Narrow Results

The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering space in the mapping class group of the total surface. We give finite presentations for the balanced superelliptic mapping class groups of closed surfaces, surfaces with one marked point, and surfaces with one boundary component. To give these presentations, we construct finite presentations for corresponding liftable mapping class groups in a different generating set from Ghaswala–Winarski’s presentation in [9].

Athreya et al. [Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J.161 (2012) 1055–1111] have shown the number of mapping class group lattice points intersecting a closed ball of radius $R$ in Teichmüller space is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichmüller space. In contrast we show the number of Dehn twist lattice points intersecting a closed ball of radius $R$ is coarsely asymptotic to $e^{\frac{h}{2}R}$. Moreover, we show the number multi-twist lattice points intersecting a closed ball of radius $R$ grows coarsely at least at the rate of $R\cdot e^{\frac{h}{2}R}$.

Concerning Johnson’s homomorphism from the Torelli group, there are previous works to define a logarithm of the homomorphism, and give some extension of the logarithm. This paper considers exponential solvable elements in the mapping class group of a surface, and defines the logarithms of such elements.

A pair $(\alpha ,\beta )$ of simple closed curves on a closed and orientable surface $S_g$ of genus $g$ is called a filling pair if the complement is a disjoint union of topological disks. If $\alpha$ is separating, then we call it as separating filling pair. In this paper, we find a necessary and sufficient condition for the existence of a separating filling pair on $S_g$ with exactly two complementary disks. We study the combinatorics of the action of the mapping class group $\mathrm{\text{Mod}}(S_g)$ on the set of such filling pairs. Furthermore, we construct a Morse function ${{ℱ}}_g$ on the moduli space ${{ℳ}}_g$ which, for a given hyperbolic surface $X$, outputs the length of the shortest such filling pair with respect to the metric in $X$. We show that the cardinality of the set of global minima of the function ${{ℱ}}_g$ is the same as the number of $\mathrm{\text{Mod}}(S_g)$-orbits of such filling pairs.

Omori and the author [R. Kobayashi and G. Omori, An infinite presentation for the mapping class group of a non-orientable surface with boundary, Osaka J. Math. 59(2) (2022) 269–314] have given an infinite presentation for the mapping class group of a compact non-orientable surface. In this paper, we give more simple infinite presentations for this group.

A $(g,n)$-decomposition of a link $L$ in a closed orientable $3$-manifold $M$ is a decomposition of $M$ by a closed orientable surface of genus $g$ into two handebodies each intersecting the link $L$ in $n$ trivial arcs. The Goeritz group of that decomposition is then defined to be the group of isotopy classes of orientation-preserving homeomorphisms of the pair $(M,L)$ that preserve the decomposition. We compute the Goeritz groups of all $(1,1)$-decompositions.

By gluing two copies of surface S_0,g+2 along g + 1 holes, we get surface S_g,1. The pillar switching is a self-homeomorphism of S_g,1 which switches two pillars of surfaces by 180° horizontal rotation. We analyze the actions of the pillar switchings on π₁S_g,1 and then give concrete expressions of the pillar switchings in terms of standard Dehn twists. The map ψ : B_g → Γ_g,1 sending the generators of B_g to the pillar switchings on S_g,1 is defined by extending the embedding B_g ↪ Γ_0,(g+1),1. We show that this map is injective by analyzing the actions of pillar switchings on π₁S_g,1. We also prove that this map induces a trivial homology homomorphism in the stable range. For the proof we use the categorical delooping method. We construct a suitable monoidal 2-functor from tile category to surface category and show that this functor thus induces a map of double loop spaces.

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus two fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic but not diffeomorphic to ${ℂ\mathbb{P}}^2\#6{\overline{ℂ\mathbb{P}}}^2$, ${ℂ\mathbb{P}}^2\#7{\overline{ℂ\mathbb{P}}}^2$, and $3{ℂ\mathbb{P}}^2\#k{\overline{ℂ\mathbb{P}}}^2$ for $k=16,17,18,19$ using combinations of such deformations, symplectic blowups, and (generalized) rational blowdown surgery. We also discuss generalizing our constructions to higher genus fibrations using $g$-nodal spherical deformations of certain singular fibers of genus $g\ge 3$ fibrations.

