A real open book not fillable by a real Lefschetz fibration

0.36 MB·English

by  Ferit Ozturk

A REAL OPEN BOOK NOT FILLABLE BY A REAL
LEFSCHETZ FIBRATION
FERI˙TO¨ZTU¨RKANDNERMI˙NSALEPCI˙
3
1
Abstract. A real 3- or 4-manifold has by definition an orientation preserv-
0
ing smooth involution acting on it. We consider Lefschetz fibrations of 4-
2
dimensionalmanifolds-with-boundaryandopenbookdecompositionsontheir

n boundary in the existence of a real structure. We prove that there is a real
a open book which cannot be filled by a real Lefschetz fibration, although it is
J filledbynon-realLefschetzfibrations.

2
2

]
T 1. Introduction and basic definitions
G
There is a very close relation between open book decompositions and Lefschetz
.
h fibrations over D2, with fibers surfaces with boundary. It is known that any open
t
a book decomposition is filled by a Lefschetz fibration if its monodromy can be fac-
m torized as a product of positive Dehn twists (work of Eliashberg, Giroux, Gompf,
[ Loi-Piergallini, Akbulut-Ozbagci etc. See e.g. [4]). Moreover, two fibrations filling

the same open book are isomorphic if and only if the two such factorizations are
1
Hurwitz equivalent [3].
v
7 In the present work we consider Lefschetz fibrations and open book decomposi-
4 tionsonmanifoldswitharealstructure. Areal structureonanoriented4-manifold
1 (respectively3-manifold)withboundary(possiblyempty)isdefinedtobeaninvolu-
5
tionwhichisorientationpreservingandthefixedpointsetofwhichisofdimension
.
1 2 (respectively 1), if it is not empty. Hence, if M is the oriented boundary of an
0 oriented 4-manifold X, a real structure on X restricts to a real structure on M.
3
We call a manifold X together with a real structure a real manifold and denote by
1
: cX the real structure on it.
v
Let (L,π) be an open book decomposition of a real 3-manifold (M,c ). Here
M
i
X L⊂M isthebindingandπ :M−L→S1isafibrationwithfibersthepagesurfaces.
r We say that (L,π) is a real open book decomposition, if ρ◦π =π◦cM|M−L where
a ρ:S1 →S1 is a reflection [6].
Let p : E → B be a genus-g Lefschetz fibration of a 4-manifold E. A real
Lefschetz fibration is a Lefschetz fibration together with a pair of real structures
c :E →E and c :B →B commuting with the fiber structure [8].
E B
Inthisarticleweansweranaturalquestionthatrelatesrealopenbooksandreal
Lefschetzfibrations. WeshowthatthereisaLefschetzfibrationthatdoesnotadmit
a real structure, while the canonical open book on its boundary is real. We also
show that this real open book cannot be filled by any real Lefschetz fibration with
the same fiber topology and with arbitrary number of singular fibers (Theorem 6).
Acknowledgements. The second author would like to thank A. Degtyarev for
helpfuldiscussions. TheworkinthisarticleissupportedbytheScientificandTech-
nological Research Council of Turkey [TUBITAK-ARDEB-109T671]. The second
1
2 FERI˙TO¨ZTU¨RKANDNERMI˙NSALEPCI˙
authorissupportedbytheEuropeanCommunity’sSeventhFrameworkProgramme
([FP7/2007-2013] [FP7/2007-2011]) under grant agreement no [258204].
2. Construction
Let E be an oriented smooth 4-manifold and B an oriented smooth surface. A
genus-g Lefschetz fibration of E is a proper smooth projection p : E → B such
that p hasonly finitelymany criticalpointsinint(E)with pairwisedistinctimages
around which one can choose complex charts such that the projection takes the
form (z ,z )→z2+z2. Moreover, the inverse image of a regular value is a closed
1 2 1 2
oriented smooth surface of genus g. It follows from the definition that if ∂B (cid:54)= ∅,
then ∂E = p−1(∂B) is a fiber bundle over ∂B. The notion of Lefschetz fibration
canbeslightlygeneralizedtocoverthecaseoffiberswithboundary. ThenE turns
into a manifold with corners and its boundary, ∂E, becomes naturally divided into
two parts: p−1(∂B) and ∂F ×B. In this case, if, in particular, B = D2, then ∂E
admits a canonical open book decomposition with binding p−1(0) and projection
p| . A real Lefschetz fibration is a fibration together with a pair of real structures
∂E
c :E →E and c :B →B commuting with the fiber structure.
