2017年度秋季総合分科会(於:山形大学)

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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 \(39n\).

SUMMARY: 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.

SUMMARY: 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 \(2\).

SUMMARY: 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 \(b_1(\partial W)<n\), the family cannot be generated by twisting \(X\) along \(W\) and varying the gluing map. As a corollary, we show that there exists no ‘universal’ 4-manifold with boundary such that any exotic family is generated by twisting along an embedded copy of the 4-manifold. Moreover, we give similar results for surgeries.

SUMMARY: This talk is concerned with symplectic surfaces in a symplectic \(4\)-disk \((D^4, \omega )\) bounded by the same transverse link in the standard contact \(3\)-sphere \((S^3, \xi _{st})\). There are some examples of transverse links (or knots) bounding more than two distinct symplectic surfaces. All these surfaces can be distinguished by the fundamental groups of their complements. In this talk, I will present a family of pairs of two distinct symplectic surfaces whose boundaries are the same transverse knot and whose complements have isomorphic fundamental groups. To tell apart the two surfaces of each pair, I take double branched covers branched along them.

SUMMARY: A \(\mathbb {C}{\rm P}^1\)-structure on a surface is an atlas modeled on \(\mathbb {C}{\rm P}^1\) with transition maps in \({\rm PSL}(2, \mathbb {C})\), and a \(\mathbb {C}{\rm P}^1\)-structure corresponds to a pair of a Riemann surface and a holomorphic quadratic differential on it. In addition, each \(\mathbb {C}{\rm P}^1\)-structure has a holonomy representation from its fundamental group into \({\rm PSL}(2, \mathbb {C})\).

In this talk, we consider a path of diverging \(\mathbb {C}{\rm P}^1\)-structures on a closed oriented surface such that, their holonomy representations converge. We discuss about its limit under the assumption that the Riemann surface structures are pinched along some loops.

SUMMARY: Ishizaka classified up to conjugation orientation preserving periodic maps of a surface which commute with a hyperelliptic involution. Here, an involution on a surface is hyperelliptic if and only if the quotient space is a sphere. In this talk, we consider maps which commute with involutions whose quotient space is a torus. We will classify up to conjugation orientation preserving irreducible periodic maps which commute with such involutions to give a complete list. It turned out that there are only finite conjugacy classes. We present a representative of each conjugacy class with certain decomposition of surfaces into fundamental domains.

SUMMARY: Let \(G=SU(n,1)\) be the isometry group of complex hyperbolic space \(X\) and \(G=KAN\) an Iwasawa decomposition. We proved local rigidity of actions of certain finitely generated subgroups \(\Gamma \) of \(AN\) on the imaginary boundary of \(X\).

SUMMARY: We prove that every lens space contains a genus one homologically fibered knot, which is contrast to the fact that some lens spaces contain no genus one fibered knot. In the proof, the discriminant of a binary quadratic form and the Chebotarev density theorem in number theory play central roles.

SUMMARY: We consider groups presented as \(G(p,q)=\langle \ x,\ y \ |\ x^p =y^q \ \rangle \), with integers \(p\) and \(q\) satisfying \(2 \leq p \leq q\). The groups are geometrically realized as the fundamental groups of Seifert fiber spaces over 2-dimensional disks with two cone points. We present rational function expressions for the spherical growth series of such groups with respect to the generating set \(\{x,y,x^{-1},y^{-1}\}\).

SUMMARY: Let \(M\) be a homology 3-sphere. Reidemeister torsion is a topological invariant of \(M\) with a representation \(\rho :\pi _1(M)\rightarrow \mathit {SL}(2;\mathbb {C})\). It gives a complex valued function on the space of conjugacy classes of \(\mathit {SL}(2;\mathbb {C})\)-irreducible representations. We show the image is a finite set for some splicing manifolds along torus knots or the figure-eight knot.

SUMMARY: A hyperbolic 3-manifold that has torus boundaries can change to a non-hyperbolic 3-manifold by filling the boundary by a solid torus. Such surgeries are called exceptional surgeries. We study the exceptional Dehn surgeries along the Mazur link and Akbulut–Yasui links.

SUMMARY: It is known that Hempel distance of Heegaard splittings has a good relationship with topology and geometry of 3-manifolds. We discuss an approach to defining Hempel distance of generalized Heegaard splittings.

SUMMARY: We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants \(J^{\pm }\) and \(\mathit {St}\) behave under the generalized connected sums.

