Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces

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CONFLUENTES MATHEMATICI

Daciberg LIMA GONÇALVES, John GUASCHI, and Miguel MALDONADO Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces

Tome 10, n^o1 (2018), p. 41-61.

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10, 1 (2018) 41-61

EMBEDDINGS AND THE (VIRTUAL) COHOMOLOGICAL DIMENSION OF THE BRAID AND MAPPING CLASS GROUPS

OF SURFACES

DACIBERG LIMA GONÇALVES, JOHN GUASCHI, AND MIGUEL MALDONADO

Abstract. We use the relations between the braid and mapping class groups of a compact, connected, non-orientable surfaceN without boundary and those of its orientable double coveringSto study embeddings of these groups and their (virtual) cohomological dimensions.

We first generalise results of [4, 14] to show that the mapping class groupM CG(N;k) ofN relative to ak-point subset embeds in the mapping class groupM CG(S; 2k) ofSrelative to a 2k-point subset. We then compute the cohomological dimension of the braid groups of all compact, connected aspherical surfaces without boundary, generalising results of [15].

Finally, if the genus ofNis at least 2, we deduce upper bounds for the virtual cohomological dimension ofM CG(N;k) that are coherent with computations of Ivanov.

1. Introduction

Let S be a compact, connected surface, and let X = {x1, . . . , xk} be a finite (possibly empty) subset ofSof cardinalityk>0. Let Top(S;X) denote the group of homeomorphisms of S for the operation of composition that leaveX invariant.

If S is orientable, let Top⁺(S;X) denote the set of elements of Top(S;X) that preserve orientation. Note that Top⁺(S;X) is a subgroup of Top(S;X) of index two. We define themapping class groupM CG(S;X)ofS relative to X by:

M CG(S;X) =

(π0Top⁺(S;X) ifS is orientable

π0Top(S;X) ifS is non orientable. (1.1) If S is orientable (resp. non orientable), the groupM CG(S;X) is thus the set of isotopy classes of Top⁺(S;X) (resp. Top(S;X)), the isotopies being relative to the setX. Iff ∈Top⁺(S;X) (resp. Top(S;X)) then we let [f] denote its mapping class inM CG(S;X). Up to isomorphism,M CG(S;X) only depends on the cardinalityk of the subsetX, and we shall often denote this group byM CG(S;k). IfX is empty, then we omit it from the notation, and shall just write Top(S), Top⁺(S),M CG(S) etc. The mapping class group may also be defined in other categories (PL category, smooth category), the groups obtained being isomorphic [5]. The mapping class group has been widely studied from different points of view – see [1, 9, 10, 24, 30]

for example.

If k∈N, there is a surjective homomorphismσ: Top(S;X)−→Σ_k defined by f(x_i) = x_(σ(f))(i) for all f ∈ Top(S;X) andi ∈ {1, . . . , k}, where Σ_k is the symmetric group onkletters. This homomorphism induces a surjective homomorphism fromM CG(S;X) to Σk that we also denote byσ, and its kernel is called thepure mapping class group ofS relative to X, denoted byP M CG(S;X).

2010Mathematics Subject Classification:57N05, 20F36, 55R80, 55P20, 20F38, 57M07, 20J06.

Keywords: Mapping class groups, surface braid groups, finite coverings, embeddings, (virtual) cohomological dimension.

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We have a short exact sequence:

1−→P M CG(S;X)−→M CG(S;X)−→^σ Σk −→1, (1.2) and iff ∈Top(S;X) then [f]∈P M CG(S;X) if and only iff fixesX pointwise.

Mapping class groups are closely related to surface braid groups. If k ∈ N, thek^th ordered configuration space Fk(S) of S is the set of all orderedk-tuples of distinct points of S that we may consider as a subspace of the k-fold product of S with itself. The group Σk acts freely on Fk(S) by permuting coordinates, and the associated quotient is the k^th unordered configuration space of S, denoted by Dk(S). The fundamental groupπ1Dk(S) (resp.π1Fk(S)), denoted byBk(S) (resp.

Pk(S)) is the braid group (resp.pure braid group) of S on k strings [8, 11], and Bk(S) andPk(S) are related by a short exact sequence similar to that of (1.2). It is well known thatBk(R²) (resp.Pk(R²)) is isomorphic to Artin’s braid group(resp.

Artin’s pure braid group) on k strings. If π : Se −→ S is a d-fold covering map, whered∈N, then for allk>1, there is a continuous map:

ϕk :Dk(S)−→Dkd(S)e (1.3) defined by ϕk(A) =π⁻¹(A) for allA ∈ Dk(S), and the induced homomorphism ϕk#:Bk(S)−→Bkd(S) on the level of fundamental groups is injective [14]. Takinge π: S² −→RP² to be the standard 2-fold (universal) covering, where S² is the 2- sphere andRP² is the real projective plane, this result was then applied in [14] to classify the isomorphism classes of the finite subgroups of B_k(RP²), and to show that B_k(RP²) andM CG(RP²;k) are linear groups for allk∈N.

If g > 0 (resp. g > 1), let S = S_g (resp. N = N_g) be a compact, connected orientable (resp. non-orientable) surface of genusg without boundary. In the non- orientable case,gis the number of projective planes in a connected sum decomposition. Ifg>1 andk>0, the orientable double coveringπ:S_g−1−→Ng induces a homomorphismφk :M CG(Ng;k)−→M CG(S_g−1; 2k), where the 2k-point subset of marked points in S is equal to the inverse image by π of a k-point subset of N. One of the main aims of this paper is to generalise the injectivity result of [14]

to this homomorphism. If k = 0 and g > 3, it was shown in [4, 22] that there exists an injective homomorphism φ : M CG(Ng) −→ M CG(Sg−1), and that via φ, M CG(Ng) may be identified with the subgroup of M CG(Sg−1) that consists of isotopy classes of symmetric homeomorphisms. In Section 3, we show that a similar result holds for allk>0, namely that the homomorphismφ_k induced byπ is injective, and that viaφ_k, M CG(N_g;k) may be identified with the subgroup of M CG(S_g−1; 2k) that consists of isotopy classes of symmetric homeomorphisms.

