一般社団法人 日本数学会 Application Server

# アブストラクト事後公開 — 2017年度秋季総合分科会(於:山形大学)

## トポロジー分科会

 特別講演 写像類群, Goldman–Turaev Lie 双代数, 柏原 Vergne 問題 河澄響矢 (東大数理) Let $\Sigma$ be a compact connected oriented surface with non-empty boundary and a framing $f$. Then a subset of the mapping class group of $\Sigma$, which includes the Torelli group, is naturally embedded into the (completed) Goldman–Turaev Lie bialgebra of $\Sigma$. A framed version of the Turaev cobracket vanishes on the image of the embedding. So we need a formal description of the Goldman–Turaev Lie bialgebra. In the genus $0$ case, the set of expansions inducing a formal description of the bialgebra is naturally bijective to the set of solutions of the Kashiwara–Vergne problem in the formulation of Alekseev–Torossian. In view of this bijection, we can formulate a Kashiwara–Vergne problem associated with $(\Sigma, f)$. The set of its solutions is non-empty except some of the genus $1$ case. This talk is based on a joint work with Anton Alekseev, Yusuke Kuno and Florian Naef. PDF 2017年度日本数学会幾何学賞受賞特別講演 $3$次元多様体のファイバー曲面とヘガード曲面 作間　誠 (広島大理) Though fiber surfaces and Heegaard surfaces have completely different natures, we can find various analogies between them. We describe the analogies from the view points of (1) the branched fibration theorem, (2) monodromy groups, (3) McShane’s identity and (4) geometric structures. PDF 特別講演 Sullivan’s coproduct on the reduced loop homology 内藤貴仁 (東大数理) In the theory of string topology initiated by Chas and Sullivan, the homology of free loop spaces of manifolds (called the loop homology) has very rich algebraic structures. The loop product is a multiplication on the loop homology and it is the most basic operation in string topology. Cohen and Godin discovered a 2-dimensional TQFT structure on the homology. Moreover, Godin showed that the loop homology is a homological conformal field theory. In this talk, I will discuss a coproduct on the reduced loop homology which is introduced by Sullivan. The coproduct and the loop product give the loop homology an infinitesimal bialgebra structure. I will explain how to construct Sullivan’s coproduct and give some computational examples. PDF 1. Signatures of surface bundles 門田直之 (大阪電通大工) The signature of a surface bundle over a surface has some restrictions, for examples, it is dividable by 4 and vanishes if the base genus is 0 or 1. Bryan and Donagi constructed examples over a genus-2 surface with non-zero signatures. However, the signatures and the genera of their examples are sporadic. In this talk, for any positive integer $n$, we give a surface bundle of fiber genus $g$ over a surface of genus 2 with signature $4n$ and a section of self-intersection 0 if $g$ is greater than or equal to $39n$. PDF 2. Stable commutator lengths of Dehn twists 門田直之 (大阪電通大工) In this talk, we give explicit factrizations of certain powers of Dehn twists as products of commutators. As a corollary, we improve upper bounds for stable commutator lengths of Dehn twists. Moreover, we show that the stable commutator length in the mapping class group is different from that in the hyperelliptic mapping class group for a surface of large genus. PDF 3. コルクとそのシャドウ複雑度 直江央寛 (東北大理) For a simply connected closed $4$-manifold $M$, any $4$-manifold exotic to $M$ is obtained from $X$ by twisting a contractible Stein domain called a cork. We study a cork from a viewpoint of the notion of shadows. A shadow of a $4$-manifold is a simple polyhedron collapsed from the manifold. By using a shadow, Costantino defined a complexity of a $4$-manifold, which is the minimum number of vertex of its shadow. We show that there are no corks with complexity zero and that there are infinitely many corks with complexity $1$ and $2$. PDF 4. Nonexistence of twists and surgeries generating exotic 4-manifolds 安井弘一 (阪大情報) It is well known that for any exotic pair of simply connected closed 4-manifolds, one is obtained by twisting the other along a contractible submanifold. In