Related papers: Families of superelliptic curves, complex braid gr…
Homology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus $g$ with one boundary component, that are double coverings of…
The Coulomb branch geometry of a 4d $\mathcal{N}=2$ SCFT is encoded in the data of a complex integrable system. In class-S, this is the Hitchin System (of ADE type) on the punctured curves $C$ on which we compactified from 6d to 4d. As we…
In this paper we define a new family of groups which generalize the {\it classical braid groups on} $\C $. We denote this family by $\{B_n^m\}_{n \ge m+1}$ where $n,m \in \N$. The family $\{ B_n^1 \}_{n \in \N}$ is the set of classical…
Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic from the braid group to the mapping class group. We prove here that this map is trivial in stable homology with…
We show that the morphisms from the braid group with n strands in the mapping class group of a surface with a possible non empty boundary, assuming that its genus is smaller or equal to n/2 are either cyclic morphisms (their images are…
We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides…
The reduced Burau representation $V_n$ of the braid group $B_n$ is obtained from the action of $B_n$ on the homology of an infinite cyclic cover of the disc with $n$ punctures. The group homology $H_*(B_n;V_n)$ of braid groups with…
Let $n\geq 1$. The pro-unipotent completion of the pure braid group of $n$ points on a genus 1 surface has been shown to be isomorphic to an explicit pro-unipotent group with graded Lie algebra using two types of tools: (a) minimal models…
We classify homomorphisms from the braid group on $n$ strands to the pure mapping class group of a nonoriantable surface of genus $g$. For $n\ge 14$ and $g\le 2\lfloor{n/2}\rfloor+1$ every such homomorphism is either cyclic, or it maps…
We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the…
We consider homologically essential simple closed curves on Seifert surfaces of genus one knots in $S^3$, and in particular those that are unknotted or slice in $S^3$. We completely characterize all such curves for most twist knots: they…
An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams $E_6, E_7$, $E_8$. These curves are…
We generalize the topological recursion of Eynard-Orantin (2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic…
The $d$-fold ($d \geq 3$) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into…
We investigate the space $C(X)$ of images of linearly embedded skeleta of simplices $X$ in $\mathbb R^n$, for two families of codimension 2 complexes, each ranging over $n$. In the first family, $X=K$ is the $(n-2)$-skeleton of the…
We show that, up to a natural equivalence relation, the only non-trivial, non-identity holomorphic maps $\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$ between unordered configuration spaces, where $m\in\{3,4\}$, are the resolving…
For any number field K with a complex place, we present an infinite family of elliptic curves defined over K such that $dim \mathbb{F}_2 Sel_2(E^F/K) \ge dim \mathbb{F}_2 E^F(K)[2] + r_2$ for every quadratic twist E^F of every curve E in…
We generalize the notion of quasielliptic curves, which have infinitesimal symmetries and exist only in characteristic two and three, to a remarkable hierarchy of regular curves having infinitesimal symmetries, defined in all…
Locally trivial bundles of $C^*$-algebras with fibre $D \otimes \mathcal{K}$ for a strongly self-absorbing $C^*$-algebra $D$ over a finite CW-complex $X$ form a group $E^1_D(X)$ that is the first group of a cohomology theory $E^*_D(X)$. In…
Complex braid groups are the natural generalizations of braid groups associated to arbitrary (finite) complex reflection groups. We investigate several methods for computing the homology of these groups. In particular, we get the Poincar\'e…