrete space S^0∨ S^0={∗,a,b}, where ∗ is the basepoint. The labelled configuration space C(;S^0∨ S^0) is a disjoint union of components {C_n∘+m∙()}_n,m∈ consisting of all configurations of m black particles and n white particles, where by black particle (resp. white particle) we mean a point in the plane with label a (resp. b). A tipical element of C(;S^0∨ S^0) is depicted in Figure <ref>.
 

  
		C_n(^*) is homotopy equivalent to the configuration space of n black particles and one white particle in the plane. Therefore H_*(C_n(^*);_p) will be the subspace of H_*(C(,S^0∨ S^0);_p) spanned by those classes that involve only n black particles and one white particle. 
	
  Let us denote by a (resp. b) the class in H_0(S^0∨ S^0) which represent a black particle (resp. a white particle). Then H_*(C(,S^0∨ S^0);_p) can be computed using Theorem <ref>. The following Proposition just identifies H_*(C_n(^*);_p) as a subspace of H_*(C(,S^0∨ S^0);_p) (for details see <cit.>):
	 Let n∈ and p be a prime. Then
 
    H_*(C_n(^*);_p)=⊕_k=0^nb_k· H_*(C_n-k();_p)

 where 
 
    b_0 b
        b_1[b,a]
       b_2[[b,a],a]
       b_3[[[b,a],a],a]
       ⋯

	
 
     The generators of H_*(C_3(^*);_2) are listed in the first column of this table:
	
    Homology class x    Degree 
    ba^3   0
    [b,a]a^2   1
    baQa   1
    [b,a]Qa   2
    [[b,a],a]a   2
    [[[b,a],a],a]   3

 


 §.§ Computation

The main result of this paragraph is the following:

	Let p be any prime and n∈ be a natural number not divisible by p. Then 
	
    H_*^S^1(C_n(^*);_p)≅ coker(Δ)

	where Δ:H_*(C_n(^*))→ H_*+1(C_n(^*)) is the BV-operator.


	This Theorem gives us an explicit description of H_*^S^1(C_n(^*);_p): by Proposition <ref> we get that any class in H_*(C_n(^*);_p) is a linear combination of monomials of the form b_k· x where x∈ H_*(C_n-k();_p). Therefore if we want to compute Δ on H_*(C_n(^*);_p) it suffices to know its value on these generators. Since 
	
    Δ(xy)=Δ(x)y+(-1)^x xΔ(y)+(-1)^x[x,y]
 
 (see <cit.>) it suffices to know how Δ acts on the brackets {b_k}_k∈ and on the classes x∈ H_*(C_n-k(^*);_p). b_k is a top dimensional class in H_*(C_k(^*);_p) therefore 
	
    Δ(b_k)=0    k∈

	On the other hand, Δ(x)=0 if n=0,1 mod p (Theorem <ref>), while if n≠ 0,1 mod p Δ(x) is given by the formulas <ref>. For example, let us suppose n=3 and p=2. The generators of H_*(C_3(^*);_2) are listed in the first column of this table:
	
    Homology class x    Degree    Δ(x)
    ba^3   0   [b,a]a^2
    [b,a]a^2   1   0
    baQa   1   [b,a]Qa+[[b,a],a]a
    [b,a]Qa   2   [[[b,a],a],a]
    [[b,a],a]a   2   [[[b,a],a],a]
    [[[b,a],a],a]   3   0

	In the last column we see the image of each generator through Δ. Therefore
	
    H_i^S^1(C_3(^*);_2)=_2  if  i=0,1,2
    
    		0  otherwise



The idea behind the proof of Theorem <ref> is to analyse the Serre spectral sequence associated to the fibration

    C_n(^*)↪ C_n(^*)_S^1→ BS^1

As usual it is useful to compare the spectral sequence to one that is better understood. To define the comparison spectral sequence we need to introduce some notation:

	Let us denote by C_(n-1)∙+∘(^*) the space F_n(^*)/Σ_n-1, where Σ_n-1 is embedded in Σ_n as the subgroup that fix n. As the notation suggests, we can think of a point of C_(n-1)∙+∘(^*) as a configuration in ^* of n-1 indistinguishable black particles and one white particle. 

S^1 acts by rotations on C_(n-1)∙+∘(^*) and we have a n-fold covering

    p: C_(n-1)∙+∘(^*)→ C_n(^*)

which colors black the white particle (see Figure <ref>). Since p is S^1-equivariant we get a map of fibrations which will be the key to prove Theorem <ref>:

    C_(n-1)∙+∘(^*)[r][d,hook]    C_n(^*)[d,hook]
        C_(n-1)∙+∘(^*)_S^1[r][d]    C_n(^*)_S^1[d]
       BS^1[r]    BS^1





	Fix a field of coefficients  for (co)homology. Then the (co)homological spectral sequence associated to  the fibration
	
    C_(n-1)∙+∘(^*)↪  C_(n-1)∙+∘(^*)_S^1→ BS^1

	degenerates at page E^3. More precisely, we have
	
    E_i,j^3=
    		H_j(C_(n-1)∙+∘(^*)/S^1)  if  i=0
    
    		0   if  i>0



	We first observe a few elementary facts:
	
		
  * S^1 acts freely on C_(n-1)∙+∘(^*), therefore the homotopy quotient C_(n-1)∙+∘(^*)_S^1 and the strict quotient C_(n-1)∙+∘(^*)/S^1 are homotopy equivalent.
		
