Diffeomorphism groups in dimensions 2 and 3

给出了2维和3维流形的微分同胚群。在2维中，主要问题是理解这种微分同胚群的代数结构，称为曲面映射类群或Teichmueller模群。该方案解决了映射类群的一些核心问题：了解Torelli群的结构、曲面的辫子群以及映射类群本身。该方案所涉及的问题要么是关于群的基本性质的关键(如Torelli群的有限可表现性问题)，要么是与其他数学分支有关的问题(如从代数K理论和拓扑量子场论的观点理解Wajnryb表示的问题)。在3维空间中，关键问题是对微分同胚群的同伦型的理解，特别是对它们的连通分支的理解。这里的主要目的是将这一领域中为数不多的结果与拓扑学的大多数主流工具联系起来，并在此基础上将这些结果推广到其他流形上。一个特别的目标是了解所有透镜空间的微分同态群，即球面之后最简单的3维流形。这一提议属于低维拓扑领域，它研究对象在2维和3维空间中可能的形状(即所谓的流形)。这些形状一百多年来一直是数学研究的焦点。在过去的几十年里，由于一种很有前途的物理“万物理论”，即所谓的弦理论，二维形状理论，即曲面，获得了额外的意义。在这个理论中，物质的基本成分不是点状粒子，而是在运动过程中扫过表面的环路(“弦”)的集合。三维形状为宇宙的大尺度结构提供了模型。在这个方向上，PI以前的一些结果已经引起了从事广义相对论工作的物理学家的注意。在2维和3维两个维度中，关键问题之一是理解这些形状的可能对称性。该提案致力于这个问题的一些主要方面。

The proposal is devoted to diffeomorphisms groups of manifolds of thedimension 2 and 3. In the dimension 2 the main problem is to understand thealgebraic structure of the groups of isotopy classes of suchdiffeomorphisms, known as mapping class groups of surfaces or Teichmuellermodular groups. The proposal addresses some of the central problems aboutthe mapping class groups: understanding the structure of the Torelli groups,braid groups of surfaces, and mapping class groups themselves. The problemsaddressed in the proposal arethe ones which either hold the key for the fundamental propertiesof the groups in question (like the problem of finite presentability of theTorelli groups) or have connections with other branches of mathematics (likethe problem of understanding the Wajnryb presentations from the point ofview of algebraic K-theory and topological quantum field theory). In thedimension 3 the key problem is the understanding of the homotopy type of thediffeomorphisms groups, especially of their connected components. The maingoal here is to relate the few available results in this area to the moremainstream tools of topology and to extend, on this basis, these results toother manifolds. A particular goal is to understand the diffeomorphismsgroups for all lens spaces, the simplest 3-manifolds after spheres.The proposal belongs to the field of low dimensional topology, whichinvestigates the possible shapes of objects (namely, the so-calledmanifolds) in dimension 2 and 3. These shapes continue to be in the focus ofthe mathematical research for more than a hundred years. In the lastdecades, the theory of 2-dimensional shapes, i.e. surfaces, acquired anadditional significance as a result of a promising physical "theory ofeverything ", namely the so-called string theory. In this theory, theelementary constituents of matter are not point-like particles, but rathercollections of loops ("strings") sweeping a surface during their motion. The3-dimensional shapes provide the model for the large-scale structure of theuniverse. Some previous results of the PI in this direction have alreadyattracted the attention of physicists working in general relativity. In bothdimensions, 2 and 3, one of the key problem is to understand the possiblesymmetries of these shapes. The proposal is devoted to some of the mainaspects of this problem.