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2004 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Surgery on Contact 3-Manifolds and Stein Surfaces

verfasst von: Burak Ozbagci, András I. Stipsicz

Verlag: Springer Berlin Heidelberg

Buchreihe : Bolyai Society Mathematical Studies

Enthalten in: Professional Book Archive

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Über dieses Buch

The groundbreaking results of the near past - Donaldson's result on Lef­ schetz pencils on symplectic manifolds and Giroux's correspondence be­ tween contact structures and open book decompositions - brought a top­ ological flavor to global symplectic and contact geometry. This topological aspect is strengthened by the existing results of Weinstein and Eliashberg (and Gompf in dimension 4) on handle attachment in the symplectic and Stein category, and by Giroux's theory of convex surfaces, enabling us to perform surgeries on contact 3-manifolds. The main objective of these notes is to provide a self-contained introduction to the theory of surgeries one can perform on contact 3-manifolds and Stein surfaces. We will adopt a very topological point of view based on handlebody theory, in particular, on Kirby calculus for 3- and 4-dimensionalmanifolds. Surgery is a constructive method by its very nature. Applying it in an intricate way one can see what can be done. These results are nicely com­ plemented by the results relying on gauge theory - a theory designed to prove that certain things cannot be done. We will freely apply recent results of gauge theory without a detailed introduction to these topics; we will be content with a short introduction to some forms of Seiberg-Witten theory and some discussions regarding Heegaard Floer theory in two Appendices.