  * The continuous map
		
    s: F_n(^*)   → F_n(^*)
    
    			(z_1,…,z_n)   ↦(z_n/z_n)^-1·(z_1,…,z_n)

		is (Σ_n-1× S^1)-equivariant, so it induces a map
		
    F_n(^*)[r,"s"] [d]    F_n(^*)[r]     C_(n-1)∙+∘(^*)
        C_(n-1)∙+∘(^*)/S^1 [urr,dashed]

		which is a section to the S^1-principal bundle π:C_(n-1)∙+∘(^*)→ C_(n-1)∙+∘(^*)/S^1. Therefore C_(n-1)∙+∘(^*)≅ S^1× C_(n-1)∙+∘(^*)/S^1.
	
In what follows we will write X instead of C_(n-1)∙+∘(^*)/S^1 to shorten the notation. From the second point we can rewrite the fibration of the statement as
	
    S^1× Xi  X→ BS^1

 where i:S^1× X→ X is the projection on the second factor. Let us work cohomologically in order to exploit the multiplicativity of the spectral sequence: the second page looks as follows:
		
    E_2=[a]/(a^2)⊗ H^*(X;)⊗ H^*(BS^1;)

		where a is a generator of H^1(S^1). The classes x∈ H^*(X)⊆ E_2^0,* are infinite cycles because they belong to the image of i^*:H^*(X)→ H^*(S^1× X). This observation implies that E_3=E_∞, because the only multiplicative generator which can have non zero differentials is a, which is a class in E_2^0,1, and therefore d_n(a)=0 for n≥ 3. Now observe that d_2(a)≠ 0: if it were zero then E_2=E_∞ but this is a contraddiction because H^*(X;) is finite dimensional. Let us call c d_2(a); with this notation we have H^*(BS^1;)≅[c] and the differential of the second page is given by
  
    d_2(a· x· c^i)= x· c^i+1
       d_2( x· c^i)=0

  for any x∈ H^*(X;) and any i∈. This implies the (dual) statement.

Now we are ready to prove Theorem <ref>:
(of Theorem <ref>)
	Consider the Serre spectral sequence with _p-coefficients associated to the fibration
	
    C_n(^*)↪ C_n(^*)_S^1→ BS^1

	We claim that E^3_p,q=0 for any p>0, q∈, therefore the spectral sequence degenerates at the third page and 
	
    H_*^S^1(C_n(^*);_p)≅ E^3_0,*≅ coker(Δ)

	where the last isomorphisms follows directly from Proposition <ref>. Consider the map of fibrations 
	
    C_(n-1)∙+∘(^*)[r,"p"][d,hook]    C_n(^*)[d,hook]
        C_(n-1)∙+∘(^*)_S^1[r][d]    C_n(^*)_S^1[d]
       BS^1[r]    BS^1

	Let us denote by (E_*,d_*) (resp. (E'_*,d'_*)) the homological spectral sequence associated to the right (resp. left) fibration . Fix a cycle x∈ E^i,j_2 with i>0, and let τ: H_*(C_n(^*);_p)→ H_*(C_(n-1)∙+∘(^*);_p) be the transfer map associated to the n-fold covering p. We are going to show that x does not survive at page E_3, i.e. page E_3 contains elements only in the first column and this proves the claim. The key property we need to prove the Theorem is: 
	
    (τ⊗ 1)∘ d_2=d'_2∘(τ⊗ 1)

	We refer to Corollary <ref> for a proof of this equation. If we assume this result, then we have that d'_2∘(τ⊗ 1)(x)=(τ⊗ 1)∘ d_2(x)=0 because x is a cycle. By the previous Lemma (τ⊗ 1)(x) does not survive at E'_3, so there exists y∈ E'_2 such that d'_2(y)=(τ⊗ 1)(x). Applying p_*⊗ 1 we obtain
	
    (p_*⊗ 1)(d'_2(y))   =(p_*⊗ 1)∘(τ⊗ 1)(x)=n· x

	But n is not zero in _p so we can divide by n and get
	
    d_2∘(p_*⊗ 1)(y/n)=(p_*⊗ 1)∘ d'_2(y/n)=x

	which is exactly what we claimed.

We end this paragraph justifying Equation <ref>:



	
	
	
	
	
	
		The equality <ref> holds.
	
	
		By Proposition <ref> d_2 and d'_2 are given by the operator Δ, so it suffices to prove the equality
		
    τΔ=Δτ

		Consider the pullback square
		
		
    S^1× C_(n-1)∙+∘(^*)[d,"1× p"] [r,"θ'"]      C_(n-1)∙+∘(^*)[d,"p"]
       S^1× C_n(^*)[r,"θ"]   C_n(^*)

		where the horizontal arrows are the S^1-actions. Let us denote by τ (resp. τ') the homological transfer associated to right (resp. left) vertical arrow. Fix a class y∈ H_*(C_n(^*)) and consider its cross product [S^1]× y∈ H_*(S^1× C_n(^*)); by the naturality of the transfer under pullbacks we get  
		
    τΔ(y)   =τθ_*([S^1]× y)
       =θ'_*τ'([S^1]× y)
       =θ'_*([S^1]×τ (y))=Δτ(y)

	
