Section problems for configuration spaces of surfaces
Lei Chen

 alpha
Sales Forecast in E-commerce using 
 Convolutional Neural Network
    
Kui Zhao

       College of Computer Science

       Zhejiang University

       Hangzhou, China

       zhaokui@zju.edu.cn
Can Wang

       College of Computer Science

       Zhejiang University

       Hangzhou, China

       wcan@zju.edu.cn

    Received: December 30, 2023/ Accepted: December 30, 2023
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In this paper we give a close-to-sharp answer to the basic questions: When is there a continuous way to add a point to a configuration of n ordered points on a surface S of finite type so that all the points are still distinct? When this is possible, what are all the ways to do it?  More precisely, let PConf_n(S) be the space of ordered n-tuple of distinct points in S.  Let f_n(S): PConf_n+1(S) →PConf_n(S) be the map given by f_n(x_0,x_1,… ,x_n):=(x_1,… ,x_n). We classify all continuous sections of f_n up to homotopy by proving the following.

1. If S=ℝ^2 and n>3, any section of f_n(S) is either “adding a point at infinity" or “adding a point near x_k". (We define these two terms in Section 2.1; whether we can define “adding a point near x_k" or “adding a point at infinity" depends in a delicate way on properties of S. )

2. If S=S^2 a 2-sphere and n>4, any section of f_n(S) is “adding a point near x_k"; if S=S^2 and n=2, the bundle f_n(S) does not have a section. (We define this term in Section 3.2)

3. If S=S_g a surface of genus g>1 and for n>1, we give an easy proof of <cit.> that the bundle f_n(S) does not have a section. 




§ INTRODUCTION

Let M be a manifold. There is a natural geometric question: How can we continuously introduce a new point on M for any collection of n distinct points on M? We denote by PConf_n(M) the pure configuration space parametrizing ordered n-tuple of distinct points on M. Let f_n(M): PConf_n+1(M) →PConf_n(M) be the map given by f_n(x_0,x_1,… ,x_n):=(x_1,… ,x_n). There is a natural action of permutation group Σ_n on PConf_n(M) by permuting the n points. Permutation group Σ_n acts on the fiber bundle f_n(M) as well. Thus we get a new fiber bundle F_n(M): PConf_n+1(M)/Σ_n →PConf_n(M)/Σ_n, given by F_n(x_0,{x_1,… ,x_n}):={x_1,… ,x_n}. The quotient PConf_n(M)/Σ_n=:Conf_n(M) is called the configuration space parametrizing unordered n-tuple of distinct points on M. In this article, we will study the existence and uniqueness of sections of  f_n(M) and F_n(M) when M is a surface.

The study of sections of configuration spaces of open manifolds goes back to the work of McDuff and Segal <cit.> <cit.>. They introduce a point “at infinity", which allows them to prove homological stability for configuration spaces. For closed manifolds, the possibility of adding a point depends on the topology of the manifold. For a manifold M with a nowhere vanishing vector field, Cantero and Palmer <cit.>, Berrick, Cohen, Wong and Wu <cit.> introduced another way to add a new point by adding a point infinitesimally near an old point using the vector field. This allows Ellenberg and Wiltshire-Gordon <cit.> to improve eventual polynomiality to immediate polynomiality of the betti numbers of PConf_n(M) for some closed manifolds M. The following figures illustrate adding a point “at infinity" and adding a point infinitesimally near an old point on the plane ℝ^2.




 We call a section s of f_n(ℝ^2) (resp. f_n(S^2)) “adding a point near x_k" if s is homotopic to an element in the collection of sections Add_n,k(ℝ^2) (resp. Add_n,k(S^2)). Informally, we assign x_0 at a sufficiently small distance to x_k along some nonvanishing vector field. See Figure <ref> for a demonstration of “adding a point near x_k". Notice that there are infinitely many homotopy classes of sections in Add_n,k(ℝ^2) and Add_n,k(S^2) and they are classified by a kind of twists or sections of a circle bundle. See Section 2.1 and Section 3.2 for formal definitions of Add_n,k(ℝ^2) and Add_n,k(S^2) respectively.

We call a section s of f_n(ℝ^2) “adding a point at infinity" if s is homotopic to an element in the collection of sections Add_n,∞(ℝ^2); see Figure <ref>. Informally, we consider ℝ^2 as S^2 missing a point ∞, we can assign x_0 at a sufficiently small distance to ∞ along some nonvanishing vector field. See Section 2.1 for a formal definition of Add_n,∞(ℝ^2). 

Let S_g be a surface of genus g and S^2 be the 2-sphere. In this paper, we will classify the sections of the fiber bundle f_n(S) for 3 cases: ℝ^2, S^2 and S_g when g>1. Here by section we mean continuous section.
The following holds:

(1) If S=ℝ^2 and n>3, any section of f_n(S) is either “adding a point at infinity" or “adding a point near x_k" for some 1≤ k≤ n.

(2) If S=S^2 and n=2, the bundle f_n(S) does not have a section. If S=S^2 and n>4, any section of f_n(S) is “adding a point near x_k" for some 1≤ k≤ n.

(3) If S=S_g a surface of genus g>1 and for n>1, the bundle f_n(S) does not have a section.  



For unordered case, we have the following corollary.

The following holds:

(1) If S=ℝ^2 and n>3, any section of F_n(S) is “adding a point at infinity";

(2) If S=S^2 and n>4 or n=2,3, the bundle F_n(S) does not have a section.


We discuss the exceptional cases when n=3 for S=S^2 in Section <ref>. Our method does not work for the case n=4 but <cit.> proved that F_4(S^2) does not have sections. The g=1 case seems to be more complicated to analyze, therefore we do not pursue here. Notice that the construction “adding a point near x_k" works for the torus as well; see Section 2.1.


It is classical that f_n(ℝ^2) admits a section. In <cit.>, Fadell showed that when n>2, the bundle f_n(S^2) admits a section. The unordered case for S=ℝ^2, i.e. (1) of Corollary <ref> has been proved by <cit.> and <cit.>. In <cit.>, they prove the case (2) of Corollary <ref>, and even stronger, they deal with the multi-section problems. All the previous proofs make use of the braid relation and the presentations of braid groups and do not imply (1) and (2) in Theorem <ref>. Our main novelty is to use the characterization of lantern relation in analyzing the canonical reduction systems. The canonical reduction system uses the Thurston classification of isotopy classes of diffeomorphisms of surfaces. This idea originated from <cit.>.
 
The ordered case for S=S_g of g>1, i.e. (3) of Theorem <ref> has been proved by <cit.>. Their proof makes heavily use of the presentations of surface braid group. We give a simpler proof using the cohomology of surface braid group and a classification theorem in <cit.>.

The structure of the paper


  * In Section 2, we introduce the construction and the main tool we use: canonical reduction system.

  * In Section 3, we reduce Theorem <ref>(1) to a more algebraic statement Theorem <ref>.

  * In Section 4, we prove Theorem <ref>, which is the main work of this paper.

  * In Section 5, we prove Theorem <ref>(2) by reducing it to Theorem <ref>(1).

  * In Section 6, we prove Theorem <ref>(3) by a classification of maps between configuration spaces of surfaces in <cit.>.

  * In Section 7, we ask further questions.

Acknowledgements
The author would like to thank Kevin Casto for telling her about the construction of adding a nearby point and thank Nir Gadish for discussion. She would also like to thank Paolo Bellingeri, Dan Margalit, 
Cihan Bahran and an anonymous referee for pointing out many references of previous works on braid groups and typos. Finally, she would like to thank her advisor Benson Farb for his extensive comments and for his invaluable support from start to finish. 





§ THE CONSTRUCTION AND BACKGROUND ON CANONICAL REDUCTION SYSTEMS

Let S be a surface and let PConf_n(S) the pure configuration space be the space of ordered n-tuple of distinct points on S. The natural embedding PConf_n(S)⊂ S^n gives the topology on PConf_n(S). Let f_n(S): PConf_n+1(S) →PConf_n(S) be the map given by f_n(x_0,x_1,… ,x_n):=(x_1,… ,x_n).

