2018年度年会(於:東京大学)

SUMMARY: A polyhedral product is a space constructed combinatorially from a given abstract simplicial complex. Its homotopy invariants like cohomology give important combinatorially defined algebras such as Stanley–Reisner rings, and it is also important in toric topology. So there have been considerable efforts to develop the homotopy theory of polyhedral products in view of both combinatorics and topology. But there is no general technique to develop the homotopy theory of polyhedral products until the fat wedge filtration has been introduced. I will present a survey of the recent development of the homotopy theory of polyhedral products based on the fat wedge filtration and its applications. This talk is based on the joint work of Kouyemon Iriye (Osaka Prefecture University).

SUMMARY: The exponential rate of the growth rate of the number of periodic points is an important invariant of a dynamical sytems. For example, it determines the convergence radius of the dynamical zeta function of the system and, for hyperbolic dynamics, it determines the topological entropy. It is natural to ask whether ‘most’ of smooth dynamical systems exhibit at most exponential growth of the number of periodic points or not. Some classical results showed that systems in a dense subset of the set of smooth maps in general dimensions and all real-analytic one-dimensional systems exhibit such tame growth. However, in 2000, Kaloshin proved that super-exponential growth is ‘abundant’ in smooth dynamics, and recently, the author found ‘abundant’ examples in real-analytic dynamics. In the first part of the talk, we survey classical results on at most exponential growth for tame cases, including hyperbolic systems. In the second part, we discuss contemporary results for wild cases.

SUMMARY: The purpose of this talk is to explain nilpotent studies in low-dimensional topology, and to introduce my resent results on this topic. In particular, we focus on the topic of Milnor–Orr invariants, higher Massey products, and tree part of the Kontsevich invariant of links. The main result is that I gave diagrammatic computation of these invariants (of appropriate degree), and computed some examples. In the nilpotent work, the nilpotent quotient of the free group plays key role. So, in this talk, I start by reviewing properties and homology of the quotient group. After that, I briefly explain the above invariants with properties, and introduce the diagrammatic computations. Here we consider a comparison with known results concerning the mapping class group. Finally, I roughly show a future plan.

SUMMARY: I have constructed the simplex \(\mathrm {MS}_n\) of \(n\)-dim. The alternative sum of numbers of \(k\)-dim-sub-simplexes \(\mathrm {MS}^k_n\) of \(n\)-simplex \(\mathrm {MS}_n\) are New Euler–Poincaré Expanded characteristic “NEPE”. I have had \(\mathrm {NEPE}=1\) for all dim. \(n\). I have obtained the table like “Pascal’s triangle” for \(\mathrm {M}^k_n\ (k=0, 1, \cdots , n)\) and we also have \(\mathrm {M}^k_n={}_{n+1}C_{k+1}\) (\({}_{n+1}C_{k+1}\): combinatorial numbers).

SUMMARY: I have constructed the cuboid \(\mathrm {M}_n\) of \(n\)-dim. The alternative sum of numbers of \(k\)-dim. sub-cuboids \(\mathrm {M}^k_n\) are “New Euler–Poincaré Expanded characteristic NEPE”. I have had \(\mathrm {NEPE}=1\) for all dim. \(n\). We can connect \(4\) dim. cuboid with hyper planes. If the complex of cubes has \(2\)-holes, then \(\mathrm {NEPE}=-1\).

SUMMARY: New possibility of topological application is explored. In this study, topology is not only regarded as a system of morphological concepts, but also tried to be interpreted as the discussion of conceptual morphology. This means that our concepts themselves are tried to be described by the topology which is a system of our concepts, as mathematical system itself is described by mathematics in the field of foundations of mathematics.

Then, new picture of the world is tried to be drawn by such a sense of topology. The world is not regarded as a priori existence, but tried to be described as the emergence from our recognition.

SUMMARY: Any exotic pair of simply connected closed 4-manifolds are related by a cork twist. Every 4-manifold can be represented by a simple polyhedron with a coloring on each region, called a shadow. Using shadows of 4-manifolds, Costantino defined a complexity of a 4-manifold, which is the minimum number of true vertices of its shadow. We have known many examples of corks having low complexities. In this talk, we will show that there also exist infinitely many corks with large complexity.

SUMMARY: In this talk, we present a signature formula for allowable Lefschetz fibrations over \(D^2\) with planar fiber by computing Maslov index appearing in Wall’s non-additivity formula.

