EXAMPLES OF NON-MINIMAL LEFSCHETZ FIBRATIONS OF GENUS 2
SUSUMU HIROSE
1. Introduction
Because of the remarkable works by Donaldson [3] and Gompf [5], Lefschetz fi- brations are the one of the most interesting objects for the study on 4-dimensional topology. We recall the definition:
Definition 1.1. A Lefschetz fibration on an oriented compact smooth 4-manifoldM is a smooth map f : M → S2 which is a submersion everywhere except at finitely many non-degenerate critical points p1, . . . , pr, near whichf identifies in orientation- peserving complex coorinates with the model map (z1, z2)7→z12+z22.
The smooth fibers off are compact oriented surfaces and diffeomorphic each other.
If the genus of the fiber is g, we call M a genus g Lefschetz fibration. In this paper, we always assume that the images q1, . . . , qk of critcal points p1, . . . , pk are distinct.
Under this assumption, each singular fiber is obtained by collapsing a simple closed curve (which we call a vanishing cycle) in the smooth fiber. If its vanishing cycle is non-separating (resp. separating), this singular fiber is called irreducible singular fiber (resp. reducible singular fiber). The monodromy of the fibration around a singular fiber is given by a right-handed Dehn twist about its vanishing cycle. The monodromy is characterised by the homomorphism from the fundamental group of S2 \ {q1, . . . , qk} to the mapping class group Mg. If we fix a system of generators for the fundamental group of S2 \ {q1, . . . , qk}, then the above homomorphism is characterised by the relation t1·t2· · ·tk =idΣg among the right-handed Dehn twist,
This research was partially supported by Grant-in-Aid for Encouragement of Young Scientists (No. 16740038), Ministry of Education, Science, Sports and Culture, Japan.
which we call a positive relation. On the other hand, any positive relation defines a Lefschetz fibration.
A 4-manifold M is called non-minimal if there is an embedded sphere whose self intersection number is−1 (this sphere is called a (−1)-curve). A section for a Lefschetz fibrationf :M →S2is called a (−1)-section, if it is a (-1)-curve. In [8], Yoshihisa Sato made the following geographical table of non-minimal genus-2 Lefschetz fibrations. In
Table 1. Possible pairs (n0, s) as geography b+2 Possible pairs (n0, s) EX
(16,2),(30,0) Type (1,1) b+2 >1 (14,3),(28,1) Type(1)
(12,4),(26,2),(40,0)
(14,3),(28,1) Type(2) b+2 = 1 n0+ 2s = 20, n0 >0, s≥0
n0+ 2s = 10, n0 >0, s≥0
this table,n0is the number of irreducible singular fibers, andsthe number of reducible singular fibers. Type (1) means that there is only one (-1) section, Type (1,1) means that there are two (-1) sections, and Type (2) means that there is one (-1) curve which intersects a fiber in two points. Examples representing (30,0), (40,0), (20,0), (28,1) and (6,2) have already been known. In this paper, we introduce examples for the case where (n0, s) = (16,2), (18,1) and a new example for (28,1).
2. Examples
Let a be a simple closed curve on Σ2, the closed orientable surface of gunus 2. A homeomorphism indicated in Figure 1 is called a right-handed Dehn twist about a.
Lickorish [6] showed that M2, the mapping class group of Σ2, is generated by the
a
Figure 1
right-handed Dehn twists about circles indicated at the left hand side of Figure 2.
The right-handed Dehn twist about i (i= 1,· · · ,5, σ) is denoted by i, for short. For
1
2 3
4
5 σ
Figure 2
elements φ1, φ2 in M2, φ1·φ2 stands for applying φ1 first and then applyingφ2, and sometimes ”·” are dropped. For abbreviation, we use φψ in place of ψ−1·φ·ψ, and φ in place of φ−1. If a product of right-handed Dehn twists i·j· · ·k is isotopic to identity, then the equation i·j· · ·k =idΣ2 is called a positive relation.
Theorem 2.1. The followings are positive relations for M2. (1) (4·321·4·32·1·23 ·4·123·σ)2 =idΣ2.
(2) (5·4·3·2·1·123)3·σ =idΣ2.
(3) (4·3·2·1)3·1·2·345·345·234·123·σ =idΣ2. (4) ((4·3·2·1)2·2·3·4·1·2·3)2σ =idΣ2.
