Simple infinite presentations for the mapping class group of a compact non-orientable surface
aa r X i v : . [ m a t h . G T ] S e p SIMPLE INFINITE PRESENTATIONS FOR THE MAPPINGCLASS GROUP OF A COMPACT NON-ORIENTABLE SURFACE
RYOMA KOBAYASHI
Abstract.
Omori and the author [6] have given an infinite presentation forthe mapping class group of a compact non-orientable surface. In this paper,we give more simple infinite presentations for this group. Introduction
For g ≥ n ≥
0, we denote by N g,n the closure of a surface obtainedby removing disjoint n disks from a connected sum of g real projective planes,and call this surface a compact non-orientable surface of genus g with n boundarycomponents. We can regard N g,n as a surface obtained by attaching g M¨obiusbands to g boundary components of a sphere with g + n boundary components, asshown in Figure 1. We call these attached M¨obius bands crosscaps . Figure 1.
A model of a non-orientable surface N g,n .The mapping class group M ( N g,n ) of N g,n is defined as the group consistingof isotopy classes of all diffeomorphisms of N g,n which fix the boundary point-wise. M ( N , ) and M ( N , ) are trivial (see [2]). Finite presentations for M ( N , ), M ( N , ), M ( N , ) and M ( N , ) ware given by [9], [1], [14] and [16] respectively.Paris-Szepietowski [13] gave a finite presentation of M ( N g,n ) with Dehn twists and crosscap transpositions for g + n > n ≤
1. Stukow [15] gave another finitepresentation of M ( N g,n ) with Dehn twists and one crosscap slide for g + n > n ≤
1, applying Tietze transformations for the presentation of M ( N g,n ) givenin [13]. Omori [12] gave an infinite presentation of M ( N g,n ) with all Dehn twistsand all crosscap slides for g ≥ n ≤
1, using the presentation of M ( N g,n )given in [15], and then, following this work, Omori and the author [6] gave aninfinite presentation of M ( N g,n ) with all Dehn twists and all crosscap slides for g ≥ n ≥
2. In this paper, we give more simple four infinite presentations of M ( N g,n ) with all Dehn twists and all crosscap slides, and with all Dehn twists andall crosscap transpositions. Mathematics Subject Classification . 20F05, 57M07, 20F65.
Key words and phrases . mapping class group, presentation.The author was supported by JSPS KAKENHI Grant Number JP19K14542.
Through this paper, the product gf of mapping classes f and g means that weapply f first and then g . Moreover we do not distinguish a simple closed curvefrom its isotopy class.We define a Dehn twist , a crosscap slide and a crosscap transposition which areelements of M ( N g,n ). For a simple closed curve c of N g,n , a regular neighborhoodof c is either an annulus or a M¨obius band. We call c a two sided or a one sided simple closed curve respectively. For a two sided simple closed curve c , we can takethe two orientations + c and − c of a regular neighborhood of c . The right handedDehn twist t c ; θ about a two sided simple closed curve c with respect to θ ∈ { + c , − c } is the isotopy class of the map described as shown in Figure 2 (a). We write t c ; θ = t c if the orientation θ is given explicitly, that is, the direction of the twist is indicatedby an arrow written beside c as shown in Figure 2 (a). For a one sided simple closedcurve µ of N g,n and an oriented two sided simple closed curve α of N g,n such that N g,n \ { α } is non-orientable when g ≥ µ and α intersect transverselyat only one point, the crosscap slide Y µ,α about µ and α is the isotopy class of themap described by pushing the crosscap which is a regular neighborhood of µ oncealong α , as shown in Figure 2 (b). The crosscap transposition U µ,α ; θ about µ and α with respect to θ ∈ { + α , − α } is defined as U µ,α ; θ = t α ; θ Y µ,α , described as shownin Figure 2 (b). Note that Y µ,α and U µ,α ; θ can not be defined when g = 1. (a) The Dehn twist t c ; θ = t c about c withrespect to θ ∈ { + α , − α } .(b) The crosscap slide Y µ,α about µ and α , and the crosscap transposition U µ,α : θ about µ and α with respect to θ ∈ { + α , − α } . Figure 2.