There is a natural action of permutation group Σ_n on PConf_n(S) by permuting the n points. Thus the quotient space Conf_n(S) is the space of unordered n-tuple of distinct points in S. Permutation group Σ_n acts on the fiber bundle f_n(S) as well. Let F_n(S): PConf_n+1(S)/Σ_n →PConf_n(S)/Σ_n be the map given by F_n(x_0,{x_1,… ,x_n}):={x_1,… ,x_n}. The subject of this section is to classify the sections of the fiber bundles f_n(ℝ^2) and F_n(ℝ^2). 




 §.§ Constructing sections

In this subsection we give constructions of sections of the fiber bundle f_n(ℝ^2). There are two cases: “adding a point near x_k" and “adding a point at infinity". These constructions originate from Berrick, Cohen, Wong and Wu <cit.>, but the idea appeared in <cit.>.



Case 1: adding a point near x_k. Define

    PConf_n,k(ℝ^2)={(v_k,x_1,...,x_n)|x_1,...,x_n  be  n  points on ℝ^2  and  v_k  be a unit vector at  x_k}.

This is the total space of a circle bundle by forgetting the vector v_k

    S^1→PConf_n,k(ℝ^2)→PConf_n(ℝ^2).

Equip ℝ^2 with the Euclidean metric. Set

    ϵ(x_1,...,x_n)=1/2min_1≤ i≠ j≤ n{d(x_i,x_j)}.

By the definition of ϵ(x_1,...,x_n), setting x_0 to be the image of the v_k-flow at time ϵ(x_1,...,x_n) from x_k gives a map:

    em_n,k(ℝ^2):PConf_n,k(ℝ^2)↪PConf_n+1(ℝ^2).
 
Composing a continuous section s:PConf_n(ℝ^2)→PConf_n,k(ℝ^2) of the fiber bundle (<ref>) with em_n,k(ℝ^2) gives a section of the fiber bundle f_n(ℝ^2). 


We denote by Add_n,k(ℝ^2) the collection of sections of f_n(ℝ^2) consisting of compositions of a section of (<ref>) with em_n,k(ℝ^2).

Notice that there are infinitely many homotopy classes of sections in Add_n,k(ℝ^2) and they are in one-to-one correspondence with the homotopy classes of sections of (<ref>).



Case 2: adding a point at infinity. Let us call the north pole of a 2-sphere the point at infinity ∞. Then ℝ^2≅ S^2-∞ through the stereographic projection. Define

    PConf_n,∞(ℝ^2)={(v_∞,x_1,...,x_n)|x_1,...,x_n  be  n  points on ℝ and  v_∞ be a unit vector at ∞}.


This is the total space of a circle bundle by forgetting the vector


    S^1→PConf_n,∞(ℝ^2)→PConf_n(ℝ^2).


Equip S^2 with the spherical metric; i.e. the metric that is induced from the standard embedding S^2⊂ℝ^3. Set

    ϵ(x_1,...,x_n)=1/2min_1≤ i≤ n{d(x_i,∞)}.

By the definition of ϵ(x_1,...,x_n), setting x_0 to be the image of the v_∞-flow at time ϵ from ∞ gives a map:

    em_n,∞(ℝ^2):PConf_n,∞(ℝ^2)↪PConf_n+1(ℝ^2).
 
Composing a continuous section s:PConf_n(ℝ^2)→PConf_n,∞(ℝ^2) of the fiber bundle (<ref>) with em_n,∞(ℝ^2) gives a section of the fiber bundle f_n(ℝ^2).

We denote by Add_n,∞(ℝ^2) the collection of sections of f_n(ℝ^2) consisting of compositions of a section of (<ref>) with em_n,∞(ℝ^2).

Notice that there are infinitely many homotopy classes of sections in Add_n,∞(ℝ^2) and they are in one-to-one correspondence with the homotopy classes of sections of (<ref>).



 §.§ Background

In this subsection we discuss some properties of canonical reduction systems and the lantern relation. Let S=S_g,p^b be a surface with b boundary components and p punctures. Let Mod(S) (reps. PMod(S)) be the mapping class group (resp. pure mapping class group) of S, i.e. the group of isotopy classes of orientation-preserving diffeomorphisms of S fixing the boundary components pointwise and punctures as a set (resp. pointwise).  By “simple closed curves", we often mean isotopy class of simple closed curves, e.g. by “preserve a simple closed curve", we mean preserve the isotopy class of a curve. 

Thurston's classification of elements of Mod(S) is a very powerful tool to study mapping class groups. We call a mapping class f∈Mod(S) reducible if a power of f fixes a nonperipheral simple closed curve. Each nontrivial element f∈Mod(S) is of exactly one of the following types: periodic, reducible, pseudo-Anosov. See <cit.> and <cit.> for more details. We now give the definition of canonical reduction system. 


 A reduction system of a reducible mapping class h in Mod(S) is a set of disjoint nonperipheral curves that h fixes as a set up to isotopy. A reduction system is maximal if it is maximal with respect to inclusion of reduction systems for h. The canonical reduction system CRS(h) is the intersection of all maximal reduction systems of h. 

For a reducible element f, there exists n such that f^n fixes each element in CRS(f) and after cutting out CRS(f), the restriction of f^n on each component is either periodic or pseudo-Anosov. See <cit.>. Now we mention three properties of the canonical reduction systems that will be used later.

CRS(h^n)=CRS(h) for any n.

This is classical; see  <cit.>.

For a curve a on a surface S, denote by T_a the Dehn twist about a. For two curves a,b on a surface S, let i(a,b) be the geometric intersection number of a and b. For two sets of curves P and T, we say that S and T intersect if there exist a∈ P and b∈ T such that i(a,b)≠ 0. Notice that two sets of curves intersecting does not mean that they have a common element.

Let h be a reducible mapping class in Mod(S). If {γ} and CRS(h) intersect, then no power of h fixes γ.




Suppose that h^n fixes γ. Therefore γ belongs to a maximal reduction system M. By definition, CRS(h)⊂ M. However γ intersects some curve in CRS(f); this contradicts the fact that M is a set of disjoint curves. 


Suppose that h,f∈ Mod(S) and fh=hf. Then CRS(h) and CRS(f) do not intersect.



By conjugation, we have that CRS(hfh^-1)=h(CRS(f)). Since hfh^-1=f, we get that CRS(f)=h(CRS(f)). Therefore h fixes the whole set CRS(f). A power of h fixes all curves in CRS(f). By Proposition <ref>, curves in CRS(h) do not intersect curves in CRS(f).


Now, we introduce a remarkable relation for Mod(S) that will be used in the proof.

There is an orientation-preserving embedding of S_0,4⊂ S and let x,y,z,b_1,b_2,b_3,b_4 be simple closed curves in S_0,4 that are arranged as the curves shown in the following figure. 


In Mod(S) we have the relation

    T_xT_yT_z=T_b_1T_b_2T_b_3T_b_4.



This is classical; see <cit.>.






§ AN ALGEBRAIC RESULT AND HOW IT IMPLIES (1) OF THEOREM <REF>

In this section we give an algebraic result about the braid groups and prove how it implies (1) of Theorem <ref>. PConf_n(ℝ^2) and PConf_n+1(ℝ^2) are both K(π,1) spaces. This can be seen by induction on n and taking the long exact sequence of homotopy groups of the fiber bundle f_n(ℝ^2). Therefore, the homotopy classes of sections of f_n(ℝ^2) only depend on the homomorphisms of the fundamental groups. Let PB_n=π_1(PConf_n(ℝ^2)) and let F_n be a free group of n generators. The fundamental groups of the fiber bundle f_n(S) gives us the following short exact sequence, i.e. the Fadell-Neuwirth short exact sequence:



    1→ F_n→ PB_n+1 PB_n→ 1.


