Hyperelliptic Lefschetz fibrations and the Dirac braid

Hyperelliptic Lefschetz fibrations and the Dirac braid joint work with Seiichi Kamada Hisaaki Endo (Tokyo Institute of Technology) Differential Topology 206 March 20, 206, University of Tsukuba March 20, 206, University of Tsukuba /36

Contents Introduction 2 Hyperelliptic Lefschetz fibrations 3 Chart descriptions 4 An invariant w 5 Proof of invariance March 20, 206, University of Tsukuba 2/36

Introduction 4-manifolds Lefschetz fibrations mapping class groups total space monodromy Elliptic surfaces w/o multiple tori Hyperelliptic Lefschetz fibrations natural generalization In this talk... We define a new invariant w for hyperelliptic Lefschetz fibrations We employ Kamada s chart description to introduce w A detailed proof of invariance of w is given Reference: H. Endo and S. Kamada, Counting Dirac braids and hyperelliptic Lefschetz fibrations, arxiv:508.07687. March 20, 206, University of Tsukuba 3/36

2 Hyperelliptic Lefschetz fibrations Lefschetz fibrations M, B : closed oriented smooth 4-manifold and 2-manifold Σ g : a closed surface of genus g, f : M B : a smooth map Def. : f is a (achiral) Lefschetz fibration (LF) of genus g () B : the set of the critical values of f, f is a fiber bundle with fiber Σ g over B (2) there exists a unique critical point p on F b := f (b), f is written as (z, z 2 ) z z 2 or z z 2 about p and b (3) no fiber contains a (±)-sphere M : total space, B : base space, f : projection, Σ g : fiber, F b : singular fiber (b ) March 20, 206, University of Tsukuba 4/36

Anatomy of an LF F b F b2 F bn F 0 M p p 2 p n f Σ g B b b 2 b n q 0 March 20, 206, University of Tsukuba 5/36

Monodromy Φ : Σ g F 0 := f (b 0 ) : an orientation-preserving diffeomorphism γ : [0, ] B : a loop based at b 0 the pull-back γ f : γ M [0, ] of f is a trivial bundle a natural bundle map φ : [0, ] Σ g M extending Φ ρ : π (B, b 0 ) π (BDiff + Σ g ) = π 0 (Diff + Σ g ) =: M g : [γ] [Φ φ ] : monodromy representation of f w.r.t. Φ γ b ρ c t ± c b 0 Dehn twist The mapping calss group M g of Σ g acts on the right. March 20, 206, University of Tsukuba 6/36

Hurwitz system of LF over S 2 π (S 2, b 0 ) = a,, a n a a n = b 0 a a 2 a n b b 2 b n ρ : a a n = t ε a t ε n an = M g (ε i = ±) (ρ(a ),..., ρ(a n )) = (t ε a,..., t ε n an ) : a Hurwitz system of f Theorem (Kas, cf. Matsumoto) : g 2 { LF f : M B of genus g }/ isomorphism = of LF : { homomorphism ρ : π (B, b 0 ) M g with ρ(a i ) a Dehn twist }/ equivalence : { Hurwitz system (tε B=S 2 a,..., t ε n an ) }/ Hurwitz equiv. & conj. March 20, 206, University of Tsukuba 7/36

Singular fibers and vanishing cycles p c h p c g h F b F 0 F b F 0 non-separating, type I ± separating, type II ± h # = n ± 0 (f) # = n± h (f) March 20, 206, University of Tsukuba 8/36

Hyperelliptic mapping class group ι : Σ g Σ g : an involution of Σ g with 2g + 2 fixed points H g := {φ M g φι = ιφ} : hyperelliptic mapping class group Σ g C C 2 C 3 C 4 C 2g 80 C 2g+ ι S h Σ g C 2 C 2h C 2h+2 C 2g t C H g t ι(c) = t C ι(c) = C ζ i := t Ci, σ h := t Sh (i =,..., 2g +, h =,..., [g/2]) March 20, 206, University of Tsukuba 9/36