Theorem 1.1. — Let k, g ∈ N, let N = Ng be a compact, connected, non- orientable surface of genusgwithout boundary, andSits orientable double covering.

The homomorphism φk : M CG(N;k) −→ M CG(S; 2k) induced by the covering π:S−→N is injective; ifg>3we have the following commutative diagram:

1 Bk(N) M CG(N;k) M CG(N) 1

1 B2k(S) M CG(S; 2k) M CG(S) 1,

ϕ_k#

τ_k

φ_k φ

e^τ^k

(1.4)

where τ_k and eτ_k are the homomorphisms induced by forgetting the markings on the sets of marked points.

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Note that in contrast with [4], Theorem 1.1 holds for allg>1, not just forg>3.

As we recall in Remark 2.4, the result of [4] does not hold if g = 2. The proof of this exceptional caseg= 2 in Theorem 1.1 will turn out to be the most difficult, in part due to the non injectivity of φ:M CG(N2)−→M CG(S1).

In Section 4, we compute the cohomological dimension of the braid groups of all compact surfaces without boundary different fromS² andRP², and we give an upper bound for the virtual cohomological dimension of the mapping class group M CG(N_g;k) for all g > 2 and k > 1. Recall that the virtual cohomological dimensionvcd(G) of a groupGis a generalisation of the cohomological dimension cd(G) ofG, and is defined to be the cohomological dimension of any torsion-free finite index subgroup of G [6]. In particular, if G is a braid or mapping class group of finite (virtual) cohomological dimension then the corresponding pure braid or mapping class group has the same (virtual) cohomological dimension. In [15, Theorem 5], it was shown that ifk>4 (resp. k>3) then vcd(Bk(S²)) =k−3 (resp.

vcd(Bk(RP²)) =k−2). These results are generalised in the following theorem, the proof being a little more straightforward due to the fact that the braid groups of Sg andNg+1are torsion free if g>1.

Theorem 1.2. — Letg, k>1. Then:

cd(B_k(S_g)) =cd(P_k(S_g)) =cd(B_k(N_g+1)) =cd(P_k(N_g+1)) =k+ 1.

We then turn our attention to the mapping class groups. If 2g+k >2, J. Harer proved that [21, Theorem 4.1]:

vcd(M CG(Sg;k)) =







4g−5 ifk= 0

k−3 ifg= 0 and k >0 4g+k−4 ifg, k >0,

and in the non-orientable case, N. Ivanov showed that [23, Theorem 6.9]:

vcd(M CG(N_g;k)) =











0 ifg= 1 andk62 k−2 ifg= 1 andk>3 k ifg= 2

2g−3 ifg>3 andk= 0 2g+k−2 ifg>3 andk>1.

(1.5)

We believe that there is a typographical error in the last case of [23, Theorem 6.9], and that ‘k= 1’ should read ‘k>1’ as above. Using the embedding ofM CG(N_g) in M CG(S_g−1) given in [4], G. Hope and U. Tillmann showed that ifg >3 then vcd(M CG(Ng))64g−9 [22, Corollary 2.2]. Using [15, Theorem 5], it was shown in [15, Corollary 6] that if k > 4 (resp. k > 3) then vcd(M CG(S²;k)) =k−3 (resp. vcd(M CG(RP²;k)) =k−2). In the case ofS², we thus recover the results of Harer. For non-orientable surfaces, our methods shed light on the cohomological structure ofM CG(Ng;k). In particular, we obtain the following result.

Corollary 1.3. — Let k > 0. The mapping class groups M CG(Ng;k) and P M CG(Ng;k)have the same (finite) virtual cohomological dimension. Further:

(a) vcd(M CG(N₂;k)) =k.

(b) ifg>3then vcd(M CG(N_g;k))64g+k−8.

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Part (a) reproves (1.5) in the case of the Klein bottle (g = 2), and part (b) is coherent with the result of (1.5) forg>3.

This paper is organised as follows. In Section 2, we recall some definitions and results about orientation-true mappings and the orientable double covering of non-orientable surfaces, we describe how we lift an element of Top(N) to one of Top⁺(S) in a continuous manner, and we show how this correspondence induces a homomorphism from the mapping class group of a non-orientable surface to that of its orientable double covering (Proposition 2.3). In Section 3, we prove Theorem 1.1 using long exact sequences of fibrations involving the groups that appear in equation (1.1) and the corresponding braid groups [3, 33]. In most cases, using [14, 20], we obtain commutative diagrams of short exact sequences, and the conclusion is obtained in a straightforward manner. The situation is however much more com- plicated in the case whereN is the Klein bottle andS is the 2-torusT, due partly to the fact that the exact sequences that appear in the associated commutative diagrams are no longer short exact. This case requires a detailed analysis, notably in the case k= 1. In Section 4, we prove Theorem 1.2 and Corollary 1.3. Finally, in an Appendix, we provide presentations of P2(T) and B2(T) that we use in the proof of Theorem 1.1.