Let D_n be the disk with n punctures {x_1,...,x_n} and D_n+1 be the disk with n+1 punctures {x_0, x_1,...,x_n}  and the forget map forgets the point x_0. We view PB_n and PB_n+1 as mapping class groups as the following:

    PB_n=PMod(D_n) and PB_n+1=PMod(D_n+1).

A simple closed curve a on D_n separates D_n into two parts: the outside of a, i.e. the component containing the boundary of D_n and the inside of a, i.e. the one not containing the boundary of D_n. We say that a surrounds x_k if x_k∈ the inside of a. The following algebraic result on the splittings of the exact sequence (<ref>) is a key ingredient in the proof of Theorem <ref>. 


Suppose that we have a section s: PB_n→ PB_n+1. Then the image s(PB_n) either preserves a simple closed curve c surrounding points {x_1,...,x_n}, or preserves a simple closed curve c surrounding {x_i,x_0} for some i∈{1,2,...,n}. 



The rest of this subsection focuses on how Theorem <ref> implies part (1) of Theorem <ref>. Let c be a curve inside D_n+1 surrounding k points. Let D_k^l be a disk with k punctures and l open disks removed. We call the boundary of the l disks the small boundary components and the original boundary of D the big boundary component. See the following figure for a geometric explanation.

Let 

    PB_k,l:=PMod(D_k^l)

 be the pure mapping class group of D_k^l. The difference between punctures and boundary components is that the Dehn twist about a puncture is trivial but the Dehn twist about a boundary component is nontrivial. The following proposition describes the centralizer of T_c. Denote the centralizer of T_c by C_PB_n+1(T_c).  


C_PB_n+1(T_c) satisfies the following exact sequence

    1→ℤ PB_k× PB_n+1-k,1→ C_PB_n+1(T_c)→ 1

where k is the number of points that c surround.


This is classical. The centralizer of T_c is the subgroup of Mod(D_n) that fixes c. The curve c separates D_n into two components: C_1 that contains the boundary and C_2 that does not contain the boundary. Since C_1 and C_2 are not homeomorphic, we have that C_PB_n+1(T_c) only contains elements that preserve C_1 and C_2. Therefore, our statement holds.




Now we are ready to prove (1) of Theorem <ref>.

 
Let g:PConf_n(ℝ^2)→PConf_n+1(ℝ^2) be a section of the fiber bundle f_n(ℝ^2). Let s=g_*: PB_n→ PB_n+1 be the induced map on the fundamental groups of g. By Theorem <ref>, the image s(PB_n) preserves a curve c that either surrounds 2 points or n points. Therefore, s(PB_n) is in the centralizer of T_c in PB_n+1 by the fact that fT_cf^-1=T_f(c).



Case 1: when c surrounds {x_0,x_k}. PB_2≅ℤ, which is generated by the Dehn twist about the boundary component. From Proposition <ref> we have

    1→ℤℤ× PB_n-1,1→ C_PB_n+1(T_c)→ 1.

Therefore C_PB_n+1(T_c)≅ PB_n-1,1. The inclusion PB_n-1,1↪ PB_n+1 is induced by gluing a 2-punctured disk inside the small boundary of D_n-1^1.



On the other hand, we have that

    π_1(PConf_n,k(ℝ^2))=PB_n-1,1.

The fundamental groups of the fiber bundle (<ref>) is the following exact sequence:

    1→ℤ PB_n-1,1→ PB_n→ 1.

Here T_d is the Dehn twist about the small boundary component. The embedding em_n,k: PConf_n,k(ℝ^2)↪PConf_n+1(ℝ^2) induces a homomorphism on the fundamental group em_n,k*:PB_n-1,1→ PB_n+1. On the mapping class group level, since T_d in PB_n-1,1 is mapped to the Dehn twist about a curve surrounding {x_0,x_k}, we know that em_n,k* is also induced by gluing a 2-punctured disk inside the small boundary of D_n-1^1. The theorem holds.



Case 2: when c surrounds {x_1,...,x_n}. Since PB_1,1≅ℤ×ℤ, which is generated by the Dehn twists about the two boundaries, we have the following exact sequence:

    1→ℤℤ×ℤ× PB_n→ C_PB_n+1(T_c)→ 1.

On the mapping class group level, PB_n× PB_1,1→ C_PB_n+1(T_c)→ PB_n+1 is induced by gluing D_1^1 outside the big boundary component of D_n. Therefore ℤ× PB_n≅ C_PB_n+1(T_c) and the generator of ℤ is mapped to T_cT_b^-1 where b is the big boundary of D_n+1.

On the other hand, we have that

    π_1(PConf_n,∞(ℝ^2))=ℤ× PB_n.

The embedding em_n,∞: PConf_n,∞(ℝ^2)↪PConf_n+1(ℝ^2) induces em_n, ∞*:ℤ× PB_n→ PB_n+1 on the fundamental groups. On the level of mapping class groups, since ℤ maps to T_cT_b^-1, we know that em_n,∞* is induced by the embedding of D_n in D_n+1 and maps the generator of ℤ to T_cT_b^-1. Therefore, em_n,∞* is induced by gluing D_1^1 outside the big boundary component of D_n as well. Our theorem holds.


The classification of the sections of the fiber bundle f_n(S) is not entirely the same as the classification of the splittings of the exact sequence (<ref>). There is an subtlety coming from the choice of base point in the fundamental groups. Therefore, we classify the splittings of the exact sequence (<ref>) up to conjugacy. In Theorem <ref>, all the choices of c is coming from a conjugacy by an element F_n; thus they decide the same sections. 





§ THE PROOF OF THEOREM <REF>

Throughout the section we prove Theorem <ref>, which implies Theorem <ref>(1). The strategy of the proof is the following. We assume that there exists a section s:PB_n→ PB_n+1, i.e. f_n(ℝ^2)_*∘ s=id. The strategy is that we first determine s(T_a) for any simple closed curve a on D_n. We first prove that the lift s(T_a) is always a multi-twist about at most two curves on D_n+1; these two curves or one curve are either trivial or isotopic to a after forgetting the point x_0. This is done by using a result of McCarthy on centralizer of pseudo-Anosov element and lantern relation. We find a generating set of PB_n consisting of Dehn twists about curves bounding two points. We then argue depending on whether s(T_a) is a multi-twist on two curves or a single twist. The main tool of this part is Proposition <ref>, characterizing lantern relation that we deduce from Thurston's construction.



 §.§ Step 1: constrain the image of s(T_c) for a simple closed curve c

 The following proposition characterizes intersection number 2 of two curves and will be used many times in the proof.

Let i(a,b)≠ 0. Then T_aT_b is a multitwist if and only if i(a,b)=2.



This result was previously obtained by Margalit <cit.> and Hamidi-Tehrani <cit.>. We give a different proof using Thurston's construction; see e.g. <cit.>. There is a subspace T of S that a,b fills, i.e. the tubular neighborhood of a∪ b. Let ⟨ T_a,T_b⟩ be the group generated by T_a and T_b in Mod(T). Thurston's theorem says that when a,b fill, there is a representation ρ: ⟨ T_a,T_b⟩→PSL(2,ℝ) such that 

    T_a→[       1 -i(a,b);       0       1 ] and 
     T_b→[      1      0; i(a,b)      1 ].

ρ(h) is parabolic if and only if h is reducible on T. We know that 

    ρ(T_aT_b)= [       1 -i(a,b);       0       1 ][      1      0; i(a,b)      1 ]=[ 1-i(a,b)^2    -i(a,b);     i(a,b)          1 ]

Since Trace(ρ(T_aT_b))=2-i(a,b)^2, we know that T_aT_b is reducible on T if and only if i(a,b)=2. By the lantern relation, we know that T_aT_b is a multitwist when i(a,b)=2.