Let c , . . . , c k , c , c ′ and d , . . . , d be simple closed curves with arrows of asurface as shown in Figure 3. M ( N g,n ) admits following relations. • – ( t c t c · · · t c k ) k +1 = t c t c ′ if k is odd, – ( t c t c · · · t c k ) k +2 = t c if k is even. • t d t d t d = t d t d t d t d .These relations are called a k -chain relation and a lantern relation respectively.Let T , Y and U ⊂ M ( N g,n ) denote the sets consisting of all Dehn twists, allcrosscap slides and all crosscap transpositions respectively. We denote by f ∗ ( θ ) theorientation of a regular neighborhood of f ( c ) induced from θ ∈ { + c , − c } , for a two IMPLE INFINITE PRESENTATIONS FOR M ( N g,n ) 3 Figure 3.
Simple closed curves c , . . . , c k , c , c ′ and d , . . . , d with arrows of a surface.sided simple closed curve c of N g,n and f ∈ M ( N g,n ). Our main results are asfollows. Theorem 1.1.
For g ≥ and n ≥ , M ( N g,n ) admits a presentation with agenerating set T ∪ Y . The defining relations are (1) t c ; θ = 1 if c bounds a disk or a M¨obius band, (2) (a) t − c ;+ c = t c ; − c for any t c ;+ c ∈ T , (3) (a) f t c ; θ f − = t f ( c ); f ∗ ( θ ) for any t c ; θ ∈ T and f ∈ T ∪ Y , (b) f Y µ,α f − = Y f ( µ ) ,f ( α ) for any Y µ,α ∈ Y and f ∈ T , (4) all the -chain relations, (5) all the lantern relations and (6) Y µ,α = t δ ( µ,α ) for any Y µ,α ∈ Y , where δ ( µ, α ) is a simple closed curve withan arrow determined by µ and α as shown in Figure 2 (b). Theorem 1.2.
For g = 2 and n ≥ , M ( N g,n ) admits a presentation with agenerating set T ∪ Y . The defining relations are (1) t c ; θ = 1 if c bounds a disk or a M¨obius band, (2) (a) t − c ;+ c = t c ; − c for any t c ;+ c ∈ T , (b) Y − µ,α = Y µ,α − for any Y µ,α ∈ Y , (3) (a) f t c ; θ f − = t f ( c ); f ∗ ( θ ) for any t c ; θ ∈ T and f ∈ T ∪ Y , (b) f Y µ,α f − = Y f ( µ ) ,f ( α ) for any Y µ,α ∈ Y and f ∈ T , (4) all the -chain relations and (5) all the lantern relations. Theorem 1.3.
For g ≥ and n ≥ , M ( N g,n ) admits a presentation with agenerating set T ∪ U . The defining relations are (1) t c ; θ = 1 if c bounds a disk or a M¨obius band, (2) (a) t − c ;+ c = t c ; − c for any t c ;+ c ∈ T , (3) (a) f t c ; θ f − = t f ( c ); f ∗ ( θ ) for any t c ; θ ∈ T and f ∈ T ∪ U , (b) f U µ,α ; θ f − = U f ( µ ) ,f ( α ); f ∗ ( θ ) for any U µ,α ; θ ∈ U and f ∈ T ∪ { U µ,α ; θ } , (4) all the -chain relations, (5) all the lantern relations and (6) U µ,α ; θ = t δ ( µ,α ) for any U µ,α ; θ ∈ U , where δ ( µ, α ) is a simple closed curvewith an arrow determined by µ and α as shown in Figure 2 (b). Theorem 1.4.
