The women in symplectic and contact geometry and topology workshop wiscon is a research collaboration conference for women rccw in the fields of contact and symplectic geometrytopology and related areas of lowdimensional topology. A framed link in the 3sphere encodes instructions for attaching 2handles to the 4 ball. A new description is given for the iinvariant of a knot and this description is used to study almost framed 3manifolds. A symplectic form on a 4manifold m is a closed twoform. Heegaard diagrams for closed 4manifolds sciencedirect.
Restricting such presentations, habiro arranged kirby calculus so that it preserves framings and linkings and moreover showed that his calculus suffices to relate all links with the same results. Indeed,noexoticpairoforientable 4 manifolds, which are related each other by gluck twist, known to exits. Pdf convex plumbings in closed hyperbolic 4manifolds. Stern geometric topology is the study of metric spaces which are locally homeomorphic to euclidean nspace rn. The main tools are the moves of kirby s calculus of framed links. Part i of the text presents the basics of the theory at the secondyear graduate level and offers an overview of current research.
Stipsicz, 4manifolds and kirby calculus, graduate studies in math. Instantons and the topology of 4 manifolds ronald j. They construct a total of 1171 distinct isometry classes of finite volume. This site is like a library, you could find million book here by using search box in the widget. Trisections of 4 manifolds department of mathematics. Calculating the homology and intersection form of a 4. Surface diagrams are a new way to specify any smooth closed orientable 4manifold by an orientable surface decorated with simple closed curves. Furthermore, we show that we can take such knots so that they are linearly independent in the the knot concordance group. This chapter presents heegaard diagrams for closed 4manifolds. We prove that there exist infinitely many topologically slice knots which cannot bound a smooth nullhomologous disk in any definite 4manifold.
Bridge trisections of knotted surfaces in 4manifolds jeffrey meiera,1 and alexander zupanb adepartment of mathematics, university of georgia, athens, ga 30602. Show that the manifold w in figure 8 is a contractible manifold, and by sugeries in its interior corresponding to zero and dot. Some topics in part 1, such as the classification of indefinite unimodular forms. Using fourdimensional cerf theory, he proved that if m and n are 3manifolds, resulting from dehn surgery on framed links l and j respectively, then they are homeomorphic if and only if l and j are related by a. A new description is given for the iinvariant of a knot and this description is used to study almost framed 3 manifolds. Refined kirby calculus for threemanifolds of first homology. Smooth structures on nonorientable fourmanifolds and free. Lefschetz fibrations of 4dimensional manifolds terry fuller department of mathematics california state university, northridge. The homotopy type of a simply connected compact 4manifold only depends on the intersection form on the middle dimensional homology. All books are in clear copy here, and all files are secure so dont worry about it. Wallace, every orientable 3manifold is the boundary of some 4 manifold 9.
The cus tomary goal is to discover invariants, usually algebraic. Realizing 4manifolds as achiral lefschetz fibrations. A link calculus for 4manifolds, in topologyoflowdimensional manifolds, proc. Pdf gluck twisting 4manifolds with odd intersection form. Eliashberg, filling by holomorphic discs and its applications, geometry of lowdimensional manifolds, 2 durham. Manifolds with boundary pdf, pdf dominic joyce, on manifolds with corners arxiv. The purpose of this paper is to study the types of framed 4manifolds that a given closed, oriented. The chapter presents the solution of the equivalence relation problem.
After a short introduction to the basic notions regarding trisections, we will see some examples and state the main theorem developed so far in this theory by abrams, gay, and kirby. Please click button to get 4 manifolds and kirby calculus book now. We can thus write the attaching map of 1handles down in the form. Lettris is a curious tetrisclone game where all the bricks have the same square shape but different content.
Part iii offers in depth a broad range of topics from current 4 manifold research. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. Gompf and stipsicz 4manifolds and kirby calculus 4, roughly from front to finish. To do this, elie cartan invented an efficient new operation which does both things at once, thus saving a great deal of labour. Gompf and andras stipsiczs 4manifolds and kirby calculus. A smooth 4 manifold is a 4 manifold with a smooth structure. From this point of view, section 2 presents the 0handle of a neutral kirby calculus, with preferred curves along which to do surgery. Recall that an scobordism is an hcobordism with vanishing whitehead torsion. Show that the manifold w in figure 8 is a contractible manifold, and by sugeries in its interior corresponding to zero. This allows an extension of the kirby calculus to rational surgeries. Kirby diagrams and the ratcliffetschantz hyperbolic 4manifolds. These curves are cyclically indexed, and each curve has a unique transverse intersection with the next. Afterwards we reprove the existence fact by constructing a folded symplectic form. Scorpans wild world of 4manifolds is to my knowledge a unique book in that it covers extensively just about everything that one would need to know to study 4manifold topology, both the topological i.
We also show that the volume spectrum of hyperbolic 4manifolds is the set of all positive integral multiples of 4 2 3. In this paper, we investigate existence of inequivalent smooth structures on closed smooth nonorientable 4manifolds building upon results of akbulut, cappellshaneson, fintushelstern, gompf and s. Stipsicz, 4 manifoldsandkirbycalculus,1999 19 lawrence c. Gompf, university of texas, austin, austin, tx and andras i.