SUMMARY: Hitchin components are the connected components of character varieties of surface groups containing Teichmüller spaces, and the subsets of Hitchin components which correspond to Teichmüller spaces are called Fuchsian loci. Recently Bonahon–Dreyer constructed a parameterization of \({\rm PSL}_n(\mathbb {R})\)-Hitchin components by using the Anosov property of elements of \({\rm PSL}_n(\mathbb {R})\)-Hitchin components and invariants of flags introduced by Fock–Goncharov, which is a parameterization by Euclidian convex polytopes. In this talk, we give an explicit description of Fuchsian loci of a pair of pants by using the Bonahon–Dreyer parameterization.

SUMMARY: As a natural extension of the period, the pointed harmonic volume for a compact Riemann surface is defined using Chen’s iterated integrals. It captures more detailed information of the complex structure. It is also one of a few explicitly computable examples of complex analytic invariants. We obtain its new value for a certain pointed hyperelliptic curve. An application of the pointed harmonic volume is presented. We explain the relationship between the pointed harmonic volume and first extended Johnson homomorphism on the mapping class group of a pointed oriented closed surface.

SUMMARY: In this talk, we discuss whether cobordism groups of Morse functions on manifolds with boundary are trivial or not.

SUMMARY: We consider singularities of bundle homomorphisms from a tangent distribution and a vector bundle of the same rank. Generic classification of the singularities for low dimensional cases are studied.

We also consider a bundle homomorphism which is induced from a Morin map. In the case a distribution is a contact structure, we give a characterization of singularities of the bundle homomorphisms by using the contact Hamiltonian vector field.

SUMMARY: We construct a form of swallowtail singularity in \(\boldsymbol {R}^3\) which uses coordinate transformations on the source and isometries on the target. As an application, we classify configurations of asymptotic curves and characteristic curves near swallowtail.

SUMMARY: I define a rotation number of a regular closed curve on a complete euclidean/hyperbolic syrface, which, together with the free homotopy class, determines a regular homotopy class. I also give a Whitney-type formula for this rotation number.

SUMMARY: Stratifolds are considered from a categorical point of view. We show among others that the category of stratifolds fully faithfully embeds into the category of \(\mathbb R\)-algebras as does the category of smooth manifolds. We prove that a variant of the Serre–Swan theorem holds for stratifolds. In particular, the category of vector bundles over a stratifold is shown to be equivalent to the category of vector bundles over an associated affine scheme although the latter is in general larger than the stratifold itself.

SUMMARY: Discovery or recognition of the right kind of algebraic structure is often important in the development of mathematical subjects. In situations where various complex kinds of algebraic structure can arise, special technology for systematically finding and treating algebraic structures would be desirable. In particular, such technology would be necessary for broad application of higher category theory, since algebraic structures of high categorical dimension are varied and can be complicated. We shall describe how concrete understanding of higher categorical coherence leads to a systematic view on some (quite general) kinds of algebraic structure. A consequently found new phenomenon concerning topological field theories is interesting in its contrast to the cobordism hypothesis.

SUMMARY: We show that the bulk-edge correspondence for two-dimensional type A topological insulators follows directly from the cobordism invariance of the index.

SUMMARY: We consider a translation invariant bounded linear self-adjoint operator (model of a Hamiltonian) on a three-dimensional lattice (bulk) and its restrictions onto two subsemigroups (edges) and their intersection (corner). We first show that, if our bulk and edges Hamiltonians have a common spectral gap, we can define a topological invariant for the gapped bulk and edges. We next show a relation between this invariant and another invariant defined for the corner.

SUMMARY: For closed simply connected manifold \(M\) of dimension \(\geq 4\), we introduce a new spectral sequence converging to the space of knots in \(M\).

SUMMARY: We prove the odd dimensional framed little disks operads is not formal as a non-symmetric operad.

SUMMARY: A complete nonsingular toric variety (called a toric manifold) is over \(P\) if its quotient by the compact torus is homeomorphic to \(P\) as a manifold with corners. Bott manifolds are toric manifolds over an \(n\)-cube \(I^n\) and blowing them up at a fixed point produces toric manifolds over \(\mbox {vc}(I^n)\) an \(n\)-cube with one vertex cut. They are all projective. On the other hand, Oda’s \(3\)-fold, the simplest non-projective toric manifold, is over \(\mbox {vc}(I^3)\). In this paper, we classify toric manifolds over \(\mbox {vc}(I^n)\) \((n\ge 3)\) as varieties and as smooth manifolds. It consequently turns out that there are many non-projective toric manifolds over \(\mbox {vc}(I^n)\) but they are all diffeomorphic, and toric manifolds over \(\mbox {vc}(I^n)\) in some class are determined by their cohomology rings as varieties.

