Generalized Lantern Relations and Planar Line Arrangements
GGeneralized Lantern Relations and Planar LineArrangements
Eriko Hironaka
Abstract.
In this paper we show that to each planar line arrangement definedover the real numbers, for which no two lines are parallel, one can write downa corresponding relation on Dehn twists that can be read off from the combi-natorics and relative locations of intersections. This gives an alternate proof ofWajnryb’s generalized lantern relations, and of Endo, Mark and Horn-Morris’daisy relations.
1. Introduction
Braid monodromy is a useful tool for studying the topology of complements ofline arrangements as is seen in work of [ ], [ ], [ ]. In this paper, we adapt braidmonodromy techniques to generate relations on Dehn twists in the mapping classgroup MCG( S ) of an oriented surface S of finite type.The study of hyperplane arrangements has a rich history in the realms of topol-ogy, algebraic geometry, and analysis (see, for example, [ ] for a survey). Whileeasy to draw, the deformation theory of real planar line arrangements L holds manymysteries. For example, there are topologically distinct real line arrangements withequivalent combinatorics [ ] [ ]. Moreover, by the Silvester-Gallai theorem [ ]there are planar line arrangements defined over complex numbers, whose combi-natorics cannot be duplicated by a real line arrangement, for example, the linesjoining flexes of a smooth cubic curve. Braid monodromy is a convenient tool forencoding the local and global topology of L .The lantern relation on Dehn twists is of special interest because it withfour other simple to state relations generate all relations in the Dehn-Lickorish-Humphries presentation of MCG( S ) (see [ ], [ ], [ ], [ ]). The lantern relationalso plays an important role in J. Harer’s proof that the abelianization of MCG( S )is trivial if S is a closed surface of genus g ≥ ] (cf., [ ], Sec. 5.1.2).Let S be an oriented surface of finite type. If S is closed, the mapping classgroup MCG( S ) is the group of isotopy classes of self-homeomorphisms of S . If S hasboundary components, then the definition of MCG( S ) has the additional conditionthat all maps fix the boundary of S pointwise. For a compact annulus A , MCG( A )is isomorphic to Z and is generated by a right or left Dehn twist centered at its core Mathematics Subject Classification.
Primary 57M27, 20F36; Secondary 32Q55.This work was partially supported by a grant from the Simons Foundation ( a r X i v : . [ m a t h . G T ] J a n ERIKO HIRONAKA cc Figure 1.
Right Dehn twist on an annulus A .curve. As illustrated in Figure 1, a right Dehn twist takes an arc on A transverseto the core curve to an arc that wraps once around the core curve turning in theright hand direction (a left Dehn twist correspondingly turns in the left direction)as it passes through c . A Dehn twist can also be thought of as rotating one ofthe boundary components by 360 ◦ while leaving the other boundary componentfixed. Each simple closed curve c on S determines a right Dehn twist on an annulusneighborhood of c , and this Dehn twist extends by the identity to all of S . Theisotopy class ∂ c of this map is the (right) Dehn twist centered at c and is an elementof MCG( S ).The original statement and proof of the lantern relation appears in Dehn’s1938 paper [ ] and relates a product of three interior Dehn twists to four boundarytwists on a genus zero surface with four boundary components. The relation wasrediscovered by D. Johnson [ ], and B. Wajnryb gave the following generalizedversion in [ ] (Lemma 17). Theorem . Let S c ,n +1 ⊂ S be a surface of genus zero with n + 1 boundary components d , d , . . . , d n . There is a collection of simple closed curves a i,j , ≤ i < j ≤ n in the interior of S c ,n +1 , so that (1) for each i, j , a i,j separates d i ∪ d j from the rest of the boundary compo-nents, and (2) there is a relation on Dehn twists ∂ ( ∂ · · · ∂ n ) n − = α , · · · α ,n α , · · · α ,n · · · α n − ,n − α n − ,n α n − ,n , (1.1) where α i,j is the right Dehn twist centered at a i,j , and ∂ i is the right Dehncentered at a curve parallel to the boundary components d i . We now generalize Theorem 1.1 in terms of line arrangements in R . Theorem . Let L be a union of n ≥ distinct lines in the ( x, y ) -plane overthe reals with distinct slopes and no slope parallel to the y -axis. Let I = { p , . . . , p s } be the intersection points on L numbered by largest to smallest x -coordinate. Foreach L ∈ L , let µ L be the number of points in I ∩ L . Let S c ,n +1 be a surface ofgenus zero and n +1 boundary components, one denoted d L for each L ∈ L , and oneextra boundary component d . Then there are simple closed curve a p k , k = 1 , . . . , s on S c ,n +1 so that the following holds: (1) each a p k separates (cid:91) p k ∈ L ∈L d L from the rest of the boundary curves; and ENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 3 L L L d d d d a a a d d a a a d d Figure 2.