contrast, we show that for each positive integer $n$, there exists an infinite family of pairwise exotic simply connected closed 4-manifolds such that, for any 4-manifold $X$ and any compact (not necessarily connected) codimension zero submanifold $W$ with $b_1(\partial W) 2$), any example of $n$-trivial classical and virtual knot by virtualizations is still missing. In this talk, we obtain an example of $n$-trivial knots by virtualizations. We also introduce a new filtration of Vassiliev-type invariants by using an unknotting operation, called Forbidden moves. We obtain $n$-trivial knots of this new degree. PDF 26. 自明な$(2,1)$ケーブル$\Gamma$多項式をもつ結び目の無限族について 滝岡英雄 (阪市大数学研) It is known that there exist many polynomial invariants for knots. For example, Alexander–Conway, Jones, $\Gamma$, $Q$, HOMFLYPT, Kauffman polynomials are well known. These polynomials of the trivial knot are one. The problem is whether there exists a non-trivial knot such that these polynomials are one. It is known that there exists such a knot for the Alexander–Conway, $\Gamma$, $Q$ polynomials. However, it is still an open problem for the other polynomial invariants. Moreover, we consider the $(p,1)$-cable versions of these polynomial invariants for an integer $p(\geq 2)$. These $(p,1)$-cable versions of the trivial knot are one. The problem is whether there exists a non-trivial knot such that these $(p,1)$-cable versions are one. It is known that there exists such a knot for the Alexander–Conway polynomial. However, it is still an open problem for the other polynomial invariants. In this talk, we show that there exist infinitely many knots such that the $(2,1)$-cable version of the $\Gamma$-polynomial for the knots is one. PDF 28. 部分的モノイドにラベルをもつ区間の配置空間 奥山真吾 (香川高専)・​島川和久 (岡山大理) A configuration space of intervals in 1-dimensional Euclidean space with partially summable labels is constructed. It is a kind of an extension of the configuration space with partially summable labels constructed by the second author and at the same time a generalization of the configuration space of intervals with labels in a based space constructed by the first author. An approximation theorem of the preceding configuration space is generalized to our case. More precisely, we construct a configuration space of intervals in $\mathbb{R}$ with labels in a partial abelian monoid $M$, and show that it is weakly homotopy equivalent to the space of based loops on the classifying space of $M$. PDF 29. Generic linear perturbations 一木俊助 (横浜国大環境情報) In his celebrated paper “Generic projections”, John Mather has shown that almost all linear projections from a submanifold of a vector space into a subspace yield a stable mapping in the nice dimensions. In this talk, an improvement of the Mather result is given. Namely, almost all linear perturbations of a smooth mapping from a submanifold of $\mathbb{R}^m$ into $\mathbb{R}^\ell$ yield a stable mapping in the nice dimensions. PDF 30. Stability of $C^\infty$ convex integrands E. B. Batista (Federal Univ. of Cariri)・​韓　呼和 (横浜国大環境情報)・​西村尚史 (横浜国大環境情報) In this talk, it is shown that the set consisting of stable convex integrands $S^n\to \mathbb{R}_+$ is open and dense in the set consisting of $C^\infty$ convex integrands with respect to Whitney $C^\infty$ topology. Moreover, it is given examples representing well why stable convex integrands are preferred. PDF 31. 