These positive relations define non-minimal genus-2 Lefschetz fibrations. (1) defines one which represents (16,2) and has at least two -1 sections. (2) defines one which represents (18,1) and has at least two -1 sections. (3) defines one which represents (18,1)and has at least one -1 section. (4) defines one which represents (28,1)of Type (1).
Remark2.2. (a) By using the result of [7], we see b+2 =b1+n0+2s5−15. From the above positive relation, we calculate b1 for each Lefschetz fibration, and show that b+2 = 1 for (1) (2) (3), and b+2 = 3 for (4).
(b) Chakiris introduced examples of genus-2 Lefschetz fibrations with reducible sin- gular fibers in [2] (see for [4] for the generalization of these examples). The author does not know the exact relationship between Chakiris’ examples and examples introduced in this paper.
In the next setion, we show the following Lemma.
Lemma 2.3. (1) (4·3·2·1)5 = 4·321·4·32·1·23·4·123·σ.
(2) (4·3·2·1)5 = 1·2·3·4·5·5·4·3·2·1.
(3) (5·4·3·2·1)3 = 345·234·123·σ.
(4) (4·3·2·1)7 = 1·2·345·345·234·123·σ.
(5) 5·4·3·2·1·345= 234·5·4·3·2·1, 5·4·3·2·1·234= 123·5·4·3·2·1.
(6) (4·3·2·1)3·4·321·4·32·1·23·4·123= 2·3·4·1·2·3·(4·3·2·1)2·2·3·4·1·2·3.
Birman and Hilden showed the following Theorem.
Theorem 2.4. [1] M2 is generated by 1,2,3,4,5 and its defining relations are:
(1) i·j =j ·i, if |i−j| ≥2, i, j = 1,2,3,4,5, (2) i·i+ 1·i=i+ 1·i·i+ 1, i= 1,2,3,4, (3) (1·2·3·4·5)6 =idΣ2,
(4) (1·2·3·4·5·5·4·3·2·1)2 =idΣ2,
(5) 1·2·3·4·5·5·4·3·2·1Ài, i= 1,2,3,4,5,
where À means ”commute with”. ¤
We call (1) (2) of the above relations braid relations. The following Lemma follows immediately from (iii) of Lemma 21 in Wajnryb’s paper [9]. These relations and (3) of Theorem 2.4 are calledchain relations.
Lemma 2.5. (2·1)6 =σ, (3·2·1)4 = 52, (4·3·2·1)10=idΣ2.
Due to Lemma 2.3, we check that positive relations in Theorem 2.1 are valid. In the following equations, by the letter settled under ”=”, we indicate which equations in Lemma 2.3 are used .
(1): idΣ2 = (4·3·2·1)10= (4·3·2·1)5·(4·3·2·1)5 =
(1)
(4·321·4·32·1·23·4·123·σ)2. (2): idΣ2 = (5·4·3·2·1)6 = (5·4·3·2·1)3·(5·4·3·2·1)3 =
(3) (5·4·3·2·1)3·345·234·123·σ =
(5)
(5·4·3·2·1·123)3·σ.
(3): idΣ2 = (4·3·2·1)10 = (4·3·2·1)3·(4·3·2·1)7 =
(4)(4·3·2·1)3·1·2·345·345·234·123·σ.
(4): idΣ2 = (4·3·2·1)10 = (4·3·2·1)5·(4·3·2·1)5 =
(1) (4·3·2·1)5·4·321·4·32·1·23·4·123·σ= (4· 3·2·1)2·(4·3·2·1)3·4·321·4·32·1·23·4·123·σ =
(6) (4·3·2·1)22·3·4·1·2·3·(4·3·2·1)2·2·3·4·1·2·3·σ.
In order to check the existence of -1 sections for each examples, we define a kind of mapping class group over the surface with distinguished disks. Let D1,· · · , Dn be disjoint 2-disks in Σ2, andHomeo+(Σ2,fix D1∪ · · · ∪Dn) be the group of orientation preserving homeomorphisms over Σ2 preserving D1 ∪ · · · ∪Dn pointwise. The set of connected components π0(Homeo+(Σ2,fix D1 ∪ · · · ∪Dn)) has a group structure induced from that ofHomeo+(Σ2,fix D1∪ · · · ∪Dn). This group is denoted byM2,n. There is a natural surjection p fromM2,n to M2 defined by forgetting distinguished disks. For an element φ ∈ M2, we choose an element of p−1(φ) and call this a lift of φ by p. Let τ1 ·τ2· · ·τk = idΣ2 be a postitive relation, then this defines a Lefshcetz fibration f :X →S2. If we find liftsτ1′, τ2′,· · ·, τk′ of τ1, τ2,· · · , τk byp:M2,n → M2
so thatτ1′ ·τ2′· · ·τk′ =t∂D1· · ·t∂Dn, then f has n sections with square−1.