For g = 2 and n ≥ , M ( N g,n ) admits a presentation with agenerating set T ∪ U . The defining relations are
R. KOBAYASHI (1) t c ; θ = 1 if c bounds a disk or a M¨obius band, (2) (a) t − c ;+ c = t c ; − c for any t c ;+ c ∈ T , (b) U − µ,α ; θ = U µ,α − ; θ for any U µ,α ; θ ∈ U , (3) (a) f t c ; θ f − = t f ( c ); f ∗ ( θ ) for any t c ; θ ∈ T and f ∈ T ∪ U , (b) f U µ,α ; θ f − = U f ( µ ) ,f ( α ); f ∗ ( θ ) for any U µ,α ; θ ∈ U and f ∈ T ∪ { U µ,α ; θ } , (4) all the -chain relations and (5) all the lantern relations. In Section 2, we prove Theorem 1.1. In Section 3, we prove Theorem 1.2. InSection 4, we prove Theorems 1.3 and 1.4. Finally, in Appendix A, we introducesome problems on presentations for M ( N g,n ).2. Proof of Theorem 1.1
We denote by T ( N g,n ) the subgroup of M ( N g,n ) generated by T , and call the twist subgroup of M ( N g,n ). Omori and the author [7] gave an infinite presentationfor T ( N g,n ) as follows. Theorem 2.1 ([7]) . For g ≥ and n ≥ , T ( N g,n ) admits a presentation with agenerating set T . The defining relations are (1) t c ; θ = 1 if c bounds a disk or a M¨obius band, (2) t − c ;+ c = t c ; − c for any t c ;+ c ∈ T , (3) f t c ; θ f − = t f ( c ); f ∗ ( θ ) for any t c ; θ and f ∈ T , (4) all the -chain relations and (5) all the lantern relations. As the corollary of Theorem 2.1, we obtain the following.
Corollary 2.2.
Fix one crosscap slide Y µ ,α ∈ Y . For g ≥ and n ≥ , M ( N g,n ) admits a presentation with a generating set T ∪ { Y µ ,α } . The defining relationsare (1) t c ; θ = 1 if c bounds a disk or a M¨obius band, (2) t − c ;+ c = t c ; − c for any t c ;+ c ∈ T , (3) f t c ; θ f − = t f ( c ); f ∗ ( θ ) for any t c ; θ ∈ T and f ∈ T ∪ { Y µ ,α } , (4) all the -chain relations, (5) all the lantern relations and (6) Y µ ,α = t δ ( µ ,α ) .Proof. For g ≥
2, we have the short exact sequence1 → T ( N g,n ) → M ( N g,n ) → h Y µ ,α | Y µ ,α i → M ( N g,n ) from the presentation for T ( N g,n ) given in Theorem 2.1 and thepresentation h Y µ ,α | Y µ ,α i . For details, for instance see [5]. (cid:3) We now prove Theorem 1.1.Let G be the group which admits the presentation as in Theorem 1.1. When g = 1, since a crosscap slide is not defined we see M ( N g,n ) = T ( N g,n ). In addition,since Y is the empty set, G is isomorphic to T ( N g,n ) by Theorem 2.1, and so M ( N g,n ). Therefore we suppose g ≥ ϕ : G → M ( N g,n ) be the natural homomorphism and ψ : M ( N g,n ) → G thehomomorphism defined as ψ ( f ) = f for any f ∈ T ∪ { Y µ ,α } . Since any relation IMPLE INFINITE PRESENTATIONS FOR M ( N g,n ) 5 of M ( N g,n ) in Corollary 2.2 is satisfied in G , we see that ψ is well-defined. Since ϕ ◦ ψ is the identity map clearly, it suffices to show that ψ ◦ ϕ is the identity map.For any t c ; θ ∈ T , it is clear that ψ ( ϕ ( t c ; θ )) = t