Hyperelliptic Lefschetz fibrations f : M B : an LF of genus g Φ : Σ g F 0 := f (b 0 ) : an orientation-preserving diffeomorphism ρ : π (B, b 0 ) M g : monodromy of f w.r.t. Φ Def. : (f, Φ) is a hyperelliptic LF Im ρ H g We can define an isomorphism = H of two hyperelliptic LFs. () (f, Φ) = H (f, Φ ) ρ, ρ is equiv. in H g up to conj. (2) (f, Φ) = H (f, Φ ) f = f as LFs (3) Im ρ = H g and f = f (f, Φ) = H (f, Φ ) We often denote (f, Φ) by f for short. March 20, 206, University of Tsukuba 0/36

3 Chart descriptions G-monodromy representations B : a closed oriented smooth 2-manifold : a finite subset of B, b 0 : a base point of B X : a set, R, S : sets of words in X X G : a group with presentation X R C := (X, R, S) M(B,, b 0 ; C) := {ρ : π (B, b 0 ) G : homomorphism ρ([l]) [s] ( s S) for every meridional loop l} b 0 l b b We can define an equivalence of two such ρ s. March 20, 206, University of Tsukuba /36

Charts Γ : a finite graph in B, edges oriented and labeled with x X Def. : the intersection word w Γ (γ) of a simple path γ w.r.t. Γ γ x y x z y Def. : Γ is a C-chart in B Γ saitisfies () and (2): () vertices of Γ : white vertices and black vertices (2) for each white vertex v, w Γ (m v ) R R ; for each black vertex v, w Γ (m v ) S m v : a (counterclockwise) meridian loop of v v : white vertex of type r w Γ (m v ) = r R v : black vertex of type s w Γ (m v ) = s S w Γ (γ) = xy x zy March 20, 206, University of Tsukuba 2/36

Charts and monodromies Γ : a C-chart in B with base point b 0 Γ : the set of black vertices of Γ Def. : the homomorphism determined by Γ ρ Γ : π (B Γ, b 0 ) G : [η] [w Γ (η)] Theorem (Kamada, Hasegawa) : For any ρ M(B,, b 0 ; C), there exists a C-chart Γ with ρ Γ =ρ. Theorem (Kamada, Hasegawa) : M(B,, b 0 ; C)/ equivalence of G-monodromies : { C-charts Γ in B }/ chart moves We use the terminology of chart description in Kamada s paper: S. Kamada, Topology Appl. 54 (2007) March 20, 206, University of Tsukuba 3/36

An example of C-chart Γ G := B 4, C := (X, R, S), X := S := {σ, σ 2, σ 3 }, R := {σ σ 3 σ σ 3, σ σ 2 σ σ 2 σ σ 2, σ 2σ 3 σ 2 σ 3 σ 2 σ 3 } b 0 γ 4 2 γ 2 γ γ 3 3 3 2 2 3 2 w Γ (γ ) = σ σ 2 σ σ 2 σ, w Γ (γ 2 ) = σ σ 3σ,... etc. March 20, 206, University of Tsukuba 4/36

Chart moves Γ, Γ : C-charts in B, b 0 : a base point of B Def. : Γ is obtained from Γ by chart move of type W Γ (B Int D) = Γ (B Int D) both Γ D and Γ D have no black vertices for a disk D embedded in B {b 0 } x x x x x empty r r March 20, 206, University of Tsukuba 5/36

Def. : Γ is obtained from Γ by chart move of transition Γ is obtained from Γ by a local replacement depicted below s s w s T w where s, s S, w X X, and s and wsw determine the same element of G the box labeled T is filled only by edges and white vertices March 20, 206, University of Tsukuba 6/36

Def. : Γ is obtained from Γ by chart move of conjugacy type Γ is obtained from Γ by a local replacement depicted below b 0 b 0 x b 0 b 0 x Def. : chart moves for C-charts the following four kinds of moves: chart moves of type W chart moves of transition chart moves of conjugacy type sending by orientation preserving diffeomorphisms of B March 20, 206, University of Tsukuba 7/36