The following lemma determines s(T_a) for any simple closed curve a on D_n. 




Let a be a simple closed curve on D_n, then s(T_a) can only be one of the following three cases:

(1) It can be a Dehn twist T_a' about a curve a' on D_n+1 such that after forgetting x_0, we have a'=a.

(2) It can be a multitwist T_a'T_c^m (i.e. a product of twists on disjoint curves) about two curves a' and c on D_n+1 for m∈ℤ, where c surrounds 2 points {x_0,x_k} and after forgetting x_0, we have that a'=a.

(3) It can be T_a'(T_a'T_a”^-1)^n, where a' and a” are disjoint on D_n+1  such that after forgetting x_0, we have that a'=a”=a.



We start with a the proof of the claim:  After forgetting x_0, any element of CRS(s(T_a)) is either a or surrounding one puncture (trivial). 

The centralizer of T_a contains T_b when a,b are disjoint curves. By injectivity of s, the centralizer of s(T_a) contains a copy of ℤ^2 when n>3. By <cit.> that the centralizer of a pseudo-Anosov element is virtually cyclic, we know that s(T_a) is not pseudo-Anosov. The injectivity of s also implies that s(T_a) is not a torsion element. Therefore we have that s(T_a) is reducible under Thurston's classification of mapping classes. Assume there exists b'∈CRS(s(T_a)) such that after forgetting x_0, we have that b is not trivial and b≠ a. Since b'∈CRS(s(T_a)), we have that a power of s(T_a) fixes b'. Also a power of any mapping class that commutes with s(T_a) fixes b as well. We break our discussion into the following two cases.
 
 
  * Case 1: If i(a,b)≠ 0, then no power of T_a fixes b. However we also have some power of s(T_a) fixes b'. This is a contradiction.

  * Case 2: If i(a,b)=0 but b≠ a, then there exists a curve c such that i(c,b)≠ 0 but i(c,a)=0. Since s(T_c) commutes with s(T_a), we know that s(T_c) preserves CRS(s(T_a)). However i(b,c)≠ 0, which shows that no power of T_c preserve b. This contradicts the fact that a power of  s(T_c) preserves b'.


By the disjointness of curves in canonical reduction system, we have that CRS(s(T_a)) contains at most 2 curves. We break the rest of the proof into 3 cases.
 
 
  * Case 1: CRS(s(T_a)) only contains one curve a'. It depends on the location of x_0, only one side of a' will contain x_0, which means only that side could s(T_a) could possibly be not identity. If a' surrounds x_0 and a surrounds more than 2 points, there is a curve b inside of a containing 2 points, therefore s(T_a) fixes CRS(s(T_b)) inside of a', which means s(T_a) does not acts as pseudo-Anosov inside a'. This proves that a power of s(T_a) is the identity on the inside. Therefore a power of s(T_a) is a power of the T_a'. Since s(T_a) is a lift of T_a, we have that a power of s(T_a)T_a'^-1 is the identity. Therefore s(T_a)=T_a'.

If a' surrounds x_0 and a surrounds 2 points, we position a as in the Figure <ref>.

By the lantern relation, we have

    T_cT_dT_b=T_aT_e.

Since T_b,T_c,T_d,T_e commutes with T_a, we have that s(T_b),s(T_c),s(T_d),s(T_e) fix a' and therefore is identity on the component that a' stay. Since s(T_a) is a product of s(T_b),s(T_c),s(T_d),s(T_e), we know that s(T_a) is also identity in the interior of a'. Therefore s(T_a)=T_a'. The case when a' does not surround x_0 is similar.


  * Case 2:  CRS(s(T_a)) only contains two curves a' and c such that c surrounds 2 points {x_0,x_k}.  On both the exterior of a' and the interior of a' pinching curve c, we know that s(T_a) is identity. Therefore we know that s(T_a) is the multi-twist on a',c. 

  * Case 3:  CRS(s(T_a)) only contains two curves a' and a” such that after forgetting x_0, both curves a' and a” become a.  On both the interior of a' and the exterior of a”, we know that s(T_a) is identity. Therefore we know that s(T_a) is the multi-twist on a',a”. 



In the following argument, we will use small letters like a,b,c,... to represent simple closed curves on D_n and small letters with a prime or double primes like a',a”,b', ... to represent the canonical reduction systems of s(T_a), s(T_b),.... If we have two curves in CRS(s(T_a)), we use a' and a”. 



 §.§ Step 2: the case of adding points at infinity

On D_n, we call a simple closed curve surrounding 2 points by a basic simple closed curve. The following lemma gives one condition for s(PB_n) to preserve a simple closed curve surrounding {x_1,...,x_n}. 


If the canonical reduction system of any basic simple closed curve does not contain a curve surrounding x_0, then s(PB_n) preserves a simple closed curve surrounding {x_1,...,x_n}.

 
 

Suppose that there exists a simple closed curve a such that CRS(a) contains a curve surrounding x_0. We call a the innermost if a surrounds k points and the canonical reduction systems of all curves surrounding k-1 points does not contains a curve surrounding x_0. Take an innermost curve a such that a surrounds k points in D_n. By the assumption of Lemma <ref>, we have that k>2. There are three cases according to Lemma <ref>.




  
  


 
  
  


 
  
 




  * Case 1: CRS(a)={a'} such that after forgetting x_0, we have a'=a. We take b and c inside D_n as in the following figure, we have the lantern relation T_bT_c=T_eT_aT_d^-1. 




Because b,c,d,e surround less points than k and a is the innermost curve, we know that CRS(s(T_b)), CRS(s(T_c)), CRS(s(T_d)) and CRS(s(T_e)) each only contains one curve not surrounding x_0 and we denote them by b',c',d',e'. Since T_e, T_a and T_d commute with each other, their canonical reduction systems would be disjoint. By Lemma <ref>, we know that s(T_eT_aT_d^-1) is also a multitwist. Therefore by Lemma <ref>, we know that i(b',c')=2. However CRS(T_b'T_c') does not contain a' because a' surround x_0 but b',c' do not. This is a contradiction.

  * Case 2: CRS(a)={a',a”} such that a” surrounds 2 points {x_0,x_k} and after forgetting x_0, we have that a'=a.


  * Case 3: CRS(a)={a',a”} such that after forgetting x_0, we have that a'=a”=a.



Case 2 and 3 can be proved using a similar argument as in Case 1. We construct the same lantern relation and use the fact that a is the innermost curve to reach a contradiction. Therefore if the canonical reduction systems of all basic simple closed curves do not contain a curve surrounding x_0, then the canonical reduction systems of any curve does not surround x_0. This is true for the center element of PB_n as well. Let c be the boundary curve of D_n. Then CRS(c)={c'} does not contain x_0. However, all Dehn twists commute with T_c which preserves c'. 


Now we introduce a generating set for PB_n. Consider the n-punctured disk D_n in Figure <ref>. Let L be a segment below all the other points. Let L_1,...,L_n be segments connecting x_1,...,x_n to the segment L. Figure <ref> is the corresponding figure for D_n+1.


For {i_1,...,i_k} a subset of {1,...,n}, let a_i_1i_2...i_k be the boundary curve of the tubular neighborhood of L∪∪_m=1^kL_i_k. Denote by A_i_1i_2...i_k the Dehn twist about a_i_1i_2...i_k.  For {i_1,...,i_k} a subset of {0,1,...,n}, let b_i_1i_2...i_k be the boundary curve of the tubular neighborhood of L∪∪_m=1^kL_i_k. Denote by B_i_1i_2...i_k the Dehn twist about  b_i_1i_2...i_k. See Figure <ref> for an example of a curve representing a_124.



The following proposition describes a generating set of the group PB_n.
 
There is a generating set of PB_n consisting of all the Dehn twists about the basic curves a_ij for 1≤ i<j≤ n.


This is classical and can be prove it by induction on the exact sequence

    1→ F_k→ PB_k+1→ PB_k→ 1.