写像の特異点集合の自己交差と不足符号数 清水達郎 (京大数理研) We give a geometric proof of that the $k$-times self-intersection of singular set of a generic smooth map from $n$-dimensional manifold $X$ to $R^p$ coincides with the corank (of Jacobian)$=k$ singular set of any generic map from $X$ to $R^{p+k-1}$ as homology classes with $Z/2$ coefficient ($(n-p+2)k>n+1$). As an application we give a description of the signature defect of framed 3-manifold from the point of view of singular sets of maps. PDF 32. Essentially weakly onesided resolving endomorphisms of the shift 那須正和 Characterizations are given for essentially weakly onesided resolving endomorphisms of subshifts. PDF 33. On the limits of onesided resolving directions of endomorphisms of subshifts of finite type 那須正和 We present the results that there exists an automorphism of a full shift having a limit of onesided resolving directions of type II or III and that no automorphism of a transitive subshift of finite type has an irrational unique non-expansive direction. PDF 34. トポロジカルエントロピーと1次元連続体の幾何学的構造 加藤久男 (筑波大数理物質) In this talk, we define a new notion of “free tracing property by free chains” on $G$-like continua and we prove that a positive topological entropy homeomorphism on a $G$-like continuum admits a Cantor set $Z$ such that every tuple of finite points in $Z$ is an $IE$-tuple of $f$ and $Z$ has the free tracing property by free chains. Also, by use of this notion, we prove the following theorem: If $G$ is any graph and a homeomorphism $f$ on a $G$-like continuum $X$ has positive topological entropy, then there is a Cantor set $Z$ which is related to both the chaotic behaviors of Kerr and Li in dynamical systems and composants of indecomposable continua in topology. This theorem implies that chaotic dynamics induce complicated topology. PDF 35. Topological rank does not increase by natural extension of Cantor minimals 下村尚司 (名経大経済) Downarowicz and Maass (2008) defined the topological rank for all homeomorphic Cantor minimal dynamical systems. This definition can be extended to all continuous surjective Cantor minimal systems. We have made it clear that taking natural extension does not increase the topological rank. PDF 36. Proximal Cantor systems with topological rank 2 are residually scrambled 下村尚司 (名経大経済) Downarowicz and Maass (2008) introduced the topological rank on all homeomorphic Cantor minimal dynamical systems. This definition can be easily extended to homeomorphic Cantor proximal dynamical systems. We consider the homeomorphic proximal Cantor dynamical systems with topological rank 2. We show that they are all residually scrambled. Evidently, such systems have at most two ergodic measures. We have obtained a necessary and sufficient condition for the unique ergodicity of these systems. In addition, we show that the number of ergodic measures of systems that are topologically mixing can be 1 and 2. Moreover, there exist examples that are topologically weakly mixing, not topologically mixing, and uniquely ergodic. Finally, we show that the number of ergodic measures of systems that are not weakly mixing can be 1 and 2. PDF 37. Hyperspaces homeomorphic to Hilbert spaces 越野克久 (神奈川大工) Let ${\rm Comp}(X)$ be the hyperspace consisting of non-empty compact subsets of a space $X$ with the Vietoris topology, and ${\rm C}(X)$ be the hyperspace of compact and connected sets in $X$, that is considered as a subspace of ${\rm Comp}(X)$. In this talk, we characterize a metrizable space $X$ whose hyperspaces ${\rm Comp}(X)$ and ${\rm C}(X)$ are homeomorphic to a non-separable Hilbert space. PDF 38. Local and end deformation theorems for uniform embeddings 矢ヶ崎達彦 (京都工繊大工芸) This talk is concerned with local and end deformation properties of spaces of uniform embeddings in metric manifolds. Using the local deformation theorem by Cernavskii and Edwards–Kirby, we show that any metric manifold with a geometric group action has the local deformation property for uniform embeddings (LD). As an example, the $\kappa$-cone end ($\kappa \leq 0$) over any compact Lipschitz metric manifold is shown to have the property (LD). We also introduce a notion of end deformation of uniform embeddings (ED) and show that the 0-cone end over any compact Lipschitz metric manifold has the property (ED). A role of uniform isotopies in uniform topology is also clarified. PDF 39. Insertion of poset-valued maps with the way-below and -above relations 山﨑 薫里 (高崎経大経済) We give insertion theorems for maps with values in bi-bounded complete and bicontinuous posets by using the way-below relation and the way-above relation. PDF