Crossed flux homomorphisms and vanishing theorems for flux groups

交叉同态与曲面映射类群的低维表示

Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces

本文对亏格$g\geq7$的紧致可定向曲面纯映射类群的$(2g+1)$维复线性表示给出完全分类，利用扭曲上同调，并证明不存在不可约表示，与$g=2$情形形成对比。

First cohomology groups of the automorphism group of a free group with coefficients in the abelianization of the IA-automorphism group

We compute a twisted first cohomology group of the automorphism group of a free group with coefficients in the abelianization $V$ of the IA-automorphism group of a free group. In particular, we show that it is generated by two crossed homomorphisms constructed with the Magnus representation and the Magnus expansion due to Morita and Kawazumi respectively. As a corollary, we see that the first Johnson homomorphism does not extend to the automorphism group of a free group as a crossed homomorphism for the rank of the free group is greater than 4.

Crossed modules as maps between connected components of topological groups

The purpose of this note is to observe that a homomorphism of discrete groups $f:Γ\to G$ arises as the induced map $π_0(\mathfrak{M})\to π_0(\mathfrak{X})$ on path components of some closed normal inclusion of topological groups $\mathfrak{M}\subseteq \mathfrak{X},$ if and only if the map $f$ can be equipped with a crossed module structure. In that case an essentially unique realization $\mathfrak{M}\subseteq \mathfrak{X}$ exists by homotopically discrete topological groups.

交叉模

Crossed modules

本文综述了交叉模的概念，指出其由群、自同构群及典范同态构成，并源于群展示关系、扩张问题、低维拓扑及非交换代数推广，且对Lie代数同样适用。

Normal closure and injective normalizer of a group homomorphism

Let $φ\colonΓ\to G$ be a homomorphism of groups. We consider factorizations $Γ\xrightarrow{f} M\xrightarrow{g} G$ of $φ$ such that either $g$ or $f$ are universal normal maps (namely, crossed modules). These two factorizations are natural generalizations of the usual normal closure and normalizer of a subgroup. Iterating these universal factorizations yield towers related, on the one hand, to hypercentral group extensions, Bousfield's localizations, and relative Schur multipliers. Dually, they extend to a relative situation, the automorphisms tower of a centerless group. Our constructions have…

论 Gottlieb 群、Drinfeld 中心与交叉模的中心

On the Gottlieb group, Drinfeld centre and the centre of a crossed module

本文引入交叉模的中心概念，利用交叉同态建立其与群上同调、Gottlieb 群、Drinfeld 中心及 Whitehead 中心的联系。