This generating set is given by Artin; e.g. see <cit.>




 §.§ Step 3: Finishing the proof of Theorem <ref>
 In this subsection, we prove Theorem <ref>. We break the proof into several cases. By Lemma <ref>, we only need to consider the case that there exists at least one basic simple closed curve a such that some element of CRS(a) surrounds x_0. We break our discussion into the following four cases.

Case 1: The canonical reduction systems of any basic curves only contain one curve a' and a' surrounds x_0.  

Let a,b,c,d be the curves in Figure <ref>. We have the lantern relation T_aT_bT_c=T_d. 


Since T_c and T_d commute, s(T_c)=T_c' and s(T_d)=T_d' also commute. Therefore s(T_a)s(T_b)=T_c'^-1T_d' is a multitwist by Lemma <ref>. By Lemma <ref>, we know that i(b',a')=2 as in Figure <ref>. Suppose that b' does not surround x_0. By the lantern relation, T_a'T_b'=T_d'T_e'T_c'^-1. Since s(T_d) and s(T_c) are commuting multicurves, s(T_d)s(T_c)^-1 is multicurve as well. Since a,b,c are basic curves, we know that s(T_d)=T_d'T_e' and s(T_c)=T_c'. By the same reason, we have that s(T_f)=T_f' in Figure <ref> and <ref>.



In the following, we prove that s(PB_n) preserves b_01. Under Notation <ref> for PB_n, we have a=a_12, b=a_23, c=a_13 and d=a_123. We also have that s(A_12)=B_012, s(A_23)=B_23 and s(A_13)=B_013. Since A_ij generates PB_n, all we need to show is that s(A_ij) preserves b_01. Since CRS(d) contains b_01, any curve disjoint from d preserves b_01. We only need to consider the curves that intersect with d. Without loss of generality, we only need to show that s(A_14), s(A_24) and s(A_34) preserve b_01. By the assumption of Case 1, we only need to show that the CRS(a_14), CRS(a_24) and CRS(a_34) are disjoint from b_01. 

Since i(a_12,a_34)=0, we have that CRS(a_12) is disjoint from CRS(a_34), which means that CRS(a_34) is disjoint from b_01.  Since s(T_f)=T_f' in Figure <ref> and Figure <ref>, CRS(a_24) is also disjoint from b_01. We have the following lantern relation:

    A_13A_34A_14=A_134.
 
The image of relation under lift s is:

    B_013B_34s(A_14)=s(A_134).
 
A_134 commutes with A_13 and A_34, thus CRS(a_134) is disjoint from b_013 and b_34. The only possible curves are b_01 and b_0134. If s(A_134)=B_0134, we have another lantern relation in D_n+1:

    B_013B_34B_014=B_0134B_01.
 
This proves that s(A_14)=B_014B_01^-1 preserving b_01=e'. If CRS(a_134) contains b_01, we also have that s(A_14) preserves b_01=e'. The case when b' surrounds x_0 follows from the same argument.


Case 2: There exists a basic simple curve a such that CRS(a) has two curves and both are isotopic to a after forgetting x_0.

Let b,c,d,e be curves in Figure <ref>. We have the lantern relation T_bT_cT_d=T_eT_a.
 

Since b,c,d,e are disjoint from a, we have that CRS(b), CRS(c), CRS(d) and CRS(e) are disjoint from {a',a”}. Therefore s(T_b)=T_b', s(T_c)=T_c', s(T_d)=T_d' and s(T_e)=T_e' as in Figure <ref>. But we also have the lantern relation T_b'T_c'T_d'=T_e'T_a'. Thus s(T_a)=T_a'. This contradicts the assumption of Case 2.

Case 3: There exists a basic simple curve a such that CRS(a) has two curves a',a” such that a' is isotopic to a and a” is trivial after forgetting x_0, and a' surrounds a”. 

We arrange a to be a_12 and a',a” to be b_01,b_012. Then we have s(A_12)=B_012B_01^k for k≠ 0 by Lemma <ref>. Without loss of generality, we only need to show that CRS(a_13) and CRS(a_23) are disjoint from b_01. First of all, we have the following lantern relation:

    A_123A_34A_124=A_12A_1234.




Since all of the curves above are disjoint from a_12, their canonical reduction systems are disjoint from a_12'=b_01 and a_12”=b_012. We have the lantern relation:

    B_0123B_34B_0124=B_012B_01234.

Since s(A_12)=B_012B_01^k, there exists at least one other curve in a_123, a_34, a_124, a_1234, whose canonical reduction system contains b_01. We break our discussion into the following four subcases depending on whether b_01 is an element in CRS(A_1234), CRS(A_123),  CRS(A_124) or  CRS(A_34), respectively.


  
Subcase 1 and 2: In the first two cases, it is clear that CRS(a_13) and CRS(a_23) are disjoint from b_01 because a_13 and a_23 are disjoint from a_123 and a_1234.



Subcase 3: By i(a_14,a_124)=0, we have that b_014∈CRS(a_14) and b_01 does not intersect CRS(a_14). Since i(a_23,a_14)=0, we have that b_014 does not intersect CRS(a_23). Suppose CRS(a_23) contains another curve z that is trivial after forgetting x_0. Since a_23 is disjoint from a_123 and a_14, we have that z has to be disjoint from b_014 and b_0123. The only possibility is that z=b_01.


Because of the disjointness of a_123 and a_12, we have that s(A_123) preserves CRS(a_12). This shows that s(A_123) preserves b_01. We have a lantern relation

    A_12A_23A_13=A_123

After applying the homomorphism s to the above relation, all of the above element except s(A_13) preserves b_01. Therefore s(A_13) fixes b_01.



Subcase 4:  Since i(a_234,a_34)=0, we have that b_234∈CRS(a_234). Since i(a_123,a_12)=0, we have that b_0123∈CRS(a_123). Therefore b_23∈CRS(a_23) and CRS(a_23) may contain another curve z that is trivial after forgetting x_0. However a_23 is disjoint from a_123 and a_234, which implies that z is disjoint from b_0123 and b_234. Therefore z can only be b_01. By the same argument as Subcase 3, we know that A_13 also fixes b_01.
Case 4: There exists a basic simple curve a such that CRS(a) has two curves a',a” such that a' is isotopic to a and a” is trivial after forgetting x_0, and a' does not surround a”. 

Let a',a” be positioned into the following Figure <ref> such that a'=b_34 and a”=b_01. If a curve c is disjoint from a_34, then s(T_c) preserves b_01. Therefore without loss of generality, we only need to show that s(A_23) and s(A_13) preserve b_01.





Since i(a_12,a_34)=0, we have that b_012∈CRS(a_12). Since i(a_124,a_12)=0, we have that b_0124∈CRS(a_124). Possibly  CRS(a_124) contains another curve z that is trivial after forgetting x_0. However b_24∈CRS(a_24) because b_234∈CRS(a_234) and i(a_234,a_24)=0. Therefore, z is disjoint from b_24 and b_012, which means z=b_01. By the same reason, we can prove that b_0123∈CRS(a_123) and s(A_123) preserves b_01.
We have the following lantern relation.

    A_123A_34A_124=A_12A_1234.

Since A_34=B_34B_01^k for nonzero k, therefore the canonical reduction system of one of the curves in the relation (<ref>) contains b_01. The rest of the discussion is similar to Case 3 by doing a case study.




 §.§ The proof of (1) of Corollary <ref>


Let B_n=π_1(PConf_n(ℝ^2)/Σ_n) and B_n,1=π_1(PConf_n+1(ℝ^2)/Σ_n). The fiber bundle F_n(S) gives the first line of the following commutative diagram:

    1[r]        PB_n+1[r][d]^f_n(ℝ^2)_*       B_n,1[r][d]^F_n(ℝ^2)_*   Σ_n   [r][d]^=   1
    
    1[r]         PB_n[r]                                B_n[r]   Σ_n  [r]              1  .