SUMMARY: We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form \(h=(h(1),n\cdots ,n)\) in Lie type \(A_{n-1}\). The main result gives an explicit presentation of the cohomology rings in terms of generators and their relations. Our presentation naturally specializes to Borel’s presentation of the cohomology ring of the flag variety, and it is compatible with the representation of the symmetric group on the cohomology constructed by J. Tymoczko.

SUMMARY: In the previous work, the speaker studied the positive and negative symplectic structures on the space of the pairs of normal distributions and found a Lagrangian submanifold with nice properties. In this talk, we extend this result to the space of the pairs of t-distributions and propose an application concerning the smoothness of the movement of a parameter.

SUMMARY: In 2011, Sang-hyun Kim and Thomas Koberda proved that, for any finite graphs \(\Lambda \) and \(\Gamma \), a full embedding of \(\Lambda \) into the extension graph \(\Gamma ^e\) of \(\Gamma \) gives rise to an embedding between the corresponding right-angled Artin groups, \(A(\Lambda ) \hookrightarrow A(\Gamma )\). Then the following natural question arises: for which graphs \(\Lambda \) and \(\Gamma \), can we reduce an embedding \(A(\Lambda ) \hookrightarrow A(\Gamma )\) into a full embedding \(\Lambda \rightarrow \Gamma ^e\)? Recently, several authors proved that the reduction is impossible for some \(\Lambda \) and \(\Gamma \). In this talk, we give a positive answer when \(\Lambda \) is the complement graph of a linear forest. In addition, we can further reduce an embedding \(A(\Lambda ) \hookrightarrow A(\Gamma )\) into a full embedding between the defining graphs, \(\Lambda \rightarrow \Gamma \), if \(\Lambda \) is the complement graph of a linear forest.

SUMMARY: Higman–Thompson groups are groups of homeomorphisms of the Cantor space which are locally orientation preserving. They are examples of finitely presented virtually simple groups. Generalizing these groups, Funar and Neretin defined signed Higman–Thompson groups. Signed Higman–Thompson groups are groups of homeomorphisms of the Cantor space which are locally orientation preserving or orientation reversing. In this talk, we give a necessary and sufficient condition for a signed Higman–Thompson group to be isomorphic to one of Higman–Thompson groups. This is based on a joint work with Javier Aramayona and Julio Aroca (Autonomous University of Madrid).

SUMMARY: It has been known that Kazhdan’s property (T) is semi-decidable and an algorithm to detect property (T) has been proposed. In this talk, I will describe an improved algorithm that exploits the symmetry on the given test group \(G\). The improved algorithm makes computer verification of property (T) for certain groups possible otherwise impossible. I will report the result of a large-scale calculation. This talk is based on a joint work with M. Kaluba and P. Nowak.

SUMMARY: In this talk, we consider the relation between the shadowing property and the limit shadowing property of topological dynamical systems. We show that for any continuous self-map \(f\) of a compact metric space, if \(f\) has the limit shadowing property, then the restriction of \(f\) to the non-wandering set satisfies the shadowing property. As an application, we prove the equivalence of the two shadowing properties for equicontinuous maps.

SUMMARY: We study conjugacy invariants for 2-dimensional diffeomorphisms with homoclinic cubic tangencies (two-sided tangencies of the lowest order) under certain open conditions. Ordinary arguments used in past studies of conjugacy invariants associated with one-sided tangencies do not work in the two-sided case. In this talk, we will present a new method which is applicable to the two-sided case.

SUMMARY: In this talk we discuss the relationship between topological entropy and Lyapunov exponents, provided that the Pesin entropy formula holds, and mention some results by experimental mathematics.

SUMMARY: Answering a question of Yang, we show that, for an ordered topological vector space \(Y\) with positive interior points, if each non-zero positive element is an order unit, then \(Y\) is isomorphic to the real line. We also provide a technique which reduces some vector-valued results to the original real-valued ones by using some Minkowski functionals.

SUMMARY: We discuss when \(C^*\)-embedding or \(C\)-embedding implies \(P\)-embedding in products of generalized metric spaces, such as \(M\)-spaces, \(\Sigma \)-spaces and semi-stratifiable spaces.

