Reducing and annular Dehn fillings

If two Dehn fillings on a simple manifold create a reducible manifold and an annular manifold respectively, then the distance between those filling slopes is known to be at most two. Moreover, Eudave-Muñoz and Wu gave infinitely many examples of manifolds admitting reducing and annular Dehn fillings at distance two. In this paper, we complement their examples to establish a complete list of simple manifolds admitting such a pair of Dehn fillings.

     

k-noncrossing and k-nonnesting graphs and fillings of Ferrers diagrams

We study the distribution of k-crossings and k-nestings and other patterns in ordered graphs by using fillings of Ferrers diagrams. The main result states that there are as many graphs without k-crossings as without k-nestings. We also show that studying equirrestrictive patterns in ordered graphs is equivalent to studying equirrestrictive matrices in fillings of Ferrers diagrams.     

Contact open books with exotic pages

We consider a fixed contact 3-manifold that admits infinitely many compact Stein fillings which are all homeomorphic but pairwise non-diffeomorphic. Each of these fillings gives rise to a closed contact 5-manifold described as a contact open book whose page is the filling at hand and whose monodromy is the identity symplectomorphism. We show that the resulting infinitely many contact 5-manifolds are all diffeomorphic but pairwise non-contactomorphic. Moreover, we explicitly determine these contact 5-manifolds.     

On symplectic cobordisms

In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are “symplectic cobordism equivalent”. Received: 26 March 2001 / Revised version: 1 May 2001 / Published online: 28 February 2002     

Examples of isolated surface singularities whose links have infinitely many symplectic fillings

For certain classes of isolated complex surface singularities, it is shown that there exist infinitely many distinct topological types of minimal symplectic fillings of the link of the singularity.     

One-dimensional minimal fillings with negative edge weights

A.O. Ivanov and A.A. Tuzhilin proposed a particular case of Gromov??s minimal fillings problem generalized to the case of stratified manifolds using weighted graphs with a nonnegative weight function as minimal fillings of finite metric spaces. In this paper we introduce generalized minimal fillings, i.e., the minimal fillings where the weight function is not necessarily nonnegative. We prove that for any finite metric space its minimal filling has the minimum weight in the class of its generalized fillings.     

Row-Strict Quasisymmetric Schur Functions

Haglund, Luoto, Mason, and van Willigenburg introduced a basis for quasisymmetric functions, called the quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as Schur functions are generated through reverse column-strict tableaux. We introduce a new basis for quasisymmetric functions, called the row-strict quasisymmetric Schur function basis, generated combinatorially through fillings of composition diagrams in much the same way as quasisymmetic Schur functions are generated through fillings of composition diagrams. We describe the relationship between this new basis and other known bases for quasisymmetric functions, as well as its relationship to Schur polynomials. We obtain a refinement of the omega transform operator as a result of these relationships.     

Annular Dehn fillings

We show that if a simple 3-manifold M has two Dehn fillings at distance , each of which contains an essential annulus, then M is one of three specific 2-component link exteriors in S ³ . One of these has such a pair of annular fillings with , and the other two have pairs with . Received: February 20, 1999.     

The functions that do not change types of minimal fillings

It is proved that a function changing distances in finite metric spaces and preserving the types of their minimal fillings has the form f (x) = kx + b. It is sufficient to assume that the types of fillings are preserved for spaces consisting of at most five points.     

Complexity of Geometric Three-manifolds

We compute for all orientable irreducible geometric 3-manifolds certain complexity functions that approximate from above Matveev's natural complexity, known to be equal to the minimal number of tetrahedra in a triangulation. We can show that the upper bounds on Matveev's complexity implied by our computations are sharp for thousands of manifolds, and we conjecture they are for infinitely many, including all Seifert manifolds. Our computations and estimates apply to all the Dehn fillings of M 6 ₁ ³ (the complement of the three-component chain-link, conjectured to be the smallest triply cusped hyperbolic manifold), whence to infinitely many among the smallest closed hyperbolic manifolds. Our computations are based on the machinery of the decomposition into ‘bricks’ of irreducible manifolds, developed in a previous paper. As an application of our results we completely describe the geometry of all 3-manifolds of complexity up to 9.     