Three lines in general position, and curves definingassociated lantern relation drawn two ways.(2) the Dehn twist ∂ L centered at d L and α p k centered at a p k satisfy ∂ (cid:89) L ∈L ∂ µ L − L = α p s · · · α p . (1.2) Remark . In Equation (1 . p , . . . , p s matters, andreflects the global (as opposed to local) combinatorics of the line arrangement. Thecurves a p k can be found explicitly (see Section 2.2, Lemma 2.1). Remark . The relations in MCG( S c ,n +1 ) give rise to relations on MCG( S )for any surface S admitting an embedding S c ,n +1 (cid:44) → S .When n = 3, Theorem 1.1 gives the standard Lantern relation ∂ ∂ ∂ ∂ = α , α , α , . The core curves for these Dehn twists and the corresponding line arrangements areshown in Figure 2. The diagram to the right is the motivation for the name of thisrelation.Here is a sketch of our proof of Theorem 1.2. First consider a great ball B ⊂ C containing all the points of intersection of L . Let CP be the projective compact-ification of C . Then the complement of B in CP is a neighborhood of the “lineat infinity” or L ∞ = CP \ C . Let ρ : C → C be a generic projection. Themonodromy of ρ over the boundary γ of a large disk in C depends only on theway L intersects L ∞ . If no lines in L are parallel to each other, then it is possibleto move the lines in L to obtain a new configuration T where all lines meet ata single point without changing any slopes, and hence the topology of CP \ B remains the same (Lemma 2.3). Thus, the monodromies over γ defined by L and T are the same. Theorem 1.2 then follows from a description of the monodromyof line arrangements on compactified fibers of a generic projection (Lemma 2.1).The monodromy can be interpreted as point pushing maps, where we keep track oftwisting on the boundary components of the compactified fibers using the complexcoordinate system of the ambient space C (Lemma 2.2).This paper is organized as follows. In Section 2.1 we recall the Moishezon-Teicher braid monodromy representation of a free group associated to a planar linearrangement. We refine the representation so that its image is the the mapping classgroup of compactified fibers in Section 2.2. In Section 2.3, we prove Theorem 1.2 ERIKO HIRONAKA using deformations of line arrangements and give further variations of the lanternrelation, including the daisy relation (Theorem 3.1).
Acknowledgments:
The author is grateful to J. Mortada and D. Margalitfor helpful discussions and comments, and to the referee for careful corrections tothe original version.