c ; θ . For any Y µ,α ∈ Y , there is f ∈ M ( N g,n ) such that f Y µ ,α f − = Y µ,α in M ( N g,n ). If f is in T ( N g,n ), since f can be represented as a word f · · · f k on T , repeating the relations (2) (a) and(3) (b) of G , we calculate ψ ( ϕ ( Y µ,α )) = ψ ( Y µ,α )= ψ ( f Y µ ,α f − )= ψ ( f · · · f k Y µ ,α f − k · · · f − )= f · · · f k Y µ ,α f − k · · · f − = Y f ··· f k ( µ ) ,f ··· f k ( α ) = Y f ( µ ) ,f ( α ) = Y µ,α . If f is not in T ( N g,n ), there exists h ∈ T ( N g,n ) such that f = hY µ ,α . Since h canbe represented as a word h · · · h k on T , repeating the relations (2) (a) and (3) (b)of G , we calculate ψ ( ϕ ( Y µ,α )) = ψ ( Y µ,α )= ψ ( f Y µ ,α f − )= ψ ( hY µ ,α Y µ ,α Y − µ ,α h − )= ψ ( h · · · h k Y µ ,α h − k · · · h − )= h · · · h k Y µ ,α h − k · · · h − = Y h ··· h k ( µ ) ,h ··· h k ( α ) = Y h ( µ ) ,h ( α ) = Y h ( Y µ ,α ( µ )) ,h ( Y µ ,α ( α )) = Y f ( µ ) ,f ( α ) = Y µ,α . Therefore we conclude that ψ ◦ ϕ is the identity map.Thus we have that G is isomorphic to M ( N g,n ), and hence the proof of Theo-rem 1.1 is completed. 3. Proof of Theorem 1.2
Any relation which is appeared in Theorem 1.2 is satisfied in M ( N g,n ) clearly.In addition, the relations (1)-(5) in Theorem 1.1 are included in the relations inTheorem 1.2. Hence it suffices to show that the relation (6) in Theorem 1.1 isobtained from relations in Theorem 1.2 when g ≥ Remark 3.1. • In the relation (3) (a) in the main theorems, if f = t c ′ ; θ ′ , | c ∩ c ′ | = 0 or 1, and the orientations θ and θ ′ are compatible, then therelation can be rewritten as a commutativity relation t c ; θ t c ′ ; θ ′ = t c ′ ; θ ′ t c ; θ or a braid relation t c ; θ t c ′ ; θ ′ t c ; θ = t c ′ ; θ ′ t c ; θ t c ′ ; θ ′ respectively. We assign thelabel (3) ′ to all the commutativity relations and all the braid relations. R. KOBAYASHI • It is known that any chain relation is obtained from the relations (3) (a),(4) and (5) in the main theorems (see [3, 11]). We assign the label (4) ′ toall the chain relations.For any Y µ,α ∈ Y , let β , γ , δ A , B , C , D and E be simple closed curves witharrows as shown in Figure 4. We calculate Y µ,α (3)(b) = t β t α t γ t β t α t β Y µ,α − t − β t − α t − β t − γ t − α t − β Y µ,α (2)(b) = t β t α t γ t β t α t β Y µ,α − t − β t − α t − β t − γ t − α t − β Y − µ,α − (2)(a) , (3)(a) = t β t α t γ t β t α t β t δ t α t δ t β t α t δ (3) ′ = t β t α t γ t α t β t α t δ t α t β t δ t α t δ (3) ′ = t β t α t γ t α t β t α t δ t α t β t α t δ t α (3) ′ = t β t α t γ t α t β t α t δ t β t α t β t δ t α (3) ′ = t β t α t γ t α t β t α t β t δ t α t δ t β t α (3) ′ = t β t α t γ t α t β t α t β t α t δ t α t β t α (3) ′ = t β t α t γ t α t β t α t β t α t δ t β t α t β (3) ′ = t β t α t γ t α t β t α t β t α t β t δ t α t β (3) ′ = t β t α t γ t α t − δ t β t δ t α