Lefschetz Fibrations

by

Patrick Naylor

A thesispresented to the University of Waterloo

in fulfillment of thethesis requirement for the degree of

Master of Mathematics

Waterloo, Ontario, Canada, 2016

c� Patrick Naylor 2016

I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis,including any required final revisions, as accepted by my examiners.

I understand that my thesis may be made electronically available to the public.

ii

Acknowledgements

This project certainly would not have been possible without the support and guidanceof many people. First, to the authors of the excellent textbooks from which I learnedmost of this material: Glen Bredon [Topology and Geometry ], Robert Gompf and AndrasStipsicz [4-Manifolds and Kirby Calculus ], and Alexandru Scorpan [The Wild World of 4-

Manifolds ], amongst many others. As well, to Terry Fuller, whose paper in 2003 [LefschetzFibrations of 4-Dimensional Manifolds ] was a truly invaluable source for this project.

I am also indebted to the Pure Mathematics faculty here at the University of Waterloofor the friendly and productive environment that made this project so enjoyable. As well,to my fellow graduate students, who were an endless source of encouragement: TyroneGhaswala, Ehsaan Hossain, and Zack Cramer, for your advice and help with all thingsmathematical. My fellow Master’s students: Sam Kim, Nick Rollick, Kari Eifler, and KyleStolcenberg for your collaboration and company this year.

Of course, I owe a great deal of thanks to my supervisor Doug Park. Your guidance andwisdom have made working on this project a pleasure, and I look forward to working withyou during my Ph.D. I would also like to thank David McKinnon for his insightful andthoughtful comments on this paper.

Lastly, to my family, and friends that I keep as close as family - you all know who you are- I am forever grateful for your love and support.

Thesis Draft

Patrick Naylor

September 12, 2016

Contents

1 Introduction 2

2 Conventions 3

3 Background Material 3

3.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 The Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Fiber Bundles and Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Covering Space Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5 4-Manifold Specifics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Lefshetz Fibrations 12

4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 The Local Topology of Lefschetz Fibrations . . . . . . . . . . . . . . . . . . 144.3 The Homotopy Exact Sequence . . . . . . . . . . . . . . . . . . . . . . . . . 154.4 Mapping Class Groups and Dehn Twists . . . . . . . . . . . . . . . . . . . . 174.5 The Monodromy of Leschetz Fibrations . . . . . . . . . . . . . . . . . . . . 204.6 Lefschetz Fibrations over S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Examples 23

5.1 Classification Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.2 The Mapping Class Group of ⌃1 . . . . . . . . . . . . . . . . . . . . . . . . 235.3 Genus One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 Topological Invariants of Ep1q . . . . . . . . . . . . . . . . . . . . . . . . . . 265.5 A Higher Genus Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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1 Introduction

The subject of 4-manifold topology has long been known to be somewhat different than thatof other dimensions. For instance, one of the most surprising facts about four dimensions isthat given a closed topological 4-manifold M , there may be infinitely many distinct smoothstructures on M . However, in dimensions other than 4, there are only ever finitely manypossible smooth structures. Much of the research done during the 1980’s and 1990’s focusedon obtaining some kind of classification result for 4-manifolds. It was conjectured relativelyearly on that complex manifolds might form building blocks for such a classification, but thisturned out not to be the case. However, another category of manifolds are the symplecticmanifolds: those admitting a closed nondegenerate 2-form. Using a type of gluing operation,the symplectic normal sum, Gompf [7] was able to show that one can produce many newsymplectic manifolds from old, and so it was thought that these manifolds might be theright candidate for classification theorems. Unfortunately this turned out not to be thecase either, but symplectic manifolds remain important class of manifold to study for theirown sake.

One might ask if it’s possible to give a purely topological description of these manifolds.Donaldson and Gompf provided a resolution to this question in two companion theoremswhich roughly state that symplectic manifolds are the ones that admit the structure of aLefschetz fibration [9]. These objects are essentially a generalization of a fiber bundle, butwith finitely many singular fibers. The ultimate goal of this thesis is to give a survey ofLefschetz fibrations, their properties, and some of the basic classification results that existat the time of writing.

In section 3, we will give a fairly brief review of the material required to work with Lefschetzfibrations. We assume a reasonable familiarity with singular homology/cohomology, butpresent some important theorems such as Poincaré duality and the Universal Coefficienttheorem. We’ll also give a review of basic covering space theory and the Euler characteristic.Last, we move to a more specialized review of 4-manifolds, giving a precise description ofthe homology of closed orientable 4-manifolds, and presenting the basics of the intersectionform.

In section 4, we present the Lefschetz fibration, and describe the topology as best wecan. Some of the proof techniques require the relatively advanced machinery of Kirbycalculus and handlebody decompositions, and so in these cases we’ll refer the reader to anappropriate reference, as such machinery is beyond the scope of this project. We’ll showthat such fibrations satisfy a homotopy exact sequence like fiber bundles, and use this tofocus our attention on the relatively nice connected cases.

Last, section 5 will give a survey of some classification results that exist for Lefschetzfibrations. Some of these results are rather technical, but we will include as many proofs

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as is reasonable. In particular, we will have a complete classiciation for the “genus one”Lefschetz fibrations, i.e., when the fibers look like a genus one surface.

2 Conventions

In this document, we’ll assume that all maps are continuous and probably smooth unlessotherwise stated. We’ll also use the following notational conventions:

I will denote the unit interval.

⌃g will denote the genus g surface.

In general, M will denote an m-manifold, X will denote a 4-manifold (or sometimesan arbitrary topological space)

3 Background Material

For the convenience of the reader, we include a review of some of the necessary prerequisitesfor the discussion of Lefschetz fibrations. Some of this material is standard; some of it isless so, but the experienced reader can probably skip this section. We will include manystandard results for convenience, but few proofs.

3.1 Basics

For our purposes, we will assume that all manifolds are connected. By a closed manifold,we mean a compact manifold with no boundary. We will also assume a reasonable grasp ofsingular homology and cohomology. If necessary, an excellent reference to learn the subjectis [1].

We recall that for for any orientable m-manifold M , we have HmpM, BM ;Zq – Z, and somake the usual definition of the fundamental class.

Definition 3.1. An orientation of an m-manifold M is a choice of generator for HmpM, BM ;Zq,which we will call the fundamental class of M , and denote rM s.

The most important tools for dealing with homology and cohomology classes are the cupand cap products, i.e.,

Y : H ipM ;Zq ˆ Hj

pM ;Zq Ñ H i`jpM ;Zq

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X : H ipM ;Zq ˆ HjpM ;Zq Ñ Hj´ipM ;Zq

We also have the Kronecker product, obtained by evaluating a cohomology class at a ho-mology class, which we will write as:

x , y : H ipM ;Zq ˆ HipM ;Zq Ñ Z

The formal development of such products is relatively involved, so we will assume that thereader has some familiarity with them. If not, there is an excellent treatment in [1]. Wewill not have much need to use these products except while stating some more advancedresults, but we’ll give a short list of properties for completeness.

Theorem 3.1. The following properties hold for the cup product whenever they make sense:

(1) Y is natural, i.e., if f : X Ñ Y then f˚p↵ Y �q “ f˚

p↵q Y f˚p�q;

(2) Let 1 P H0pX;Zq denote the class of the augmentation cocycle which takes each 0-

simplex to 1 P Z. Then we have ↵ Y 1 “ 1 Y ↵ “ ↵.