U(1) Chern–Simons theory is quantized canonically on manifolds of the form , where Σ is a closed orientable surface. In particular, we investigate the role of mapping class group of Σ in the process of quantization. We show that, by requiring the quantum states to form representation of the holonomy group and the large gauge transformation group, both of which are deformed by quantum effect, the mapping class group can be consistently represented, provided the Chern–Simons parameter k satisfies an interesting quantization condition. The representations of all the discrete groups are unique, up to an arbitrary sub-representation of the mapping class group. Also, we find a k↔1/k duality of the representations.

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph, respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result, we are able to develop a Nielsen–Thurston classification for elements in the right-angled Artin group. Our analogy spans both the algebra regarding subgroups of right-angled Artin groups and mapping class groups, as well as the geometry of the extension graph and the curve graph. On the geometric side, we establish an analogue of Masur and Minsky's Bounded Geodesic Image Theorem and their distance formula.

We continue to study the algebraic property of the linear representation of the mapping class group of a closed oriented surface of genus 2 constructed by V. F. R. Jones. We consider the perturbation of the representation at the involved parameter t=1. This perturbation naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of the associated graded quotients. Our arguments follow the same line of our previous paper which dealt with the perturbation at t=-1. However, the obtained results may still suggest a new aspect of the representation.

A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehornoy.

For a surface F bounding a handlebody H, we look at simple closed curves on F which intersect every disk in the handlebody, at least n times (called n-closed curves). We give a finite criterion for a curve to be n-closed. Using this, we derive a sufficiency condition for a Heegaard splitting to be strongly irreducible. We then look at further intersection properties of curves with disk families in H. In particular, we look at the effects of Dehn twists on n-closed curves, and using a finite fixed disk collection as a coordinate system, give heuristics and a counting formula for measuring the number of intersections of the resulting curves, with disks in H. In a certain instance, this yields a partial "grading" on the Dehn twist quandle with respect to the degree of n-closedness.

Consider a surface braid group of n strings as a subgroup of the isotopy group of homeomorphisms of the surface permuting n fixed distinguished points. Each automorphism of the surface braid group (respectively, of the special surface braid group) is shown to be a conjugate action on the braid group (respectively, on the special braid group) induced by a homeomorphism of the underlying surface if the closed surface, either orientable or non-orientable, is of negative Euler characteristic. In other words, the group of automorphisms of such a surface braid group is isomorphic to the extended mapping class group of the surface with n punctures, while the outer automorphism group of the surface braid group is isomorphic to the extended mapping class group of the closed surface itself.

We derive a formula expanding the bracket with respect to a natural deformation parameter. The expansion is in terms of a two-variable polynomial algebra of diagram resolutions generated by basic operations involving the Goldman bracket. A functorial characterization of this algebra is given. Differentiability properties of the star product underlying the Kauffman bracket are discussed.

It is shown that for the braid group B_n(M) on a closed surface M of nonnegative Euler characteristic, Out(B_n(M)) is isomorphic to a group extension of the group of central automorphisms of B_n(M) by the extended mapping class group of M, with an explicit and complete description of Aut(B_n(S²)), Aut(B_n(P²)), Out(B_n(S²)) and Out(B_n(P²)).

Braid groups and mapping class groups have many features in common. Similarly to the notion of inverse braid monoid, inverse mapping class monoid is defined. It concerns surfaces with punctures, but among given n punctures, several can be omitted. This corresponds to braids where the number of strings is not fixed. In the paper we give the analogue of the Dehn–Nilsen–Baer theorem, propose a presentation of the inverse mapping class monoid for a punctured sphere and study the word problem. This shows that certain properties and objects based on mapping class groups may be extended to the inverse mapping class monoids. We also give analogues of Artin presentation with two generators.

Let K be a two-bridge knot or link in S³. Then K is also denoted as the four-plat, b(p, q) to indicate its association with some rational number p/q. The lens space L = L(p, q) admits an isometry τ of order two, such that the quotient space L modulo the involution τ is an orbifold whose exceptional set is K and whose underlying space is S³. In this paper, the mapping class groups of these orbifolds are classified. While these groups can be found as a result of Mostow's Rigidity Theorem, this paper calculates the generators and relations of the groups and the proof does not rely on this strong theorem for the majority of cases.

We observe that the determinant of the representation provides a little restriction for the structure of the graded quotients introduced in both [2] and [3] that any one of them does not contain the trivial 1-dimensional summand.

A necessary and sufficient algebraic condition for a diffeomorphism over a surface embedded in S³ to be induced by a regular homotopic deformation is discussed, and a formula for the number of signed pass moves needed for this regular homotopy is given.