Global fixed points of mapping class group actions and a theorem of Markovic

A periodicity criterion and the section problem on the Mapping Class Group

Some years ago, V. Markovic proved that there is no section of the Mapping Class Group for a closed surface of genus g larger than 5 (in the case of homeomorphims) and more recently generalized this result with D. Saric to the case where g is larger than 1. We will state a periodicity criterion and will use it to simplify some of the arguments given by Markovic and Saric in the proof of their theorem. The periodicity criterion tells us that a homeomorphism of a connected surface must be periodic if the set of connected periodic open sets generates the topology of the surface.

On the non-realizability of braid groups by homeomorphisms

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 $n$ marked points in the interior as a set. We also give a new proof of a result of Markovic that the mapping class group of a closed surface cannot be geometrically realized as a group of homeomorphisms.

Putman-Wieland 猜想的代数几何应用

Applications of the algebraic geometry of the Putman-Wieland conjecture

本文给出 Putman-Wieland 猜想工作的两个应用：加强了 Marković-Tošić 关于虚拟映射类群作用的结果，并证明映射类群在 $H$-等变同调分量上的虚拟作用具有非酉像。

每个映射类中曲面辛同胚不动点的精确界

A sharp bound on fixed points of surface symplectomorphisms in each mapping class

本文研究了曲面辛同胚不动点数量问题，通过 Floer 同调计算与扭曲系数方法，得到了每个映射类中不动点的精确下界，将 Poincaré-Birkhoff 定理推广至任意曲面和映射类。

Generating mapping class groups with elements of fixed finite order

We show that for any $k$ at least $6$ and $g$ sufficiently large, the mapping class group of a surface of genus $g$ can be generated by three elements of order $k$. We also show that this can be done with four elements of order $5$. We additionally prove similar results for some permutation groups, linear groups, and automorphism groups of free groups.