t β t α t β t δ t α t β = t β t α t γ t α t − δ t − α t − β t β t α t β t δ t α t β t α t β t δ t α t β (3) ′ = t β t α t γ t − δ t − α t δ t − β t α t β t α t δ t α t β t α t β t δ t α t β (3) ′ = t β t α t γ t − δ t − α t − β t δ t α t β t δ t α t δ t β t α t β t δ t α t β (3) ′ = ( t β t α t γ t − α t − β )( t β t α t − δ t − α t − β )( t δ t α t β ) , (3)(a) , (4) ′ = t A t B t C t D (1) = t A t B t C t E t α t − α (2)(a) , (5) = t δ ( µ,α ) . Therefore the group presented in Theorem 1.2 is isomorphic to the group pre-sented in Theorem 1.1, and so M ( N g,n ). Thus we finish the proof of Theorem 1.2.4. Proof of Theorems 1.3 and 1.4
We first note that Theorems 1.3 and 1.4 can be shown by the arguments similarto Sections 2 and 3 respectively. However we prove these by giving isomorphismsfrom the groups presented in Theorems 1.3 and 1.4 to the groups presented inTheorems 1.1 and 1.2 respectively.Let H be the group which admits the presentation as in either Theorems 1.3or 1.4. When g = 1, since U is the empty set, the presentation of H is same IMPLE INFINITE PRESENTATIONS FOR M ( N g,n ) 7 Figure 4.
Simple closed curves µ , α , β , γ , δ A , B , C , D and E of M ( N g,n ).to the presentation of the group presented in Theorems 1.1 and 1.2. Hence H isisomorphic to M ( N ,n ). Therefore we suppose that g ≥ η : H → M ( N g,n ) be the natural homomorphism and ν : M ( N g,n ) → H thehomomorphism defined as ν ( t c ; θ ) = t c ; θ and ν ( Y µ,α ) = t − α ; θ U µ,α ; θ for any t c ; θ ∈ T and Y µ,α ∈ Y . If ν is well-defined, µ is the inverse map of η clearly. So it sufficesto show well-definedness of ν , that is, we show that the correspondence ν ( Y µ,α ) = t − α ; θ U µ,α ; θ does not depend on the choice of θ ∈ { + α . − α } , and that any relation of M ( N g,n ) is satisfied in H .First we show that t − α ;+ α U µ,α ;+ α = t − α ; − α U µ,α ; − α in H . By the relations (2) (a)and (3) (a) of H , we calculate U µ,α ;+ α t α ;+ α U − µ,α ;+ α = t U µ,α ;+ α ( α );( U µ,α ;+ α ) ∗ (+ α ) = t α ; − α = t − α ;+ α , and so t − α ;+ α U µ,α ;+ α = U µ,α ;+ α t α ;+ α . Hence by the relations (2) (a) and (3) (b) of H , we calculate t α ; − α t − α ;+ α U µ,α ;+ α = t α ; − α U µ,α ;+ α t α ;+ α = t α ; − α U µ,α ;+ α t − α ; − α = U t α ; − α ( µ ) ,t α ; − α ( α );( t α ; − α ) ∗ (+ α ) = U U µ,α ; − α ( µ ) ,U µ,α ; − α ( α );( U µ,α ; − α ) ∗ ( − α ) = U µ,α ; − α U µ,α ; − α U − µ,α ; − α = U µ,α ; − α , and so t − α ;+ α U µ,α ;+ α = t − α ; − α U µ,α ; − α . Therefore we conclude that the correspon-dence ν ( Y µ,α ) = t − α ; θ U µ,α ; θ does not depend on the choice of θ ∈ { + α . − α } .Next we show that any relation appeared in Theorems 1.1 and 1.2 is satisfied in H . The relations (1), (2) (a), (3) (a) with f ∈ T , (4) and (5) in Theorems 1.1 and R. KOBAYASHI H clearly. By the