A survey of some topics around the 11/8-conjecture is given.
We give a survey for the Fukumoto-Furuta invariant, which is defined for a pair of a closed 3-manifold and a spin 4-orbifold bounded by it. In certain cases, such as Seifert rational homology 3-spheres, we can consider it as a spin homology cobordism invariant for a 3-manifold by choosing a "canonical" spin 4-orbifold. The proof is based on the orbifold 10/8 theorem.
This is a survey talk on recent developments in the global theory of singularities of differentiable maps. We give several remarkable results concerning differentiable structures of manifolds and singularities of differentiable maps. A generalization of the theory of Morse functions to generic maps is also discussed.
In this talk, we study some topological properties of stable maps of 4-manifolds into a surface whose regular fibers are disjoint union of 2-spheres. If a closed 4-manifold admits such a map, then it bounds a 5-manifold. Each regular fiber of such a map of the 4-sphere into the plane is a homotopy ribbon 2-link. And any spun 2-knot of a classical knot can be realized as a component of a regular fiber of such a map.
Let S2 be the oriented 2-sphere, S01⊂S2 its equator and R2 the oriented plane. We denote by F(S2, R2; S01) the space of all fold maps of S2 to R2 such that the singular set coincides with S01. Here, a fold map is a smooth map with only fold singularities. In 1970's, Eliashberg proved that F(S2, R2; S01) is homotopic to disjoint union of four circles. In his paper, he only mentioned that this is an application of his elegant theory, the homotopy principle for fold maps. In this talk, we introduce another method to determine the number of connected components of F(S2, R2; S01). Comparing to Eliashberg's method, our method is concrete, combinatorial.
We explain the shape of singular fibers of degenerations of Riemann surfaces. Then we discuss some topics on deformations of degenerations.
A family of A'Campo's divide knots yielding "at most" graph manifolds by Dehn surgery is constructed. The proof is by a sequence of Kirby calculus and related to the resolution of a singularity of a complex curve. This is an extension of the author's recent work on J. Berge's family of lens surgery. The motivation is a question by M. Ue in topology on 4-manifolds.
In this talk I will address the problem of representing 2-dimensional homology classes of simply connected, piecewise linear (PL) 4-manifolds with topologically embedded 2-spheres. The first theorem states that each such class can be represented by a relatively simple codimension 0 submanifold.
Theorem 1. If W is a compact, simply connected, PL 4-manifold, then each element ofH2(W) can be represented by a compact PL submanifold M⊂W such that M consists of a Mazur-like contractible 4-manifold with a single 2-handle attached.
A compact contractible PL 4-manifold is Mazur-like if it has a handle
decomposition in which there is one 0-handle, no handles of index greater than 2,
and the attaching map for the ith 2-handle is homotopic to the loop represented
by the ith 1-handle.
The manifold M represents a specified element of
Theorem 1 is the main ingredient in the proof of the following result.
Theorem 2. If W is a compact, simply connected, PL submanifold of S4, then each element ofH2(W) can be represented by a locally flat topological embedding of S2.
Both Theorems 1 and 2 are false without the hypothesis that W is compact.
For studying various 4-dimensional topological objects, Lefschetz fibrations of 4-manifolds, 2-dimensional braids, algbraic curves, etc., we need to treat monodromy representations. However it is usually hard to classify the monodromy representations or even to decide whether two given representations are isomorphic or not. Here we introduce the notions of rack and chart and show how they appear in studying monodromy representations. This is based on a series of researchs by Professor Yukio Matsumoto and myself, and other collaborators.
In my talk, we investigate properties of minimal charts which we need to prove that there is no minimal chart with exactly seven white vertices.
Quandle cohomology theory was developed to define invariants, called quandle cocycle (knot) invariants, for classical knots and knotted surfaces in state-sum form. The quandle cohomology theory is a modification of rack cohomology theory. The cocycle knot invariants are analogous in their definitions to the Dijkgraaf-Witten invariants of triangulated 3-manifolds with finite gauge groups. In this paper, we give a brief review with a few new progresses on quandle cocycle invariants, on two aspects: (1) constructions and computations, and (2) applications.
In two-dimensional knot theory, Alexander's theorem and Markov's theorem were proved. The former insists that every oriented surface-link is ambient isotopic to the closure of a surface braid, and the latter gives the necessary and sufficient condition for closures of two surface braids to be ambient isotopic to the same surface-link. At present, however, there exist few results about relation between surface-links and surface braids. In this talk, we show that there exist non-ribbon surface braids whose closures are ribbon surface-links.
The braid index of a surface-knot F is the minimal number among the degrees of all simple surface braids whose closures are ambient isotopic to F. We prove that an S2-knot which is the connected sum of k copies of the spun (2; p)-torus knot has the braid index k+2. To prove it, we use colorings of surface-knots by quandles and give lower bounds of the braid index of surface-knots.
The projections of pseudo-ribbon 2-knots are immersions of S2 into R3 whose self-intersection sets consist of only double points. Their inverse images are circles on S2. We show a necessary and sufficient condition to realize as an immersion in R3 when we arbitrarily decide a configuration of circles on S2 and how they intersect.