E B
It is known that around a singular fiber, Lefschetz fibrations are determined by
the monodromy that is a single positive Dehn twist around a simple closed curve,
thevanishingcycle. Algebraically,LefschetzfibrationsoverB =D2 canbeencoded
by the factorization (t ,...,t ) of the monodromy f =t ◦...◦t (because of
a1 an an a1
thecompositionnotationtheorderisreversed)along∂D2 intoaproductofpositive
DehntwistsconsidereduptoHurwitzequivalence. Namely,twosuchfactorizations
are called Hurwitz equivalent if one can get from one factorization to the other by
a finite sequence of Hurwitz moves:
(...,t ,t ,...)→(...,t−1◦t ◦t ,t ,...),
ai ai+1 ai ai+1 ai ai
(...,t ,t ,...)→(...,t ,t ◦t ◦t−1 ,...);
ai ai+1 ai+1 ai+1 ai ai+1
and possibly a global conjugation.
In what follows, we consider a genus-1 Lefschetz fibration p : X → D2 with
exactly two singular fibers. Choose a base point d ∈ ∂D2 and consider a basis
(γ ,γ ) of π (D2\{critical values},d) obtained by connecting the base point d to
1 2 1
the positively oriented simple loops, each surrounding the corresponding critical
value once. Fix an identification of F = p−1(d) with T2 = R2/Z2. Denote by
d
a, the class on T2 of (1,0) ∈ R2 and by b, the class of (0,1) and consider the
curves u,v represented respectively by 3a+5b and a on T2. We assume that the
monodromy along γ and γ are given respectively by the positive Dehn twists t
1 2 u
and t . In other words, the corresponding singular fibers are obtained from T2 by
v
pinching the curves u and v. The total monodromy of the fibration, thus, becomes
the composition f =t ◦t .
v u
Recall that f (cid:55)→ f defines an isomorphism from the mapping class group
∗
Map(T2) (the identity component of the space of diffeomorphisms) of the torus
to the group of automorphisms of H (T2,Z) ∼= Za⊕Zb. The latter is isomorphic
1
to
(cid:26) (cid:20) (cid:21) (cid:20) (cid:21) (cid:27)
1 1 1 0 [t ][t ][t ]=[t ][t ][t ]
SL(2,Z)= [t ]= ,[t ]= : a b a b a b
a 0 1 b −1 1 ([t ][t ])6 =id
a b
A REAL OPEN BOOK NOT FILLABLE BY A REAL LEFSCHETZ FIBRATION 3
where[t ]referstothematrixrepresentationoftheautomorphismt . Withrespect
◦ ◦∗
(cid:20) (cid:21)
−39 25
to the above presentation, [f]=[t ][t ] is given by the matrix .
v u −25 16
Proposition 1. p:X →D2 does not admit a real structure.
Proof: Suppose that p : X → D2 admits a real structure. That is to say, there
exist real structures c : X → X and c : D2 → D2 such that p◦c = c ◦
X D2 X D2
p. Therefore, the critical points as well as their images are invariant under the
action of the real structures c and c , respectively, so either they are both real
X D2
or they are interchanged by the real structures. By definition of real Lefschetz
fibrations, the decomposition (t ,t ) associated to (γ ,γ ) is Hurwitz equivalent
u v 1 2
to a decomposition (t ,t ) associated to (c (γ ),c (γ )). Although, the
cX(v) cX(u) D2 2 D2 1
positions of (γ ,γ ) can be arbitrary with respect to the real structure c , by
1 2 D2
conjugation by a power of f (such a conjugation preserves Hurwitz classes) and by
moving, if necessary, the base point on the boundary, we can assume that we have
only the two positions shown in Figure 1. In the former case (the case shown on
the left in Figure 1), the fiber F can be endowed with a real structure c by pulling
d
a real structure on a real fiber between the two real singular fibers. Up to isotopy,
we can assume that vanishing cycles u, v are invariant under the action of the real
structure c. Lemma 2 concerns the intersection number of invariant curves and
prohibits this case, (since the intersection number of u and v is equal to 5).