SUMMARY: It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this talk, we show that there exist infinitely many achiral 1-cusped hyperbolic 3-manifolds not homeomorphic to any amphicheiral null-homologous knot complement in any closed achiral 3-manifold.

SUMMARY: I will report on our recent study of chirally cosmetic surgery, that is, a pair of Dehn surgeries on a knot producing homeomorphic 3-manifolds with opposite orientations. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredients are the original and the \(SL(2,C)\) version of Casson invariants. As an application, we give a complete classification of chirally cosmetic surgeries on two bridge knots of genus one.

SUMMARY: Epstein–Penner has proved that each finite-volume cusped complete hyperbolic manifold admits a canonical decomposition into ideal polyhedra. I proved that the canonical decompositions of hyperbolic fibered two-bridge link complements are layered with respect to the fiber structures. On the other hand, Agol has shown that every pseudo-Anosov mapping torus of a surface, punctured along the singular points of the stable and unstable foliations, admits a canonical “veering” layered triangulation. In this talk, we completely determine, for each hyperbolic fibered two-bridge link, whether the canonical decomposition of its complement is veering with respect to the fiber structure.

SUMMARY: Let us glue a 3-manifold and the complement of a trivial tangle along bounding 4-punctured spheres. We propose that this construction can be regarded as an analogue of a Dehn filling. We will show an analogous result for Thurston’s hyperbolic Dehn surgery theorem.

SUMMARY: In this talk, we investigate three geometrical invariants of knots, the height, the trunk and the representativity. First, we give a counterexample for the conjecture which states that the height is additive under connected sum of knots. Next, we show that the representativity is bounded above by a half of the trunk. We also define the trunk of a tangle and show that if a knot has an essential tangle decomposition, then the representativity is bounded above by half of the trunk of either of the two tangles. Finally, we remark on the difference among Gabai’s thin position, ordered thin position and minimal critical position. We also give an example of a knot which bounds an essential non-orientable spanning surface, but has arbitrarily large representativity.

SUMMARY: We define and compare several natural ways to compute the bridge number of a knot diagram. We study bridge numbers of crossing number minimizing diagrams, as well as the behavior of diagrammatic bridge numbers under the connected sum operation. For each notion of diagrammatic bridge number considered, we find crossing number minimizing knot diagrams which fail to minimize bridge number. Furthermore, we construct a family of minimal crossing diagrams for which the difference between diagrammatic bridge number and the actual bridge number of the knot grows to infinity.

SUMMARY: Murakami introduced the Gordian distance as the least crossing-changes to transform one knot into another. Based on a matrix operation analogous to the crossing-change, he also introduced the algebraic Gordian distance between Seifert matrices. We consider the restrictions when the algebraic Gordian distance is one and improve a result of Kawauchi that if two matrices have algebraic Gordian distance one, then their Alexander polynomials have a certain relation. We give new answers to a question of Jong, showing that some Alexander polynomials cannot be realized by distance one matrices if a corresponding quadratic equation does not have an integer solution.

SUMMARY: Let \(M=\Sigma (a_1,a_2,a_3)\) be a Brieskorn homology \(3\)-sphere. Here \(2\leq a_1<a_2<a_3\) are pairwise coprime integers. Further we suppose that \(a_1=2\), or \(a_1,a_2,a_3\) are odd integers. We write \(\tau _\rho (M)\) to Reidemeister torsion of \(M\) for an irreducible representation \(\rho :\pi _1(M)\rightarrow \mathit {SL}(2;\mathbb {C})\). Now we consider the set \(RT(M)=\{\tau _\rho (M)\}\subset \mathbb {R}\) of all values, which is a finite set of real numbers.

In this talk, we would like to discuss the problem that how strong \(RT(M)\) is as an invariant for Brieskorn homology \(3\)-spheres and show \(RT(M)\) determines \(M\).

SUMMARY: Recently, J. Greene and J. Howie gave intrinsic characterizations of alternating links in terns of a pair of definite spanning surfaces. These answer the Fox problem which asked what non-diagrammatic properties characterize alternating links. In this talk, we give a characterization of alternating link exteriors in terms of cubed complexes.

SUMMARY: We call the link obtained from a ball of sepaktakraw by replacing each annulus piece with a circle the sepatakraw link. The sepaktakraw link is an alternating link with rich symmetry. In this talk, we introduce sevearl properties of the sepaktakraw link.

