4-manifolds and kirby calculus pdf form

In many applications, it is important to extract the antisymmetric part of the product tensor. Some topics in part 1, such as the classification of indefinite unimodular forms. One of the largest censuses of finite volume hyperbolic 4 manifolds was constructed by j. Let x 0 and x 1 be smooth, open, homeomorphic 4manifolds. 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. Recall that a symplecticformon a smooth manifold is a 2form.

This paper provides an extension of his result for manifolds of first homology groups of odd prime orders. We can thus write the attaching map of 1handles down in the form. Icerm women in symplectic and contact geometry and. Manifolds with boundary pdf, pdf dominic joyce, on manifolds with corners arxiv. Part iii offers in depth a broad range of topics from current 4 manifold research. 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. Many interesting manifolds admit such decompositions and kirbys techniques have led to a deeper understanding of a number of these. Bridge trisections of knotted surfaces in 4manifolds.

Manifold, in mathematics, a generalization and abstraction of the notion of a curved surface. They construct a total of 1171 distinct isometry classes of finite volume. We show that any 4manifold, after surgery on a curve, admits an achiral lefschetz fibration. Part i of the text presents the basics of the theory at the secondyear graduate level and offers an overview of current research.

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. Gompf and stipsicz 4manifolds and kirby calculus 4, roughly from front to finish. Afterwards we reprove the existence fact by constructing a folded symplectic form. A symplectic form on a 4manifold m is a closed twoform. We also study complex curves in complex 4manifolds, announcing preliminary results related to ongoing work with peter lambertcole. May 31, 2015 in many applications, it is important to extract the antisymmetric part of the product tensor.

Scorpans wild world of 4 manifolds is to my knowledge a unique book in that it covers extensively just about everything that one would need to know to study 4 manifold topology, both the topological i. Stipsicz, 4 manifoldsandkirbycalculus,1999 19 lawrence c. There are kirby diagrams for 3 and 4manifolds which consist of framed links corresponding to 1 and 2handles attached to a single 0handle. Calculating the homology and intersection form of a 4. The focus in this paper is to explicitly construct kirby diagrams for a certain collection of finite volume hyperbolic 4 manifolds. Any such diagram will give a unique closed manifold since the 3 and 4 handles that close the 012handlebody are unique. In this talk i will introduce the topic and discuss the aim of the seminar. Download pdf 4 manifolds and kirby calculus graduate. Constructing a fake 4 manifold by gluck construction to a.

The differential calculus on a smooth manifold x is defined as that over a ring c. S2 are related to each other by surgeries of their core spheres, correponding to zero and dot exchanges. The purpose of this paper is to study the types of framed 4 manifolds that a given closed, oriented. This site is like a library, you could find million book here by using search box in the widget. This is a negative definite intersection form, and therefore cp2. Topology of surface diagrams of smooth 4manifolds pnas. Part iii offers in depth a broad range of topics from current 4manifold research. In mathematics, 4 manifold is a 4 dimensional topological manifold. Monopoles and threemanifolds, volume 10 of new mathematical monographs. However, as powerful as these invariants are, they cannot distinguish all 4manifolds. The cus tomary goal is to discover invariants, usually algebraic. Refined kirby calculus for threemanifolds of first homology. Gompf showed that a basic cut and paste operation, a symplectic normal sum, could be performed on symplectic. This allows an extension of the kirby calculus to rational surgeries.

Part ii is devoted to an exposition of kirby calculus, or handlebody theory on 4manifolds. Pdf convex plumbings in closed hyperbolic 4manifolds. 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. The main ingredient there will be achiral lefschetz. Stern geometric topology is the study of metric spaces which are locally homeomorphic to euclidean nspace rn. Topology and its applications 46 1992 167172 167 northholland links and polar flows on 4manifolds michael d. There are kirby diagrams for 3 and 4 manifolds which consist of framed links corresponding to 1 and 2handles attached to a single 0handle. Now the ordinary calculus of r n can be patched together via the transition functions to allow one to do calculus on all of m. Stipsicz, 4manifolds and kirby calculus, graduate studies in mathematics, 20. Furthermore, we show that we can take such knots so that they are linearly independent in the the knot concordance group. Bridge trisections of knotted surfaces in 4manifolds jeffrey meiera,1 and alexander zupanb adepartment of mathematics, university of georgia, athens, ga 30602.