d
γ γ
1 2
γ
2 x
d
x x
γ
1 x
Figure 1. Twopossiblepositionsof(γ ,γ )onbaseD2 regarding
1 2
the real structure reflection with respect to horizontal.
Now assume that critical points are exchanged by the real structure (this is the
case depicted on the right in Figure 1). Let c : T2 → T2 be the real structure
conjugate to the inherited real structure on the base fiber F . Then, by the as-
d
sumption we have c(u) = v; as a consequence, u+v and u−v are elements of ±
eigenspaces Hc and, respectively, Hc of c : H (T2,Z) → H (T2,Z). Thus, the
+ − ∗ 1 1
primitive forms of the classes of u+v and u−v form a basis, respectively of Hc
+
and Hc. Moreover, for any real structure c, the intersection number of the bases
−
of Hc and Hc is at most 2 which is violated by our choice of u and v. (cid:3)
+ −
Lemma 2. The number of intersection points of two invariant curves on a real
torus can only be 0, 1 or 2.
Proof: The real structure restricted to an invariant curve defines an action on the
curve. This action can be the identity, a reflection or an antipodal involution;
let us call those curves a real curve, a reflection curve and an antipodal curve
4 FERI˙TO¨ZTU¨RKANDNERMI˙NSALEPCI˙
respectively. Recall that up to equivariant diffeomorphisms, the real structure on
a torus are distinguished by the number of real components that can be 0,1 or 2.
for each real structure we have
• if c has no real components, then there exist two c-equivariant isotopy
classes of antipodal curves and no classes of other types;
• if c has one real component, then T2 contains a unique c-equivariant iso-
topy class of non-contractible reflection curves, a unique class of antipodal
curves, and a unique real curve;
• ifchastworealcomponents,thenT2containsnoc-equivariantisotopyclass
of antipodal curves, a unique class of reflection curves, and two classes of
real curves.
Theresultfollowsfromthefollowingobservations: areflectioncurvehastworeal
points, so the intersection with a real curve must be at 0, 1 or 2 points. Moreover,
areflectioncurvemustintersectanantipodalcurveat0or2points,andrealcurves
and antipodal curves are disjoint. (cid:3)
Proposition 3. Let M =∂X; then M as a surface bundle is real.
Proof: By [7, Proposition 4], it is enough to check whether its monodromy is real,
i.e. itadmitsadecompositionintoaproductoftworealstructures. Realstructures
on T2 correspond to involutive elements of determinant −1 in GL(2,Z). Here we
(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
−39 25 5 −3 −120 77
have [f] = = , a product of two involutive
−25 16 8 −5 −187 120
elements of determinant −1. (cid:3)
Remark 1. In [7], an explicit classification of real elements of SL(2,Z) as well as a
particulardecompositionofaproductoftworealstructuresforeachconjugacyclass
isgiven. Accordingto[7,Proposition4],ahyperbolicelement,i.e. anelementwith
absolute value of trace greater than 2, of SL(2,Z) is real if and only if its cutting
periodcycle [a ,...,a ]isodd-bipalindromic. Forhyperbolicmatrices,thecutting
1 2k
periodcycletogetherwiththesignofthetraceisacompleteinvariantofconjugacy
classesanditcanberecoveredfromthetraceandtheperiodictailofthecontinued
fraction expansion of the slope of the eigenvectors. In our specific example the
cutting period cycle of f is found to be [1313] which is odd-bipalindromic, i.e. up
to cyclic ordering, it has two palindromic pieces 1, 313 of odd length.