Acknowledgements. The first-named author was partially supported by FAPESP- Fundação de Amparo a Pesquisa do Estado de São Paulo, Projeto Temático Topolo- gia Algébrica, Geométrica 2012/24454-8. He and the second-named author would like to thank the ‘Réseau Franco-Brésilien en Mathématiques’ for financial support for their respective visits to the Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, Université de Caen Normandie, from the 2^nd to the 15^th of September 2016, and to the Instituto de Matemática e Estatística, Universidade de São Paulo, from the 9^th of July to the 1^st of August 2016. The third-named author was supported by the Consejo Nacional de Ciencia y Tecnología to visit the Lab- oratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, Université de Caen Normandie, from the 1^stof February 2014 to the 31^stof January 2015. During that period, he was also partially supported by the Laboratorio Internacional Solomon Lefschetz (LAISLA).

2. Orientation-true mappings and the orientable double covering Let M, N be manifolds with base points x0, y0 respectively. Following [29], a pointed mapf : (M, x0)−→(N, y0) is calledorientation trueif the induced homomorphism

f_∗:π1(M, x0)−→π1(N, y0)

sends orientation-preserving (resp. orientation-reversing) elements to orientation- preserving (resp. orientation-reversing) elements. In other words, the map f is orientation-true if for all α∈ π1(M, x0), either α and f∗(α) are both orientation preserving or they are are both orientation reversing. In the case of a branched coveringf :M −→N it follows by [16, Proposition 1.4] thatf is orientation-true.

In what follows, we will be interested in the case of maps between surfaces. We will start with the case of non-orientable surfaces.

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Lemma 2.1. — Every homeomorphismf :N−→N of a non-orientable surface is orientation true. Consequently, the subgroup of all orientation-preserving loops is invariant with respect tof∗.

Lemma 2.1 is an obvious consequence of [16, Proposition 1.4], but we shall give a more direct proof.

Proof of Lemma 2.1. — Letx₀∈N be a base point and considerα∈π₁(N, x₀), which we represent by a loop h:S¹ −→N. We fix a local orientationO₁ at x₀, and we consider the orientation on N induced by f, denoted by O₂. Moving the orientation O₁ along the path h by a finite sequence of small open sets U_i, the images f(U_i) may be used to transport the local orientation O₂ along the path f◦h.

Consider the mapsh, f◦h:S¹−→N and the tangent bundleT N ofN. Pulling back by these maps, we obtain bundles over S¹ that are homeomorphic, and that are the trivial bundle if the loop is orientation-preserving or the twisted bundle if

the loop is orientation-reversing.

Letg >1, and letNg be as defined in the introduction. The unique orientable double covering of Ng may be constructed as follows (see [27] for example). Let the orientable surface Sg−1 be embedded in R³ in such a way that it is invariant under the reflections with respect to the xy-, yz- and xz-planes. Consider the involutionJ :S_g−1−→S_g−1defined byJ(x, y, z) = (−x,−y,−z). The orbit space S_g−1/hJi is homeomorphic to the surface N_g, and the associated quotient map π:S_g−1−→N_gis a double covering.

To simplify the notation, from now on, we will drop the subscripts g and g− 1 from N and S respectively unless there is risk of confusion, so S will be the orientable double covering ofN. As indicated previously, the mapπgives rise to a map on the level of configuration spaces that induces an injective homomorphism Bk(N)−→B2k(S) of the corresponding braid groups, and this allows us to study the braid groups of a non-orientable surface in terms of those of its orientable double covering [14].

The following result is an immediate consequence of Lemma 2.1 and is basically contained in [22, Key-Lemma 2.1].

Lemma 2.2. — Let f : N −→ N be a homeomorphism of a non-orientable surface, and letπ :S −→N be the orientable double covering. Thenf admits a lift, and the number of lifts is exactly two.

Proof. — Using Lemma 2.1, we know that f#|π₁(S)(π1(S))⊂π1(S), where we identify π1(S) with a subgroup of π1(N). By basic properties of covering spaces, the map f lifts to a map f⁰ : S −→ S, and since S −→ N is a double covering,

there are two lifts.

The lifts off are in one-to-one correspondence with the grouphJiof deck trans- formations. There is thus a natural way to choose a lift in a continuous manner simply by choosing ˜f to be the lift of degree 1 (the other lift is of degree−1 since J is of degree−1). Letρ: Top(N)−→Top⁺(S) denote this choice of lift. We may use Lemma 2.2 to compare mapping class groups of orientable and non-orientable surfaces.

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Proposition 2.3. — Let N be a non-orientable surface, and let S be its orientable double covering. Then there is a homomorphismφ:M CG(N)→M CG(S) such that the following diagram commutes:

Top(N) ^ρ //

Top⁺(S)

M CG(N) ^φ //M CG(S).

(2.1)

Further, if the genus ofN is greater than or equal to3 thenφis injective.

Proof. — If f, g ∈ Top(N) are isotopic homeomorphisms then an isotopy between them lifts to an isotopy between the orientation-preserving homeomorphisms ρ(f) and ρ(g) of S. This proves the first part of the statement. For the second part, suppose that the genus of N is greater than or equal to 3. Let M CG^±(S) denote the extended mapping class group of S consisting of the isotopy classes of all homeomorphisms ofS, and letChJibe the subgroup ofM CG^±(S) defined by:

ChJi=

[f]∈M CG^±(S)

f ∈Top(S), and there existsf⁰'f such thatf⁰◦J =J◦f⁰

, where f⁰ ' f means that f⁰ is isotopic to f. By [4], M CG(N) ∼= ChJi/h[J]i.