In this talk we introduce a conjecture about local monodromies of a fibration from the Fermat type surface to a line. Here we made computer experiments using software Monomie.
Recently we often hear the words like Numerical Analysis, Numerical method, Numerical Calculation or Numerical Computation, Numerical Simulation, Computational Science and so on. The progress of computer hardware technology during the last few decades is very fast and eminent. For example, in some research institutes it is not so difficult for people to use computers of T(Tera)FLOPS class. TFLOPS means that the computer executes 1012 times basic floating operation, i.e. addition, reduction, multiplication, or division, per second. The large computing power has realized the idea to solve the partial differential equations (PDE's) numerically, exactly speaking, to obtain an approximate solution by calculating a discretized model of the original PDE.
Computational Fluid Dynamics, which is often called CFD in abbreviation, is one of the examples.
Seeing the situation above, people may think that so called numerical methodology including CFD has realized a great progress. Of course it is not wrong. But from the viewpoint of mathematics or mathematical manner of thinking, there are still many problems left open. Some of them are not only important from the practical viewpoint but also interesting from the theoretical viewpoint. Here we would like to show a few example of such problems.
This is a brief survey of surgery theory. The first part describes the classical surgery theroy. The second part describes the developments in the last 10 years mainly concerning controlled surgery therory.
In this lecture, I will introduce the terms to understand the following statement:
Theorem 1 Let M be an open, connected, oriented 3-manifold. Let Mˆ denote its Freudenthal compactification. Then, there exist a 3-fold simple branched covering p : Mˆ → S3 such that p maps the end space E(M) of M homeomorphically onto a tame subset T of S3. The 3-fold branched covering p | M : M → S3−T is simple, and the branching set is a locally finite disjoint union of strings (properly embedded arcs).
I will apply the theorem to the proof that the Whitehead contractible 3-manifold is a 2-fold covering of R3 branched over the string shown in the Figure.
I will also explore the possibility of embedding a non trivial string (R3, K) in the trivial knot (S3, U). I will give some examples. Since the complementary space in S3 of the image of R3 under the embedding is a continuum, I will show that some well known snake-like continua appear as these residual spaces. I will give a concrete descriptions of the p-adic solenoids and the Whitehead continuum. I will show that this last is homeomorphic to Bing's snake-like continuum without end points.
A fibering structure over CP1 will be introduced to the degree n Fermat surface Vn. The exact positions and the types of all the sigular fibers of this fibration will be determined on the assumption that n≠1 (mod 6). In the excluded case n≡1 (mod 6), we have no general result. We have in this case, however, a recipe for computing the types and the positions of all the singular fibers in the fibration Vn→CP1 each time a concrete number n is given.
The new construction of Smooth (4k-1,6k)-Knots shows that their unknotting number is 1.
The first half of this talk will survey a recent work of G. Perelman on the geometization of 3-manifolds, which implies a topological classification of 3-manifolds. Then, knowing the topology of 3-manifolds, we discuss a possible direction for understanding very deep mathematical structures behind the world where the 3-manifold plays.
By virtue of recent work of Perelman on the geometrisation conjecture for 3-manifolds, it has turned out to be the case that to study the topology of 3-manifolds, what remains to be done is to understand hyperbolic 3-manifolds completely. Naturally, the study of hyperbolic 3-manifolds is closely related to that of Kleinian groups. In this talk, we should like to discuss the present situation and the recent progress (y compris the speaker's own work) in this field and exhibit the prospects.
Braid foliations on surfaces are introduced and studied by Bennequin, and independently by Birman and Menasco. In this talk, we study braid foliations on Seifert surfaces of genus one bounded by knots of genus one, and we give a characterization of braid foliations on such surfaces.
In this talk I want to review R. Hamilton's papers on the Ricci flow in dimension 4, and hope to discuss some general (unstable) features of the Ricci flow in this dimension.
In the former half of my talk, I explain a brief survey on some fundamental open problems on the Kontsevich invariant of knots and the LMO invariant of 3-manifolds from the viewpoint of the classification problems of knots and 3-manifolds. In the latter half of the talk, I explain the loop expansion of the Kontsevich invariant, which is related to one of the open problems. Further, I give a result on a formula to calculate the 2-loop polynomial of knots, which presents the 2-loop part of the loop expansion.
How many Reidemeister moves do we need for unknotting a given diagram of the trivial knot? Hass and Lagarias gave an upper bound. We give an upper bound for deforming a diagram of a split link to be disconnected. On the other hand, the absolute value of the writhe gives a lower bound of the number of Reidemeister I moves for unknotting. That of a complexity of knot diagram "cowrithe" works for Reidemeister II, III moves. We give an example of an infinite sequence of diagrams Dn of the trivial knot with an O(n) number of crossings such that the author expects the number of Reidemeister moves needed for unknotting it to be O(n2). However, writhe and cowrithe do not prove this.
We explain the definition of a category B– of bottom tangles in homology handlebodies.
We give a set of generators of B– as a monoidal category.
We will give a very rough description of a functor defined on B–
associated to the quantized enveloping algebra