Every splitting of F_n(ℝ^2)_* induces a splitting of f_n(ℝ^2)_*. Therefore, we only need to study the extension of a splitting of f_n(ℝ^2)_* to a splitting of F_n(ℝ^2)_*. Let ϕ:B_n→ B_n,1 be a splitting of F_n(ℝ^2)_*. Let x∈ PB_n and e∈ B_n. We have that 

    ϕ(exe^-1)=ϕ(e)ϕ(x)ϕ(e)^-1.

Denote by C_e the conjugation of e on PB_n. Therefore, we have the following diagram:

    PB_n[r]^C_e[d]^ϕ|_PB_n        PB_n[d]^ϕ|_PB_n
    
    PB_n+1[r]^C_ϕ(e)           PB_n+1.


By Theorem <ref>, there are two possibilities of ϕ|_PB_n: 



(1) ϕ fixes a simple closed curve c surrounding {x_k,x_0}



(2) ϕ fixes a simple closed curve c surrounding {x_1,...,x_n}.


We claim that ϕ|_PB_n fixes a simple closed curve surrounding {x_1,...,x_n}. To prove this claim, we assume the opposite that ϕ|_PB_n  fixes a simple closed curve c surrounding {x_k,x_0}. There exists an element e∈ B_n such that e permutes punctures k and j≠ k. Since c is the only curve that ϕ(PB_n) fixes, we have that c is the only curve that ϕ(C_e(PB_n))=PB_n fixes, which contradicts that C_ϕ(e)(ϕ(PB_n)) also fixes ϕ(e)(c) surrounding {x_j,x_0}. Therefore, ϕ|_PB_n fixes a simple closed curve surrounds {x_1,...,x_n}. In this case, the section is adding a point at infinity.





§ THE CASE WHEN S IS THE 2-SPHERE S^2

In this subsection we give a construction of sections of the fiber bundle f_n(S^2).



 §.§ Nonexistence of a continuous section for n=2

We prove a more general result on the sections of the fiber bundle f_n(S^2) for n=2. Let S^2k be 2k-dimensional sphere for k>0 integer.  Let x_1,x_2 be two distinct points in S^2k. The following is classical; see <cit.>.

The following fiber bundle

    S^2k-{x_1,x_2}→ PConf_3(S^2k) PConf_2(S^2k)

does not have a continuous section.


Suppose that there is a continuous map s:PConf_2(S^2k)→PConf_3(S^2k) such that f_2(S^2k)∘ s=identity. Then after post-composing with a forgetful map to the last coordinate, we obtain a map f: PConf_2(S^2k)→ S^2k. We denote by p_i:PConf_2(S^2k)→ S^2k the projection to the ith component. Let

    g_i:PConf_2(S^2k) PConf_2(S^2k)⊂ S^2k× S^2k.


Let ⊂ S^2k× S^2k be the diagonal subspace in the product. Let []∈ H^2k(S^2k× S^2k,ℚ) be the Poincaré dual of . By the Thom isomorphism, there is an exact sequence for the computation of cohomology:

    0→ℚ H^2k(S^2k× S^2k,ℚ)→ H^2k(S^2k× S^2k-;ℚ)→ 0.

Let c∈ H^2k(S^2k;ℚ) be the fundamental class and c_i=p^*_i(c). The image of diagonal is the Thom class c_1+c_2. Therefore in H^2k(S^2k× S^2k-;ℚ), we have c_1+c_2=0. This means that 

    H^2k(S^2k× S^2k-;ℚ)=ℚc_1.


Suppose that f^*(x)=kc_1 for k an integer. Therefore we will have g_i^*([])=kc_1+c_i. Since the image of g_i misses the diagonal , we have that g_1^*([])=kc_1+c_1=0 and g_2^*([])=kc_1+c_2=kc_1-c_1=0. Since c_1 is a generator of H^2k(S^2k× S^2k-;ℚ), we have that k+1=0 and k-1=0. These two formulas cannot be satisfied at the same time.







 §.§ Constructing sections when n>2

Define

    PConf_n,k(S^2)={(v_1,x_1,...,x_n)|x_1,...,x_n  be  n  points in  S^2  and  v_k  be a unit vector at  x_k}.

This is the total space of a circle bundle by forgetting the vector:

    S^1→PConf_n,k(S^2)→PConf_n(S^2).



For n>2, the fiber bundle (<ref>) is a trivial bundle.



S^1-bundle is classified by Euler class, i.e. a second cohomology class of the base. We investigate H^2(PConf_n(S^2);ℤ) first. There is a graded-commutative ℚ-algebra [G_ij] defined in <cit.>, where the degree of the generators G_ij is 1. By Totaro <cit.>, there is a spectral sequence E_2^p,q=H^p((S^2)^n;ℚ)[G_ij]^q converging to H^*(PConf_n(S^2);ℚ). Since we only compute H^2, the differential involved is d_2:E_2^0,1=H^0(S^2;ℚ)[G_ij]→ E_2^2,0=H^2(S^2;ℚ). Let [_ij]∈ H^2(S^2;ℚ) be the Poincaré dual of _ij⊂ S^2. By <cit.>, the differential d_2(G_ij)=[_ij]. Let p_i:(S^2)^n→ S^2 be the projection to the ith coordinate and [S^2]∈ H^2(S^2;ℤ) be the generator of H^2(S^2;ℤ). Therefore we have that

    H^2(PConf_n(S^2);ℤ)=⊕_i=1^nℤp_i^*[S^2]/(p_i^*[S^2]+p_j^*[S^2])≅ℤ/2,

which is generated by p_k^*[S^2] and we have that 2p_k^*[S^2]=0. The circle bundle (<ref>) is induced from the circle bundle 

    S^1→PConf_1,1(S^2)→ S^2

by the projection to the kth coordinate. The bundle (<ref>) is the unit tangent bundle over S^2. Since the Euler characteristic of S^2 is 2, the Euler class of (<ref>) is eu=2[S^2]∈ H^2(S^2;ℤ). Therefore the Euler class of (<ref>) is p_k^*[eu]=2p_k^*[S^2]=0∈ H^2(PConf_n(S^2);ℤ).


Equip S^2 with the spherical metric; i.e. the metric that is induced from the standard embedding S^2⊂ℝ^3. Let 

    ϵ(x_1,...,x_n)=1/2min_1≤ i<j≤ n{d(x_i,x_j)}.

Set x_0 to be the image of the v_k-flow at time ϵ from x_k; that is

    em_n,k(S^2):PConf_n,k(S^2)↪PConf_n+1(S^2)
 

Composing a continuous section s:PConf_n(S^2)→PConf_n,k(S^2) of the fiber bundle (<ref>) with em_n,k(S^2) gives a section of the fiber bundle f_n(S^2). 

We denote by Add_n,k(S^2) the collection of sections of f_n(S^2) consisting of compositions of a section of (<ref>) with em_n,k(S^2).

Notice that there are infinitely many homotopy classes of sections in Add_n,k(S^2) and they are classified by sections of (<ref>).



A special section for n=3.  Since there is a unique Mobius transformation ϕ(x_1,x_2,x_3) that transforms (0,1,∞) to any ordered three points (x_1,x_2,x_3). we have that 

    PConf_3(S^2)      PSL(2,ℂ).

We can assign any new point x_0=ϕ(x_1,x_2,x_3))(a) such that a≠ 0,1,∞.



 §.§ The proof of (2) of Theorem <ref>

In this subsection we prove (2) of Theorem <ref>. Let S_0,n a sphere with n punctures. Let Diff(S_0,n) be the orientation-preserving diffeomorphism group of S_0,n fixing the n punctures pointwise. While the following is surely known to experts, we could not find this statement or a proof in the literature. I am thus incluing it for completeness. We believe that it follows from Earle-Eells <cit.> in the punctured case.