Let π: ChJi −→ ChJi/h[J]i denote the quotient map. Let f ∈ Top(N). Then φ([f]) = [ρ(f)] by diagram (2.1). By Lemma 2.2 and the comment that follows it,f admits exactly two lifts, ρ(f) andJ◦ρ(f), the first (resp. second) preserving (resp. reversing) orientation. Nowρ(f)◦J is an orientation-reversing lift off, from which we conclude that J ◦ρ(f) = ρ(f)◦J. Hence [ρ(f)] ∈ ChJi ∩M CG(S), and so Im(φ)⊂ ChJi ∩M CG(S). On the other hand, π

ChJi∩M CG(S) : ChJi ∩ M CG(S)−→M CG(N) is an isomorphism using the proof of [22, Key-Lemma 2.1].

It then follows thatπ◦φ= Id_{M CG(N}₎, in particularφis injective.

In the case of surfaces with marked points, there is another continuous way to choose a lift. For example, given a finite subset X ={x1, . . . , xk} ofN, denote its preimage under πby:

Xe ={x⁰₁, x⁰⁰₁, x⁰₂, x⁰⁰₂, . . . , x⁰_k, x⁰⁰_k},

where{x⁰_i, x⁰⁰_i}=π⁻¹(x_i) for all 16i6k. Pick a subset ofXe that contains exactly one element of {x⁰_i, x⁰⁰_i}, denoted by x_i, for all 1 6i 6k. Iff ∈ Top(N;X) then the restriction of f toX is a given permutation of X, and ifj∈ {1, . . . , k}is such that f(x₁) =x_j, we may definefeto be the unique lift of f such that f(xe ₁) =x_j. The two choices forxj correspond to the two possible lifts off.

Remark 2.4. — Suppose that the genus of N is 2, in which caseN is the Klein bottleK. IfT is the torus, the homomorphismφ:M CG(K)−→M CG(T) is not injective, and it may be described as follows. Set π1(K) =

α, β|αβαβ⁻¹ . The group M CG(K) is isomorphic toZ2⊕Z2 [26], and its elements are given by the mapping classes of the following automorphisms:

(α7−→α β7−→β,

(α7−→α β7−→αβ,

(α7−→α β7−→β⁻¹,

(α7−→α

β7−→αβ⁻¹. (2.2)

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Letπ:T −→Kbe the orientable double covering ofKand letπ1(T) =ha, b|[a, b]i, so that π#(a) = α and π#(b) = β². Given a map f : K −→ K such that the induced homomorphism on the fundamental group is given byf#(α) =α^r,f#(β) = α^uβ^v, if v is odd, then f lifts to a map from T to T, and there are exactly two lifts. The matrices of the induced homomorphisms of these lifts on π1(T) are (^r₀⁰_v) and ^−r₀ _v⁰

. Observe that the determinant of one of these two matrices is positive. Identifying M CG(T) with SL(2,Z), we conclude that φsends the first (resp. second) two automorphisms of equation (2.2) to the matrix (^{1 0}_{0 1}) (resp. to

−1 0 0 −1

). In particular, the second part of Proposition 2.3 does not hold in this case.

3. Embeddings of mapping class groups

As in all of this paper, the surfaces N and S under consideration are compact and without boundary, andπ:S−→N is the double covering defined in Section 2.

If X is a finite k-point subset of N, Xe = π⁻¹(X) and f ∈ Top(N;X), then we define a map ρk : Top(N, X)−→Top⁺(S,X) bye ρk(f) =ρ(f), where we consider f to be an element of Top(N) andρis as defined in Section 2. By Proposition 2.3, the mapρ_k induces a homomorphismφ_k :M CG(N;X)−→ M CG(S;X) definede byφ_k([f]) = [ρ_k(f)]. On the other hand, the mapψ_k : Top(N)−→D_k(N) (resp.

ψe_2k : Top⁺(S)−→ D_2k(S)) defined by ψ_k(f) = f(X) for all f ∈ Top(N) (resp.

ψe_2k(h) =h(X) for alle h∈Top⁺(S)) is a locally-trivial fibration [3, 28] whose fibre is Top(N;k) (resp. Top⁺(S; 2k)). The long exact sequence in homotopy of these fibrations gives rise to the following exact sequences:

Bk(N)−→M CG(N;X)−→^τ^k M CG(N), Bk(S)−→M CG(S;X)e −→e^τ^k M CG(S), whereτk, τek are induced by suppressing the markings onX andXe respectively [3].

In Theorem 1.1, we will see that the injectivity ofφgiven by Proposition 2.3 carries over to that of the homomorphism φk between mapping class groups of marked surfaces and holds forall g>1, in contrast to the non injectivity in the caseg= 2 described in Remark 2.4.

Let S be a compact, connected surface without boundary, and let l ∈ N. We define the following subgroups of Top(S) and Top(S;X):

(a) ifS is orientable and X ⊂S is a subset of cardinalityl, let:

Top⁺_F(S;l) ={f ∈Top⁺(S)|f(x) =xfor allx∈X}

={f ∈Top⁺(S;X)|[f]∈P M CG(S;l)}.

(b) ifS is orientable,π:S−→N is the orientable double covering, andX ⊂N is a subset of cardinalityl, let:

Top⁺

Fe(S; 2l) =n

fe∈Top⁺(S)

fe(π⁻¹(x)) =π⁻¹(x) for all x∈Xo . (c) ifS is non orientable andX ⊂S is a subset of cardinalityl, let:

Top_F(S;l) ={f ∈Top(S)|f(x) =xfor allx∈X}

={f ∈Top(S;X)|[f]∈P M CG(S;l)}.

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Observe that if S is orientable (resp. non orientable), Top⁺_F(S; 1) = Top⁺(S; 1) (resp. Top_F(S; 1) = Top(S; 1)). Before giving the proof of Theorem 1.1, we state and prove the following two results.