SUMMARY: Let \(\Gamma \) be a chart. For each label \(k\), we denote by \(\Gamma _k\) the union of edges of label \(k\) and their vertices. Let \(\Gamma \) be a chart. If \(\Gamma \) has exactly 8 white vertices and \(\Gamma _m \cap \Gamma _{m+1}\) has 2 white vertices, \(\Gamma _{m+1} \cap \Gamma _{m+2}\) has 4 white vertices, \(\Gamma _{m+2} \cap \Gamma _{m+3}\) has 2 white vertices, then \(\Gamma \) is called a chart of type (2,4,2). In this talk we study for a minimal 5-chart of type (2,4,2) with a lens of type 1 which has less than or equal to 14 black vertices.

SUMMARY: We consider a surface-knot in the form of a simple branched covering over an oriented surface-knot \(F\), which is called a covering surface-knot over \(F\). Such a surface-knot is presented by a certain graph called a chart on a surface diagram of \(F\). For a covering surface-knot, an addition of 1-handles with chart loops is a simplifying operation which deforms the chart to a union of free edges and 1-handles with chart loops. Here, we obtain properties of such simplifications.

SUMMARY: A ribbon surface-tangle is a compact surface in upper four-space with no minimal points such that the boundary presents a trivial link. Any surface-link is the closure of some ribbon surface-tangle. We introduce an equivalence relation among ribbon surface-tangles, called a root-equivalence, and prove that the closures of two ribbon surface-tangles present the same surface-link if and only if they are root-equivalent. As an application, we see that the double of a surface-link is well-defined up to stable equivalence. We also study several properties of the double.

SUMMARY: For an ordered \(n\)-component link diagram \(D\), we construct a virtual link diagram \(D(m_1,m_2,\ldots ,m_n)\) which is obtained from \(D\) by multiplexing of the crossings of \(D\), where \(m_i\) is an integer. If two link diagrams \(D\) and \(D'\) are equivalent, then \(D(m_1,m_2,\ldots ,m_n)\) and \(D'(m_1,m_2,\ldots ,m_n)\) are equivalent as welded links. Since an invariant of \(D(m_1,m_2,\ldots ,m_n)\) is that of \(D\), we try to find new invariants of \(D\) via \(D(m_1,m_2,\ldots ,m_n)\).

SUMMARY: In the talk, we will consider a relationship between an invariant and local moves for virtual knots. We show that two virtual knots have the same writhe polynomial if and only if they are related by a finite sequence of certain local moves.

SUMMARY: There are many relations between graph theory and knot theory. In particular, certain knot invariants have been expressed in terms of graph invariants. As an example, the interior polynomial is an invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to a part of the HOMFLY polynomial of a naturally associated link. We also establish some other, more basic properties of this new notion. This leads to new identities involving the original interior polynomial.

SUMMARY: For the set of virtual knots \(\mathcal {V}\), we define a relation \(\leq \) as follows. Let \(K,K'\) be virtual knots. We write \(K'\leq K\) if for any virtual knot diagram \(D\) of \(K\), we obtain a virtual knot diagram of \(K'\) by replacing several real crossings of \(D\) with virtual crossings.

In this talk, we show that \((\mathcal {V},\leq )\) is a partially ordered set. We also show that any finite subset of \(\mathcal {V}\) has an upper bound with respect to this order.

SUMMARY: We develop a diagrammatic computation of the Milnor invariant of links, in terms of central group extensions and unipotent Magnus embeddings. As a corollary, we compute the invariants of the Milnor link and of some links. We also powerfully extend the higher invariant, by mproving indeterminacy therein.

SUMMARY: We focus on the cohomology of the \(k\)-th nilpotent quotient of the free group, \(F/F_k\). This paper describes all the group 2-, 3-cocycles in terms of Massey products, and gives expressions for some of the 3-cocycles. We also give simple proofs of some of the results on Milnor invariants and the Johnson–Morita homomorphisms.

SUMMARY: We will discuss the asymptotic behavior of the twisted Alexander invariants for higher-dimensional representations of a knot group and the relation to the asymptotic behavior of the Reidemeister torsion. We will focus metabelian representations of a knot group into SL\((2;\mathbb {C})\). It is known that metabelian representations are related to exceptional surgeries along a knot. We will see the relation to our previous result on the asymptotic behavior of the Reidemeister torsion for the resulting manifold obtained by an exceptional surgery along a twist knot.

SUMMARY: In 1990, Vassiliev introduced a filtered space of knot invariants via a standard unknotting operation, called crossing change. In 2000, Goussarov, Polyak, and Viro introduced another degree and filtration via another unknotting operation, called virtualization, for classical and virtual knots. In these theories, a notion of \(n\)-trivialities has played a significant role. However, for an integer \(n\) (\(> 2\)), 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.

SUMMARY: 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.

SUMMARY: 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 \(M\).

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: Characterizations are given for essentially weakly onesided resolving endomorphisms of subshifts.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.