(3) Y is associative, i.e., ↵ Y p� Y �q “ p↵ Y �q Y �.

(4) ↵ Y � “ p´1q

degp↵qdegp�q� Y ↵.

Similar properties hold for the cap product.

One of the most important tools for us will be the Poincaré Duality theorem, as we willdeal almost exclusively with oriented closed 4-manifolds. In later sections, we will expresslyuse this theorem to give a concrete description of the homology and cohomology of suchmanifolds.

Theorem 3.2 (Poincaré Duality). Suppose M is a closed orientable manifold of dimensionm. Then the map

XrM s : HkpM ;Zq Ñ Hm´kpM ;Zq

f fiÑ f X rM s

is an isomorphism for all k. The inverse of this map will be denoted PD : HkpM ;Zq Ñ

Hm´kpM ;Zq.

In fact the theorem holds in any coeffient ring, but we will only be concerned with coefficientsin Z. For completeness of this section, we’ll state another important theorem which lets uscompute the homology of product spaces relatively easily.

Theorem 3.3 (Geometric Künneth Theorem). There is a natural exact sequence:

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0 ›Ñ pH˚pXq b H˚pY qqn ›Ñ HnpX ˆ Y q ›Ñ pH˚pXq ‹ H˚pY qqn´1 ›Ñ 0

which splits (not naturally). Here, rpA˚q ˚ pB˚qsn “

Ài`j“n Tor1pAi, Bjq. The coefficients

can be taken in any PID.

We won’t give any treatment of the Tor functor, as its definition is standard and can befound in any standard algebra book, or [1]. As an easy example, we can compute thehomology of the torus:

Example: Consider the torus T 2“ S1

ˆ S1. Then using the Künneth Theorem, wehave:

H0pT 2q “ H0pS1

q b H0pS1q “ Z b Z

“ ZH1pT 2

q “ rH0pS1q b H1pS1

qs ‘ rH1pS1q b H0pS1

qs ‘ rH0pS1q ˚ H0pS1

qs

“ rZ b Zs ‘ rZ b Zs ‘ rZ ˚ Zs

“ Z ‘ Z

Similarly, we find that H2pT 2q “ Z, and it is easy to see that this formula also gives

HnpT 2q “ 0 for n ° 2.

The last result we will have need of is the Universal Coefficient theorem; it has a few forms,but for us this will be sufficient.

Theorem 3.4 (The Universal Coefficient Theorem). For singular homology/cohomology,the following sequence is split exact:

0 ›Ñ ExtpHn´1pX,A;Zqq ›Ñ HnpX,A;Zq ›Ñ HompHnpX,A;Zq,Zq ›Ñ 0

Again, this holds for more general coefficient groups than Z, but this will be adequate. Fora nice proof of this result, the reader can consult [1][p. 282].

3.2 The Euler Characteristic

The Euler characteristic is a commmonly used numerical invariant one can assign to topo-logical spaces with finite homology.

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Definition 3.2. For any topological space X with finite homology, we define the Eulercharacteristic of X, denoted �pXq to be:

�pXq “

8ÿ

i“0

p´1q

ibi

where bi is the ith Betti number of X, i.e., the free rank of the ith homology of X. Byassumption, only finitely many terms are nonzero, so this sum is finite.

Since homology is invariant under homotopy type, so too is the Euler characteristic. Forsufficiently nice spaces, we can actually compute the Euler characteristic without the ho-mology.

Proposition 3.1. If X is a finite CW-complex, then:

�pXq “

8ÿ

i“0

p´1q

iki

where ki is the number of i-cells in any given decomposition. In particular, �pXq is inde-pendent of the choice of cell decomposition.

Proof. Fix a cell decomposition of X, and consider the associated sequence of abeliangroups:

¨ ¨ ¨ ›Ñ 0Bn`1›››Ñ Cn

Bn›Ñ Cn´1

Bn´1›››Ñ ¨ ¨ ¨

B1݄ C0

B0݄ 0

Set Zi “ ker Bi and Bi “ im Bi`1. Then we have the exact sequences from homology:

0 ›Ñ Zi ›Ñ CiBi›Ñ Bi´1 ›Ñ 0

0 ›Ñ BiBi`1›››Ñ Zi

⇡i›Ñ Hi ›Ñ 0

From the additivity of rank for short exact sequences of abelian groups, we get:

ki “ rankpCiq “ rankpZiq ` rankpBi´1q

rankpZiq “ rankpBiq ` rankpHiq

and so substituting we obtain:

8ÿ

i“0

p´1q

iki “

8ÿ

i“0

p´1q

irrankpHiq ` rankpBiq ` rankpBi´1qs

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“

8ÿ

i“0

p´1q

iprankpHiqq ` rankpB´1q “ �pXq

This lets us easily calculate the Euler characteristic of some nice spaces. For instance,one can easily see that �pSn

q “ 1 ` p´1q

n, �pT 2q “ 0, �pIq “ 1 by using the obvious cell

decompositions. Basic properties of homology show that the Euler characteristic is additiveunder disjoint unions, and one can also show that �pM ˆ Nq “ �pMq�pNq. Another easyresult that we will use later is the formula for connected sums: for two connected closedm-manifolds M and N , we have:

�pM#Nq “ �pMq ` �pNq ´ �pSmq

3.3 Fiber Bundles and Fibrations

We recall the definition and basic properties of fiber bundles, and more generally of fibra-tions.

Definition 3.3. A fiber bundle consists of the following topological spaces: B (the base),E (the total space), F (the fibre), and a continuous surjection ⇡ : E Ñ B. We require thatfor all x P E, there is an open neighborhood U Ä B of ⇡pxq (a local trivialization) suchthat there is a homeomorphism � : ⇡´1

pUq Ñ UˆF , where the following diagram commutes:

⇡´1pUq U ˆ F

U

⇡

�

proj1

We often assume that the base space is path connected. One can easily see that the fibersabove each point must be homeomorphic. One can also define a smooth fiber bundle, inwhich all maps are smooth, and the fibers above any two points are diffeomorphic.

There is another, more general notion called a fibration. While the definition is fairlyabstract, it generalizes the properties of fiber bundles.

Definition 3.4. A map p : Y Ñ B is called a fibration if it satisfies the homotopy liftingproperty for all topological spaces X, that is, the following diagram can always be completed:

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X ˆ t0u Y

X ˆ I B

f0

p

f

f

If it satisfies the homotopy lifting property for all disks Dn, it is called a Serre fibration.

Example: A projection map p : B ˆF Ñ B is a fibration, as one can easily see that givenany maps f : X ˆ t0u Ñ B ˆ F and g : X ˆ I Ñ B, the diagram can be completed withthe map hpx, tq : fpx, tq ˆ qgpx, 0q, where q : B ˆ F Ñ F is the projection in the secondcoordinate. Hence, any trivializable fiber bundle is a fibration; in fact, all fiber bundles are.The following result shows that fibrations are indeed a generalization of fiber bundles, butthe proof is somewhat technical; [1][p. 453] gives a full proof.

Theorem 3.5. For a map p : Y Ñ B, the property of being a Serre fibration is a localproperty in B.