4 An invariant w Three explicit Cs ➊ C for M 0,2g+2 G = M 0,2g+2 : the mapping class group of S 2 with 2g + 2 points C 0 := (X 0, R 0, S 0 ), X 0 := {ξ, ξ 2,..., ξ 2g+ }, R 0 := {r (i, j) i j > } {r 2 (i) i =,..., 2g} {r 3, r 4 }, S 0 := {l 0 (i) ± i =,..., 2g + } {l ± h h =,..., [g/2]} r (i, j) := ξ i ξ j ξ i ξ r 2 (i) := ξ i ξ i+ ξ i ξ j ( i j > ), (i =,..., 2g), i+ ξ i ξ i+ r 3 := ξ ξ 2 ξ 2g+ ξ 2g+ ξ 2 ξ, r 4 := (ξ ξ 2 ξ 2g+ ) 2g+2, l 0 (i) := ξ i (i =,..., 2g + ), l h := (ξ ξ 2 ξ 2h ) 4h+2 (h =,..., [g/2]). M 0,2g+2 = X 0 R 0 (Magnus) March 20, 206, University of Tsukuba 8/36

Vertices of types l 0 (i) ±, r (i, j), r 2 (i) i i j i i j = j i i j i i+ i+ i i i+ Vertices of types r 3, r 4, l h 2g + 2g + 2g + 2g + 2g + 2h 2h 2h March 20, 206, University of Tsukuba 9/36

Three explicit Cs ➋ C for B 2g+2 (S 2 ) G = B 2g+2 (S 2 ) : the braid group of S 2 with 2g + 2 strands C := ( X, R, S), X := {x, x 2,..., x 2g+ }, R := { r (i, j) i j > } { r 2 (i) i =,..., 2g} { r 3 }, S := { l 0 (i) ± i =,..., 2g + } { l ± h r (i, j) := x i x j x i x j ( i j > ), r 2 (i) := x i x i+ x i x i+ x i x i+ (i =,..., 2g), r 3 := x x 2 x 2g+ x 2g+ x 2 x, l 0 (i) := x i (i =,..., 2g + ), l h := (x x 2 x 2h ) 4h+2 (h =,..., [g/2]). h =,..., [g/2]} B 2g+2 (S 2 ) = X R (Fadell Van Buskirk) Vertices of types l 0 (i) ±, r (i, j), r 2 (i), r 3, l h in C charts are similar to those of types l 0 (i) ±, r (i, j), r 2 (i), r 3, l h in C 0 charts March 20, 206, University of Tsukuba 20/36

Three explicit Cs ➌ C for H g G = H g : the hyperelliptic mapping class group of Σ g Ĉ := ( ˆX, ˆR, Ŝ), ˆX := {ζ, ζ 2,..., ζ 2g+ }, ˆR := {ˆr (i, j) i j > } {ˆr 2 (i), ˆr 5 (i) i =,.., 2g} {ˆr 3, ˆr 4 }, Ŝ := {ˆl 0 (i) ± i =,..., 2g + } {ˆl ± h h =,..., [g/2]} ˆr (i, j) := ζ i ζ j ζ i ζ j ( i j > ), ˆr 2 (i) := ζ i ζ i+ ζ i ζ i+ ζ i ζ i+ (i =,..., 2g), ˆr 3 := (ζ ζ 2 ζ 2g+ ζ 2g+ ζ 2 ζ ) 2, ˆr 4 := (ζ ζ 2 ζ 2g+ ) 2g+2, ˆr 5 (i) := [ζ i, ζ ζ 2 ζ 2g+ ζ 2g+ ζ 2 ζ ] (i =,..., 2g + ), ˆl 0 (i) := ζ i (i =,..., 2g + ), ˆl h := (ζ ζ 2 ζ 2h ) 4h+2 (h =,..., [g/2]). H g = ˆX ˆR (Birman Hilden) March 20, 206, University of Tsukuba 2/36