So far we have constructed a T2-fibered Lefschetz fibration cannot be real while
the T2 bundle at its boundary is real. Now we show that a similar construction
can be cooked up to obtain a real open book at the boundary.
Denote by p˘ : X˘ → D2 the Lefschetz fibration with boundary obtained from
p : X → D2 above by taking out a neighborhood of a section. Note that such a
section always exists for genus 1 Lefschetz fibrations [5]. The regular fibers of p˘
aretoriwithoneboundarycomponent. Thefibrationp˘hasnorealstructure, since
otherwise p would have one. However, we have:
Proposition 4. The canonical open book of M˘ =∂X˘ admits a real structure.
Proof: AsaconsequenceofLemma1in[6]itisenoughtoshowthatthemonodromy
ofp˘| isreal. Notethatthemonodromyofp˘| isanelementofMap(T2(cid:114)intD2,∂),
M˘ M˘
A REAL OPEN BOOK NOT FILLABLE BY A REAL LEFSCHETZ FIBRATION 5
thegroupofrelativeisotopyclassesoforientationpreservingdiffeomorphismswhich
are identity on the boundary.
The crucial observation is that being real in Map(T2) is equivalent to being real
in Map(T2(cid:114)intD2,∂). Namely, there is a short exact sequence
0→(cid:104)t (cid:105)→Map(T2(cid:114)intD2,∂)→Map(T2)→0
∂
given by the central extension of Map(T2), where t is the Dehn twist along the
∂
boundarycomponent. Obviously,imagesofrealelementsofMap(T2(cid:114)intD2,∂)are
real. For the converse, suppose im[f] = [c][c(cid:48)] for some [f] ∈ Map(T2 (cid:114)intD2,∂)
andrealstructures[c],[c(cid:48)]∈GL(2,Z). Ifnecessary,byreplacingcbyc◦g andc(cid:48) by
g−1◦c(cid:48) forsomediffeomorphismg,wecanassumethatcandhencec(cid:48) leaves∂T2 in-
variant. Wehavet =c(cid:48)t−1c(cid:48). Thust−1cisarealstructurewhichwedenotebyc(cid:48)(cid:48).
∂ ∂ ∂
As[f]=[c][c(cid:48)][t ]k,thenforsomeintegerk,[f]=[c][c(cid:48)][c(cid:48)◦c(cid:48)(cid:48)]k =[c][c(cid:48)(cid:48)◦(c(cid:48)◦c(cid:48)(cid:48))k−1].
∂
Being conjugate to a real structure, c(cid:48)(cid:48)◦(c(cid:48)◦c(cid:48)(cid:48))k−1 is a real structure. Thus, [f] is
real. (cid:3)
Withsimilarhands-onapproachastheoneabove,weprovethefollowingfurther
result.
Proposition 5. Up to isomorphism preserving the identification T2 → F , there
d
areexactlytwoLefschetzfibrationswithexactlytwosingularfibersfillingthecanon-
ical open book on M˘; moreover, neither of the fibrations admit a real structure.
Proof: Let u(cid:48),v(cid:48) be two simple curves on T2 such that tv(cid:48) ◦tu(cid:48) = f. We have
(cid:20) (cid:21)
−39 25
[f] = [t ][t ] = . Suppose v(cid:48) is the curve defined by αa+βb, then
v u −25 16
(cid:20) (cid:21)
1−αβ α2
a simple calculation gives [tv(cid:48)] = −β2 1+αβ . All Dehn twists are repre-
sented by matrices whose traces have absolute value equal to 2. Therefore, from
[tu(cid:48)] = [t−v(cid:48)1f], we get the identity 25α2+25β2−55αβ = 25. This quadratic Dio-
phantineequationhassolutions(±1,0),(0,±1)and(thetransposeof)thesevectors
(cid:20) (cid:21)
−3 5
multiplied from left by the powers of the matrix S = (see e.g the step-
−5 8
by-step computation of the online application [2]). Note that u and v are in the
solution set. Note also that u(cid:48),v(cid:48), the curves with class b and 5a+8b respectively,
are in the solution set too. The pairs (tu,tv) and (tu(cid:48),tv(cid:48)) are not Hurwitz equiva-
lent. Indeed in that case there would be an invertible matrix K with determinant
1, such that either
K−1[t−u1][tv][tu]K =[tu(cid:48)] and K−1[tu]K =[tv(cid:48)]
or
K−1[t−v1][tu][tv]K =[tu(cid:48)] and K−1[tv]K =[tv(cid:48)].