Exotic Stein fillings with arbitrary fundamental group

Let G be a finitely presentable group. We provide an infinite family of homeomorphic but pairwise non-diffeomorphic, symplectic but non-complex closed 4-manifolds with fundamental group G such that each member of the family admits a Lefschetz fibration of the same genus over the two-sphere. As a corollary, we also show the existence of a contact 3-manifold which admits infinitely many homeomorphic but pairwise non-diffeomorphic Stein fillings such that the fundamental group of each filling is isomorphic to G. Moreover, we observe that the contact 3-manifold above is contactomorphic to the link of some isolated complex surface singularity equipped with its canonical contact structure.     

Tight contact structures with no symplectic fillings

We exhibit tight contact structures on 3-manifolds that do not admit any symplectic fillings. Oblatum 7-XII-2000 & 14-XI-2001?Published online: 9 April 2002     

<Emphasis Type="Italic">k</Emphasis>-noncrossing and <Emphasis Type="Italic">k</Emphasis>-nonnesting graphs and fillings of ferrers diagrams

We give a correspondence between graphs with a given degree sequence and fillings of Ferrers diagrams by nonnegative integers with prescribed row and column sums. In this setting, k-crossings and k-nestings of the graph become occurrences of the identity and the antiidentity matrices in the filling. We use this to show the equality of the numbers of k-noncrossing and k-nonnesting graphs with a given degree sequence. This generalizes the analogous result for matchings and partition graphs of Chen, Deng, Du, Stanley, and Yan, and extends results of Klazar to k > 2. Moreover, this correspondence reinforces the links recently discovered by Krattenthaler between fillings of diagrams and the results of Chen et al.     

Non degenerating Dehn fillings on genus two Heegaard splittings of knots′ complements

It is Thurston's result that for a hyperbolic knot K in S~3, almost all Dehn fillings on its complement result in hyperbolic 3-manifolds except some exceptional cases. So almost all produced 3-manifolds have the same geometry. It is known that its complement in S~3, denoted by E(K), admits a Heegaard splitting. Then it is expected that there is a similar result on Heegaard distance for Dehn fillings. In this paper, Dehn fillings on genus two Heegaard splittings are studied. More precisely, we prove that if the distance of a given genus two Heegaard splitting of E(K) is at least 3, then for any two degenerating slopes on ?E(K), there is a universal bound of their distance in the curve complex of ?E(K).     

Asymptotic Formulas for Precision of Discrete Spectral Problems

We find principal terms in the power expansion, with respect to the step of a square grid, of the eigenvalue error for a discrete analogue of spectral problems for elliptic operators of the second and fourth order. We use the compactness of a bounded set in a Hilbert space, which gives the mean convergence of piecewise-constant fillings of grid eigenfunctions and the weak convergence of these fillings for difference derivatives. This, in turn, allows one to prove that eigenfunctions of the initial problems belong to the corresponding Sobolev spaces.     

Tiling space with the aid of the holomorph

The notion of a factorization of a group is generalized and a method is presented for obtaining new factorizations from old ones. The results are applied to obtain new fillings of the lattice spaces Z, Z ⊕ Z and the cube.     

Embedding fillings of contact 3-manifolds

In this survey article, we describe different ways of embedding fillings of contact 3-manifolds into closed symplectic 4-manifolds.     

The additivity criterion for finite metric spaces and minimal fillings

A new additivity criterion for finite metric spaces is obtained. The criterion is based on properties of minimal fillings in the sense of M. Gromov     

Bijections between pattern-avoiding fillings of Young diagrams

The pattern-avoiding fillings of Young diagrams we study arose from Postnikov's work on positive Grassmann cells. They are called -diagrams, and are in bijection with decorated permutations. Other closely-related fillings are interpreted as acyclic orientations of some bipartite graphs. The definition of the diagrams is the same but the avoided patterns are different. We give here bijections proving that the number of pattern-avoiding filling of a Young diagram is the same, for these two different sets of patterns. The result was obtained by Postnikov via a recurrence relation. This relation was extended by Spiridonov to obtain more general results about other patterns and other polyominoes than Young diagrams, and we show that our bijections also extend to more general polyominoes.     

Filling by holomorphic curves in symplectic 4-manifolds

We develop a general framework for embedded (immersed) -holomorphic curves and a systematic treatment of the theory of filling by holomorphic curves in 4-dimensional symplectic manifolds. In particular, a deformation theory and an intersection theory for -holomorphic curves with boundary are developed. Bishop's local filling theorem is extended to almost complex manifolds. Existence and uniqueness of global fillings are given complete proofs. Then they are extended to the situation with nontrivial -holomorphic spheres, culminating in the construction of singular fillings.