2. Real line arrangements and relations on Dehn twists
In this section, we analyze line arrangements L in the complex plane definedby real equations and the monodromy on generic fibers under linear projections C \ L → C . A key ingredient is B. Moishezon and M. Teicher description of the monodromy aselements of the pure braid group. (See, for example, [ ] and [ ].) We generalizethis braid monodromy by studying the action of the monodromy not only on genericfibers of ρ , but also on their compactifications as genus zero surfaces with boundary.This leads to a proof of Theorem 1.2.The ideas in this section can be generalized to more arbitrary plane curves.An investigation of the topology of plane curve complements using such generalprojections appears in work of O. Zariski and E. van Kampen [ ]. We leave thisas a topic for future study. In this section we recall the Moishezon-Teicher braid monodromy associatedto a real line arrangement. For convenience we choose Euclidean coordinates ( x, y )for C so that no line is parallel to the y -axis, and no two intersection points havethe same x -coordinate. For i = 1 , . . . , n , let L i be the zero set of a linear equationin x and y with real coefficients: L i = { ( x, y ) ; y = m i x + c i } m i , c i ∈ R and assume that the lines are ordered so that the slopes satisfy: m > m > · · · > m n . Let I = I ( L ) = { p , . . . , p s } ⊂ C be the collection of intersections points of L ordered so that the x -coordinates are strictly decreasing.Let ρ : C → C be the projection of C onto C given by ρ ( x, y ) = x . For each x ∈ C , let F x = ρ − ( x ) \ L . The y -coordinate allows us to uniformly identify F x with the complement in C of n points L i ( x ), where { ( x, L i ( x )) } = ρ − ( x ) ∩ L i . Thus, we will think of F x as a continuous family of copies of C minus a finite setof points, rather than as a subset of C .Let x ∈ R be greater than any point in ρ ( I ). Then there is a natural map γ : [0 , → C \ ρ ( I )from arcs based at x to a braid on n strands in C parameterized by { L i ( γ ( t )) : i = 1 , . . . , n } . ENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 5 p g p g p g p f p f p f x Figure 3.
Simple loop generators for π ( C \ ρ ( I )).Since two homotopic arcs give rise to isotopic braids, and a composition of arcsgives rise to a composition of braids, we have a homomorphism β : π ( C \ ρ ( I ) , x ) → B ( S , n + 1)from the fundamental group to the spherical braid group on n + 1 strands.The (braid) monodromy of ( C , L ) with respect to the projection ρ and base-point x is the homomorphism σ L : π ( C \ ρ ( I ) , x ) → MCG( F x ) , (2.1)given by the composition of β and the braid representation B ( S , n + 1) → MCG( S ,n +1 ) = MCG( F x ) , from the braid group to the mapping class group on a genus zero surface with n + 1punctures.We now study the image of simple generators of π ( C \ ρ ( I ) , x ) in MCG( F x ).By a simple loop in π ( C \ ρ ( I ) , x ), we mean a arc of the form (cid:96) p = f p g p f − p ,where p ∈ ρ ( I ), (cid:15) p > g p : [0 , → C \ ρ ( I ) t (cid:55)→ p + (cid:15) p e πit and f p is a arc from x to p + (cid:15) p whose image is in the upper half plane except atits endpoints. Since π ( C \ ρ ( I ) , x ) is generated by simple loops, it is enough tounderstand the monodromy in the image of these elements.In order to describe the monodromy of (cid:96) p we study how F x is transformed as x follows its arc segments g p and f p First we look at g p . Let L j , L j , . . . , L j k bethe lines in L that pass through p . We can assume by a translation of coordinatesthat p = 0, and L j r is defined by an equation of the form y = m r x where m > m > · · · > m k . Then as t varies in [0 , L j r with ρ − ( g p ( t )) is given by L j r ( g p ( t )) = ( (cid:15) p e πit , m r (cid:15) p e πit ) . The other lines in L locally can be thought of as having constant slope, hencetheir intersections with ρ − ( g p ( t )) retain their order and stay outside a circleon F g p ( t ) enclosing L j ( g p ( t )) , . . . , L j k ( g p ( t )) (see Figure 4). Let a loc p ⊂ F p + (cid:15) be this circle. The restriction of ρ to C \ L defines a trivial bundle over theimage of f p . Thus a loc p determines a simple closed curve a p on F x separating L j ( x ) , . . . , L j k ( x ) from the rest of the L j ( x ). ERIKO HIRONAKA p Figure 4.