relation (2) (b) of H , we calculate ν ( Y − µ,α ) = ( t − α ; θ U µ,α ; θ ) − = ( U µ,α ; θ t α ; θ ) − = t − α ; θ U − µ,α ; θ = t − α − ; θ U µ,α − ; θ = ν ( Y µ,α − ) . Hence the relation (2) (b) in Theorem 1.2 is satisfied in H . By the relations (2) (a)and (3) (a) of H , we calculate ν ( Y µ,α t c ; θ Y − µ,α ) = ( t − α ; θ U µ,α ; θ ) t c ; θ ( t − α ; θ U µ,α ; θ ) − = ( t α ; θ ′ U µ,α ; θ ) t c ; θ ( t α ; θ ′ U µ,α ; θ ) − = t t α ; θ ′ U µ,α ; θ ( c );( t α ; θ ′ U µ,α ; θ ) ∗ ( θ ) = t t − α ; θ U µ,α ; θ ( c );( t − α ; θ U µ,α ; θ ) ∗ ( θ ) = t Y µ,α ( c );( Y µ,α ) ∗ ( θ ) = ν ( t Y µ,α ( c );( Y µ,α ) ∗ ( θ ) ) , where θ ′ is the inverse orientation of θ . Hence the relations (3) (a) with f ∈ Y inTheorems 1.1 and 1.2 are satisfied in H . By the relations (3) (a) and (3) (b) of H ,we calculate ν ( f Y µ,α f − ) = f ( t − α ; θ U µ,α ; θ ) f − = ( f t α ; θ f − ) − ( f U µ,α ; θ f − )= t − f ( α ); f ∗ ( θ ) U f ( µ ) ,f ( α ); f ∗ ( θ ) = ν ( Y f ( µ ) ,f ( α ) ) . Hence the relations (3) (b) in Theorems 1.1 and 1.2 are satisfied in H . By therelation (6) of H , we calculate ν ( Y µ,α ) = ( t − α ; θ U µ,α ; θ ) = ( U µ,α ; θ t α ; θ )( t − α ; θ U µ,α ; θ )= U µ,α ; θ = t δ ( µ,α ) = ν ( t δ ( µ,α ) ) . Hence the relation (6) in Theorem 1.1 is satisfied in H . Therefore we conclude thatany relation of M ( N g,n ) is satisfied in H .So we obtain well-definedness of ν , and hence it follows that H is isomorphic to M ( N g,n ). Thus we complete the proof of Theorems 1.3 and 1.4. Appendix A. Problems on presentations for M ( N g,n ) . In this appendix, we introduce three problems on presentations for M ( N g,n ) asfollows.In the main theorems, we can reduced relations. For example, in the rela-tion (3) (a) in Theorems 1.1 and 1.2 (resp. Theorems 1.3 and 1.4), T ∪ Y (resp.
T ∪ U ) can be reduced to a finite number of Dehn twists and one crosscap slide(resp. a finite number of Dehn twists and one crosscap transposition). Similarly,
IMPLE INFINITE PRESENTATIONS FOR M ( N g,n ) 9 in the relation (3) (b) in Theorems 1.1 and 1.2 (resp. Theorems 1.3 and 1.4), T (resp. T ∪ { U µ,α ; θ } ) can be reduced to a finite number of Dehn twists (resp. afinite number of Dehn twists by adding the relation U µ,α ;+ α = U t α ;+ α ( µ ) ,α ; − α ). Inaddition, we can reduced the relation (6) in Theorem 1.1 (resp. Theorem 1.3) toone relation Y µ ,α = t δ ( µ ,α ) (resp. U µ ,α ; θ = t δ ( µ ,α ) ) for some pair ( µ , α ).On the other hand, the relation of the mapping class group of an orientable surfacecorresponding to the relation (3) (a) with f ∈ T in the main theorems can be re-duced to only all the commutativity relations and all the braid relations. However,in the non-orientable case, we do not know whether or not the same holds. So wehave a natural problem as follows. Problem A.1.
