Shuya CAI Hao LI

Abstract Motivated by the work of Birman about the relationship between mapping class groups and braid groups, the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper. Their construction is based on the exact sequence given by the fibration M →F(M/G,n). The conclusion is closely connected with the braid group of the quotient space. Comparing with the situation without the group action, there is a big difference when the quotient space is T2.

Keywords Equivariant mapping class group, Orbit braid group, Evaluation map,Center of braid group

1 Introduction

Braid groups of the plane were defined by Artin [1] in 1925, and further studied in [2–3]. Braid groups of surfaces were studied by Zariski [21], and were later generalized using the definition from Fox[9]. Bellingeri simplified presentations of braid groups and pure braid groups on surfaces and showed some propertities of surface pure braid groups in [4].

Let M be a closed surface,with a finite set P of n distinguished points in M. BnM (respectively, FnM) denotes all homeomorphisms between M which preserve the set P (respectively,each point in P) and preserve the orientation if M is oriented. π0(FnM) is the set of all path connected components of FnM, and the n-th mapping class group denoted by Mod(M,n) is the set of all path connected components of BnM. The algebraic structure of the mapping class group is of great importance in the theory of Riemann surfaces. Connections between the mapping class group and the braid group of closed surfaces have been studied, in order to help to find the generators and relations of their mapping class groups.

The(pure)braid group of M on n strands is denoted by BnM (PnM). Let Sgbe an oriented closed surface of genus g,with g ≥0. In 1969,Birman[5–6]gave the basic relationship between mapping class groups and braid groups on Sg.

Theorem 1.1(see [5]) Let i?: π0(FnSg) →π0(F0Sg) be the homomorphism induced by inclusion FnSg?F0Sg. Then

Theorem 1.2(see [5]) Let i?: Mod(Sg,n) →Mod(Sg,0) be the homomorphism induced by inclusion BnSg?B0Sg. Then

Using above results, Birman obtained a full set of generators for the mapping class groups of n-punctured oriented 2-manifolds. Then she computed the mapping class group of the npunctured sphere and gave relations in the mapping class group of torus in a new way.

Denote the nonorientable closed surface of genus k as Nkwith k ≥1. Sk?1is its orientable double covering and π : Sk?1→Nkis a covering map. The induced homomorphism between braid group ?n:Bn(Nk)→B2n(Sk?1)is injective(see [13])on the level of fundamental group.The homomorphism between mapping class groups φn:Mod(Nk,n)→Mod(Sk?1,2n)induced by π is also injective. Furthermore, if k ≥3, then we have a commutative diagram of the following form:

where ψnandare the homomorphisms induced by inclusions BnNk?B0Nkand B2nSk?1?B0Sk?1(see [14]).

In this paper, we study the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M which admits a group action. Let G be a discrete group and act on M freely and properly. Consider a finite set P ={x1,··· ,xn}of n arbitrarily chosen points on M with different orbits. DefineM (respectively,M) to be the group of all G-homeomorphisms f : M →M which satisfy f(GP) = GP (respectively, f(Gxi) =Gxi,i=1,··· ,n)and preserve the orientation if M is oriented. These two groups are endowed with compact-open topology. π0(M,id)is the set of all path connected components ofM.The equivariant mapping class group denoted by MG(M,n) is defined to be the set of all path connected componentsM.

Let M be a connected topological manifold of dimension at least 2 with an effective action of a finite group G. The orbit configuration space of n ordered points in the G-space M is defined as

with subspace topology and G(x) denotes the orbit of x. The notion of the orbit configuration space was first defined in [20]. Since then, this subject, with respect to the algebraic topology (especially cohomology) and relative topics of orbit configuration spaces, has been further developed.

We prove the following results in this article.

Theorem 1.3(Theorems 3.1–3.3) Let: π0(M) →π0(M) be the homomorphism induced by inclusionM ?M. Then

When M/G is T2,

Theorem 1.4(Theorem 3.4) Let M be a closed surface. Let

be the homomorphism induced by inclusion BGnM ?BG0M. Then

When M/G is T2,

This paper is organized as follows. In Section 2,we introduce the centers of pure braid groups on oriented and nonorientable closed surfaces. Section 3 is the main part of this paper. We establish the relationship between the orbit braid group and the equivariant mapping class group on the closed surface which admits a free and proper group action. The proofs of Theorems 1.3–1.4 are given in this section. The conclusion is closely connected with the quotient space.And comparing with the situation without the group action, there is a big difference when the quotient space is T2.