Proposition 3.1. — Letk>1, and consider the homomorphism ψe_2k#:π₁(Top⁺(T))−→B_2k(T)

induced by the mapψe2k defined above. Thenπ1(Top⁺(T))∼=Z², and Im(ψe2k#) = Z(B2k(T))⊂P2k(T).

Proof. — First, by [18, Theorem 2, p. 63]), π1(Top⁺(T))∼=Z². Secondly, taking Top⁺(T) to be equipped with IdT as its basepoint, a representative loop of an element γ∈ π₁(Top⁺(T),Id_T) is a path in Top⁺(T) from Id_T to itself. It follows from the definition of ψe_2k that ψe_2k#(γ)∈ P_2k(T). It remains to show that Im(ψe2k#) =Z(B2k(T)). Since Im(ψe2k#)⊂P2k(T), the homomorphismψe2k# coin- cides with the homomorphism of [3, Theorem 1]. Using the exact sequence of that theorem and [3, Corollary 1.3], we see that Im(ψe2k#) =ha1, b1i, where a1 and b1

are the generators ofP2k(T) defined in [2, Theorem 5]. But by [31, Proposition 4.2], these two elements generate the centre ofB2k(T) as required.

The first part of the following lemma generalises results of [19].

Lemma 3.2. —

(a) LetSbe a compact, connected orientable (resp. non-orientable) surface without boundary for whichπ₁(Top⁺(S; 1)), (resp.π₁(Top(S; 1))) is trivial, and letl >1. Then π1(Top⁺_F(S;l)), (resp.π1(Top_F(S;l))) is trivial. In particu- lar,π₁(Top⁺_F(T;l))(resp.π₁(Top_F(K;l))) is trivial for alll>1.

(b) π₁(Top⁺

Fe(T; 2l))is trivial for alll>1.

Proof. —

(a) Assume first that S is orientable. We prove the result by induction on l. If l = 1 then the result holds by the hypothesis, using the fact that Top⁺_F(S; 1) = Top⁺(S; 1). Suppose by induction that the result holds for somel>1. The map Top⁺_F(S;l)−→S\ {x1, . . . , xl}given by evaluating an element of Top⁺_F(S;l) at the pointx_l+1is a fibration with fibre Top⁺_F(S;l+1).

Taking the long exact sequence in homotopy of this fibration and using the fact that π₂(S\ {x1, . . . , x_l}) is trivial, it follows that the homomorphism π1(Top⁺_F(S;l+ 1))−→π1(Top⁺_F(S;l)) induced by the inclusion of the fibre in the total space is injective, and since π₁(Top⁺_F(S;l)) is trivial by the induction hypothesis,π₁(Top⁺_F(S;l+ 1)) is too. This proves the first part of the statement ifS is orientable. IfS is non orientable, it suffices to replace Top⁺_F(S;l) by Top_F(S;l) in the proof of the orientable case. The second part of the statement is a consequence of the first part and [18, Corollary, p. 65] and [19, Theorem 4.1].

(b) Let l>0. The map Top⁺

Fe(T; 2l)−→D₂(T\ {x₁, . . . , x2l)}that to a homeomorphism f ∈ Top⁺

Fe(T; 2l) associates the set {f(x_2l+1), f(x_2l+2)} is a fibration whose fibre is Top⁺

Fe(T; 2l+ 2). If l = 0 then Top⁺

Fe(T; 2l) is just Top⁺(T). Taking the long exact sequence in homotopy, and using the fact

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that D2(T \ {x1, . . . , x2l}) is a K(π,1), we obtain the following exact sequence:

1→π1(Top⁺

Fe(T; 2l+ 2))−−−−−→⁽e^q^2l+2⁾^# π1(Top⁺

Fe(T; 2l))→B2(T\ {x1, . . . , x2l}), (3.1) where the homomorphism (qe2l+2)# is induced by the map

qe_2l+2: Top⁺

Fe(T; 2l+ 2)−→Top⁺

Fe(T; 2l) that forgets the marking on the last two points. Ifπ1(Top⁺

Fe(T; 2l)) is trivial for somel >1 then clearly π1(Top⁺

Fe(T; 2l+ 2)) is also trivial. So applying induction on l, it suffices to prove the result for l = 1. It follows from Proposition 3.1 that the map π₁(Top⁺

Fe(T))−→ B₂(T) sends π₁(Top⁺

Fe(T)) isomorphically onto the centre ofB₂(T). It follows from exactness of (3.1) thatπ1(Top⁺

Fe(T; 2)) is trivial, and this completes the proof of the lemma.

Remark 3.3. — If S is a surface that satisfies the hypotheses of Lemma 3.2(a) then S is different from the 2-sphere [28, lines 2–3, p. 303] and the real projective plane [19, Theorem 3.1], soS is an Eilenberg Mac Lane space of typeK(π,1).

Proof of Theorem 1.1. — Consider the double coveringπ:S−→N, fix ak-point subsetX ofN, and letXe =π⁻¹(X). By the comments preceding the statement of the theorem, we obtain the following commutative diagram of fibrations:

Top(N;X) //

ρ_k

Top(N) ^ψ^k //

ρ

Dk(N)

ϕ_k

Top⁺(S;Xe) //Top⁺(S) ^ψe^2k //D_2k(S),

(3.2)

where ρis as defined in Section 2, and ϕk : Dk(N)−→D2k(S) is given by equation (1.3). The left-hand square clearly commutes because ρ_k is the restriction of ρto Top(N;k). We claim that the right-hand square also commutes. On the one hand, π(ψe_2k ◦ρ(f)) =π◦ρ(f)(X) =e f ◦π(X) =e f(X), using the fact that ρ(f) is a lift of f, so ψe_2k◦ρ(f) =ρ(f)(Xe)⊂π⁻¹(f(X)) =ϕ_k◦ψ_k(f). Conversely, if y∈π⁻¹(f(X)), then there existsx∈X such thatπ(y) =f(x). Ifxe∈Xe is a lift of xtheny∈ {ρ(f)(x), Je ◦ρ(f)(x)}e ={ρ(f)(x), ρ(fe )(J(x))} ⊂e ρ(f)(X), which provese the claim. Let Φ₁:π₁Top(N)−→π₁Top⁺(S) denote the homomorphism induced byρon the level of fundamental groups. We now take the long exact sequence in homotopy of the commutative diagram (3.2). The form of the resulting commutative diagram depends on the genus g of N, hence we consider the following three cases.