Monodromy defined by g p with the real part of L drawn in.Next we notice that lifting over f p defines a mapping class on F x . This isbecause there is a canonical identification of F x and F x for any x ∈ R \ ρ ( I ) givenby the natural ordering of L ∩ ρ − ( x ) by the size of the y -coordinate from largestto smallest. Thus f p determines a braid on n strands and corresponding mappingclass β p ∈ MCG( F cx ). We have shown the following. Lemma . Let (cid:96) p = f p g p f − p . The element σ L ( (cid:96) p ) in MCG ( F cx ) is the Dehntwist α p centered at a p = β − p ( a loc p ) . Proof.
By the above descriptions of the fibers above the arcs f p and g p , wecan decompose σ L ( (cid:96) p ) as α p = β − p ◦ σ p ◦ β p , where σ p is a right Dehn twist centered at a loc p = β p ( a p ). In this section, we define themonodromy representation of π ( C \ L , y ) into MCG( F cx ), where F cx is a com-pactification of F x as a compact surface with boundary.As before choose coordinates for C , and let L = ∪ ni =1 L i be a planar linearrangement defined over the reals with distinct slopes. Assume all points of in-tersection I have distinct x -coordinates. Let (cid:15) > (cid:15) radius disks N (cid:15) ( p ) around the points p ∈ ρ ( I ) are disjoint. Let δ > δ radiustubular neighborhoods N δ ( L i ) around L i are disjoint in the complement of (cid:91) p ∈ ρ ( I ) ρ − ( N (cid:15) ( p )) . ENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 7
Let D be a disk in C containing all points of ρ ( I ) in its interior, and having x onits boundary. Let N ∞ be a disk centered at the origin of C so that C × N ∞ contains L ∩ ρ − ( D ).For each x ∈ C \ ρ ( I ), let F cx = ρ − ( x ) ∩ ( C × N ∞ \ ∪ ni =1 N δ ( L i )) ⊂ F x \ N δ ( L i ) . For each x ∈ D and i = 1 , . . . , n , let d i ( x ) = ∂N δ ( L i ) ∩ F x . We are now ready to define the monodromy on the compactified fibers σ c L : π ( F x , y ) → MCG( F cx ) . Let η be the inclusion homomorphism η : MCG( F cx ) → MCG( F x ) , that is, the homomorphism induced by inclusion F cx ⊂ F x . Then we would like tohave a commutative diagram π ( C \ ρ ( I ) , y ) σ c L (cid:47) (cid:47) σ L (cid:40) (cid:40) (cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80)(cid:80) MCG( F c ) η (cid:15) (cid:15) MCG( F ) . The kernel of η is generated by Dehn twists centered at the boundary componentsof F cx (Theorem 3.18, [ ]). Thus, in order to describe σ c L , we need to understandwhat twists occur near boundary components in the monodromy associated to thearcs g p and f p defined in Section 2.1.Consider the simplest case when L ⊂ C is a single line defined by y = L ( x ) = mx. Let N δ ( L ) be the tubular neighborhood around L in C N δ ( L ) = { ( x, L ( x ) + y ) : | y | < δ } . Then N δ ( L ) ∩ F g ( t ) is a disk centered at L ( g ( t )) of radius δ . The boundary ∂N δ ( L )is a trivial bundle over C \ δ ( ρ ( I )) with trivialization defined by the framing of C by real and purely imaginary coordinates.Now assume that there are several lines L j , . . . , L j k meeting above p ∈ ρ ( I ).Let L be a line through p with slope equal to the average of those of L j , . . . , L j k ,and let δ > N δ ( L ) ∩ F g p ( t ) contains d j ( g p ( t )) , . . . , d j k ( g p ( t )), butno other boundary components of F cg p ( t ) , for all t . Let d p ( t ) be the boundarycomponent of F g p (0) given by d p ( t ) = ∂N (cid:15) ( L ) ∩ F g p ( t ) . Let d p = d p (0) and d j i = d j i ( g p (0)).Then looking at Figure 4, we see that the points L j ( g p ( t )) , . . . , L j k ( g p ( t )) ro-tate as a group 360 ◦ in the counterclockwise direction as t ranges in [0 , F g p (0) enclosed by d p isthe composition of a clockwise full rotation of d p and a counterclockwise rotationaround d j , . . . , d j k . It can also be thought of as moving the inner boundary compo-nents d j ( g p (0)) in a clockwise direction while leaving all orientations of boundarycomponents fixed with respect to the complex framing of C . ERIKO HIRONAKA
Figure 5.