Give an infinite presentation of M ( N g,n ) whose relations are moresimple for g ≥ and n ≥ . It is known that M ( N g,n ) can not be generated by only either Dehn twists orcrosscap slides for g ≥ n ≥ M ( N g,n ) can be generated by only crosscap transpo-sitions for g ≥ n ≥
0. So we have a natural problem as follows.
Problem A.2.
Give an infinite presentation of M ( N g,n ) with all crosscap trans-positions for g ≥ and n ≥ . It is known that M ( N , ) and M ( N , ) are trivial (see [2]). In addition, for g ≥ n ≤
1, a simple finite presentation of M ( N g,n ) was given in [9, 1, 14, 16, 13].Moreover, Omori and the author [6] gave a finite presentation of M ( N g,n ) for g ≥ n ≥
2. However this presentation is very complicated. On the other hand, thereis a simple finite presentation for the mapping class group of a compact orientablesurface (see [4]). So we have a natural problem as follows.
Problem A.3.
Give a simple finite presentation of M ( N g,n ) with Dehn twists andcrosscap slides, or with Dehn twists and crosscap transpositions (resp. with crosscaptranspositions) for g ≥ (resp. g ≥ ) and n ≥ . References [1] J. S. Birman, D. R. J. Chillingworth,
On the homeotopy group of a non-orientable surface ,Proc. Cambridge Philos. Soc. (1972), 437–448. Erratum: Math. Proc. Cambridge Philos.Soc. (2004), no. 2, 441.[2] D. B. A. Epstein, Curves on -manifolds and isotopies , Acta Math. (1966), 83–107.[3] S. Gervais, Presentation and central extensions of mapping class groups , Trans. Amer. Math.Soc. (1996), no. 8, 3097–3132.[4] S. Gervais,
A finite presentation of the mapping class group of a punctured surface , Topology (2001), no. 4, 703–725.[5] D.L. Johnson, Presentations of groups , Second edition. London Mathematical Society StudentTexts, . Cambridge University Press, Cambridge, 1997.[6] R. Kobayashi, G. Omori, An infinite presentation for the mapping class group of a non-orientable surface with boundary , arXiv:1610.04999 [math.GT].[7] R. Kobayashi, G. Omori,
An infinite presentation for the twist subgroup of the mapping classgroup of a compact non-orientable surface , arXiv:2009.02022 [math.GT].[8] M. Le´sniak, B. Szepietowski,
Generating the mapping class group of a nonorientable surfaceby crosscap transpositions , Topology Appl. (2017), 20–26.[9] W. B. R. Lickorish,
Homeomorphisms of non-orientable two-manifolds , Proc. Cambridge Phi-los. Soc. (1963), 307–317.[10] W. B. R. Lickorish, On the homeomorphisms of a non-orientable surface , Proc. CambridgePhilos. Soc. (1965), 61–64. [11] F. Luo, A presentation of the mapping class groups , Math. Res. Lett. (1997), no. 5, 735–739.[12] G. Omori, An infinite presentation for the mapping class group oorientable surface , Algebr.Geom. Topol. (2017), no. 1, 419–437.[13] L. Paris, B. Szepietowski, A presentation for the mapping class group of a nonorientablesurface , Bull. Soc. Math. France (2015), no. 3, 503–566.[14] M. Stukow,
Dehn twists on nonorientable surfaces , Fund. Math. (2006), no. 2, 117–147.[15] M. Stukow,
A finite presentation for the mapping class group of a nonorientable surface withDehn twists and one crosscap slide as generators , J. Pure Appl. Algebra (2014), no. 12,2226–2239.[16] B. Szepietowski,
A presentation for the mapping class group of the closed non-orientablesurface of genus 4 , J. Pure Appl. Algebra (2009), no. 11, 2001–2016.
Department of General Education,National Institute of Technology, Ishikawa College,Tsubata, Ishikawa, 929-0392, Japan
E-mail address ::