D1 D2
1′
σ′ 4′
3′21
3′2
1′23
2′3 Figure 3
(1): Let 1, 4, 321, 32, 23, 123, σ ∈ M2 be lifted to 1′, 4′, 3′21, 3′2, 2′3, 1′23, σ′ ∈ M2,2 by p:M2,2 → M2as indicated in Figure 3, then (4′·3′21·4′·3′2·1′·2′3·4′·1′23·σ′)2 =t∂D1·t∂D2. This equation is checked by observing the action of the left hand side elements on arcs properly embedded in Σ2\(D1 ∪D2). For other relations, we can check in the same method as this relation.
1′ 3′ 5′
2′ 4′
1′23
σ′ D1
D2 Figure 4
(2): Let 1, 2, 3, 4, 5, 123, σ ∈ M2 be lifted to 1′, 2′, 3′, 4′, 5′, 1′23, σ′ ∈ M2,2 by p:M2,2 → M2 as indicated in Figure 4, then (5′·4′·3′·2′·1′·1′23)3·σ′ =t∂D1·t∂D2.
1′
2′ 3′ σ′
4′ D1
3′′45
3′45
2′34 1′23
Figure 5
(3): As indicated in Figure 5, let 1, 2, 3, 4, 234, 123, σ ∈ M2 be lifted to 1′, 2′, 3′, 4′, 2′34, 1′23, σ′ ∈ M2,1 by p : M2,1 → M2, and 345 be lifted to 3′45 and 3′′45. Then (4′·3′·2′·1′)3·1′ ·2′·3′45·3′′45·2′34·1′23·σ′ =t∂D1.
(4): As indicated in Figure 5, let 1, 2, 3, 4,σ ∈ M2 be lifted to 1′, 2′, 3′, 4′,σ′ ∈ M2,1
byp:M2,1 → M2. Then, ((4′·3′·2′·1′)2·2′·3′ ·4′ ·1′·2′·3′)2·σ′ =t∂D1. 3. Proof of Lemma 2.3
In the following equations, we indicate places used braid relation with underlines.
(1): At first, we rewrite (4321)5 into the form, for which we can apply the chain relation (21)6 =σ.
(4321)5 =
= 43214321432143214321 = 43214321432143243121 = 43214321432143423121
= 43214321432134323121 = 43214321432134232121 = 4321432143213423(21)2
= 4321432143234123(21)2 = 4321432142324123(21)2 = 4321432142342123(21)2
= 4321432142341213(21)2 = 4321432142341231(21)2 = 4321432124341231(21)2
= 4321432123431231(21)2 = 4321432123413231(21)2 = 4321432123412321(21)2
= 43214321234123(12)3(21)6 = 43214321234123(12)3σ.
We rewite 43214321234123(12)3 as follows.
43214321234123121212 = 43214321234121321121
= 43214321234212321121 = 43214321234213231121
= 43214321234213211321 = 43214321234231211321
= 43214321234232121321 = 432143212342322·12321
= 432143212342322·12321 = 432143212324322·12321
= 432143213234322·12321 = 432143213243422·12321
= 43214321324322·4·12321 = 43214321342322·4·12321
= 43214321343·232 ·4·12321 = 4321432343·1·232 ·4·12321
= 4321423243·1·232 ·4·12321 = 4321243423·1·232 ·4·12321
= 4·32123 ·4·323 ·1·232 ·4·12321.
Since 32123 = 31213 = 13231 = 12321 = 321, 323 = 232 = 32, 232 = 323 = 23, 12321 = 13231 = 31213 = 32123 = 123, we get the equation which we need.
(2): At first, we show (4321)4 = 1234(321)4.
4321432143214321 = 4321432143421321 = 4321432134321321 = 4321432134(321)2
= 4321432341(321)2 = 4321423241(321)2 = 4321243421(321)2 = 4321234321(321)2
= 4321234(321)3 = 4312134(321)3 = 1432341(321)3 = 1423241(321)3
= 1423421(321)3 = 1243421(321)3 = 1234(321)4.
Because of the chain relation (321)4 = 52, (4321)4 is equal to 123455. Therefore, (4321)5 = 1234554321.
(3): At first, we rewrite (54321)3 in order to apply the chain relation (21)6 =σ.