The mapping class ∂ d p . Figure 6.
The monodromy defined by g p .Figure 5 illustrates the Dehn twist ∂ d p centered at a simple closed curve parallelto d p and Figure 6 shows the monodromy σ c L ( g p ) in the case when L is a union of4 lines meeting at a single point p . In both figures, the middle picture illustratesthe fiber F g p (0 . half way around the circle traversed by g p . From the abovediscussion, we have σ c L ( g p ) = ( ∂ d ∂ d ∂ d ∂ d ) − ∂ d p . More generally we have the following lemma.
Lemma . Let L j , . . . , L j k be the lines meeting above p , and let g p : [0 , → C \ L t (cid:55)→ p + (cid:15)e πit . Then the monodromy on F cg p (0) defined by g p is given by σ c L ( g p ) = ( ∂ d j · · · ∂ d jk ) − ∂ d p . To finish our proof of Theo-rem1.2 we analyze the effect of deforming a line arrangement.Let L = n (cid:91) i =1 L i be a finite union of real lines in the Euclidean plane, R with no two lines parallel.Let T be the complexified real line arrangement with all n lines intersecting at asingle point p . Let ρ : C → C be a generic projection, and let D ⊂ C be a disk ofradius r centered at the origin containing ρ ( I ) and ρ ( p ) in its interior. Let γ : [0 , → C t (cid:55)→ re πit . ENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 9 x xx x x x N 2
Figure 7.
Two representatives of γ in π ( C \ ρ ( I )). Lemma . The monodromies σ c L ( γ ) and σ c T ( γ ) are the same. Proof.
Let D ∞ = N ∞ × C ∩ ρ − ( ∂D ) . Then D ∞ \ T and D ∞ \ L are isomorphic as fiber bundles over γ and hence themonodromies over γ defined by L and T are the same up to isotopy. Proof of Theorem 1.2.
By Lemma 2.3, σ c L ( γ ) = σ c T ( γ ). Figure 7 gives anillustration of two equivalent representations of the homotopy type of γ .By Lemma 2.1 and Lemma 2.2, we have σ c T ( γ ) = ( ∂ d · · · ∂ d n ) − ∂ d p . Let p , . . . , p s be the elements of I numbered by decreasing x -coordinate. Then foreach i = 1 , . . . , s , we have σ c L ( f p i g p i f − p i ) = ( ∂ d j · · · ∂ d jk ) − α p i where α p i is the pullback of d p i along the arc f p i . Thus, σ c L ( γ ) = ( ∂ µ d · · · ∂ µ s d s ) − α p s · · · α p , where µ i is the number of elements in I ∩ L i .To show that Theorem 1.1 follows from Theorem 1.2, we need to show that theordering given in Equation (1 .
1) can be obtained by a union of lines L satisfying theconditions. To do this, we start with a union of lines T intersecting in a single point.Let L , . . . L n be the lines in T ordered from largest to smallest slope. Translate L in the positive x direction without changing its slope so that the intersections ofthe translated line L (cid:48) with L , . . . , L n have decreasing x -coordinate. Continue foreach line from highest to lowest slope, making sure with each time that the shifting L creates new intersections lying to the left of all previously created ones.More generally, we can deform the lines through a single point T to one ingeneral position L so that the only condition on the resulting ordering on the pairsof lines is the following. A pair ( i, j ) must preceed ( i, j + 1) for each 1 ≤ i < j ≤ n .Thus, we have proved the following restatement of Theorem 1.1. Theorem . Let { p , . . . , p s } be an ordering of the pairs ( i, j ) , ≤ i < j ≤ n ,so that for all i , the sequence ( i, i + 1) , ( i, i + 2) , . . . , ( i, n ) is strictly decreasing. Then there a lantern relation of the form ∂ ( ∂ · · · ∂ n ) n − = α p · · · α p s . Figure 8.