Corollary 3.1. Any fibre bundle is a fibration.

It is easy to prove that the Euler characteristic multiplies over fiber bundles. In fact, withreasonable assumptions the same is true for fibrations. This result is proven in [14][p. 481];we will not do so here.

Theorem 3.6. Suppose p : Y Ñ B is a fibration orientable over a field, with fiber Fand with base B path connected. Then if �pBq,�pF q are defined, so is �pEq, and we have�pEq “ �pBq ¨ �pF q.

One of the most useful properties of fibrations is the induced homotopy exact sequence.

Theorem 3.7. If p : Y Ñ B is a fibration and y0 P Y, b0 “ ppy0q, and F “ p´1pb0q, then

taking y0 as the base point of Y and of F and b0 as the base point of B, we have the exactsequence:

¨ ¨ ¨ ›Ñ ⇡npF q

i7›Ñ ⇡npY q

p#››Ñ ⇡npBq

B7›Ñ ⇡n´1pF q ›Ñ ¨ ¨ ¨

¨ ¨ ¨ ›Ñ ⇡1pY q ›Ñ ⇡1pBq ›Ñ ⇡0pF q ›Ñ ⇡0pF q ›Ñ ⇡0pBq

where i7, p7 are the induced maps on homotopy group, and B7 is the “connecting homomor-phism.” While the 0th homotopy is a pointed set rather than a group, the sequence is stillexact as maps on sets there.

In the interest of moving forward, we will also leave this proof to the reader. One can alsoconsult [1][p. 453] for a proof.

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3.4 Covering Space Theory

Most of this material is standard, but we will have occasional need of some basic results.We’ll summarize the important classical results without proof for the convenience of thereader. For a complete treatment of this subject, one can consult Chapter 3 of Bredon’sexcellent textbook [1].

We recall the following definitions:

Definition 3.5. A covering space of a topological space X is a space X together with acontinuous map p : X Ñ X such that for each x P X, there is an open neighbourhood Usuch that each path component of p´1

pUq is mapped homeomorphically onto U by p.

It is a standard result that the sets p´1ptxuq have the same cardinality for each x P X, and

so we refer to this as the number of sheets of the covering space. In the language of theprevious section, we see that a covering space is just a fiber bundle with a discrete fiber.The importance of covering spaces comes from results that relate the fundamental groupof the base space to that of the cover; for instance we have:

Theorem 3.8. Let pX, pq cover X, x0 P X, and x0 “ ppx0q. Then the induced homomor-phism p˚ : ⇡1pX, x0q Ñ ⇡1pX,x0q is injective.

A cover X of a space X is called a universal cover if X is simply connected.

Theorem 3.9. Suppose that X has a universal cover. Then for any subgroup G of ⇡pXq,there is a covering space X such that p˚p⇡pXqq “ G.

The question of when X has a universal cover has a precise technical resolution givingconditions on X, but this won’t be of much consequence. All manifolds, for instance, enjoythis property.

3.5 4-Manifold Specifics

This material is somewhat less standard, so we will give a better exposition.

We’ll begin by giving a concrete description of the homology of orientable closed 4-manifolds,which will be indispensible. If X is a closed orientable manifold, the homology modulesare finitely generated, so write HipX;Zq – Fi ‘ Ti, where Fi is free, and Ti is the torsionsubmodule. The following lemma is from [12].

Lemma 3.1. Let X be a closed oriented 4-manifold. The homology and cohomology mod-

9

ules are:H0pX;Zq – Z H0

pX;Zq – ZH1pX;Zq – F1 ‘ T1 H1

pX;Zq – F1

H2pX;Zq – F2 ‘ T1 H2pX;Zq – F2 ‘ T1

H3pX;Zq – F1 H3pX;Zq – F1 ‘ T1

H4pX;Zq – Z H4pX;Zq – Z

Proof. By Poincaré duality, it suffices to compute about half the list. Since X is connected,we get H0pXq – Z, and since X is closed and oriented, we get H4pX, BXq – H4pXq – Z.

To get the rest, we use the Universal Coefficient theorem; since the sequence splits, we have:

H ipXq – ExtpHi´1pXq,Zq ‘ HomZpHipXq,Zq – Fi ‘ Ti´1

The last isomorphism follows from the fact that ExtpH,Zq is isomorphic to the torsionsubmodule of H if H is finitely generated, and similarly that HompH,Zq is isomorphic tothe free submodule of H if H is finitely generated [10][p. 196]. Now, with Poincaré duality,we have:

F3 ‘ T2 – H3pXq – H1pXq – F1 ‘ T1

F2 ‘ T1 – H2pXq – H2pXq – F2 ‘ T2

F1 ‘ T0 – H1pXq – H3pXq – F3 ‘ T3

But this immediately gives us that T1 – T2, T3 – T0 “ 0 and F1 – F3, which finishes theproof.

Remark: We note that the only source of torsion in the homology of a 4-manifold origi-nates in H1pXq; in the special case that X is simply connected, the homology/cohomologywill be particularly easy to calculate.

Now we’ll give a brief review of intersection forms on a 4-manifolds, mostly by following[12]. We begin with some definitions:

Definition 3.6. For a closed oriented 4-manifold X, its intersection form is the symmetricbilinear form:

QX : H2pX;Zq ˆ H2

pX;Zq Ñ Z

given by:QXp↵,�q “ x↵ Y �, rXsy

where rXs P H4pX;Zq denotes the fundamental class of X.

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Note that since both the cup product and the Kronecker product are bilinear, QX is alsobilinear. Moreover, since dimX “ 4, the cup product is symmetric, and so this does define asymmetric bilinear form. One can define the intersection form on manifolds with boundary,as in [9], but for our purposes this is sufficient.

Since X is closed, we can use Poincaré duality to define QX on H2pXq ˆ H2pXq, by

QXp↵,�q “ xPDp↵q Y PDp�q, rXsy

It is clear that if either ↵ or � has torsion, QXpa, bq “ 0. Hence, if T denotes the torsionZ-submodule of H2pXq (since X is closed, its homology/cohomology is finitely generatedso this is possible) we can consider QX as just defined on H2pXq{T . Since this is a freeZ-module, say of rank r, we can choose a basis for H2pXq{T and represent QX as a r ˆ rmatrix with respect to this basis.

Following [8], we now make some appropriate definitions for this situation.

Definition 3.7. Suppose A is a free Z-module with symmetric bilinear form Q. Let P bethe matrix of Q with respect to some basis for A. Then:

‚ The rank of Q is the rank of A as a Z-module. Note that in the case that A “ H2pXq,this is exactly the second Betti number of X.

‚ Consider P as a matrix over R or Q and diagonalize. Let b`2 be the number of

positive eigenvalues, and b´2 be the number of negative eigenvalues. The signature of

Q is defined as �pQq “ b`2 ´ b´

2 .

‚ If an orientation for X has been chosen, we usually write �pXq instead of �pQXq.

Since this section deals with intersection forms, one might hope to relate this discussion tothe actual intersection of submanifolds. In fact, this is the case. We define the intersectionproduct more generally by:

Definition 3.8. Let M be an m-manifold. The intersection product is defined:

‚ : HipM ;Zq ˆ HjpM ;Zq Ñ Hi`j´mpX;Zq

by PDpa ‚ bq “ PDpaq Y PDpbq.

