Lifting Braids

可提升辫与着色辫群胚

Liftable braids and the coloured braid groupoid

本文针对圆盘的分歧覆盖，将辫群的提升同态推广到着色辫群胚到映射类群胚的映射，并刻画了每个着色辫的提升，恢复了经典结果。

Lifting Branched Covers to Braided Embeddings

Braided embeddings are embeddings to a product disc bundle so the projection to the first factor is a branched cover. In this paper, we study which branched covers lift to braided embeddings, which is a generalization of the Borsuk-Ulam problem. We determine when a braided embedding in the complement of branch locus can be extended over the branch locus in smoothly (or locally flat piecewise linearly), and use it in conjunction with Hansen's criterion for lifting covers. We show that every branched cover over an orientable surface lifts to a codimension two braided embedding in the piecewise l…

在实射影平面RP²上提升辫子与非定向勒夫谢茨纤维化

Lifting braids over $\mathbb{RP}^2$ and non-orientable Lefschetz fibrations

本文研究在RP²上提升辫子与非定向勒夫谢茨纤维化，证明次数≥5的简单分支覆盖下非定向曲面的自微分同胚可提升，且对应勒夫谢茨纤维化可分解为简单覆盖与投影的复合。

Braid groups and discrete diffeomorphisms of the punctured disk

We show that the group cohomology of the diffeomorphisms of the disk with $n$ punctures has the cohomology of the braid group of $n$ strands as the summand. As an application of this method, we also prove that there is no cohomological obstruction to lifting the "standard" embedding $\mathrm{Br}_{2g+2}\hookrightarrow \mathrm{Mod}_{g,2}$ to a group homomorphism between diffeomorphism groups.

Representing Dehn twists with branched coverings

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 to have disconnected boundary. As a consequence, any allowable Lefschetz fibration on B^2 is a branched covering of B^2 x B^2.