Line arrangement, and associated arrangement ofcurves (n=6).
3. Applications
Although it is known that all relations on the Dehn-Lickorish-Humphries gen-erators can be obtained from the braid, chain, lantern and hyperelliptic relations,there are some other nice symmetric relations that come out of line arrangementsthat are not trivially derived from the four generating ones. We conclude this paperwith a sampling.
Consider the line arrangements given in Figure 8. Aspointed out to me by D. Margalit, this relation was recently also discovered by H.Endo, T. Mark, and J. Van Horn-Morris using rational blowdowns of 4-manifolds[ ]. We follow their nomenclature and call this the daisy relation .Let S c ,n +1 denote the compact surface of genus 0 with n + 1 boundary com-ponents. Consider the configuration of simple closed curves shown in Figure 8.Let d , . . . , d n be the boundary components of S c ,n +1 . Let d be the distinguishedboundary component at the center of the arrangement, and let d , d , . . . , d n bethe boundary components arranged in a circle (ordered in the clockwise direc-tion around d ). Let a ,k be a simple closed loop encircling d and d k , where k = 0 , , , . . . , n . Let ∂ i be the Dehn twist centered at d i , and let α ,k be the Dehntwist centered at a ,k . Theorem . [Daisy relation] For n ≥ , the Dehn twists on S c ,n +1 satisfythe relation ∂ ∂ n − ∂ · · · ∂ n = α , α ,n · · · α , where ∂ i is the Dehn twist centered at the boundary component d i , and α ,j is theDehn twist centered at curves a ,j . When n = 3, Theorem 3.1 specializes to the usual lantern relation. Proof.
We associate the boundary component d i with L i for i = 1 , . . . , n ,and d with the “line at infinity”. Theorem 1.2 applied to the line arrangement inFigure 8 gives: ∂ ( ∂ · · · ∂ n ) − = R p n . . . R p where p , . . . , p n are the intersection points of the line arrangement L ordered bylargest to smallest x -coordinate. For this configuration, p k gives rise to R p k = ( ∂ ∂ k +1 ) − α ,k +1 , ENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 11 L L L d d d d a a a a d L Figure 9.
Alternate drawing of the daisy configuration (n=4). q L L L n L p p n (cid:239) s n (cid:239) p n (cid:239) s L Figure 10.
Configuration of lines giving rise to the doubled daisy relation.for k = 1 , . . . , n −
1. Noting that the loop that separates d ∪ · · · ∪ d n from d ∪ d can be written as a , , we have R p n = ( ∂ · · · ∂ n − ) − α , yielding the desired formula. Remark . Let β : B ( S , n + 1) → MCG( S ,n +1 )be the braid representation from the spherical braid group to the mapping classgroup. Recall the relation R in B ( S , n + 1) given by( σ )( σ − σ σ ) · · · ( σ − σ − · · · σ − n − σ n σ n − · · · σ ) = σ · · · σ n − σ n σ n − · · · σ . = 1This induces a relation R (cid:48) in MCG( S ,n +1 ). The daisy relation can be consideredas the lift of R (cid:48) under the inclusion homomorphism η . As a final example, we consider a configura-tion of n ≥ n − d , . . . , d n be the boundary components of S c ,n +1 . The boundarycomponent d i is associated to the line L i for i = 1 , . . . , n , and d is the boundary d d d d d d Figure 11.
The doubled daisy relation for n = 5. c n (cid:239) d n (cid:239) d d d d a d n d d Figure 12.