One can easily check that this satisfies the usual axioms of associativity, and in the casethat M is a 4-manifold, commutativity. This product is indeed dual to the intersectionform, as we have:

QXpa, bq “ xPDpaq Y PDpbq, rXsy “ xPDpa ‚ bq, rXsy “ x1, a ‚ by

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by defintion of the Poincaré dual.

Now we have the following highly useful fact, proved in [1][p. 372].

Fact 3.1. Suppose M is an oriented m-manifold, and N ,K are oriented n- and k-manifolds,respectively. If N and K intersect transversely, then we have:

rK \N s “ rN s ‚ rKs P Hn`k´mpM ;Zq

Hence, we have PDprN \Ksq “ PDprN sq Y PDprKsq, and so in a sense, the cup productis dual to the transverse intersection of submanifolds.

The signature turns out to be a finer numerical invariant than the Euler characteristic, butit also obeys some nice properties. For instance, it is clear that if X denotes X with theopposite orientation, then �pXq “ ´�pXq. More importantly, the signature is additiveunder connected sums [12], i.e.,

�pX1#X2q “ �pX1q ` �pX2q

To finish this section, we will compute �pCP2q, as we will need it later. For a plethora of

other calculations, [12][p. 120-126] has an excellent treatment.

Example: Indeed, we recall that CP2 has Betti numbers b0 “ 1, b2 “ 1, b4 “ 1, andno other homology. More importantly, H2pCP2

q “ ZrtCP1us, where rtCP1

us denotes thehomology class of the projective line [12]. Since any two projective lines intersect in exactlyone point, we conclude that QCP2 “ r`1s, i.e., �pCP2

q “ 1 [12][p. 124].

Remark: We note that this example alone shows the utility of the signature. The abovecalculation shows that CP2 cannot be oriented-homotopy equivalent to CP2, while the Eulercharacteristic only gives �pCP2

q “ �pCP2q “ 3.

4 Lefshetz Fibrations

We now move to the core topic of this thesis: the Lefschetz fibration. These spaces area kind of fibration that generalizes the idea of a fiber bundle; as we will show, they areessentially a fiber bundle except possibly at a finite number of singular points. As mentionedin the introduction, some of these results will have to be taken on faith at this point, astheir proofs lie well beyond the scope of this project. However, a reader familiar with Kirbycalculus should be able to fill in any details. As much as possible, we’ll point out exactlywhich facts are required, and reference a specific resource of [9] to consult.

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4.1 Definitions

Definition 4.1. Suppose X is a compact, connected, oriented, smooth 4-manifold. A Lef-shetz fibration of X is a map ⇡ : X Ñ B, where B is a compact connected oriented surface,⇡´1

pBBq “ BX, and such that each critical point of ⇡ lies in IntX and has a local coordinatechart (consistent with the orientation of X) of the form ⇡pz1, z2q “ z21 ` z22. We’ll usuallydenote the set of critical values by C, and the set of regular values by B˚.

Remark: For our purposes, we will assume that all critical points of ⇡ lie in distinctfibers of ⇡. One could perturb the critical points slightly to achieve this, but in general,there is no obvious canoncial way. Many authors in the literature simply include this aspart of the definition [7], and we will follow this convention.

While it doesn’t look like it, these fibrations do behave much like fiber bundles. To provethis, recall the following theorem due to Ehresmann [5].

Theorem 4.1 (Ehresmann). If f is a smooth mapping between smooth manifolds M,Nwhich is a proper map and a surjective submersion, then f is a fiber bundle.

With it, we can prove:

Proposition 4.1. Let ⇡ : X Ñ B be a Lefshetz fibration, and let C be the collection ofcritical values of ⇡. Then ⇡ :“ ⇡|⇡´1pB˚q is a fiber bundle.

Proof. We have:

(1) ⇡ is a submersion, by definition.

(2) We claim that ⇡ is a surjection. To see this, note first that since X is compact, therecan be only finitely many critical points of ⇡, since they are isolated. Hence ⌃zCis still path connected. However, note that ⇡ is an open map, and so im ⇡ is open.However, X is compact, so Xz⇡´1

pCq is still compact, and so im ⇡ is closed. Hence,im ⇡ is clopen, and so ⇡ is surjective.

(3) Lastly, ⇡ is certainly a proper map, and so ⇡ must also be proper.

By Ehresmann’s theorem, we conclude that ⇡ is indeed a fiber bundle.

By Sard’s theorem, the regular fibers are all orientable compact 2-manifolds. If we knowthat the fibers are connected (which we will prove is the case under reasonable conditionsin the next section) then by the classification of compact connected 2-manifolds, the fibersare some genus g surface ⌃g. We will refer to this as a genus g Lefshetz fibration, and focusour attention almost exclusively on this case.

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4.2 The Local Topology of Lefschetz Fibrations

To obtain a better understanding of the topology of a Lefschetz fibration, we’ll try to givea description of the behavior near a singular fiber, following [7]. For the moment, supposethat ⇡ : X Ñ D2 is a Lefshetz fibration with just one critical value p P D2. Then we can findlocal coordinate charts where ⇡pz1, z2q “ z21 ` z22 , with ⇡p0, 0q “ p. Generically, a nearbyfiber has the form ⇡´1

ptq for some small t P R (adjusting coordinate charts if necessary).Hence, the fibers are diffeomorphic to the submanifold of C2 described by:

S “ tx21 ` x22 ´ y21 ´ y22 “ t, x1y1 ` x2y2 “ 0; xi, yi P Ru

Intersecting this with a copy of R2 (i.e., the x1, x2 plane), we obtain a copy of S1 given bytx21 `x22 “ t;xi P Ru, and if we let t approach 0, this circle shrinks to a single point. On theother hand, no such shrinking happens along the other axes. We see that ⇡´1

ppq can bethought of as obtained from a generic fiber by “pinching” it along some loop. We formallyintroduce some terminology to describe this situation:

Definition 4.2. The curve on ⌃g which determines the singular fiber at a point will becalled the vanishing cycle for that fiber. If the curve separates ⌃g into two surfaces ofnonzero genus, it will be called a separating cycle; otherwise it will be called a nonseparatingcycle.

Figure 1: On the left, a separating cycle. On the right, a non-separating cycle.

Remark: The vanishing cycle of a singular fiber, say ⇡´1ppq, actually completely de-

termines the topology of X near ⇡´1ppq. In fact, any two singular fibers described by

non-separating cycles will have diffeomorphic neighbourhoods in X, as will two singularfibers described by separating cycles which separate ⌃g into two surfaces of the same genus.Unfortunately, the proof of this fact relies on handlebody descriptions, so we will simplytake it as given and refer the more experienced reader to [7]. This fact will be convenientfor us, as there are relatively few different loops on ⌃g.

Following the convention of most authors, we will also assume that all Lefschetz fibra-tions are relatively minimal, that is, no singular fiber contains an embedded sphere ofself-intersection ´1. We note that if a separating cycle separates a fiber ⌃g into compo-nents of genus g and 0, then it will fail to be relatively minimal. Indeed, let ⌃[S2 be some

14

singular fiber separated into genus g and 0 parts, and ⌃1 be a nearby regular fiber of genusg. For notational convenience, denote the copy of S2 by S.

Figure 2: An illustration of ⌃1,⌃, and S (in genus 2).

