Mapping class groups are linear

On the Matrices of Central Linear Mappings

We show that a central linear mapping of a projectively embedded Euclidean $n$-space onto a projectively embedded Euclidean $m$-space is decomposable into a central projection followed by a similarity if, and only if, the least singular value of a certain matrix has multiplicity $\ge 2m-n+1$. This matrix is arising, by a simple manipulation, from a matrix describing the given mapping in terms of homogeneous Cartesian coordinates.

Weak Linear Mappings -- A Survey

There are various concepts of structure preserving mappings in geometry. It is the aim of the present paper to give a survey on geometrical characterizations of some of those mappings. We discuss the results for projective spaces in some detail and report on generalizations to other spaces.

On cubic-linear polynomial mappings

In the field of the Jacobian conjecture it is well-known after Druzkowski that from a polynomial "cubic-homogeneous" mapping we can build a higher-dimensional "cubic-linear" mapping and the other way round, so that one of them is invertible if and only if the other one is. We make this point clearer through the concept of "pairing" and apply it to the related conjugability problem: one of the two maps is conjugable if and only if the other one is; moreover, we find simple formulas expressing the inverse or the conjugations of one in terms of the inverse or conjugations of the other. Two nontri…

Linear representations of hyperelliptic mapping class groups

Let $p:S\to S_g$ be a finite $G$-covering of a closed surface of genus $g\geq 1$ and let $B$ its branch locus. To this data, it is associated a representation of a finite index subgroup of the mapping class group $\operatorname{Mod}(S_g\smallsetminus B)$ in the centralizer of the group $G$ in the symplectic group $\operatorname{Sp}(H_1(S,{\mathbb Q}))$. They are called \emph{virtual linear representations} of the mapping class group and are related, via a conjecture of Putman and Wieland, to a question of Kirby and Ivanov on the abelianization of finite index subgroup of the mapping class grou…

映射类群的维数不超过 $3g-3$ 的线性表示

Linear representations of the mapping class group of dimension at most $3g-3$

本文分类了亏格 $g$ 曲面（最多一个穿孔或边界分支）的映射类群在维数不超过 $3g-3$ 的表示，证明任何这样的表示是维数 $2g$ 或 $2g+1$ 的表示（由曲面或其单位切丛的（上）同调作用给出）与平凡表示的直和，并推论该范围内黎曼曲面模空间上的任何线性系统具有代数几何起源。

On quadratic polynomial mappings from the plane into the $n$ dimensional space

We show that, up to linear equivalence, there are only finitely many polynomial quadratic mappings $F:\mathbb{C}^2\to\mathbb{C}^n$ and $F:\mathbb{R}^2\to\mathbb{R}^n$. We list all possibilities.