This is a negative definite intersection form, and therefore cp2. In mathematics, the kirby calculus in geometric topology, named after robion kirby, is a method for modifying framed links in the 3sphere using a finite set of moves, the kirby moves. S2 are related to each other by surgeries of their core spheres, correponding to zero and dot exchanges. The aim of this paper is to announce a uniqueness theorem for these objects within a fixed homotopy class that turns out to be similar to. There are kirby diagrams for 3 and 4manifolds which consist of framed links corresponding to 1 and 2handles attached to a single 0handle. The main tools are the moves of kirbys calculus of framed links. In dimension 4, freedman proved that in the case of good fundamental group, such an scobordism has a topological product structure. Symplectic structures on the cotangent bundles of open 4.
Topologically slice knots that are not smoothly slice in. One of the largest censuses of finite volume hyperbolic 4 manifolds was constructed by j. Are there kirby diagrams for manifolds with boundaries. Introduction a hyperbolic manifold is a riemannian manifold of constant. The mid1990s saw a number of developments which made symplectic 4 manifolds natural candidates to be the building blocks of all smooth 4 manifolds. A calculus for framed links in s 3 robion kirby university of california, department of mathematics, berkeley, ca. Gompf showed that a basic cut and paste operation, a symplectic normal sum, could be performed on symplectic.
We also study complex curves in complex 4manifolds, announcing preliminary results related to ongoing work with peter lambertcole. We also show that these surgered manifolds admit nearsymplectic structures, and prove more generally that achiral lefschetz fibrations with sections. Stipsicz, 4 manifolds and kirby calculus, graduate studies in mathematics, 20. Instantons and the topology of 4manifolds ronald j.
Let x 0 and x 1 be smooth, open, homeomorphic 4manifolds. Kirby calculus is a little bit more complicated than in the threedimensional case. A famous theorem of michael freedman implies that the homeomorphism type of the manifold only depends on this intersection form, and on a invariant called the kirbysiebenmann invariant, and moreover that every combination of. The purpose of this paper is to study the types of framed 4 manifolds that a given closed, oriented. Hirsch department of mathematics, university of california, berkeley, ca 94720, usa xiaolu wang department of mathematics, university of maryland, college park, md 20742, usa received 3 april 1991 abstract hirsch, m. This book presents the classical theorems about simply connected smooth 4manifolds. The topology of 4manifolds, volume 74 of lecture notes in mathematics. In mathematics, 4 manifold is a 4 dimensional topological manifold. Recall that a symplecticformon a smooth manifold is a 2form. We describe rules for building 2d theories labeled by 4manifolds. Let us now define what is called the intersection form, which.
It is thus interesting to see whether gluck twists can produce an exotic pair of orientable 4 manifolds. May 31, 2015 in many applications, it is important to extract the antisymmetric part of the product tensor. On the other hand, many families of knotted 2spheres do not change s4 byglucktwistinge. We show that any 4manifold, after surgery on a curve, admits an achiral lefschetz fibration. Many interesting manifolds admit such decompositions and kirbys techniques have led to a deeper understanding of a number of these. Lp, which says that any connected smooth 4manifold is determined. A famous theorem of michael freedman implies that the homeomorphism type of the manifold only depends on this intersection form, and on a. Download pdf 4 manifolds and kirby calculus graduate. Stipsicz, 4manifolds and kirby calculus, graduate studies in mathematics, 20. An extended set of diagrams and moves are used for describing 4 manifolds. Topology of surface diagrams of smooth 4manifolds pnas. Manifold, in mathematics, a generalization and abstraction of the notion of a curved surface.
The neutral metric appears to be ideally suited to 2handle constructions in which the framing is tracked by the null cone on the associated tori. There are kirby diagrams for 3 and 4 manifolds which consist of framed links corresponding to 1 and 2handles attached to a single 0handle. Stipsicz, 4manifoldsandkirbycalculus,1999 19 lawrence c. Icerm women in symplectic and contact geometry and. In this talk i will introduce the topic and discuss the aim of the seminar.
The focus in this paper is to explicitly construct kirby diagrams for a certain collection of finite volume hyperbolic 4 manifolds. Part iii offers in depth a broad range of topics from current 4manifold research. In many applications, it is important to extract the antisymmetric part of the product tensor. Topology and its applications 46 1992 167172 167 northholland links and polar flows on 4manifolds michael d. Broken lefschetz fibrations on smooth 4manifolds were first introduced in ref.
Graduate studies in mathematics publication year 1999. Constructing a fake 4 manifold by gluck construction to a. This paper provides an extension of his result for manifolds of first homology groups of odd prime orders. Monopoles and threemanifolds, volume 10 of new mathematical monographs. However, as powerful as these invariants are, they cannot distinguish all 4manifolds. Any such diagram will give a unique closed manifold since the 3 and 4 handles that close the 012handlebody are unique.
The main ingredient there will be achiral lefschetz. Constructing a fake 4manifold by gluck construction to a. In other words, smooth 4manifolds can have in nitely many exotic copies, that is, smooth 4manifolds which are homeomorphic but not di eomorphic to the given model. Wang, links and polar flows on 4manifolds, topology. Bridge trisections of knotted surfaces in 4manifolds.
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