Then we have (denoting the intersection product as multiplication):

0 “ rSs ¨ r⌃1s “ rSs ¨ prSs ` r⌃sq “ rSs

2` rSs ¨ r⌃s “ rSs

2` 1

by standard intersection theory arguments (S and ⌃ have just one point in common). ThusrSs

2“ ´1, and so the fibration is not relatively minimal.

With the above fact in hand, it is an easy matter to classify all possible vanishing cycles,and hence distinct kinds of singular fibers up to diffeomorphism. For a genus g Lefschetzfibration, there are tg{2u distinct separating cycles, and 1 non-separating cycle, for a totalof 1 ` tg{2u.

4.3 The Homotopy Exact Sequence

We would like to say something about the connectedness of the fibers of a Lefshetz fibration.To do this, we’ll prove some results about homotopy exact sequences, similar to the case ofregular fibrations.

Theorem 4.2. Let ⇡ : X Ñ B be a Lefschetz fibration with fiber F . Then the mapsF ãÑ X Ñ B induce an exact sequence:

⇡1pF q Ñ ⇡1pXq Ñ ⇡1pBq Ñ ⇡0pF q Ñ 0

where the map ⇡1pBq Ñ ⇡0pF q is exact as a function between sets.

Before we prove this, we give the desired corollary:

15

Corollary 4.1. We can always assume that the fibers of a Lefschetz fibration are connected.

Proof. Suppose that the regular fibers of our Lefschetz fibration are not connected. Thenwe note that the image of ⇡1pXq must be some finite index subgroup of ⇡1pBq (F is acompact 2-manifold, and hence has only finitely many connected components), and so bystandard covering space theory, there is a corresponding covering space p : B Ñ B, withp˚p⇡1pBqq “ ⇡˚p⇡1pXqq. Since X is connected, we can lift ⇡ to a map ⇡ : X Ñ B, and viewthis as our new Lefschetz fibration. Locally, it will act the same, but will have connectedfibers [9][p. 290].

Indeed, since this new map is still a Lefschetz fibration, we have an exact sequence (surjec-tivity at ⇡1pBq follows from the construction of the covering space):

⇡1pF q ›Ñ ⇡1pXq ⇣ ⇡1pBq ›Ñ ⇡0pF q ›Ñ 0

By exactness, we conclude that ⇡0pF q “ t0u and so F is 0-connected, i.e., connected.

In fact, in most cases we will not even need to do anything. If B is simply connected, i.e.,if B “ D2 or B “ S2 , then the exact sequence immediately shows that ⇡0pF q “ t0u, andso no modification is necessary. We will focus all of our attention on this case.

Corollary 4.2. If ⇡ : X Ñ B is a Lefschetz fibration, and B is simply connected, then thefibers of ⇡ are connected.

Now we’ll give a proof of the theorem.

Proof of Theorem 4.2. ⇡ : X Ñ B is a fiber bundle away from the singular values, andso as a fibration satisfies the homotopty lifting property. We would like to define a map� : ⇡1pBq Ñ ⇡0pF q in the following way:

Given a loop ↵ : r0, 1s Ñ B which avoids critical values and any x P ⇡´1p↵p0qq, the

homotopy lifing properties (⇡ is a fiber bundle there) guarantees a lift ↵ : r0, 1s Ñ X with↵ “ ⇡ ˝ ↵. We set �p↵q “ ↵p1q, as we would in the case of a fiber bundle. While ⇡ is notcompletely a fiber bundle, the situation can still be resolved.

If ↵ avoids the critical values in B, we will have no problems. If it does pass through acritical value, say p P B, we will use a local homotopy to adjust ↵ away from p. To checkthat this is well defined, it sufficess to check that ↵ can’t change connected components asit passes through p. However, we saw in the previous section that ⇡´1

ppq is obtained froma regular fiber ⌃g by collapsing an embedded copy of S1; this will not connect previouslydisconnected components of the fiber. Hence we conclude that � is well defined, and so wedo have a map � : ⇡1pBq Ñ ⇡0pF q.

16

To check exactness at ⇡1pBq, we notice that lifts of a loop in ⇡1pBq lift to a loop in X (i.e.,are in the kernel of �) if and only if the lift begins and ends in the same component of afiber.

Similarly, to check exactness at ⇡1pXq, we note that a loop in ⇡1pBq is constant if and onlyif its lift is contained in a single fiber.

4.4 Mapping Class Groups and Dehn Twists

For the time being, we continue to assume that ⇡ : X Ñ D2 is a Lefschetz fibration withonly one singular fiber, say ⇡´1

ppq. In the definition of a Lefshetz fibration, we required thesingular fibers to be in the interior of X, so the boundary BX is actually a ⌃g bundle overS1. Hence after we make an identification ' of ⇡´1

p1q Ä X with ⌃g (considering S1Ä C),

it takes the form:

BX “

⌃g ˆ I

p pxq, 0q „ px, 1q

where : ⌃g Ñ ⌃g is a homeomorphism (that depends on our choice of the identificationof the reference regular fiber). This map will be called the monodromy of the fibration.It turns out that these ideas give us a nice way to translate questions about Lefschetzfibrations into questions in group theory, which offers a significant advantage. To give aformal treatment of this situtation, we’ll take a brief foray into the land of mapping classgroups.

Definition 4.3. For a smooth manifold M , we define the mapping class group of M ,denoted ModpMq, to be the group of all isotopy classes of orientation preserving diffeomor-phisms of M , i.e.,

ModpMq :“Diffeo`

pMq

Isotopy

For a fiber bundle, we are interested in the kinds of diffeomorphisms one can apply to thefibers.

Definition 4.4. The mapping class group of a smooth fiber bundle ⇡ : E Ñ B is definedto be the mapping class group of F .

For a fixed identification ' of a fiber F over a base point of B, we can define the monodromyrepresentation map : ⇡1pBq Ñ ModpF q in the following way. For a loop ↵ : I Ñ B, thepullback bundle ⇡↵ : ↵˚

pEq Ñ I induces a diffeomorphism ↵ : ⇡´1↵ p0q Ñ ⇡´1

↵ p1q. Using

17

our reference map ', we thus obtain an element p↵q P ModpF q. It is easy to check thata homotopy of loops in B will induce an isotopy between diffeomorphisms on F , and so is well defined.

Remark: Since we usually write the concatenation of paths left to right, we modifiy thegroup structure on ModpF q to be 1 ‹ 2 “ 2 ˝ 1 so that is indeed a homomorphism.This won’t come up much, but being pedantic never hurt anyone.

For a genus g Lefschetz fibration ⇡ : X Ñ B, we define its monodromy representation tobe that of the associated fiber bundle ⇡|⇡´1pB˚q.

Remark: This result is well beyond the scope of this thesis, but in 1996, Matsumotoproved that for genus at least two, the monodromy representation : ⇡1p⌃˚

q Ñ ModpF q

completely determines the Lefshetz fibration up to isomorphism, where an isomorphism ofLefschetz fibrations is defined below. The result can even be extended to genus one if thebases spaces have nonempty boundary, or if the sets of critical values are nonempty. For acomplete account of this interesting result, the reader can consult [11] [Theorem 2.4].