(a) Suppose that g > 3. Then by [20], π1Top(N) and π1Top⁺(S) are trivial, and we obtain the commutative diagram of (1.4). Since ϕk# (resp. φ) is injective by [14] (resp. by Proposition 2.3), the injectivity ofφ_k follows from the 5-Lemma.

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(b) Suppose that g = 1. Using [14, 33], we obtain the following commutative diagram of short exact sequences:

1 Z/2Z

| {z }

π₁Top(RP²)

Bk(RP²) M CG(RP²;k) 1

1 Z/2Z

| {z }

π₁Top⁺(S²)

B2k(S²) M CG(S²; 2k) 1,

Φ₁

ϕ_k# φ_k

(3.3)

where in both cases,Z/2Zis identified with the subgroup generated by the full-twist braid of the corresponding braid group. Since ϕk# is injective by [14], so is Φ1, and a routine diagram-chasing argument shows thatφk is injective.

(c) Finally suppose that g = 2, so that N is the Klein bottle K and S is the torusT. Letk>1. We claim that Ker(φ_k)⊂P M CG(K;k).

To prove this, letX ={x₁, . . . , x_k}, and fori= 1, . . . , k, lety_i∈π⁻¹(x_i).

Letσ:M CG(K;k)−→ Σ_k and eσ:M CG(T; 2k)−→Σ_2k denote the usual homomorphisms onto the symmetric groups of X and Xe respectively as described in equation (1.2). From the geometric construction ofρk, if 1 6 i, j6k andσ([f])(xi) =xj then

σ([ρe k(f)])(yi)∈π⁻¹(xj).

If [f]∈Ker(φk), where f ∈Top(K, X), then [ρ_k(f)] =φ_k([f]) = [Id_S].

In particular, eσ([ρ_k(f)]) = Id_Σ_2k, so eσ([ρ_k(f)])(y_i) = y_i for all 1 6i 6 k, and it follows thatσ([f])(x_i) =x_i, whence [f]∈P M CG(K;k) as claimed.

It thus suffices to prove that the restriction φ_k

P M CG(K;k) of φ_k to P M CG(K;k) is injective. Now assume that k>2. Let

p_k:F_k(K)−→F_k−1(K) (resp. q_k: Top_F(K;k)−→Top_F(K;k−1)) be the map given by forgetting the last coordinate (resp. the marking on the last point). Let

D⁽²⁾_2k(T) =F2k(T)/Z2× · · · ×Z2

| {z }

ktimes

,

let B⁽²⁾_2k(T) = π₁D⁽²⁾_2k(T). Let ep_2k: D_2k⁽²⁾(T) −→ D_2(k−1)⁽²⁾ (T) be the map given by forgetting the last two coordinates, and let

qe2k: Top⁺

Fe(T; 2k)−→Top⁺

Fe(T; 2(k−1))

be the map defined in the proof of Lemma 3.2(b). Then we have the following commutative diagram whose rows are fibrations:

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Top⁺

Fe(T; 2(k−1)) //Top⁺(T) e^δ^2(k−1)//D⁽²⁾_2(k−1)(T)

Top⁺

Fe(T; 2k) //

eq_2k

OO

Top⁺(T) e^δ^2k //D⁽²⁾_2k(T) ep_2k

OO

Top_F(K;k) //

ρ_k|TopF(K;k)

OO

q_k

Top(K)

ρ

OO

δk //F_k(K)

ϕb_k

OO

p_k

Top_F(K;k−1) //

ρ_k−1|TopF(K;k−1)

Top(K)

ρ

δk−1 //F_k−1(K)

ϕbk−1

Top⁺

Fe(T; 2(k−1)) //Top⁺(T) e^δ^2(k−1)//D⁽²⁾_2(k−1)(T).

(3.4)

The map δ_k: Top(K) −→ F_k(K) (resp. eδ_2k: Top⁺(T) −→ D_2k⁽²⁾(T)) is defined by:

δk(f) = (f(x1), . . . , f(xk)) (resp. eδ2k(fe) = fe(π⁻¹(x1)), . . . ,fe(π⁻¹(xk)) ) (3.5) for allf ∈Top(K) (resp. for allfe∈Top⁺(T)), and the mapϕbk: Fk(K)−→

D⁽²⁾_2k(T) is defined by

ϕbk(v1, . . . , vk) = (π⁻¹(v1), . . . , π⁻¹(vk))

for all (v1, . . . , vk)∈Fk(K). Note also that the diagram remains commutative if we identify the corresponding terms of the first and last rows. Taking the long exact sequence in homotopy of the diagram (3.4) and applying Lemma 3.2, we obtain the following commutative diagram whose rows are exact:

