In this article we survey, and make a few new observations about, the surprising connection between sub-monoids of mapping class groups and interesting geometry and topology in low-dimensions.
This article is an expository overview of work by the author characterizing tightness of a closed contact 3-manifold in terms of arbitrary open book decompositions thereof.The intent is to provide a "user's guide" of the theory.53D10; 57M25, 57R65 0 Diff C .†; @/.The relation between these objects is given most completely by the Giroux correspondence theorem: Theorem 1.1 (Giroux correspondence [11]) There is a 1-1 correspondence between the set of contact structures on M up to isotopy, and the set of open book decompositions of M up to stabilization.The stabilization operation referred to in the above theorem is just the plumbing of a Hopf band to the surface.More precisely: Definition 1.2 Let .†; '/ be an open book decomposition of M , and ˛a properly embedded arc in †.Denote by † 0 the result of adding a 1-handle to † with attaching
We extend the Eliashberg-Thurston theorem on approximations of taut oriented $C^2$-foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented $C^{1,0}$-foliations, where by $C^{1,0}$ foliation, we mean a foliation with continuous tangent plane field. These $C^{1,0}$-foliations can therefore be approximated by weakly symplectically fillable, universally tight, contact structures. This allows applications of $C^2$-foliation theory to contact topology and Floer theory to be generalized and extended to constructions of $C^{1,0}$-foliations.
The purpose of this note is to explain a combinatorial description of closed smooth oriented 4-manifolds in terms of positive Dehn twist factorizations of surface mapping classes, and further explore these connections. This is obtained via monodromy representations of simplified broken Lefschetz fibrations on 4-manifolds, for which we provide an extension of Hurwitz moves that allows us to uniquely determine the isomorphism class of a broken Lefschetz fibration. We furthermore discuss broken Lefschetz fibrations whose monodromies are contained in special subgroups of the mapping class group; namely, the hyperelliptic mapping class group and in the Torelli group, respectively, and present various results on them which extend or contrast with those known to hold for honest Lefschetz fibrations. Lastly, we show that there are 4-manifolds admitting infinitely many pairwise nonisomorphic relatively minimal broken Lefschetz fibrations with isotopic regular fibers.
We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz fibrations over the 2-sphere obtained by Endo and Nagami to that for Lefschetz fibrations over arbitrary closed oriented surface. We then show two theorems on stabilization of Lefschetz fibrations under fiber summing with copies of a typical Lefschetz fibration as generalizations of a theorem of Auroux.
We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold groups.
We review a braid theoretic self-linking number formula and study its applications.
Let M be a closed, oriented, connected 3–manifold and .B; / an open book decomposition on M with page † and monodromy ' . It is easy to see that the first Betti number of † is bounded below by the number of S S –factors in the prime factorization of M . Our main result is that equality is realized if and only if ' is trivial and M is a connected sum of copies of S S . We also give some applications of our main result, such as a new proof of the fact that if the closure of a braid with n strands is the unlink with n components then the braid is trivial.
This is an expository article on the study of topology of Stein/symplectic fillings of contact 3-manifolds. 57R17; 55R55, 57R651 Fillings of contact manifolds 1.1 What is a Stein manifold?Definition 1.1 A Stein manifold is an affine complex manifold, ie a complex manifold that admits a proper holomorphic embedding into some C N .An excellent reference for Stein manifolds in the context of symplectic geometry is the recent book of Cieliebak and Eliashberg [18].In the following we give an equivalent definition of a Stein manifold.Definition 1.2An almost-complex structure on an even-dimensional manifold X is a complex structure on its tangent bundle TX , or equivalently a bundle map J W TX !TX with J ı J D id TX .The pair .X; J / is called an almost complex manifold.It is called a complex manifold if the almost complex structure is integrable, meaning that J is induced via multiplication by i in any holomorphic coordinate chart.Example 1.3 The sphere S n admits an almost complex structure if and only if n 2 f2; 6g.It is easy to see that S 2 is indeed complex, but currently it is not known whether or not S 6 admits a complex structure.Let W X !R be a smooth function on an almost complex manifold .X; J /.We set d C WD d ı J (which is a 1-form) and hence !WD dd C is a 2-form which is skew-symmetric (by definition).In general, ! may fail to be J-invariant, ie the condition !.J u; J v/ D ! .u;v/ may not hold for an arbitrary almost complex structure J .