Definition 4.5. An isomorphism of Lefschetz fibrations ⇡ : X Ñ B and ⇡1 : X 1Ñ B1 is a

pair of diffeomorphisms f : X Ñ X 1 and g : B Ñ B1 which commute with ⇡ and ⇡1.

Now we’d like to actually describe the monodromy around a singular fiber in some way, i.e.,the representation � : ⇡1pD2

ztpuq Ñ ModpF q. However, we note that the monodromy iscompletely determined by the image of a generator of ⇡1pS1

q “ Z, and so we will often referto this single diffeomorphism as the monodromy. As it turns out, this diffeomorphism hasa very nice geometric interpretation which comes from the corresponding vanishing cycle.To do this, we define the notion of a Dehn twist, complete with some illustrations.

Definition 4.6. Suppose that F is an oriented surface. A right handed Dehn twist DC :F Ñ F on a circle C in an oriented surface F is a diffeomorphism obtained by cutting Falong C, twisting 360˝ to the right, and regluing. One can view this by means of the diagrambelow.

Figure 3: The Dehn twist with F oriented as a section of a cylinder.

More formally, choose a normal neighbourhood N of C in F , and an identification � of Nwith S1

ˆ I. Let d : S1ˆ I Ñ S1

ˆ I be the right handed twist map dp✓, tq “ p✓ ` 2⇡t, tq(we can similarly define a left handed twist). Then the Dehn twist DC : F Ñ F is defined

18

as D “ �´1˝ d ˝ � in N , and the identity elsewhere. The twist map d can be visualized in

Figure 4 below:

Figure 4: The twist map d on the annulus.

It turns out that for genus g surfaces, the mapping class group Modp⌃gq can be generatedsolely by Dehn twists. A classical theorem of Dehn in 1920 (proved independently byLickorish in 1967 [6]) is the following:

Theorem 4.3 (Dehn-Lickorish). For g • 0, the mapping class group Modp⌃gq is generatedby Dehn twists about the 3g ´ 1 nonseparating simple closed curves in Figure 5.

Remark: In fact, in 1979, Humphries was able to show [6] that Modp⌃gq is generatedby just the Dehn twists about the Humphries generators a1, ..., ag´1, c1, ..., cg,m1,m2, andthat 2g ` 1 is the minimal number of generators (if all generators are required to be Dehntwists).

Figure 5:

These results are not the focus of this thesis, but give context for the appearance of Dehntwists in the Lefschetz fibration construction. For proofs of either of the results above orfurther reading on mapping class groups one can consult Chapter 4 of [6].

To conclude this section, we will give two properties of Dehn twists that are relativelyfundamental and will be used in the next section. Neither result is particularly hard toprove, and we refer the reader to [6].

Proposition 4.2. Let F be an oriented surface. Let ↵ and � be two isotopy classes ofsimple closed curves. Then D↵ is isotopic to D� if and only if ↵ “ �. In other words, Dehntwists determine their generating cycle up to isotopy.

19

Proposition 4.3. Let F be an oriented surface. For any � P ModpF q and any isotopyclass of a closed simple curve ↵, we have � ˝ D↵ ˝ �´1

“ D�p↵q.

4.5 The Monodromy of Leschetz Fibrations

We are now in a position to describe the monodromy representation of a Lefschetz fibration⇡ : X Ñ D2 with just one singular fiber. Unfortunately, the required proof techniquesrely on the machinery of handlebody descriptions and Kirby calculus, and so we will notinclude it. The more interested and knowledgable (or just brave) reader can consult [9]for a discussion of this machinery (in particular, Exercise 8.2.4). In the spirit of movingforward, we’ll just record it as a magnificent fact.

Fact: The monodromy of the Lefschetz fibration ⇡ : X Ñ D2 with one singular pointis given by a right handed Dehn twist about the vanishing cycle for that fiber. [9][Ex.8.2.4]

Remark: The keen reader may note that while left-handed Dehn twists make perfectsense, they don’t occur as the monodromy of Lefschetz fibrations. This is because ourdefinition of a Lefschetz fibration required orientation-preserving charts. If one relaxes thisrestriction it is possible to have left-handed twists [7]; these kinds of Lefschetz fibrationsare known as achiral, and we will not mention them again.

With this description of the local monodromy of Lefschetz fibrations, we can move on toconsidering more general Lefschetz fibrations over D2. Suppose that ⇡ : X Ñ D2 is aLefschetz fibration with µ singular fibers at p1, ...., pµ P D2. Choose small disjoint disksV1, ..., Vµ Ä ⌃ with pi P Vi. Then for each i, ⇡|⇡´1pViq is a Lefschetz fibration over Vi andso can be described completely by some vanishing cycle in a nearby regular fiber. We willcombine this information to obtain a global description in the following way.

Choose some nonsingular fiber based at p0 with another small disjoint disk V0 containingp0, and choose the labeling of the pi to be compatible with some counterclockwise orderingcentered at p0 for convenience. Choose paths si in the bases from p0 to pi which are disjoint(except at p0). Since each singular fiber Fi “ ⇡´1

ppiq is described by a nearby vanishingcycle on a nonsingular fiber, say �i, we have a way of transporting these to the commonreference fiber F0, and hence via some identification to the standard surface ⌃g.

In fact, we can prove the following:

Proposition 4.4. For a Lefschetz fibration ⇡ : X Ñ D2 with µ singular fibers as above,the global monodromy about BD2 is given by the product D�1 ‹ ¨ ¨ ¨ ‹ D�µ , where D�i is aDehn twist about a nearby vanishing cycle �i for pi.

Proof. Let C be the set of critical values of ⇡, and let � P ⇡1pD2zCq be the curve around

20

BD2. Then � is homotopic to �1 as in Figure 6 below.

Figure 6:

However, we can easily obtain this global monodromy by following �1. For each singularfiber ⇡´1

ppiq, we obtain a monodromy of the form s´1i ˝D�i ˝ si, where D�i is a Dehn twist

about some nearby vanishing cycle, and by abuse of notation, we write si P Modp⌃gq forthe isotopy class of the map ⌃g Ñ ⌃g obtained by identifying the images sip0q and sip1q.As noted in the previous section on Dehn twists, the conjugate of a Dehn twist is still aDehn twist, and so the monodromy about �1 is the composition D�1

µ˝ ¨ ¨ ¨ ˝ D�1

1, where �1

i

are vanishing cycles on the regular fiber ⇡´1pp0q (the same cycle up to isotopy).

On the other hand, we see that an ordered collection D�1 , ..., D�µ of Dehn twists on ⌃ alsodetermines a Lefschetz fibration on D2. Indeed, as noted earlier, a Dehn twist determines itsgenerating circle up to isotopy, so we can always canonically produce a Lefschetz fibration⇡ : X Ñ D2 together with arcs s1, ..., sµ.

However, while there is a correspondence, we note that Lefschetz fibrations over D2 are notin one to one correspondence with ordered sequences of Dehn twists on ⌃g. The sequenceobtained in Proposition 4.4 depends implicitly on the choices of indices made during theconstruction, as well as the choice of identification of the regular fiber, and the arcs used.Having reduced the classification of Lefschetz fibrations into a group theoretic problemin the area of mapping class groups, we will do our best to investigate these equivalenceclasses.