However, by straightforward calculation one can show that there exists no such K.
Note also that the matrices S and [f] commute; hence for any pair (x,y) such
that f = t ◦t , the factorization (Skx,Sky) is Hurwitz equivalent (with a fixed
y x
identification) to the factorization (x,y) for every k ∈Z.
As a consequence, we have two Hurwitz equivalence classes given by the pairs
(tu,tv)and(tu(cid:48),tv(cid:48)). Tofinish,notethatthenumberofintersectionpointsofu(cid:48) and
v(cid:48) is 5, so the proof of Proposition 1 applies to (u(cid:48),v(cid:48)) to show that the fibrations
6 FERI˙TO¨ZTU¨RKANDNERMI˙NSALEPCI˙
defined by the decomposition (tu(cid:48),tv(cid:48)) is not real. (cid:3)
Theorem 6. The canonical open book on M˘ cannot be filled by any real Lefschetz
fibration with the same fiber topology and with arbitrary number of singular fibers.
Proof: We will show that M˘ cannot be filled by a (real or non-real) Lefschetz
fibration with the number of singular fibers different from 2. Recall that
Map(T2(cid:114)intD2,∂)={[t ],[t ]:[t ][t ][t ]=[t ][t ][t ]}.
a b a b a b a b
Therefore, the homomorphism
deg : Map(T2(cid:114)intD2,∂) → Z
t ,t (cid:55)→ 1
a b
is well-defined; hence the number of Dehn twists that may constitute a given ele-
menthequalsdeg(h). Inourcase,deg(f)=2,soitcannotadmitafactorizationas
a product of an arbitrary number of positive Dehn twists other than 2. Moreover,
all possible factorizations into a product of two Dehn twists are shown above to be
non-real. (cid:3)
Remark 2. Elements admitting a factorization into a product of two Dehn twists
will be studied in [1]. There the classification of such factorizations up to Hurwitz
equivalence are presented and their relation to real structures are also elaborated.
References
[1] DegtyarevA.,SalepciN.ProductsofpairsofDehntwistsandmaximalrealLefschetzfibra-
tions.NagoyaMath.J.(toappear)
[2] DarioAlpern’stwo-integer-variableequationsolver,http://www.alpertron.com/QUAD:HTM
[3] Kas, A. On the handlebody decomposition associated to a Lefschetz pencil. Pacific J. Math.
89(1980),89-104.
[4] Loi, A.; Piergallini, R. Compact Stein surfaces with boundary as branched covers of B4.
Invent.Math.143(2001),no.2,325348.
[5] Moishezon, B. Complex surfaces and connected sums of complex projective planes. Lecture
notesinMath.603SpringerVerlag(1977).
[6] O¨ztu¨rk,F.,Salepci,N.Realopenbooksandrealcontactstructures,preprint.
[7] Salepci, N. Real elements in the mapping class group of T2, Topology and its Applications
157(2010)2580–2590.
[8] Salepci, N. Real Lefschetz fibrations. Th`ese, Universit´e Louis Pasteur, Strasbourg, 2007.
Universit´eLouisPasteur.InstitutdeRechercheMath´ematiqueAvanc´ee(IRMA),Strasbourg,
2007
Bog˘azic¸iU¨niversitesi,DepartmentofMathematics,TR-34342Bebek,I˙stanbul,Turkey
E-mail address: [email protected]
InstitutCamilleJordan,Universite´LyonI,43,Boulevarddu11Novembre191869622
Villeurbanne Cedex, France
E-mail address: [email protected]