Inhaltsverzeichnis

Frontmatter
1. Introduction
Abstract
The intense interest of 4-manifold topologists in symplectic geometry and topology might have the following explanation. The success of the classification of higher (≥ 5) dimensional manifolds relies heavily on the famous “h-cobordism theorem”, in which the “Whitney trick” plays a fundamental role. The Whitney trick asserts that (under favorable conditions) the algebraic and geometric intersection numbers of two submanifolds can be made equal by isotoping one of them. In other words, by isotopy we can get rid of “excess intersections”, which are present in the geometric picture but are invisible for algebra. After eliminating these intersections “algebra will govern geometry”, and the smooth classification problem of manifolds can be translated into some (nontrivial) algebraic questions.
Burak Ozbagci, András I. Stipsicz
2. Topological Surgeries
Abstract
After the short Prelude given in the introductory chapter we begin our discussion by reviewing the smooth constructions behind contact and Stein surgeries. We assume that the reader is familiar with basics in differential topology as given, for example, in [72]. Standard facts regarding singular homology and cohomology theory will also be used without further explanation. The manifolds appearing in these notes are all assumed to be smooth (i.e., C ∞-) manifolds, possibly with nonempty boundary. The general discussion of handlebodies will be followed by a short overview of Dehn surgeries in dimension three, and an outline of Kirby calculus concludes the chapter. For more details about the ideas and constructions sketched here, see [66].
Burak Ozbagci, András I. Stipsicz
3. Symplectic 4-Manifolds
Abstract
In this section we recall some general facts about symplectic manifolds. Then we give a short discussion of Moser’s method, which is applied in the proof of numerous fundamental statements discussed in the text. The chapter concludes with a short review on what is known about the classification of symplectic 4-manifolds. For a more detailed treatment of symplectic geometry and topology the reader is advised to turn to [111]; here we restrict our attention mostly to the 4-dimensional case.
Burak Ozbagci, András I. Stipsicz
4. Contact 3-Manifolds
Abstract
This chapter is devoted to the recollection of basic facts about contact manifolds. As before, we start with the general case, but very quickly specialize to 3-manifolds. To understand the topology of contact 3-manifolds we consider submanifolds and the contact structures near them. The contact version of Darboux’s theorem says that every point in a contact 3-manifold has a neighborhood which is standard regardless of the contact structure. Then we consider knots which are always tangent or always transverse to the contact planes and examine their classical invariants. It turns out that the contact structures near these types of knots are essentially unique. For an arbitrary surface embedded in a contact 3-manifold we look at the characteristic foliation induced by the contact structure to extract information. It is typical to move a surface by a small isotopy to modify its characteristic foliation to get a generic picture and/or to eliminate certain type of singularities. As it turns out, the characteristic foliation determines the contact structure near the surface. A more complete treatment of the ideas and theorems collected here can be found in e.g. [1, 39, 56, 57].
Burak Ozbagci, András I. Stipsicz
5. Convex Surfaces in Contact 3-Manifolds
Abstract
When trying to do surgery on contact 3-manifolds we need to understand contact structures in neighborhoods of embedded surfaces. As we already pointed out in Chapter 4, for a given surface Σ ⊂ (Y, ξ) the characteristic foliation F Σ determines the contact structure near Σ. But it is not easy to describe or relate characteristic foliations. It turns out that the same information can be captured by certain configurations of curves on the surface at hand once the surface is in a special position with respect to the contact structure. This theory has been developed and fruitfully applied by Giroux and Honda in various circumstances in 3-dimensional contact geometry. For the sake of completeness, in this Chapter we recall the fundamental definitions and results regarding convex surfaces and dividing sets. These statements will be used in our study of contact Dehn surgery in Chapter 11. For a more detailed introduction to the subject see [43, 76].
Burak Ozbagci, András I. Stipsicz
6. Spin c Structures on 3- and 4-Manifolds
Abstract
Spin c structures turn out to be very useful tools in understanding homotopic properties of contact structures. In addition, gauge theoretic invariants — such as Seiberg-Witten and Ozsváth-Szabó invariants — are defined for spin c 3- and 4-manifolds. This chapter is devoted to the review of spin c structures — with a special emphasis on the 3- and 4-dimensional case. Throughout this chapter we will assume that the reader is familiar with the basics of the theory of characteristic classes. (For an excellent reference see [116].) For a more complete treatment of spine structures the reader is advised to turn to [113].
Burak Ozbagci, András I. Stipsicz
7. Symplectic Surgery
Abstract
After these preparatory chapters now we are ready to describe the surgery scheme in the symplectic category. First we will deal with the general cutand-paste operation and then examine the handle attachment procedure in detail. The chapter concludes with the description of a version of surgery which will be useful in the contact setting, see Chapter 11.
Burak Ozbagci, András I. Stipsicz
8. Stein Manifolds
Abstract
In this chapter we interpret the Weinstein handle attachment in the Stein category, leading us to Eliashberg’s celebrated theorem. To put this result in the right perspective, we first recall rudiments of Stein manifold theory. The chapter concludes with a discussion about surfaces in Stein manifolds. For a more detailed treatment of this topic the reader is advised to turn to [70].
Burak Ozbagci, András I. Stipsicz
9. Open Books and Contact Structures
Abstract
Recently Giroux [63] proved a central result about the topology of contact 3-manifolds. He showed that there is a one-to-one correspondence between contact structures (up to isotopy) and open book decompositions (up to positive stabilization/destabilization) on a closed oriented 3-manifold. This chapter is devoted to the introduction of relevant notions and also some parts of the proof of this beautiful correspondence.
Burak Ozbagci, András I. Stipsicz
10. Lefschetz Fibrations on 4-Manifolds
Abstract
In the light of recent results it turns out that both closed symplectic 4-manifolds and Stein surfaces admit a purely topological description in terms of Lefschetz fibrations and Lefschetz pencils. In this chapter we give the necessary definitions and sketch this topological descriptions of symplectic and Stein manifolds. In the discussion we include achiral Lefschetz fibrations as well; these more general objects are useful in viewing open book decompositions as boundaries of certain achiral Lefschetz fibrations. The chapter concludes with some applications of Lefschetz fibrations in various low dimensional problems.
Burak Ozbagci, András I. Stipsicz
11. Contact Dehn Surgery
Abstract
Now we are in the position to describe the contact version of the smooth surgery scheme we started our notes with. This method provides a rich and yet to be explored source of all kinds of contact 3-manifolds. The approach to 3-dimensional contact topology we outline here was initiated by Ding and Geiges [16, 17], see also [18, 19]. Using contact surgery diagrams — and applying achiral Lefschetz fibrations — we will make connection to Giroux’s theory on open book decompositions, and we will also show a way to determine homotopic properties of the contact structures under examination. We begin by reviewing the classification of tight structures on S l × D 2 due to Honda — this is the result which allows us to define contact surgery diagrams.
Burak Ozbagci, András I. Stipsicz
12. Fillings of Contact 3-Manifolds
Abstract
This chapter is devoted to the study of fillability properties of contact 3-manifolds. After having the necessary definitions we will see different types of fillings, and give a family of tight, nonfillable contact structures. The construction of these latter examples utilizes contact surgery, while tightness is proved by computing contact Ozsváth-Szabó invariants (see Chapter 14). In the last section we will concentrate on topological restrictions a contact 3-manifold imposes on its Stein fillings.
Burak Ozbagci, András I. Stipsicz
13. Appendix: Seiberg-Witten Invariants
Abstract
In this chapter we recall basic definition, notions and results of Seiberg-Witten gauge theory. The introduction is not intended to be complete, we rather describe arguments most frequently used in the text. We also review a variant of the theory for 4-manifolds with contact type boundary, which setting turns out to be very useful in the study of contact topological problems. The last section is devoted to a discussion centering around the adjunction inequality. For a more complete discussion of the topics appearing in this chapter the reader is advised to turn to [21, 119, 126, 149].
Burak Ozbagci, András I. Stipsicz
14. Appendix: Heegaard Floer Theory
Abstract
The topological description of contact structures as open book decompositions provides the possibility of defining contact invariants which (at least partially) can be computed from surgery diagrams. In this appendix we outline the construction of such invariants — for a complete discussion the reader is referred to the original papers of Ozsváth and Szabó [135, 136, 137, 138]. To set up the stage, first we discuss Ozsváth-Szabó homology groups of oriented, closed 3-manifolds (together with maps induced by oriented cobordisms). The definition of the group HF(Y) for a 3-manifold Y will rely on some standard constructions in Floer homology. After presenting the surgery triangles for this theory, we outline the definition of the contact Ozsváth-Szabó invariants and verify some of the basic properties of this very sensitive invariant. A few model computations are also given.
Burak Ozbagci, András I. Stipsicz
15. Appendix: Mapping Class Groups
Abstract
In this appendix we summarize some basic facts regarding algebraic properties of mapping class groups. After discussing the presentation of these groups we recall the equivalence between certain words in some mapping class groups and geometric structures discussed in earlier chapters. We close this chapter with some theorems making use of those connections.
Burak Ozbagci, András I. Stipsicz
Backmatter
Metadaten
Titel
Surgery on Contact 3-Manifolds and Stein Surfaces
verfasst von
Burak Ozbagci
András I. Stipsicz
Copyright-Jahr
2004
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-10167-4
Print ISBN
978-3-642-06184-4
DOI
https://doi.org/10.1007/978-3-662-10167-4