Drawing of the general doubled daisy configuration.component associated to the “line at infinity”. Let a i,j be the loop in Figure 12encircling d i ∪ d j and no other boundary component. Let c be the loop encircling d , . . . , d n − in Figure 12 (or, when n = 5, d , d , and d in Figure 11). Theorem . Let ∂ i be the right Dehn twist centeredat d i , α i,j the right Dehn twist centered at a i,j , and β the right Dehn twist centeredat c . Then ∂ ∂ n − ∂ · · · ∂ n − ∂ n − n = α n − ,n α n − ,n · · · α ,n β α ,n α ,n − · · · α , Proof.
Theorem 2 applied to the line arrangement in Figure 10 gives theequation ∂ ( ∂ · · · ∂ n ) − = R s n − · · · R s R q R p n − · · · R p , where R p k = ( ∂ ∂ k +1 ) − α ,k +1 R q = ( ∂ · · · ∂ n − ) − βR s k = ( ∂ n ∂ k +1 ) − α k +1 ,n . (As one sees from Figure 10 and Figure 12, the order of R p n − and R q may beinterchanged.) Putting these together yields the desired formula. ENERALIZED LANTERN RELATIONS AND PLANAR LINE ARRANGEMENTS 13
References
1. B. Artal, J. Ruber, J. Cogolludo, and M. Marco,
Topology and combinatorics of real linearrangements , Comp. Math. (2005), no. 6, 1578–1588.2. J. Birman,
Mapping class groups of surfaces , Braids (Santa Cruz, CA, 1986), vol. 78, Amer.Math. Soc., Providence, RI, 1988.3. D. Cohen and A. Suciu,
Braid monodromy of plane algebraic curves and hyperplane arrange-ments , Comm. Math. Helv. (1997), 285–315.4. R. Cordovil, The fundamental group of the complement of the complexification of a realarrangement of hyperplanes , Adv. App. Math. (1998), 481–498.5. M. Dehn, Die gruppe der abbildungsklassen , Acta Math. (1938), 135–206.6. H. Endo, T. E. Mark, and J. Van Horn-Morris, Monodromy substitutions and rational blow-downs , J. Topol. (2011), no. 1, 227–253.7. P. Erd¨os and R. Steinberg, Three point collinearity , American Mathematical Monthly (1944), no. 3, 169–171.8. B. Farb and D. Margalit, A primer on mapping class groups , Princeton University Press,2011.9. J. Harer,
The second homology group of the mapping class group of an orientable surface ,Invent. Math. (1983), no. 2, 221–239.10. E. Hironaka, Abelian coverings of the complex projective plane branched along configurationsof real lines , Mem. of the A.M.S. (1993).11. D. Johnson,
Homeomophisms of a surface which act trivially on homology , Proc. of the Amer.Math. Soc. (1979), 119–125.12. M. Matsumoto, A simple presentation of mapping class groups in terms of Artin groups ,Sugaku Expositions (2002), no. 2, 223–236.13. B. Moishezon and M. Teicher, Braid group technique in complex geometry. I. Line arrange-ments in C P , Braids (Santa Cruz, CA, 1986), Contemp. Math., vol. 78, Amer. Math. Soc.,Providence, RI, 1988, pp. 425–555.14. P. Orlik and H. Terao, Arrangement of hyperplanes , Grundlehren der math. Wissenschaften,vol. 300, Springer-Verlag, Berlin, 1992.15. G. Rybnikov,
On the fundamental group of the complement of a complex hyperplane arrange-ment. , DIMACS: Technical Report (1994), 33–50.16. E. van Kampen,
On the fundamental group of an algebraic curve , Am. Jour. Math. (1933),255–260.17. B. Wajnryb, A simple presentation for the mapping class group of an orientable surface ,Israel J. Math. (1983), 157–174.18. , Mapping class group of a handlebody , Fund. Math. (1998), 195–228.
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