Related papers: Point pushing, homology, and the hyperelliptic inv…

We consider the Birman-Hilden inclusion $\varphi\colon\mathfrak{Br}_{2g+1}\to\Gamma_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $\varphi$ is stably trivial in homology with…

Hyperelliptic mapping class groups are defined either as the centralizers of hyperelliptic involutions inside mapping class groups of oriented surfaces of finite type or as the inverse images of these centralizers by the natural…

In this paper, we use a result of Dahmani to show that the Euler class of some power subgroup (the subgroup normally generated by a fixed power of Dehn twist about a non-separating curve) is trivial inside the mapping class group of once…

We introduce a formula for the action of Dehn twists on the HOMFLY-PT type skein module of a surface. As an application of the formula to mapping class group, we give an embedding from the Torelli group of a surface $\Sigma_{g,1}$ of genus…

We calculate the Dehn twist action on the spaces of conformal blocks of a not necessarily semisimple modular category. In particular, we give the order of the Dehn twists under the mapping class group representations of closed surfaces. For…

We prove that rational homology of the Torelli group of genus g is infinite dimensional, provided g>6. This means that rational homology of the Torelli space of genus g>6 is infinite dimensional. The Torelli groups with marked points are…

Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as a generalization of the mapping class group. Using torsion invariants, we show that the abelianization of…

By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The extension problem in Torelli topology is the problem of determining when a diffeomorphism of compact…

We apply the theory of finite-type invariants of homology 3-spheres to investigate the structure of the Torelli group. We construct natural cocycles in the Torelli group and show that the lower central series quotients of the Torelli group…

Herein we prove that if $M$ is a compact oriented Riemann surface of genus $g$, and $M^{[n]}$ is the classifying space of $n$ distinct, unordered points on $M$, then the kernel of the map $\pi_1(M^{[n]})\to H_1(M)$ is generated by…

The period mapping assigns to each rank n, marked metric graph Gamma a positive definite quadratic form on H_1(Gamma). This defines maps Phi* and Phi on Culler--Vogtmann's outer space CV_n, and its Torelli space quotient T_n, respectively.…

By evaluating the Burau representation at t=-1, we obtain a symplectic representation of the braid group. We define the congruence subgroups of the braid group to be the preimages of the principal congruence subgroups of the symplectic…

Let $X_n$ be a cycle of $n$ projective lines, and $\bT_n$ a symplectic torus with $n$ punctures. Using the theory of spherical twists introduced by Seidel and Thomas (2001), I will define an action of the pure mapping class group of $\bT_n$…

We apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds. Johnson defined a well-known homomorphism from the Torelli group of a compact surface. Morita later showed that every homology…

We determine the action of the Torelli group on the equivariant cohomology of the space of flat SL(2,C) connections on a closed Riemann surface. We show that the trivial part of the action contains the equivariant cohomology of the even…

Torelli's theorem is proven by the study of the convolution product of the intersection cohomology sheaf of the thetadivisor.

A function from configuration space to moduli space of surface may induce a homomorphism between their fundamental groups which are braid groups and mapping class groups of surface, respectively. This map $\phi: B_k \rightarrow…

We classify up to conjugacy the group generated by a commuting pair of a periodic diffeomorphism and a hyperelliptic involution on an oriented closed surface. This result can be viewed as a refinement of Ishizaka's result on classification…

This article studies the group generated by automorphisms of the projective space of dimension $n$ and by the standard birational involution of degree $n$. Every element of this group only contracts rational hypersurfaces, but in odd…

Let $X$ be a complex smooth quasi-projective variety with a fixed epimorphism $\nu\colon\pi_1(X)\twoheadrightarrow H$, where $H$ is a finitely generated abelian group with $\mathrm{rank}H\geq 1$. In this paper, we study the asymptotic…