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# アブストラクト事後公開 — 2018年度年会(於:東京大学)

## トポロジー分科会

 特別講演 ポリヘドラル・プロダクトのホモトピー論 岸本大祐 (京大理) 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). msjmeeting-2018mar-10i001 PDF 特別講演 滑らかな力学系の周期点の個数の増大度 浅岡正幸 (京大理) 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. msjmeeting-2018mar-10i002 PDF 特別講演 3次元トポロジーの冪零的研究 野坂武史 (東工大) 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. msjmeeting-2018mar-10i003 PDF 1. $n$次元単体の構成とオイラーポアンカレ標数の拡張と組合せの数 岩山瑛子 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). PDF 2. $n$次元直方体の三角形とオイラーポアンカレ標数の拡張 岩山瑛子 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 $\mathrm{NEPE}=-1$. PDF 3. トポロジーが持つ応用可能性について 小笠原義仁 (早大複雑系高等研) 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. PDF 4. Infinitely many corks having large shadow-complexities 直江央寛 (東北大理) 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. PDF 5. 種数0のLefschetzファイバー空間の符号数について 宮村　旭 (東工大理) 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. PDF 6. パンツのHitchin成分のFuchs跡について 稲垣友介 (阪大理) 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. PDF 7. Pointed harmonic volume and its relation to extended Johnson homomorphism 田所勇樹 (木更津工高専) 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. PDF 8. 境界付き多様体上のモース関数の同境群 山本卓宏 (東京学大教育) In this talk, we discuss whether cobordism groups of Morse functions on manifolds with boundary are trivial or not. PDF 9. 接分布と同階数ベクトル束の間の束準同型の特異点 土田　旭 (北大理)・​佐治健太郎 (神戸大理) 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. PDF 10. スワローテイルの $SO(3)$-標準形と幾何的模様 佐治健太郎 (神戸大理) 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. PDF 11. 非球面的曲面における正則閉曲線の回転数について 山崎正之 (岡山理大理) 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. PDF 12. 階層体の圏とSerre–Swanの定理 栗林勝彦 (信州大理)・​青木稔樹 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. PDF 13. Higher theories of algebraic structures 松岡拓男 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. PDF 14. バルクエッジ対応と指数のコボルディズム不変性 林　　晋 (MathAM-OIL) We show that the bulk-edge correspondence for two-dimensional type A topological insulators follows directly from the cobordism invariance of the index. PDF 15. バルクエッジコーナー対応 林　　晋 (MathAM-OIL) 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. PDF 16. The space of knots in a manifold and the right operadic module of configuration spaces 森谷駿二 (阪府大) For closed simply connected manifold $M$ of dimension $\geq 4$, we introduce a new spectral sequence converging to the space of knots in $M$. PDF 17. Non-formality of the odd dimensional framed little disks operads 森谷駿二 (阪府大) We prove the odd dimensional framed little disks operads is not formal as a non-symmetric operad. PDF 18. $n$-cube の頂点カット上のトーリック多様体の分類 鍬田英也 (近畿大工高専)・​蓮井　翔 (阪府大高等教育)・​枡田幹也 (阪市大理)・​朴　善正 (阪市大理) 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. PDF 19. 正則半単純なヘッセンバーグ多様体のコホモロジー環について 阿部　拓 (阪市大数学研)・​堀口達也 (阪大情報／阪市大数学研)・​枡田幹也 (阪市大理) 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. PDF 20. t分布のシンプレクティック幾何・接触幾何 森　淳秀 (大阪歯大歯) 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. PDF 21. Embeddability of the right-angled Artin groups on the complement graphs of linear forests 片山拓弥 (広島大理) 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. PDF 22. 符号付きトンプソン群の同型問題 加藤本子 (東大数理) 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). PDF 23. Kazhdan’s property (T) and semidefinite programming 小沢登高 (京大数理研) 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. PDF 24. On the shadowing and limit shadowing properties 川口徳昭 (東大数理) 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. PDF 25. 3次接触をもつ2次元微分同相写像のmoduli 橋本　忍 (首都大東京理工) 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. PDF 26. 位相的エントロピーとリアプノフ指数の関係 平出耕一 (愛媛大理)・​松岡千博 (阪市大工) 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. PDF 27. Questions on monotone operators for vector-valued maps 山﨑薫里 (高崎経大経済) 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. PDF 28. Three embeddings and their implications in products of generalized metric spaces 矢島幸信 (神奈川大工)・​平田康史 (神奈川大工) 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. PDF 29. Achiral 1-cusped hyperbolic 3-manifolds not coming from amphicheiral null-homologous knot complements 鄭　仁大 (近畿大理工)・​市原一裕 (日大文理)・​谷山公規 (早大教育) 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. PDF 30. 対掌矯飾的手術とキャッソン不変量 市原一裕 (日大文理)・​伊藤哲也 (阪大理)・​斎藤敏夫 (上越教育大) 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. PDF 31. 双曲的ファイバー二橋絡み目補空間の標準的分割の veering 構造 阪田直樹 (広島大理) 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. PDF 32. タングルの貼り合わせにおける双曲性 吉田建一 (京大理) 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. PDF 33. Height, trunk and representativity of knots R. Blair (California State Univ.)・​小沢　誠 (駒澤大総合) 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. PDF 34. The incompatibility of crossing number and bridge number for knot diagrams R. Blair (California State Univ.)・​A. A. Kjuchukova (Univ. of Pennsylvania)・​小沢　誠 (駒澤大総合) 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. PDF 35. A pair of Seifert matrices that cannot have algebraic Gordian distance one Jie Chen (東北大情報) 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. PDF 36. Brieskornホモロジー$3$球面の$\mathit{SL}(2;\mathbb{C})$-既約表現のReidemeister torsionについて 北野晃朗 (創価大理工) Let $M=\Sigma(a_1,a_2,a_3)$ be a Brieskorn homology $3$-sphere. Here \$2\leq a_1