SUMMARY: Let \(P,P^{\prime }\) be spherical curves. Suppose that \(P,P^{\prime }\) are reduced spherical curves. Then the following conditions are piarwise equivalent.
(A) \( P^{\prime }\) is obtained from \(P\) by applying a sequence of deformations of RI, RIII and ambient isotopy.
(B) \( P^{\prime }\) is obtained from \(P\) by applying a sequence deformations of RIII, \(\alpha \), \(\beta \) and ambient isotopy.

SUMMARY: Vassiliev introduced filtered invariants of knots using crossing changes (1990), called finite type invariants. For the finite type invariants, Ohyama introduced a notion of n-triviality (1990) and Taniyama generalized it to obtain a notion of n-similarity (1992). Goussarov, Polyak, and Viro introduced universal finite type invariants of virtual knots using virtualization (2000). We mimicked their ideas, and defined finite type invariants of virtual knots and introduced a notion that corresponds to n-similarity, using forbidden moves (J. Math. Soc. Japan). In this talk, we give infinitely many examples of n-similar pairs of virtual knots by forbidden moves and show that every invariant of Goussarov, Polyak, and Viro is an invariant of us.

SUMMARY: A \(C_n\)-move is a family of local moves on knots and links, which gives a topological characterization of finite type invariants of knots. We extend the \(C_n\)-move to (long) virtual knots by using the lower central series of the pure virtual braid, and call it a virtual \(C_n\)-move. We then prove that for long virtual knots a virtual \(C_n\)-equivalence generated by virtual \(C_n\)-moves is equal to \(n\)-equivalence, which is an equivalence relation on (long) virtual knots defined by Goussarov–Polyak–Viro. Moreover we directly prove that two long virtual knots are not distinguished by any finite type invariants of degree \(n-1\) if they are virtual \(C_n\)-equivalent, for any positive integer \(n\).

SUMMARY: Two handlebody-links are HL-homotopic if they are transformed from one to the other by a sequence of self-crossing changes of their handles. For 3-component handlebody-links, we construct a bijection between the set of the HL-homotopy classes and the set of the equivalence classes of the set of tuples of one 3-dimensional matrix and three matrices with respect to some relations.

SUMMARY: The Dijkgraaf–Witten invariant is a topological invariant for compact oriented 3-manifolds in terms of a finite group and its 3-cocycle. The invariant is a state sum invariant constructed by using a triangulation, likewise the Turaev–Viro invariant. In this talk, we discuss an extention of the Dijkgraaf–Witten invariants to cusped 3-manifolds. We show that the Dijkgraaf–Witten invariants distinguish some pairs of orientable cusped hyperbolic 3-manifolds with the same hyperbolic volumes and Turaev–Viro invariants. We also give an example of a pair of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and homology groups, meanwhile with the distinct Dijkgraaf–Witten invariants.

SUMMARY: The colored Jones polynomial is a \(q\)-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A \(q\)-series called the tail of a knot \(K\) is obtained as the limit of the \(\mathfrak {sl}_2\) colored Jones polynomials \(\{J_n(K;q)\}_n\) (\(n\to \infty \)). We give two explicit formulae of the tail of the \(\mathfrak {sl}_3\) colored Jones polynomials colored by \((n,0)\) for the \((2,2m)\)-torus link. These two expressions of the tail derive Andrews–Gordon identities for the \(\mathfrak {sl}_3\) false theta function.

SUMMARY: Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk. We derive some formulas for \(A_1\) and \(A_2\) clasped web spaces by graphical calculus using skein theory. These formulae are colored version of the skein relation, a twist formula, and a bubble skein expansion formula. We calculate the \(\mathfrak {sl}_2\) and \(\mathfrak {sl}_3\) colored Jones polynomials of \(2\)-bridge knots and links explicitly using the twist formula.

SUMMARY: We define a functor \(\mathcal {Q}\) from the category of multiple conjugation biquandles to that of multiple conjugation quandles. We show that for any multiple conjugation biquandle \(X\), there is a one-to-one correspondence between the set of \(X\)-colorings and that of \(\mathcal {Q}(X)\)-colorings diagrammatically for any handlebody-link and spatial trivalent graph.

SUMMARY: The minimal coloring number of a \(\mathbb {Z}\)-colorable link is the minimal number of colors for non-trivial \(\mathbb {Z}\)-colorings on diagrams of the link. We determine the minimal coloring number for any \(\mathbb {Z}\)-colorable links.

SUMMARY: We calculate the twisted Alexander polynomials of \((-2,3,2n+1)\)-pretzel knots associated to their holonomy representations.