The first ambiguity clearly corresponds to cyclic permutations of pD�1 , ..., D�µq. In a similarfashion, if we change our choice of regular fiber, we will simply conjugate every Dehn twistby some fixed element of Modp⌃gq. However, this will preserve the overall product, and thenet result will just conjugate the product by some element of Modp⌃gq. The last operationpossible is called an elementary transformation, and we will describe it in more detail here.It is not difficult (albeit messy to draw) to see that one can get between different choicesof arcs tsiu by way of composing some number of the moves illustrated in Figure 7.

However, via the simple homotopy of curves below, we see that these moves correspond

21

Figure 7: Elementary transformations.

to changing the ordered collection of Dehn twists p..., D�i , D�i`1 , ...q into p..., D�i`1 , D´1�i`1

‹

D�i ‹ D�i`1 , ...q, and so the overall product is also preserved.

Figure 8: The homotopy of paths which shows that �11�

12�

13 » �1�3�

´13 �2�3.

In fact, these moves completely characterize the correspondence. Another result that wewill take as given is the following:

Fact 4.1. Two Lefschetz fibrations are isomorphic if and only if it is possible to get betweenthe associated ordered collection of monodromies by way of elementary transformations,together with an inner automorphism of Modp⌃gq (i.e., conjugation by some fixed element).

This is proved in a series of exercises in [9][p. 298], but relies again on the machinery ofKirby calculus.

While most of these problems are still intractable by hand, they do lend themselves tocomputer assisted group theoretic resolutions. We will, however, be able to come to someconclusions in the next section.

4.6 Lefschetz Fibrations over S2

As mentioned, S2 provides another compact simply connected base for reasonably simpleLefschetz fibrations. Suppose that ⇡ : X Ñ S2 is such a fibration. To understand it,we split S2 as D2

1 [D22, such that D2

1 contains all the singular fibers of the fibration, sayF1, ..., Fµ determined by vanishing cycles �1, ..., �µ, and ⇡ is just a genus g fiber bundle over

22

D22. Then as noted in the previous section, the monodromy about BD2

1 is described by theassociated composition of Dehn twists:

D�1 ‹ ¨ ¨ ¨ ‹ D�µ “ D�µ ˝ ¨ ¨ ¨ ˝ D�1

On the other hand, this fibration must be trivial on the boundary BD21, so as to extend to

the trivial fibration ⌃g ˆD22. Unforuntately, the ability to extend fibrations isn’t perfect in

all genera. With a result of Earle and Eells [4], one can prove [9][p. 300] that the extensionis actually unique in genus at least two. In genus one, one can extend the fibration, buttwo extensions may not be isomorphic. They will, however, be related by multiplicity-1logarithmic transformations on the last fiber, a fact which we will not give any expositionof, but mention for completeness. In any case, the classification of Lefschetz fibrations overS2 has essentially been reduced to a group theoretic problem in mapping class groups.

Via our correspondence result of the last section, we have [7]:

Proposition 4.5. For any fixed g • 2, there is a one-to-one correspondence between genusg Lefschetz fibrations over S2 and relations of the form D�1 ‹ ¨ ¨ ¨ ‹ D�µ in Modp⌃gq, mod-ulo the the relations discussed above, i.e., elementary transformations, and conjugation inModp⌃gq.

5 Examples

5.1 Classification Results

The previous discussion has been strikingly absent of actual examples of Lefschetz fibrations;we will give several now. While we have a characterization of Lefschetz fibrations of S2 asthe equivalence classes of trivial words in Modp⌃gq, this is still rather difficult to classify.We will be able to give a complete classification of all genus one Lefschetz fibrations over S2

(in the sense that we can state it). A partial classification exists in genus two, which is morecomplicated but still feasible to write down. Not much is known [7] in genera higher thantwo; unfortunately many known presentations of Modp⌃gq have relations which containboth right and left handed twists, so the monodromy approach is less helpful.

5.2 The Mapping Class Group of ⌃1

In this section, we’ll give a short derivation of the mapping class group of ⌃1.

Lemma 5.1. The torus has mapping class group Modp⌃1q “ SLp2,Zq.

23

Proof. Our sketch of the proof of this result will follow [6]; we’ll leave some of the detailsto the reader. We define a map

� : Modp⌃1q Ñ SLp2,Zq

by �p�q “ �˚, where �˚ is the induced map on H1p⌃1q. Indeed, this is well defined sinceany two representatives of an equivalence class will be homotopic, and so will induce thesame map on homology. In fact, since any representative � is invertible, we have:

�˚ P AutpH1p⌃1;Zqq “ GLp2,Zq

We actually have that �˚ P SLp2, Zq, but this follows from an argument involving algebraicintersection numbers that we will leave to the reader [6].

To prove that � is onto, suppose that M P SLp2,Zq. Then M is certainly a linear home-omorphism compatible with the Z2 lattice in R2, and so we obtain a map on ⌃1, say �M ,and �pr�M sq “ M .

To prove that � is injective, we suppose that �pr�sq “ I. Let ↵ and � be representatives ofthe homotopy classes defined by the vectors r1, 0s and r0, 1s, i.e., the longitude and meridianloops. Then we have �p↵q „ ↵ and �p�q „ � (where „ denotes homotopy equivalence),and by standard results (Prop 1.10, 1.11 of [6]), this homotopy extends to an isotopy, whichitself extends a map on ⌃1. But this shows that � is isotopic to the identity map on ⌃1, so� is injective.

Remark: We recall that SLp2,Zq has a relatively nice group presentation: it can bepresented as:

SLp2,Zq “ x↵1,↵2;↵41 “ 1,↵2

1 “ p↵1↵2q

3y

where ↵1 and ↵2 are identified via:

↵1 “

„0 ´11 0

⇢, ↵2 “

„1 10 1

⇢

One can easily check that these relations hold, but to show that ↵1 and ↵2 actually generateSLp2,Zq is somewhat computation heavy. More importantly, these generators and relationscorrespond to the “correct” loops in ⌃1, in the sense that ↵1 and ↵2 are identified with righthanded Dehn twists about the curves drawn below.

This result is fairly well known, so we’ll omit it.

24

Figure 9: ⌃1 with appropriate loops.

5.3 Genus One

Having computed Modp⌃1q above, we note that the relation p↵1↵2q

6n holds, and so foreach n we have a Lefschetz fibration ⇡n : Xn Ñ S2. The careful reader will note thatthere may be more than one, since we were only guaranteed unique extensions to S2 wheng • 2. However, this turns out not to be a problem, as Moishezon proved [7] that the globalmonodromy of any nontrivial genus one Lefshetz fibration is in fact equivalent (modulo theallowed moves) to one of these relations, and so any elliptic fibration is isomorphic to some⇡n.

On the other hand, we have a concrete description of elliptic surfaces via algebraic geometry.One can consider two generic cubics p0 and p1 on CP2, which intersect in 9 points, sayP1, ..., P9. Then we can define a map f : CP2

ztP1, ..., P9u Ñ CP1 in the following way: forQ P CP2

ztP1, ..., P9u, take the unique cubic t0p0 ` t1p1 passing through Q (by standardlemmas) and set fpQq “ rt0, t1s. We can’t define f at P1, ..., P9, but by blowing up CP2

each each point, we can extend f to a fibration ⇡ : CP2#9CP2Ñ CP1. We usually

denote Ep1q “ CP2#9CP2 when we endow it with this fibration structure. This works withpolynomials of different degree, so we summarize the situation with the following lemmafrom [9][p. 69]:

Lemma 5.2. The manifold CP2#d2CP2 admits a singular fibration to CP1“ S2, where

the generic fiber is a complex curve of genus 12pd ´ 1qpd ´ 2q.