Symplectic 4-manifolds, Stein domains, Seiberg-Witten theory and mapping class groups ANDRÁS I STIPSICZIn this survey we review some aspects of the connection between mapping class groups, symplectic 4-manifolds, Stein domains and Lefschetz fibrations on them.
A bundle with base B and fibre F aspherical closed surfaces has a section if and only if the action :π_1(B)→Out(π_1(F)) factors through Aut(π_1(F)) and a cohomology class is 0. We simplify and make more explicit the latter condition. We also show that the transgression d^2_2,0 in the homology LHS spectral sequence of a central extension is evaluation of the extension class. Examples with hyperbolic fibre and no section (based on ideas of Endo) added.
By use of a variety of techniques (most based on constructions of quasipositive knots and links, some old and others new), many smooth 3-manifolds are realized as transverse intersections of complex surfaces in complex 3-space with strictly pseudoconvex 5-spheres. These manifolds not only inherit interesting intrinsic structures (eg, they have canonical Stein-fillable contact structures), they also have extrinsic structures of a knot-theoretical nature (eq, the 3-sphere arises in infinitely many distinct ways). This survey is not comprehensive; a number of questions are left open for future work.
It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing phenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzburg models.One important application to knot homologies is the existence of "colored differentials" that relate homological invariants of knots colored by different representations. Based on this structure, we formulate a list of properties of the colored HOMFLY homology that categorifies the colored HOMFLY polynomial. By calculating the colored HOMFLY homology for symmetric and anti-symmetric representations, we find a remarkable "mirror symmetry" between these triply-graded theories.
We give elementary constructions of manifold with corner structures and associative gluing maps on compactifications of spaces of infinite, half infinite, and finite Morse flow lines.
Let $p:E -> B$ be a principal fibration with classifying map $w:B -> C$. It is well-known that the group $[X,\Omega C]$ acts on $[X,E]$ with orbit space the image of $p_#$, where $p_#: [X,E] -> [X,B]$. The isotropy subgroup of the map of $X$ to the base point of $E$ is also well-known to be the image of $[X, \Omega B]$. The isotropy subgroups for other maps $e:X -> E$ can definitely change as $e$ does. The set of homotopy classes of lifts of $f$ to the free loop space on $B$ is a group. If $f$ has a lift to $E$, the set $p_#^{-1}(f)$ is identified with the cokernel of a natural homomorphism from this group of lifts to $[X, \Omega C]$. As an example, $[X,S^2]$ is enumerated for $X$ a 4-complex.
By studying the example of smooth structures on CP^2#3(-CP^2) we illustrate how surgery on a single embedded nullhomologous torus can be utilized to change the symplectic structure, the Seiberg-Witten invariant, and hence the smooth structure on a 4-manifold.
The broken genera are orientation preserving diffeomorphism invariants of closed oriented 4-manifolds, defined via broken Lefschetz fibrations. We study the properties of the broken genera invariants, and calculate them for various 4-manifolds, while showing that the invariants are sensitive to exotic smooth structures.
The notion of a Bing cell is introduced, and it is used to define invariants, link groups, of 4-manifolds. Bing cells combine some features of both surfaces and 4-dimensional handlebodies, and the link group \lambda(M) measures certain aspects of the handle structure of a 4-manifold M. This group is a quotient of the fundamental group, and examples of manifolds are given with \pi_1(M) not equal to \lambda(M). The main construction of the paper is a generalization of the Milnor group, which is used to formulate an obstruction to embeddability of Bing cells into 4-space. Applications to the A-B slice problem and to the structure of topological arbiters are discussed.
Elementary geometric arguments are used to compute the group of homotopy classes of maps from a 4-manifold X to the 3-sphere, and to enumerate the homotopy classes of maps from X to the 2-sphere. The former completes a project initiated by Steenrod in the 1940's, and the latter provides geometric arguments for and extensions of recent homotopy theoretic results of Larry Taylor. These two results complete the computation of all the cohomotopy sets of closed oriented 4-manifolds and provide a framework for the study of Morse 2-functions on 4-manifolds, a subject that has garnered considerable recent attention.
