Cohomological structure of the mapping class group and beyond

The inclusion of configuration spaces of surfaces in Cartesian products, its induced homomorphism, and the virtual cohomological dimension of the braid groups of S^2 and RP^2

Let M be a surface, perhaps with boundary, and either compact, or with a finite number of points removed from the interior of the surface. We consider the inclusion i: F\_n(M) --\textgreater{} M^n of the nth configuration space F\_n(M) of M into the n-fold Cartesian product of M, as well as the induced homomorphism i\_\#: P\_n(M) --\textgreater{} (π\_1(M))^n, where P\_n(M) is the n-string pure braid group of M. Both i and i\_\# were studied initially by J.Birman who conjectured that Ker(i\_\#) is equal to the normal closure of the Artin pure braid group P\_n in P\_n(M). The conjecture was late…

Homology of the curve complex and the Steinberg module of the mapping class group

By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the homotopy type of a bouquet of spheres. Here, we give the first explicit homologically nontrivial sphere in the curve complex and show that under the action of the mapping class group, the orbit of this homology class generates the reduced homology of the curve complex.

An obstruction to embedding right-angled Artin groups in mapping class groups

For every orientable surface of finite negative Euler characteristic, we find a right-angled Artin group of cohomological dimension two which does not embed into the associated mapping class group. For a right-angled Artin group on a graph $\gam$ to embed into the mapping class group of a surface $S$, we show that the chromatic number of $\gam$ cannot exceed the chromatic number of the clique graph of the curve graph $\mathcal{C}(S)$. Thus, the chromatic number of $\gam$ is a global obstruction to embedding the right-angled Artin group $A(\gam)$ into the mapping class group $\Mod(S)$.

辛群、映射类群与有界上同调的稳定性

Symplectic Groups, Mapping Class Groups and the Stability of Bounded Cohomology

本研究探讨映射类群与辛群的有界上同调稳定性，证明映射类群有界上同调不稳定、Mumford-Morita-Miller类稳定多项式无界，辛群2次有界上同调非等距稳定并给出相关范数计算与估计。

带穿孔的可定向曲面的映射类群的真正几何维数

The proper geometric dimension of the mapping class group of an orientable surface with punctures

本文证明了带穿孔可定向闭曲面全映射类群的真正作用紧上分类空间维数等于其虚拟上同调维数，并推得全球面辫群的真正几何维数。

带孔洞的非可定向曲面的映射类群的proper几何维数

The proper geometric dimension of the mapping class groups of non-orientable surfaces with punctures

本文研究带孔洞非可定向曲面完全映射类群的proper几何维数，证明其与虚拟上同调维数在除有限低复杂度情形外一致，并给出例外情形的显式界。

映射类群的真几何维数

The proper geometric dimension of the mapping class group

该研究证明闭曲面映射类群存在维数与其虚上同调维数相等的真作用紧生成分类空间，推进了几何群论相关问题的研究。

The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary

We give two proofs that appropriately defined congruence subgroups of the mapping class group of a surface with punctures/boundary have enormous amounts of rational cohomology in their virtual cohomological dimension. In particular we give bounds that are super-exponential in each of three variables: number of punctures, number of boundary components, and genus, generalizing work of Fullarton-Putman. Along the way, we give a simplified account of a theorem of Harer explaining how to relate the homotopy type of the curve complex of a multiply-punctured surface to the curve complex of a once-pun…

Automorphisms of free groups with boundaries

The automorphisms of free groups with boundaries form a family of groups A_{n,k} closely related to mapping class groups, with the standard automorphisms of free groups as A_{n,0} and (essentially) the symmetric automorphisms of free groups as A_{0,k}. We construct a contractible space L_{n,k} on which A_{n,k} acts with finite stabilizers and finite quotient space and deduce a range for the virtual cohomological dimension of A_{n,k}. We also give a presentation of the groups and calculate their first homology group.

Cohomology of the hyperelliptic Torelli group

Let SI(S_g) denote the hyperelliptic Torelli group of a closed surface S_g of genus g. This is the subgroup of the mapping class group of S_g consisting of elements that act trivially on H_1(S_g;Z) and that commute with some fixed hyperelliptic involution of S_g. We prove that the cohomological dimension of SI(S_g) is g-1 when g > 0. We also show that H_g-1(SI(S_g);Z) is infinitely generated when g > 1. In particular, SI(S_3) is not finitely presentable. Finally, we apply our main results to show that the kernel of the Burau representation of the braid group B_n at t = -1 has cohomological dim…

理解Teichmüller空间的良圆变形收缩

Understanding the well-rounded deformation retraction of Teichmüller space

本文研究闭曲面Teichmüller空间的映射类群等变变形收缩，证明其与SL(n,Z)的良圆变形收缩在正确对偶性定义下类似，并给出曲线复形圈表示非平凡同调类的必要条件。

Thurston 骨架的等变形变收缩

An equivariant deformation retraction of the Thurston spine

本文证明了闭可定向曲面 Teichmüller 空间存在映射类群等变的形变收缩到 Thurston 骨架，其维数等于映射类群的虚上同调维数。

Finiteness properties for a subgroup of the pure symmetric automorphism group

Let F_n be the free group on n generators, and PΣ_n be the group of automorphisms of F_n which send each generator to a conjugate of itself. Let K_n be the kernel of the homomorphism from PΣ_n to PΣ_{n-1} induced by mapping one of the free group generators to the identity. We show that K_n has cohomological dimension n-1, and that the ith cohomology groups are infinitely generated for all i between 2 and n-1. It follows that K_n is not finitely presentable for n>2.

曲面群的典范表示

Canonical representations of surface groups

该研究利用霍奇理论与算术方法，证明满足映射类群作用下共轭类轨道有限且维数小于√(g+1)的曲面群表示像有限，解答了相关问题并将方法应用于多个猜想。