A famous theorem of michael freedman implies that the homeomorphism type of the manifold only depends on this intersection form, and on a. Are there kirby diagrams for manifolds with boundaries. Show that the manifold w in figure 8 is a contractible manifold, and by sugeries in its interior corresponding to zero. Lettris is a curious tetrisclone game where all the bricks have the same square shape but different content.

The topology of 4manifolds, volume 74 of lecture notes in mathematics. A new description is given for the iinvariant of a knot and this description is used to study almost framed 3manifolds. This book presents the classical theorems about simply connected smooth 4manifolds. To do this, elie cartan invented an efficient new operation which does both things at once, thus saving a great deal of labour. Pdf gluck twisting 4manifolds with odd intersection form. The mid1990s saw a number of developments which made symplectic 4 manifolds natural candidates to be the building blocks of all smooth 4 manifolds. Broken lefschetz fibrations on smooth 4manifolds were first introduced in ref. An extended set of diagrams and moves are used for describing 4 manifolds. Wang, links and polar flows on 4manifolds, topology. Constructing a fake 4manifold by gluck construction to a. 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.

In dimension 4, freedman proved that in the case of good fundamental group, such an scobordism has a topological product structure. Stipsicz, 4 manifolds and kirby calculus, graduate studies in mathematics, 20. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. Gompf, university of texas, austin, austin, tx and andras i. 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. Let us now define what is called the intersection form, which. Kirby calculus is a little bit more complicated than in the threedimensional case. More generally, we can prove, using kirby calculus 7 and some twisting manipulations, that a n. We also show that these surgered manifolds admit nearsymplectic structures, and prove more generally that achiral lefschetz fibrations with sections. Lefschetz fibrations of 4dimensional manifolds terry fuller department of mathematics california state university, northridge. We also show that the volume spectrum of hyperbolic 4manifolds is the set of all positive integral multiples of 4 2 3. 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. A link calculus for 4manifolds, in topologyoflowdimensional manifolds, proc. A new description is given for the iinvariant of a knot and this description is used to study almost framed 3 manifolds. 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. 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. Gompf and andras stipsiczs 4manifolds and kirby calculus. K bounds a disk in a punctured standard 4manifolds i. A framed link l in s 3 is a finite, disjoint collection of smoothly imbedded. 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. The chapter presents the solution of the equivalence relation problem. Topologically slice knots that are not smoothly slice in. 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. Surface diagrams are a new way to specify any smooth closed orientable 4manifold by an orientable surface decorated with simple closed curves. All books are in clear copy here, and all files are secure so dont worry about it.

These curves are cyclically indexed, and each curve has a unique transverse intersection with the next. 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. Kirby diagrams and the ratcliffetschantz hyperbolic 4manifolds. The main tools are the moves of kirbys calculus of framed links. A framed link in the 3sphere encodes instructions for attaching 2handles to the 4 ball.

Smooth structures on nonorientable fourmanifolds and free. Stipsicz, 4manifoldsandkirbycalculus,1999 19 lawrence c. Please click button to get 4 manifolds and kirby calculus book now. Introduction a hyperbolic manifold is a riemannian manifold of constant. Indeed,noexoticpairoforientable 4 manifolds, which are related each other by gluck twist, known to exits. Instantons and the topology of 4manifolds ronald j. We prove that there exist infinitely many topologically slice knots which cannot bound a smooth nullhomologous disk in any definite 4manifold.

Graduate studies in mathematics publication year 1999. On the other hand, many families of knotted 2spheres do not change s4 byglucktwistinge. Lp, which says that any connected smooth 4manifold is determined. This chapter presents heegaard diagrams for closed 4manifolds. Instantons and the topology of 4 manifolds ronald j. There are also various tricks to modify surgery diagrams. Recall that an scobordism is an hcobordism with vanishing whitehead torsion. Part ii is devoted to an exposition of kirby calculus, or handlebody theory on 4 manifolds. Trisections of 4 manifolds department of mathematics. Eliashberg, filling by holomorphic discs and its applications, geometry of lowdimensional manifolds, 2 durham. Stipsicz, 4manifolds and kirby calculus, graduate studies in math. Wallace, every orientable 3manifold is the boundary of some 4 manifold 9. 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.