Related papers: Ordering pure braid groups on closed surfaces
For every compact surface $S$ of finite type (possibly with boundary components but without punctures), we show that when $n$ is sufficiently large there is no lift $\sigma$ of the surface braid group $B_n(S)$ to $\operatorname{Diff}(S,n)$,…
We consider a set of toric Calabi-Yau varieties which arise as deformations of the small resolutions of type A surface singularities. By careful analysis of the heuristics of B-brane transport in the associated GLSMs, we predict the…
We show that any homologically non-trivial Dehn twist of a compact surface F with boundary is the lifting of a half-twist in the braid group B_n, with respect to a suitable branched covering p : F -> B^2. In particular, we allow the surface…
Moduli spaces of points on $n$-spheres carry natural actions of braid groups. For $n=0$, $1$, and $3$, we prove that these symmetries extend to actions of mapping class groups of positive genus surfaces, by establishing exceptional…
For a finite Thurston type ordering < of the braid group B_{n}, we introduce a new normal form of a dual positive braid which we call the C-normal form. This normal form extends Fromentin's rotating normal form and the author's C-normal…
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…
We provide explicit and unified formulae for the normalized 3-cocycles on arbitrary finite abelian groups. As an application, we compute all the braided monoidal structures on linear Gr-categories.
Consider an orientable compact surface in three dimensional Euclidean space with minimum total absolute curvature. If the Gaussian curvature changes sign to finite order and satisfies a nondegeneracy condition along closed asymptotic…
In this paper, we will show that the projection $\text{Homeo}^+(D^2_n)\to B_n$ does not have a section; i.e. the braid group $B_n$ cannot be geometrically realized as a group of homeomorphisms of a disk fixing the boundary point-wise and…
We construct braided versions $sV_{br}$ of the Brin-Thompson groups $sV$ and prove that they are of type $F_\infty$. The proof involves showing that the matching complexes of colored arcs on surfaces are highly connected.
In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with…
We prove that any closed simply-connected smooth 4-manifold is 16-fold branched covered by a product of an orientable surface with the 2-torus, where the construction is natural with respect to spin structures. In particular this solves…
Global results are proved about the way in which Boyland's forcing partial order organizes a set of braid types: those of periodic orbits of Smale's horseshoe map for which the associated train track is a star. This is a special case of a…
We provide new group presentations for surface braid groups which are positive. We study some properties of such presentations and we solve the conjugacy problem in a particular case.
We classify incompressible, boundary-incompressible, nonorientable surfaces in punctured-torus bundles over $S^1$. We use the ideas of Floyd, Hatcher, and Thurston. The main tool is to put our surface in the "Morse position" with respect to…
We study the topological space of left-orderings of the braid group, and its subspace of Nielsen-Thurston orderings. Our main result is that no Nielsen-Thurston ordering is isolated in the space of braid orderings. In the course of the…
Recently, Chang--Haiden--Schroll shows that the braid group action on full exceptional collections in a triangulated category is not transitive but has infinitely many orbits in general. Their proof is based on a geometric model and the…
We describe the fundamental groups of ordered and unordered $k-$point sets in the n-dimensional complex space $C^n$ generating an affine subspace of fixed dimension.
A bi-order on a group $G$ is a total, bi-multiplication invariant order. A subset $S$ in an ordered group $(G,\leqslant)$ is convex if for all $f\leqslant g$ in $S$, every element $h\in G$ satisfying $f\leqslant h \leqslant g$ belongs to…
Combining the results by Birman and Goldberg, it was proved the normal closure of the pure braid group of the disk $P_n(D)$ in the pure braid group of the torus $P_n(T)$ is the commutator subgroup $[P_n(T),P_n(T)]$. In this paper we are…