2 The Centers of Pure Braid Groups on Closed Surfaces

First,we will briefly recall the definition and properties of braid groups. Let M be a smooth manifold. The configuration space of n ordered points in M, denoted by F(M,n) is defined as:

There is a natural action of the symmetric group Σnon the space F(M,n), given by permuting the coordinates. The configuration space of n unordered points in M is the quotient space:

Following Fox and Neuwirth [9], the n-th pure braid group Pn(M)(respectively, the n-th braid group Bn(M)) is defined to be the fundamental group of F(M,n) (respectively, of C(M,n)).

If m,n(m>n)are positive integers,we can define a homomorphism θ?:Pm(M)→Pn(M)induced by the projective θ :F(M,m)→F(M,n) defined by

In [8], Fadell and Neuwirth study the map θ, and show that it is a locally trivial fibration. The fiber over a point (x1,··· ,xn) of the base space is F(M ?{x1,··· ,xn},m ?n). Applying the associated long exact homotopy sequence, we obtain the pure braid group exact sequence of Fadell and Neuwirth:

The following short exact sequence is proved to be true where n ≥3 if M = S2, n ≥2 if M =RP2and n ≥1 referred to [7]:

Let Sgbe the oriented closed surface of genus g. Birman computed all the centers of pure braid groups on oriented closed surface in the following theorem.

Theorem 2.1(see [5–6])

Let Nkbe the nonorientable closed surface of genus k. Paris and Rolfsen obtained the following theorem.

Theorem 2.2(see [18]) If k ≥2, then Z(Pn(Nk))=1.

ProofWe apply an induction on the number n of strands. For n=1,

which is a finitely generated group with a single defining relation. Thus we obtain that Z(P1(Nk))=1, if k ≥2 (see [17]). The Fadell-Neuwirth fibration gives us the exact sequence

where π1(Nk?{x1,··· ,xn}) is a free group for n ≥1. Suppose Z(Pn(Nk)) = 1. Since p?is surjective, θ?(Z(Pn+1(Nk))) ?Z(Pn(Nk)) = 1. Hence Z(Pn+1(Nk)) lies in the group of ker θ?=im δ?, which is a free group. Thus Z(Pn+1(Nk))=1.

The pure braid group of the projective plane possesses non-trivial center. In [10] and [11],the following theorem has been proved.

Theorem 2.3If n ≥2, then Z(Pn(RP2)) is cyclic of order 2.

For the proofs of the main theorems, we have to describe the generator of Z(Pn(RP2))geometrically and algebraically. Next we will give the generator in a different way. The specific presentation of Pn(RP2) is given as follows.

Theorem 2.4(see [12]) The group Pn(RP2) admits the following presentation:

? Generators: Bij, 1 ≤i

? Relations:

We view the projective plane as the quotient space of the sphere. Then the sketches of Bijand ρiare given in Figures 1–2.

From the relations (b) and (d), each generator Bijin Pn(RP2) can be presented by ρiand ρj:

Figure 1

Figure 2

With this presentation and relation (b), we obtain

ρiρjρiρj=ρjρiρjρi, 1 ≤i,j ≤n.

When n = 1, P1(RP2) = π1(RP2) = Z2, which is an Abelian group. Next we consider n ≥2.

Lemma 2.1For n ≥2, τn=τn1···τnnlies in the center of Pn(RP2), where

ProofAccording to relation (d), τniρk=ρkτnifor k

Thus we obtain

From the above results, we obtain

Lemma 2.2For n ≥2, the center of Pn(RP2) is generated by τn.

ProofWe apply an induction on the number n of strands. For n = 2,P2(RP2) is the quaternion group with the presentation

where π1(RP2?{x1,··· ,xn}) is a free group for n ≥2. Suppose the center of Pn(RP2) is generated by τn. Since θ?is surjective and θ?(τn+1)=τn, Z(Pn+1(RP2)) is generated by τn+1and generators of ker θ?= im δ?, which is a free group. Thus the center of Pn+1(RP2) is just generated by τn+1.

τncan be represented by h(t) for t ∈[0,1]:

The components of h(t) are plotted in Figure 3.

3 The Relationship Between Orbit Braid Groups and Equivariant Mapping Class Groups on Compact Closed Surfaces

Let M be a compact closed surface and G be a discrete group acting on M freely and properly. Since M is compact, G is finite. Let x1,··· ,xndenote n fixed but arbitrarily chosen points on M, satisfying GxiGxj=?for 1 ≤ij ≤n.

Figure 3

and preserve orientation if M is oriented.

These two groups are to be endowed with compact-open topology. We define equivariant mapping class groups as follows:

Since the action of G on M is free, the (pure) orbit braid group of M is isomorphic to the(pure)braid group of the quotient space(see [16]), which is an oriented or nonorientable closed surface. Hence we only need to consider the (pure) braid group of the quotient space. Similar to the situation without group actions, we define the pure evaluation map as follows.