For n>2, we have that 

    BDiff(S_0,n)≅ K( PMod(S_0,n),1)



We only need to prove that the homotopy group π_k(Diff(S_0,n))=0 for k>0. For n=0, by Smale <cit.>, Diff(S^2)≃SO(3). By fiber bundle

    Diff(S_0,n+1)→Diff(S_0,n)→ S_0,n,

we deduce that Diff(S_0,1)≃SO(2) and Diff(S_0,2)≃SO(2). The long exact sequence of homotopy groups of the fiber bundle (<ref>) is 

    1→π_1(Diff(S_0,3))→π_1( Diff(S_0,2))→π_1(S_0,2)→PMod_0,3→PMod_0,2→ 1.

However we know that PMod_0,3=1 (see <cit.>), we get that π_1(Diff(S_0,3))=0 and also π_i(Diff(S_0,3))=0 for i>1. The other cases are the same.

Let PB_n(S^2)=π_1(PConf_n(S^2)). Now we are ready to prove (2) of Theorem <ref>.

Let 

    S_0,n+1→UDiff(S_0,n+1)BDiff(S_0,n+1)

be the universal S_0,n+1-bundle in the sense that any S^2 bundle with n+1 sections 

    S_0,n+1→ E→ B

is the pullback from u_n+1 by a continuous map f:B→BDiff(S_0,n+1). By Proposition <ref>, BDiff(S_0,n+1)≅ K(PMod(S_0,n+1),1). This means that UDiff(S_0,n+1) is also a K(π,1)-space. Therefore S_0,n+1-bundles are determined by their monodromy representations and the sections of an S_0,n+1-bundle are also determined by the maps on fundamental groups. A splitting of the following exact sequence gives us a section of the fiber bundle f_n(S^2).

    1→ F_n→ PB_n+2(S^2) PB_n+1(S^2)→ 1.


We have the following diagram:

    S_0,n+1[r][d]   PConf_n+1(S_0,1)[r]^f_n(S_0,1)[d]    PConf_n(S_0,1)[d]
    
    S_0,n+1[r]    PConf_n+2(S^2)[r]^f_n+1(S^2)[d]^p_n+1   PConf_n+1(S^2)[d]^p_n+1
       S^2[r]   S^2.


By the long exact sequence of homotopy groups of the fiber bundle 

    PConf_n(S_0,1)→PConf_n+1(S^2)→ S^2,

we have that PB_n+1(S^2)=PB_n/Z where Z denotes the center of PB_n and is generated by the Dehn twist about the boundary of D_n; see <cit.>. Therefore a section of f_n(S_0,1) induced from a section of f_n+1(S^2) satisfies that f_n(S_0,1)_* maps the center to the center.



Since S_0,1≈ℝ^2, the section problem for f_n(S_0,1) has been fully discussed in Section 2. Every section of f_n+1(S^2) induces a section of f_n(ℝ^2), thus we could use the classification of sections of f_n(ℝ^2) to study the sections of f_n+1(S^2). Let s:PB_n+1(S^2)→ PB_n+2(S^2) be a splitting of f_n+1(S^2)_* such that f_n+1(S^2)_*∘ s=id. By (1) of Theorem <ref>, we break the discussion into the following two cases according to the sections of f_n(S_0,1). 



Case 1: the section of f_n(S_0,1) is adding a point near x_k. In this case, s(PB_n+1(S^2)) fixes a curve c around {x_0,x_k}. Then the image lies in the stabilizer of c. The stabilizer of c in PMod(S_0,n+2) is PMod(D_n)≅ PB_n. The boundary of D_n is c surrounding {x_1,...,x_k-1,x_k+1,...,x_n+1}. On the other hand by Proposition <ref> the circle bunlde

    S^1→PConf_n,k(S^2)→PConf_n(S^2)

is trivial, we have that 

    π_1(PConf_n+1,k(S^2))≅ℤ×π_1(PConf_n+1(S^2))≅ℤ× PB_n+1(S^2)≅ PB_n.

The last isomorphism is coming from the splitting of the following exact sequence; see <cit.>.

    1→ Z→ PB_n→ PB_n/Z→ 1.


Since the ℤ component of π_1(PConf_n+1,k(S^2)) is mapped to the Dehn twist about a curve d surrounding {x_0,x_k}, it means that d also surrounds {x_1,...,x_k-1,x_k+1,...,x_n+1}. Therefore we have that f_n+1(S^2) is adding a point near x_k.



Case 2: the section of f_n(S_0,1) is adding a point near ∞. In this case, s(PB_n+1(S^2)) fixes a curve c around {x_1,...,x_n}. Then the image lies in the stabilizer of c. The stabilizer of c in Mod(S_0,n+2) is PMod(D_n)≅ PB_n. The boundary of D_n is c surrounding {x_1,...,x_n}. On the other hand by Proposition <ref> the circle bunlde

    S^1→PConf_n+1,n+1(S^2)→PConf_n+1(S^2).

is trivial, we have that 

    π_1(PConf_n+1,n+1(S^2))≅ℤ×π_1(PConf_n+1(S^2))≅ℤ× PB_n+1(S^2)≅ PB_n.

Since the ℤ component of π_1(PConf_n+1,k(S^2)) is mapped to the Dehn twist about a curve d surrounding {x_0,x_n+1}, it means that d also surrounds {x_1,...,x_n}. Therefore we have that f_n+1(S^2) is adding a point near x_n+1.




 §.§ The unordered case



By the same argument as the proof of (1) of Corollary <ref>, we show that none of the sections of 

    f_n(S^2): PConf_n+1(S^2)→PConf_n(S^2)

can be extended to a section of 

    F_n(S^2): PConf_n+1(S^2)/Σ_n→PConf_n(S^2)/Σ_n.





 §.§ The exceptional cases

For the special cases n=3, we have the following classification.

There is a unique section for the fiber bundle f_3(S^2) up to homotopy. There is no section for the bundle F_3(S^2).


By Proposition <ref>, we have that BDiff(S_0,3)≅ K(PMod(S_0,3),1). Since PMod(S_0,3)=1, the classifying space BDiff(S_0,3) is contractible. Therefore every S^2-bundle with 3 sections is a trivial bundle. Thus f_3(S^2) is a trivial bundle. Therefore, a section of f_3(S^2) is determined by a map PConf_3(S^2)→ S_0,3. Since S_0,3≅ K(F_2,1), a map PConf_3(S^2)→ S_0,3 up to homotopy is determined by Hom(PB_3(S^2),F_2) up to conjugation. However PB_3(S^2)=PB_2/Z=1 implying that Hom(PB_3(S^2),F_2)=1. Therefore, there is a unique section up to homotopy. For the unordered case F_3(S^2), let Mod(S_0,3,1) be the mapping class group of S^2 fixing a set of 3 points and a set of 1 point. There is an exact sequence

    1→PMod(S_0,4)→Mod(S_0,3,1)→Σ_3 → 1.

Since π_1(PConf_3(S^2)/Σ_3)≅Σ_3 and π_1(S_0,3)≅PMod(S_0,4), we have that Mod(S_0,3,1)=π_1(PConf_4(S^2)/Σ_3). Therefore, the section of F_3(S^2) is the determined by the splittings of the exact sequence (<ref>).

Let Diff(S_0,3,1) be the orientation-preserving diffeomorphism group of S^2 fixing a set of 3 points and a set of 1 point. By definition there is a map ρ: Diff(S_0,3,1)→Mod(S_0,3,1) which induces isomorphism on π_0. A version of the Nielsen Realisation Theorem (e.g. <cit.> and <cit.>) tells us that a finite subgroup of Mod(S_0,3,1) has a lift to Diff(S_0,3,1). However every finite subgroup of Diff(S_0,3,1) is cyclic because Diff(S_0,3,1) fixes a point. Therefore every finite subgroup of Mod(S_0,3,1) is cyclic. Since Σ_3 is noncyclic, (<ref>) does not split.