Vertices of types ˆr 3, ˆr 5 (i) 2g + 2g + 2g + 2g + 2g + 2g + i i 2g + 2g + Vertices of types ˆl 0 (i) ±, ˆr (i, j), ˆr 2 (i), ˆr 4, ˆl h in Ĉ charts are similar to those of types l 0 (i) ±, r (i, j), r 2 (i), r 4, l h in C 0 charts Z 2 B 2g+2 (S 2 ) M 0,2g+2 : central extension Z 2 H g π M 0,2g+2 : central extension March 20, 206, University of Tsukuba 22/36

Charts and Hyperelliptic Lefschetz fibrations Γ : a Ĉ-chart in B with base point b 0 Γ : the set of black vertices of Γ ρ Γ : π (B Γ, b 0 ) H g : the homomorphism determined by Γ Propositon (Kamada E.) : (f, Φ) : hyperelliptic LF of genus g over B, g 2 ρ : π (B, b 0 ) H g : a monodromy representation of (f, Φ) = there exists a Ĉ-chart Γ which satisfies ρ Γ = ρ. Theorem (Kamada E.) : g 2 { hyperelliptic LFs (f, Φ) of genus g over B}/ = H : { monodromies ρ : π (B ) H g }/ equivalence for H g : { Ĉ-charts Γ }/ chart moves March 20, 206, University of Tsukuba 23/36

An invariant w (f, Φ) : hyperelliptic LF of genus g over B ρ : π (B, b 0 ) H g : a monodromy representation of f ˆΓ : a Ĉ-chart in B which satisfies ρˆγ = ρ. Def. : w(ˆγ) : # of white vertices of type ˆr ± 4 included in ˆΓ mod 2 w(f, Φ) := w(ˆγ) Theorem (Kamada E.): w is invariant under chart moves for odd g Γ : a C 0 -chart in B with base point b 0 Def. : w(γ) : # of white vertices of type r ± 4 included in Γ mod 2 (f, Φ), ρ, ˆΓ : as above We can construct a C 0 -chart Γ by changing ˆΓ locally Γ corresponds to ρ 0 := π ρ : π (B, b 0 ) M 0,2g+2 w(γ) = w(ˆγ) in Z 2 March 20, 206, University of Tsukuba 24/36

Examples Hurwitz system (for odd g) W := (ζ,..., ζ 2g+, ζ 2g+,..., ζ ) g+ W := (ζ,..., ζ 2g+ ) 2g+2 We obtain hyperelliptic LFs f, f of genus g over S 2 Both f and f have 2(g + )(2g + ) non-separating fibers, and they do not have separating fibers they have the same Euler characteristic and the same signature Drawing Ĉ-charts corresponding to f and f, we obtain w(f) = 0 and w(f ) = f and f are not isomorphic. By a theorem of Usher, the total spaces of f and f are homeomorphic but not diffeomorphic to each other. March 20, 206, University of Tsukuba 25/36

5 Proof of invariance We will show the invariance of w for C 0 -charts. It is obvious that w is invariant under chart moves of conjugacy and orientation preserving diffeomorphisms It suffices to show the invariance under chart moves of type W and chart moves of transition Invariance under chart move of type W Prop. : w is invariant under chart moves of type W We need a lemma. March 20, 206, University of Tsukuba 26/36

Lem. : For a C 0 -chart below, the box labeled T k can be filled only with edges and white vertices of types r (i, j) ±, r 2 (i) ±, r ± 3. 2g + k 2g + 2g + 2g + T k k 2g + 2g + Proof : The Dirac braid := (x x 2 x 2g+ ) 2g+2 is included in the center of of B 2g+2 (S 2 ) We can fill the box labeled T k with C-chart Change all labels x i of edges into ξ i to obtain a C 0 -chart March 20, 206, University of Tsukuba 27/36

Proof of Prop. : It suffices to show that w(γ) = 0 for every C 0 -chart Γ in S 2 without black vertices. Consider the chart move of type W below k k 2g + 2g + 2g + r 4 2g + T k r 4 2g + 2g + k By Lemma, the box labeled T k can be filled only with edges and white vertices of types r (i, j) ±, r 2 (i) ±, r ± 3. March 20, 206, University of Tsukuba 28/36