1 π1Top⁺(T) B⁽²⁾_2(k−1)(T) π0Top⁺

Fe(T; 2(k−1)) M CG(T) 1 1 π1Top⁺(T) B_2k⁽²⁾(T) π0Top⁺

Fe(T; 2k) M CG(T) 1

1 π₁Top(K) P_k(K) P M CG(K;k) M CG(K) 1

1 π1Top(K) P_k−1(K) P M CG(K;k−1) M CG(K) 1

1 π₁Top⁺(T) B⁽²⁾_2(k−1)(T) π₀Top⁺

Fe(T; 2(k−1)) M CG(T) 1,

(e^δ^2(k−1)⁾^# e^∂^2(k−1) e^τ^2(k−1)

(e^δ^2k⁾^# e^∂^2k

(e^p^2k⁾^#

e^τ^2k

(e^q^2k⁾^#

(δ_k)_# Φ1

∂_k (^ϕb^k⁾^#

(p_k)_#

τ_k

(q_k)_#

φk|P M CG(K;k) φ

(δk−1)#

Φ1

∂k−1

(^ϕb^k−1⁾^#

τk−1

φk−1|P M CG(K;k−1) φ

(e^δ^2(k−1)⁾^# e^∂^2(k−1) e^τ^2(k−1)

(3.6)

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where∂e_2(k−1),∂e2k, ∂kand∂_k−1are the boundary homomorphisms associated with the corresponding fibrations of (3.4),eτ_2(k−1), eτ2k,τ_k−1 andτk are the restrictions of these homomorphisms to the given groups, (qe2k)# and (qk)#

are the maps induced by qe2k and qk respectively on the level of π0, and (eδ_2(k−1))#, (eδ2k)#, (δk)# and (δ_k−1)# are the homomorphisms induced by eδ_2(k−1),eδ_2k, δ_k and δ_k−1 on the level of π₁. The homomorphism φis given by Proposition 2.3, and (ϕb_k)# is the restriction of ϕ_k# to P_k(K), so is injective by [14]. We remark that π₁Top(K)∼=Z [19, Theorem 4.1] (resp.

π1Top⁺(T)∼=Z² [18, Theorem 2]). Once more, the diagram (3.6) remains commutative if we identify the corresponding groups of the first and last rows. In particular, the second and third columns yield respectively:

(ep_2k)_#◦(ϕb_k)_#= (ϕb_k−1)_#◦(p_k)_# (3.7) (qe_2k)_#◦φ_k

P M CG(K;k) =φ_k−1

P M CG(K;k−1) ◦(q_k)_#. (3.8) We recall at this point that our aim is to show that the homomorphism φk

P M CG(K;k) is injective. Note that by diagram (3.6), (ϕbk)#◦(δk)# = (eδ_2k)#◦Φ1, and it follows from the exactness of (3.6) and the injectivity of (ϕb_k)# that Φ1 is also injective. We claim that the restriction

∂e_2k Ker((

ep_2k)_#) : Ker((ep_2k)_#)−→Ker((eq_2k)_#) (3.9) is an isomorphism. Indeed:

(i) It is well defined: if x∈Ker((pe2k)#), the commutative diagram (3.6) gives (qe2k)_#◦∂e2k(x) =∂e_2(k−1)◦(pe2k)_#(x) = 1, so∂e2k(x)∈Ker((qe2k)_#).

(ii) Suppose thatx∈Ker(e∂2k

Ker((

ep_2k)_#)). By exactness of the second row of the diagram (3.6), there existsz∈π1Top⁺(T) such that (eδ2k)#(z) = x, and by commutativity of the same diagram,

(eδ_2(k−1))#(z) = (pe2k)#◦(eδ2k)#(z) = (ep2k)#(x) = 1.

But (eδ_2(k−1))#is injective, soz= 1, whencex= 1. Thus∂e_2k _Ker((

ep2k)#)

is injective too.

(iii) Lety∈Ker((eq_2k)_#). Using commutativity and exactness of the second row of the diagram (3.6), we have y ∈ Ker(eτ_2k), so there exists w ∈ B_2k⁽²⁾(T) such that ∂e2k(w) = y, and thus (pe2k)#(w) ∈ Ker(∂e_2(k−1)).

Hence there existsz∈π1Top⁺(T) such that

(pe2k)#(w) = (eδ_2(k−1))#(z) = (pe2k)#◦(eδ2k)#(z).

It follows thatw(eδ_2k)_#(z⁻¹)∈Ker((pe_2k)_#), and that

∂e2k(w(eδ2k)#(z⁻¹)) =∂e2k(w) =y

by exactness of the second row of (3.6), which shows that∂e2k|_Ker((

ep_2k)_#)

is surjective.

In a similar manner, one may show that the restriction

∂_k

_Ker((p_k₎_#₎ : Ker((p_k)_#)−→Ker((q_k)_#) (3.10)

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is also an isomorphism, and ifw∈Ker((qk)#) then (qe2k)#(φk(w)) =φ_k−1((qk)#(w)) = 1

by equation (3.8), and soφk(w)∈Ker((qe2k)#). We thus obtain the following commutative diagram of short exact sequences:

1 //Ker((q_k)_#) //

φ_k

Ker((qk)#)

P M CG(K;k) ^(q^k⁾^# //

φ_k|P M CG(K;k)

P M CG(K;k−1) //

φk−1|P M CG(K;k−1)

1

1 //Ker((qe2k)#) //π0Top⁺

Fe(T; 2k)

(e^q^2k⁾^#

//π0Top⁺

Fe(T; 2(k−1)) //1, (3.11) the second arrow in each row being inclusion. We claim thatφ_k

_Ker((q_k₎_#₎ is injective. This being the case, if φ_k−1

P M CG(K;k−1) is injective then φk

P M CG(K;k) is also injective by the commutativity and exactness of the rows of (3.11). By induction onk, to complete the proof in the caseg= 2, it will thus suffice to prove that the homomorphism φ1:M CG(K; 1) −→

M CG(T; 2) is injective, which we shall do shortly (note thatP M CG(K; 1) = M CG(K; 1), so we may remove the restriction symbol from φ₁). We first prove the claim. Lety ∈Ker((q_k)_#) be such thatφ_k(y) = 1. From the isomorphism (3.10), there exists a uniquex∈Ker((p_k)_#) such that∂_k(x) =y.

