Mathematical Sciences: Topology

9306240 Raymond Frank Raymond在主G-丛的基础上改进了广义Seifert纤维的理论，使其包括任意李群G。应用程序和例子说明了一般理论与G是交换或幂零的情况有何不同，并将形成最终的形式。Seifert Fiberings的技巧和结果将被用来研究常曲率伪黎曼流形及其框架丛的空间形式问题。欧拉特征为零的椭圆曲面支持4维几何。关于这些几何和等距类的模的Teichmueller理论将被构造和确定。彼得·斯科特计划在三个主要领域开展工作：首先，他将继续研究3-流形的拓扑刚性。其次，他计划在三维Poincare对偶群领域开展工作。这里的长期目标是证明任何这样的群来自一个3-流形，但他将把他的注意力限制在那些‘应该’对应于Seifert纤维空间或Haken流形的群。第三，他计划将他的工作扩展到更高的维度，生成3-流形的特征子流形。流形是像圆、球体和甜甜圈这样的自然几何对象，可以用与相同维度的欧几里德空间相同类型的坐标在局部进行描述。它们的多样性和复杂性随着维度的增加而迅速增加，但一些最棘手的问题已经出现在第三和第四维度中。我们生活在一个三维的世界里，如果考虑到时间的话是四维的，这使得这一点不仅仅是纯粹的学术兴趣。宇宙学家不知道哪个3维或4维流形构成了物理宇宙，因此研究可能发生的可能性有足够的动机。这两位研究人员正在巧妙地利用几何和代数来阐明这个问题。他们的工作还将丰富其他人可用于研究低维流形的工具。***

9306240 Raymond Frank Raymond has refined a theory for generalized Seifert fiberings modelled on principal G-bundles to include arbitrary Lie groups G. Applications and examples illustrating how the general theory differs from the case where G is abelian or nilpotent have been developed and will be put into final form. Techniques and results from Seifert fiberings will be used to investigate space form problems for the pseudo-Riemannian manifolds of constant curvature and their frame bundles. Elliptic surfaces whose Euler characteristic are zero support 4-dimensional geometries. A Teichmueller theory for these geometries and the moduli of isometry classes will be constructed and determined. G. Peter Scott plans to work in three main areas: Firstly, he will continue his work on the topological rigidity of 3-manifolds. Secondly, he plans to work in the area of 3-dimensional Poincare duality groups. The long term aim here is to show that any such group comes from a 3-manifold, but he will restrict his attention to those groups which 'ought' to correspond to Seifert fiber spaces or to Haken manifolds. Thirdly, he plans to extend to higher dimensions his work generating the characteristic submanifold of a 3-manifold. Manifolds are natural geometric objects like circles and spheres and doughnuts that can be described locally by the same type of coordinates as a Euclidean space of the same dimension. Their variety and complexity increases rapidly as the dimension increases, but some of the most intractable problems arise already in dimensions three and four. That we live in a world of three dimensions, or four if time is considered, makes this of more than purely academic interest. Cosmologists do not know which 3- or 4-manifold constitutes the physical universe, so studying the possibilities that can occur has ample motivation. These two investigators are making ingenious use of geometry and of algebra to shed light on the question. Their work w ill also enrich the arsenal of tools available to others for studying low-dimensional manifolds. ***