Definition 3.1

Lemma 3.1The pure evaluation map εGhas a local cross section.

ProofLet a=([a1],··· ,[an])be an arbitrary point in F(M/G,n)and aibe the representative of each coordinate for i=1,··· ,n. Since the action of G on M is free and proper, there exist disjoint open sets {Ui:ai∈Ui,i=1,··· ,n} such that Ui∩gUj=?for every ge ∈G.We consider an open neighborhood U(a) of the point a ∈F(M/G,n), where U(a) is defined by

U(a)={([u1],··· ,[un]):ui∈Ui}.

Let u = ([u1],··· ,[un]) be an arbitrary point in U(a). Choose λuto be a G-equivariant homeomorphism from M to M such that

(i) λu(x)=x, when

(ii) λumaps each gUiinto itself homeomorphically.

(iii) λufixes the points on the boundary of gUi.

(iv) λu(ai) and uiare in the same orbit.Then the map

defined by χ(u)=λuis the required cross section.

Using Lemma 3.2, we obtain an exact sequence as follows:

Moreover,each Htcan induce the unique map from M/G to M/G, since it preserves G-action.There is no harm in denoting the induced map by Ht. Choose any element β ∈Pn(M/G)which can be represented by a loop (β1,··· ,βn) with βi(0) = βi(1) = xifor each i = 1,··· ,n. Use Htand β to construct ψ :I2→Fn(M/G) by

Thus, H(0,t)=H(1,t)represents α, while H(s,0)=H(s,1)represents β. So H|?I2represents αβα?1β?1. Therefore,

Since β ∈Pn(M/G) is arbitrary, it follows

The proof of Lemma 3.3 is completed.

Suppose G is a finite group of order l, acting freely and properly on M. The natural projection map p:M →M/G is a regular covering map with the exact sequence

and G is naturally isomorphic to the group of covering transformation. Furthermore, M/G is a closed compact surface whose Euler characteristic χ(M/G) satisfies the formula

If M is the oriented surface Sg,

If M is the nonorientable surface Nk,

We conclude the following theorem.

Theorem 3.1Letbe the homomorphism induced by inclusionThen

Consider M is S2. Because of (3.2), the only nontrivial group G1 that acts freely on the sphere is Z2. When G is trivial and n ≥3, the discussion is the same as that of [5] and [6].And since Z(P1(S2)) = Z(P2(S2)) = 1, kerPn(S2). We only discuss the situation where the orbit space is RP2.

Lemma 3.4If G=Z2acts on S2antipodally, then for n ≥1,

ProofSince Z(Pn(RP2)) is generated by τnfrom Theorem 2.4, we only need to proveτn)=1. Define Ht:S2→S2by

where the ith component of τncan be represented by [Ht(xi)], i = 1,··· ,n. Clearly τn=([Ht]). Thus we obtain τn∈ker

As for the nonorientable surface RP2, G must be trivial because of (3.2).

Lemma 3.5If G is trivial and M is RP2, then Z(Pn(RP2))?ker

ProofSince Z(Pn(RP2)) is generated by τnfrom Theorem 2.4, we only need to prove(τn)=1. View RP2as the quotient space of S2. Define Ht:RP2→RP2by

Next we consider the situation where the quotient space M/G is T2. From (3.2), M can be T2or Klein bottle N1. The following lemma rules out one situation.

Lemma 3.6The Klein bottle N1can not be a cover of the torus T2.

Proofπ1(N1) is isomorphic to the non-abelian group 〈x,y | x2= y2〉 while π1(T2) is isomorphic to the Abelian group Z ⊕Z. Suppose N1is a cover of T2, there exists an injective:

π1(N1)→π1(T2).

Since the subgroup of an Abelian group is also an Abelian group,the injective map is impossible.

Remark 3.2This lemma is well known and we just give a proof here.

We can plot the components of f(t) and g(t) in Figures 4–5 when n is 4.

Define Z2-action on T2by

Figure 4

Figure 5

where f(0) = f(1) = g(0) = g(1) = (x1,··· ,xn). According to the construction of, there exists the G-homeomorphism Ht:T2→T2for t ∈[0,1], such that H0=H1=id and

Define the map H(xi):I →T2by H(xi)(t)=Ht(xi). Then we obtain that p?([H(xi)])=ambr.Thus ambr∈im p?.

is called the evaluation map.

Since the evaluation map ηGis the pure evaluation εGassociated with the projective map F(M/G,n)→F(M/G,n)/Σn, we obtain the following lemma.