By (3.7), we have (pe_2k)_#◦(ϕb_k)_#(x) = (ϕb_k−1)_#◦(p_k)_#(x) = 1, so (ϕb_k)_#(x)∈ Ker((pe2k)#). On the other hand,∂e2k◦(ϕbk)#(x) = φk◦∂k(x) =φk(y) = 1 by commutativity of the diagram (3.6), hence (ϕb_k)_#(x) ∈ Ker(∂e_2k). So (ϕb_k)_#(x) = 1 by the isomorphism (3.9). But (ϕb_k)_#is the restriction ofϕ_k#

toPk(K), so is injective [14]. Thusx= 1,y= 1, and thereforeφk

Ker((q_k)#)

is injective as claimed.

It thus remains to show that the homomorphism φ1: M CG(K; 1) −→

M CG(T; 2) is injective. Suppose that w ∈ Ker(φ₁). Taking k = 1, the second and third rows of diagram (3.6) become:

1 //Z(P₁(K))^(δ¹⁾^#//

Φ1

P₁(K) ^∂¹ //

ϕ1#

M CG(K; 1) ^τ¹ //

φ₁

M CG(K)

| {z }

Z2⊕Z2

φ

//1

1 //Z(P₂(T))⁽e^δ²⁾^# //B₂(T) e^∂² //M CG(T; 2) e^τ² //M CG(T)

| {z }

SL(2,Z)

//1.

(3.12) Since φ◦τ1(w) = eτ2◦φ1(w) = 1, we see that τ1(w) ∈ Ker(φ), and so by Remark 2.4, τ₁(w) is either equal to the mapping class of the identity ofM CG(K), in which case we say thatw is oftype one, or is equal to the mapping class of the automorphism

(α7−→α

β 7−→αβ, in which case we say that wis oftype two. Recall that

P₁(K) = α, β

αβαβ⁻¹

, (3.13)

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Z(P1(K)) =Z(β²), and every element ofP1(K) may be written (uniquely) in the formα^rβ^s, wherer, s∈Z. By the Appendix,B2(T) =hx, y, a, b, σiand Z(P1(T)) =Z(a)⊕Z(b), where the given generators ofB2(T) are described geometrically, and are subject to the relations of Theorem A.2. The elements a and b are represented by pairs of parallel strings, and they generate the centre ofB₂(T), the elementsxandyare braids whose second string is fixed, and σ is the standard Artin generator that exchanges the two basepoints.

The groupB₂(T) contains

P2(T) =F2(x, y)⊕Z(a)⊕Z(b) (3.14) as an index two subgroup, whereF2(x, y) is the free group on{x, y}. The homomorphismϕ1#is given byϕ1#(α) =a⁻¹x²andϕ1#(β) =yσ⁻¹. Using the relations of Theorem A.2, one may check thatϕ1#(β²) =b.

First assume thatw ∈ Ker(φ1) is of type one. Then w ∈ Ker(τ1), and by exactness of the first row of (3.12), there exists w⁰ ∈ P₁(K) such that

∂₁(w⁰) =w, and sow⁰=α^rβ^s, wherer, s∈Zare unique. If w⁰∈ β²

then

∂₁(w⁰) = 1 by exactness of the first row of (3.12), and the conclusion clearly holds. So suppose that eithersis odd orr6= 0. Then

ϕ_1#(w⁰) = (a⁻¹x²)^r(yσ⁻¹)^s=

(x^2ra^−rb^s/2 ifsis even

x^2rya^−rb^(s−1)/2σ⁻¹ ifsis odd, (3.15) using the fact thatZ(B₂(T)) =ha, bi. Since (eδ₂)_#◦ϕ_1#(w⁰) =φ₁◦∂₁(w⁰) = φ₁(w) = 1, by observing the induced permutation ofϕ_1#(w⁰), we conclude that s must be even, so ϕ_1#(w⁰) ∈ P₂(T), and that ϕ_1#(w⁰) ∈ ha, bi by exactness of the second row of (3.12). Using the decomposition (3.14), it follows thatr= 0, which yields a contradiction. So ifw∈Ker(φ1) is of type one then w= 1. Now suppose that w ∈Ker(φ1) is of type two. Consider the basepoint-preserving homeomorphismh of K illustrated in Figure 3.1.

Observe that τ1([h]) = τ1(w), so by exactness of the first row of (3.12), there exists w⁰ ∈ P1(K) such that w = [h]∂1(w⁰). Let r, s ∈ Z be such that w⁰ =α^rβ^s. Nowh lifts to an orientation-preserving homeomorphism

α

β β

α h

−−−−−−−→ α

β β

h(α) =α

h(β)

Figure 3.1. The homeomorphismhofK.

eh∈Top⁺(T,X) that is illustrated in Figure 3.2, wheree Xe ={xe1,xe2}is the inverse image of the basepoint ofKunder π. It follows thateh=ρ1(h), and hence [eh] = [ρ1(h)] = φ1([h]). On the other hand, relative to the chosen generators of P₂(T), the generatorx is represented by the loopc based at ex₁, anda is represented by two loops, namelyc and a parallel copy c⁰ of c