(54321)3 = 543215432154321 = 543215432543121 = 543215435423121
= 543215453423121 = 543214543423121 = 543214534323121 = 54321453423·(21)2
= 54324534123·(21)2 = 54345234123·(21)2 = 53435234123·(21)2
= 53453234123·(21)2 = 53452324123·(21)2 = 53452342123·(21)2
= 53452341213·(21)2 = 53452341231·(21)2 = 35452341231·(21)2
= 34542341231·(21)2 = 34524341231·(21)2 = 34523431231·(21)2
= 34523413231·(21)2 = 345234123·(21)3 = 345234123(12)3(21)6
= 345234123(12)3σ.
We rewrite 345234123(12)3.
345234123121212 = 345234121321212 = 345234212321121 = 345234213231121
= 345234231211321 = 345234232121321 = 3452342322 ·12321
= 3452343232·12321 = 3452434232·12321 = 3452432432 ·12321
= 3454232432·12321 = 34543 ·23432 ·12321.
Since 34543 = 35453 = 53435 = 54345, 23432 = 24342 = 42324 = 43234, 12321 = 13231 = 31213 = 32123, we obtain the equation which we need.
(4): By virtue of (2), we show (4321)7 = (4321)2 ·(4321)5 = (4321)21234554321. We rewrite the last word as follows.
43214321(1234554321) = 432143(1234554321)21 = 432431(1234554321)21
= 434231(1234554321)21 = 343231(1234554321)21 = 342321(1234554321)21
= 3423(1234554321)2121 = 342312345543(12)3(21)6 = 342312345543(12)3σ.
In the last equation, we used the chain relation (21)6 = σ. Next, we rewrite 342312345543(12)3.
342312345543121212 = 342132345543121212 = 342123245543121212
= 341213455423121212 = 341234554123121212 = 312434554123121212
= 312343554123121212 = 132345534123121212 = 123245534123121212
= 123455234123121212
In the above proof for (3), we showed 234123(12)3 = 43 ·23432 ·12321. We apply this equation to the double underlined part. Then we get,
12345543·23432 ·12321 = 12·(34543)2·23432 ·12321 = 12·(345)2·234·123.
(5): In the following rewriting process, we use equations ab = b·bab = b·ab, ab = aba ·a=ba ·a.
54321345= 543234511 = 543231 451 = 54323451 = 543345221 = 54332 4521 = 543234521
= 5423453321 = 542343 5321 = 5424 345321 = 542345321 = 5234544321 = 5235 454321
= 525 3454321 = 523454321 = 23454321,
54321234 = 543223411 = 543221 341 = 543212341 = 5431234221 = 5431232 421
= 54313 23421 = 543123421 = 5412343321 = 54124 34321 = 5414 234321 = 541234321
= 51234321 = 123554321 = 15 2354321 = 12354321.
(6): In the following rewriting process, we use braid relations and equations ab = b·bab=b·ab,ab=aba·a=ba ·a.
43214321432143214321234123σ = 43214321434213214321234123σ
= 4321432143423214321234123σ = 432143214343214321234123σ
= 432143243431214321234123σ = 432143243432124321234123σ
= 432143243432142321234123σ = 43214324343214321234123σ
= 43421324343214321234123σ = 34323124343214321234123σ
= 34232124343214321234123σ = 34231214343214321234123σ
= 34231243431214321234123σ = 34231243432124321234123σ
= 34231243432142321234123σ = 34231243432143231234123σ
= 34231243432143213234123σ = 3423124343214321234123σ
= 3243412343214321234123σ = 3234312343214321234123σ
= 3234132343214321234123σ = 3234123243214321234123σ
= 3234123423214321234123σ = 3234123432314321234123σ
= 3234123432134321234123σ = 3234123432143421234123σ
= 3234123432143212434123σ = 3234123432143212343123σ
= 323412343214321234123σ= 232412343214321234123σ
= 234212343214321234123σ= 234121343214321234123σ
= 234123431214321234123σ= 234123432124321234123σ
= 234123432142321234123σ= 234123432143231234123σ
= 234123432143213234123σ= 234123(4321)22324123σ
= 234123(4321)22342123σ = 234123(4321)2234123σ.
Acknowledgments. The author would like to express his gratitude to Professors Yoshihisa Sato and Haruko Nishi for helpful discussions and comments, and to Pro- fessor Hisaaki Endo for introducing him the papers [2] and [4].
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Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga, 840–8502 Japan
E-mail address: [email protected]