Use this move repeatedly to obtain Γ below from Γ Θ r ± 4 r ± 4 Γ Θ 2 r ε 4 r ε 4 Γ 2 Cancel pairs of white vertices of type r ± 4 to obtain Γ 2 from Γ n := # of white vertices of type r ε 4 in Γ 2 Θ 2 Γ 3 w Replace all white vertices of type r4 ε l 0 (i) ε to obtain Γ 3 from Γ 2 with black vertices of type the box labeled Θ, Θ 2 is filled only with edges and white vertices of types r (i, j) ±, r 2 (i) ±, r ± 3. March 20, 206, University of Tsukuba 29/36

Change all labels ξ i of edges into x i to obtain a C-chart Γ 3 The intersection word w is εn = (x x 2 x 2g+ ) ε(2g+2)n, which represents of B 2g+2 (S 2 ) n must be even because represents an element of order two Thus we have w(γ) = w(γ ) = w(γ 2 ) = n = 0 and this completes the proof. March 20, 206, University of Tsukuba 30/36

Invariance under chart move of transition Prop. : w is invariant under chart moves of transition if g is odd We divide chart moves of transition for C 0 -charts into two cases. ➊ s, s {l 0 (i) ± i =,..., 2g + } (l 0 (i) := ξ i ) ➋ s, s {l ± h h =,..., [g/2]} (l h := (ξ ξ 2 ξ 2h ) 4h+2 ) s s w s T w We will give a proof of Prop. only for Case ➋. The proof for Case ➊ is omitted. March 20, 206, University of Tsukuba 3/36

Proof for Case ➋ : Assume s = s = (ξ ξ 2 ξ 2h ) 4h+2. φ := [w], τ h := [(ξ ξ 2 ξ 2h ) 4h+2 ] M 0,2g+2 a relation φτ h φ = τ h in M 0,2g+2 from the chart above pr : Σ g Σ g /I = S 2 : the projection a i := pr(c i ), b h := pr(s h ) F : S 2 S 2 : an orientation preserving diffeo. representing φ (b h )F is isotopic to b h We can assume F fixes b h pointwise because g is odd b h a a 2h a 2h+2 a 2g+ φ is represented by a word w in ξ ±,..., ξ± 2h, ξ± 2h+2,..., ξ± 2g+ March 20, 206, University of Tsukuba 32/36

We divide the box labeled T into three boxes labeled T, Θ, Θ w w Θ w l 2 (h) T = l 2 (h) T w Θ w w The box labeled Θ is filled only with edges and white vertices The box labeled Θ is the mirror image of Θ Since w is a word in ξ ±,..., ξ± 2h, ξ± 2h+2,..., ξ± 2g+, the box labeled T is filled with copies of (a) and (b) below March 20, 206, University of Tsukuba 33/36

Ω i+ Ω 2h Ω Ω 2 Ω i (a) 2h 2h 2h j j (b) 2h 2h 2h (a) corresponds to ξ i (i =,..., 2h) 2h k + k Ω k = k k 2 (b) corresponds to ξ j (j = 2h + 2,..., 2g + ) Ω k for k = 2,..., 2h is depicted above Ω is depicted below k k 2h k + k k k 2 March 20, 206, University of Tsukuba 34/36

2h 2h 2h 2 2h 3 3 2 Ω = 2h 2h 2 2h 2 4 3 2 2h 2h 2 3 2 2h 2h 2h 2 2h 3 3 2 2h 2h 2h 2 4 3 2 The box labeled T is filled only with edges and white vertices of types r (i, j) ± and r 2 (i) ± March 20, 206, University of Tsukuba 35/36

# of white vertices of type r ± 4 included in the box labeled Θ is equal to that for the box labeled Θ The box labeled T is filled with edges, white vertices of types r (i, j) ±, r 2 (i) ±, r ± 3, and an even number of white vertices of types r ± 4 By virtue of the invariance under chart moves of type W, any subchart filling the box labeled T has this property We thus proved our main theorem. Owari The speaker and his collaborator were partially supported by JSPS KAKENHI: Grant Numbers 2540079, 25400082 for H. Endo Grant Numbers 234005, 2628703 for S. Kamada March 20, 206, University of Tsukuba 36/36