For the special cases n=4, we have the following classification.

The sections of fiber bundle f_4(S^2) correspond to the splittings of the exact sequence

    1→ F_3 → PB_5(S^2) F_2→ 1

up to conjugation.


We have the following Birman exact sequence; see <cit.>.

    1→π_1(S_0,3)→ PB_4(S^2) PB_3(S^2)→ 1.

Since PB_3(S^2)=1, we have that PB_4(S^2)=π_1(S_0,3)≅ F_2. By Proposition <ref>, the sections of f_4(S^2) is determined by the splittings of the following Birman exact sequence up to conjugaction.

    1→π_1(S_0,4)→ PB_5(S^2) PB_4(S^2)→ 1.
 



§ THE CASE WHEN S=S_G A CLOSED SURFACE OF GENUS G>1

In this section, we prove Theorem <ref>(3).  Let S_g^n be the product of n copies of S_g. There is a natural embedding PConf_n(S_g)⊂ S_g^n. Let p_i:PConf_n(S_g)→ S_g be the projection onto the ith component. Denote by _ij≈ S_g^n-1⊂ S_g^n the ijth diagonal subspace of S_g^n; i.e., _ij consists of points in S_g^n such that the ith and jth coordinates are equal. Let H_i:=p_i^*H^1(S_g;ℚ) and let [S_g] be the fundamental class in H^2(S_g;ℚ). Now, we display the computation of H^*(PConf_n(S_g);ℚ) from <cit.>. 

(1) For g>1 and n>0,

    H^1( PConf_n(S_g);ℚ)≅ H^1(S_g^n;ℚ)≅⊕_i=1^nH_i.

(2)We have an exact sequence

    1→⊕_1≤ i<j≤ nℚ[G_ij]  H^2(S_g^n;ℚ)≅⊕_i=1^nℚp_i^*[S_g]⊕⊕_i≠ j H_i⊗ H_j H^2(PConf_n(S_g);ℚ),

where ϕ(G_ij)=[_ij]∈ H^2(S_g^n;ℚ) is the Poincaré dual of the diagonal _ij.



See <cit.>.


Let {a_k,b_k}_k=1^g be a symplectic basis for H^1(S_g;ℚ). For 1≤ i,j ≤ m, we denote 

    M_i,j=∑_k=1^n p_i^*a_k ⊗ p_j^*b_k-p_i^*b_k⊗ p_j^*a_k.




The diagonal element [_ij]=p_i^*[S_g]+p_j^*[S_g]+M_ij∈⊕_i=1^nℚp_i^*[S_g]⊕⊕_i≠ j H_i⊗ H_j≅ H^2(S_g^n;ℚ). 



See <cit.>.

The following lemma is the classification of homomorphisms π_1(PConf_n(S_g))→π_1(S_g) from <cit.>.

Let g>1 and n>0. Let R: π_1(PConf_n(S_g))→π_1(S_g) be a homomorphism. The followings hold:



(1)If R is surjective, then R=A∘ p_i* for some i and A an automorphism of π_1(S_g).
 


(2)If Image(R) is not a cyclic group, the homomorphism π_1(PConf_n(S_g))→π_1(S_g) factors through p_i* for some i.




See <cit.>.

Now, we are ready to prove (3) of Theorem <ref>.

Suppose that there is a map s:PConf_n(S_g)→PConf_n+1(S_g) such that f_n(S_g)∘ s=identity. Then after post-composing with a forgetful map of the last coordinate, we obtain a map f: PConf_n(S_g)→ S_g. We denote 


    g_i:PConf_n(S_g) PConf_2(S_g)⊂ S_g× S_g

Let ⊂ S_g× S_g be the diagonal subspace and []∈ H^2(S_g× S_g;ℚ) be the Poincaré dual of . Let f^*:H^1(S_g)→ H^1(PConf_n(S_g)) and f_*:π_1(PConf_n(S_g))→π_1(S_g) be the induced map on cohomology and the fundamental groups. By Lemma <ref>, either f_* factors though a forgetful map p_i* or Image(f_*)≅ℤ. We break the proof into two cases according to the image of f_*.



Case 1: Image(f_*)≅ℤ. There are two subcases:



(1) If f^*=0, then g_i^*([])=p_i^*[S_g]≠ 0. This contradicts the fact that the image of g_i misses .



(2) If f^*≠ 0, then Imf^*≅ℤ because f_* has image ℤ on the fundamental groups. We assume that there exists a symplectic basis {a_k,b_k}_k=1^g for H^1(S_g;ℚ) such that f^*(a_i)=0 for any i≠ 1 and f^*(b_i)=0 for any i. Let f^*(a_1)=(x_1,x_2,...,x_n)≠ 0∈⊕_i=1^nH_i≅ H^1(PConf_n(S_g);ℚ). Assume without loss of generality that x_1≠ 0. Therefore for k≠ 1 by Lemma <ref>, we have that

    g_k^*([])=p_k^*[S_g]+∑_i=1,i≠ k^nx_i⌣ p_k^*b_1∈⊕_i=1^nℚp_i^*[S_g]⊕⊕_i≠ j H_i⊗ H_j≅ H^2(S_g^n;ℚ).

The coordinate x_1⊗ p_k^*b_1 is not zero, therefore g_k^*([])≠ 0. This contradicts the fact that the image of g_i misses .



Case 2: f_* factors though the forgetful map p_i*. Without loss of generality, we assume that i=1. We have that

    g_2^*([])=f^*[S_g]+p_2^*[S_g]+∑_k f^*a_k ⌣ p_2^*b_k-f^*b_k⌣ p_2^*a_k.

Since Image(f^*)⊂Image(p_1^*), we have that g_2^*([]) only has nonzero terms in ℚG_12⊕ H_1⊗ H_2. The fact that g_2 misses  implies

    f^*[S_g]+p_2^*[S_g]+∑_k f^*a_k ⊗ p_2^*b_k-f^*b_k⊗ p_2^*a_k=λ ([_12])∈ℚp_1^*[S_g]⊕ℚp_2^*[S_g]⊕ H_1⊗ H_2.

 The coefficient of p_2^*[S_g] tells us that λ=1. Therefore we have that f^*[S_g]=p_1^*[S_g] and 

    ∑_k (f^*a_k-p_1^*a_k) ⊗ p_2^*b_k-(f^*b_k-p_1^*b_k)⊗ p_2^*a_k=0∈ H_1⊗ H_2

By the property of tensor product, we know that f^*a_k-p_1^*a_k=0 and f^*b_k-p_1^*b_k=0. However in this case, if we look at the map g_1:PConf_n(S_g)S_g× S_g. We have that 

    g_1^*([])=f^*[S_g]+p_1^*[S_g]+∑_k f^*a_k ⌣ p_1^*b_k-f^*b_k⌣ p_1^*a_k=2p_1^*[S_g]-2g p_1^*[S_g]=(2-2g) p_1^*[S_g]≠ 0.

This contradicts the fact that the image of g_1 misses .





§ FURTHER QUESTIONS

In this section we list a few further questions. Let m,n be two positive integers. Let (x_1,...,x_n)∈PConf_n(S) for any manifold S. Let the permutation group Σ_m acts on PConf_n+m(S) by permuting the last m points. We have the following fiber bundle:

    PConf_m(S-{x_1,...,x_n})/Σ_m→PConf_n+m(S)/Σ_mPConf_n(S).

Here denote by f_n+m,n(S) the forgetful map that forgets the first n points. A section of the fiber bundle (<ref>) is called a multi-section.

Classify the continuous sections of the fiber bundle (<ref>) up to homotopy for S a surface.


Classify the continuous sections of the fiber bundle (<ref>) up to homotopy for any manifold S.





















California Institute of Technology 

Department of Mathematics 

Pasadena, CA 91125, USA

E-mail: chenlei1991919@gmail.com