Remark: We note that if any fibration ⇡ : X Ñ S2 had all fibers diffeomorphic to ⌃1,we would have �pXq “ �pS2

q�p⌃1q “ 0. On the other hand, we can calculate the Eulercharacteristic of Ep1q using the connected sum formula from Section 3.2 (CP2 is compactand connected):

�pCP2#CP2q “ �pCP2

q ` �pCP2q ´ �pS4

q “ 3 ` 3 ´ 2 “ 4

�prCP2#CP2s#CP2

q “ �pCP2#CP2q ` �pCP2

q ´ �pS4q “ 4 ` 3 ´ 2 “ 5

After 7 more iterations, we find �pEp1qq “ 12, and so Ep1q must have fibers which are notdiffeomorphic to ⌃1.

25

We can also produce more elliptic fibrations with an operation called the fiber sum.

Definition 5.1. Let ⇡ : X Ñ C and ⇡1 : X 1Ñ C 1 be two genus g fibrations, and let

F Ä X,F 1Ä X be two regular fibers. Identify neighbourhoods of F and F 1 with F ˆ D2

and F 1ˆ D2, and choose any diffeomorphism h : F Ñ F 1. The fiber sum X#FX2 is the

manifold pXzF ˆ D2q [ pX 1

zF 1ˆ D2

q, where : BpF ˆ D2q Ñ BpF 1

ˆ D2q is given by

“ h ˆ (complex conjugation) : F ˆ S1Ñ F 1

ˆ S1.

While this does depend on several choices made during the construction, it does produce aunique object in the following case [9][p. 72].

Definition 5.2. The elliptic surface Epnq is defined inductively as Epnq :“ Epn´1q#FEp1q.

In fact, while the following fact is not obvious, it is true [9][p. 72].

Fact 5.1. Epnq can be given a complex structure. Since complex manifolds always comewith a canonical choice of orientation, this implies that Epnq is orientable for each n.

Hence, by our discussion above, we have two different descriptions of Epnq (we know wehave a nice correspondence between Epnq and Xn above by computing Euler characteristics,for example [9][p. 74]). The first doesn’t rely on any algebraic geometry, which is rathernice. We’ll finish our discussion of these spaces by computing the rest of the invariants ofEp1q.

5.4 Topological Invariants of Ep1q

The invariants of Epnq are computable, but not by particularly elementary methods. We’llcompute the invariants of Ep1q, since this doesn’t require any further understanding of thefiber sum. Since CP2 is closed and connected, it follows that Ep1q is as well. By the factabove, it is also orientable, and so its homology and cohomology has the form described inlemma 3.1. We recall the two following standard facts:

Proposition 5.1. CPn is simply connected for each n.

Proposition 5.2. If M1,M2 are two connected m-manifolds with m • 3, then ⇡1pM1#M2q “

⇡1pM1q ‹ ⇡1pM2q, where ‹ denotes the free product.

This implies that ⇡1pEp1qq “ ⇡1pCP2q ‹ ¨ ¨ ¨ ‹ ⇡1pCP2

q “ 1, and so Ep1q is simply con-nected.

Then we have:

26

H0pEp1qq – Z

H4pEp1qq – Z

by lemma 3.1. Since Ep1q is simply connected, we also get:

H1pEp1qq – 0

H3pEp1qq – 0

and lastly since H2pEp1qq is free and �pEp1qq “ 12:

H2pEp1qq – Z12´2“ Z10

By Poincaré duality, the cohomology modules are:

H0pEp1qq – Z

H1pEp1qq – 0

H2pEp1qq – Z10

H3pEp1qq – 0

H4pEp1qq – Z

To get the signature of Ep1q, we use the results in Section 3.5. Since �pCP2q “ 1, �pCP2

q “

´�pCP2q “ ´1, and signature is additive under connected sums, we obtain:

�pEp1qq “ �pCP2q ` 9�pCP2

q “ ´8

5.5 A Higher Genus Example

Unfortunately, higher genus examples are much more complicated, and less is currentlyknown. However, there have been some recent partial classification results [7]. Let D1, ..., D2g`1

be right handed Dehn twists about the non-separating curves ↵1, ...,↵2g`1 in the figure be-low.

Then one can prove (with considerable effort) that the following relations hold in Modp⌃gq

(we’ll omit the ‹ notation and just write composition as just words in Modp⌃gq):

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Figure 10:

pD1D2 ¨ ¨ ¨D22g`1 ¨ ¨ ¨D2D1q

2“ 1

pD1D2 ¨ ¨ ¨D2gq

2p2g`1q“ 1

pD1D2 ¨ ¨ ¨D2g`1q

2g`2“ 1

These correspond to Lefschetz fibrations over S2, which, following [7], we’ll denote byXp1q, Xp2q, Xp3q. Each can be shown to be complex. In genus two, Chakiris [3] (and laterSmith [13]) were able to prove that any holomorphic Lefschetz fibration with only non-separating cycles is a fiber sum of one of these three, which is certainly interesting progresstowards a complete classification.

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References

[1] Bredon, G. E. (1993). Topology and geometry. New York: Springer-Verlag.

[2] Burns, K., & Gidea, M. (2005). Differential geometry and topology: With a view todynamical systems. Boca Raton: CRC Press.

[3] Chakiris, K. (1978) The monodromy of genus 2 pencils [Dissertation]. Columbia Uni-versity.

[4] Earle, C., Eells, J. (1969) A fiber bundle description of Teichmüller theory. J. Diff.Geom., 19-43.

[5] Ehresmann, Charles. (1950) Les connexions infinité esimales dans un espace fibré dif-férentiable, Colloque de Topologie, Bruxelles, 29-55.

[6] Farb, B., & Margalit, D. (2012). A primer on mapping class groups. Princeton Univer-sity Press.

[7] Fuller, T. (2003) Lefschetz Fibrations of 4-Dimensional Manifolds. Cubo, A Mathe-matical Journal.

[8] Ghaswala, T. (2012) Intersection Forms on Fermat Hypersurfaces in CP3 [Thesis].University of Waterloo, Canada.

[9] Gompf, R. E., & Stipsicz, A. (1999). 4-manifolds and Kirby calculus. Providence, RI:American Mathematical Society.

[10] Hatcher, A. (2002). Algebraic topology. Cambridge: Cambridge University Press.

[11] Matsumoto, Y. (1996) Lefschetz Fibrations of genus two - a topological approach.Proceedings of the 37th Taniguchi Symposium on Topology and Teichmmuller Spaces,World Scientific, 123-148.

[12] Scorpan, A. (2005). The wild world of 4-manifolds. Providence, RI: American Mathe-matical Society.

[13] Smith, I. (1999) Lefschetz fibrations and the Hodge bundle. Geom. Topol., 211-233.

[14] Spanier, E. H. (1966). Algebraic topology. New York: McGraw-Hill.

[15] Tel, Jan-Willem. (2015) Lefschetz fibrations and symplectic structures [Dissertation].Utrecht University, Netherlands.

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