Homotopy theory, loop spaces,group cohomology, and configuration spaces

本提案中概述的项目是针对同伦理论、群论和功能空间拓扑之间的相互作用。马丁的编发小组发挥了核心作用。主要项目如下：(1)继续完成Barratt有限指数猜想的程序；(2)计算具有不同表示选择的离散群的群上同调；(3)进一步研究函数空间的上同调、群上同调以及与函数空间拓扑、位形空间和超平面排列的联系；(4)分析了同伦群与其他结构的性质重叠，如Artin的纯辫群的降中心级数所附的李代数，以及由位形空间的高同伦群所得到的李代数；(5)详细分析了某些协代数同态群及其与辫群的联系，以及同伦理论。这里项目的主要目标涉及在空间中移动的圆的几何形状。这些圆圈可以被认为是行星物体在轨道上相互围绕的运动。这些轨道几何结构之间的相互作用在数学和数学物理中都存在。关于这些轨道的精确信息为若干学科提供了卓有成效的数学应用。这个项目的最终目标是测量学科中不同的量，并分析具体有用的答案和计算。

Frederick R. CohenDMS-0072173The projects outlined in this proposal are directed toward the interplay between homotopy theory, group theory, and the topology of function spaces. Artin's braid group plays a central role. The main projects are as follows: (1) continuation of a program to finish Barratt's finite exponentconjecture, (2) computations of the group cohomology for certain discrete groups with varying choices of representations, (3) a further investigation of the connections between thecohomology of function spaces, group cohomology, as wellas the connections with the topology of function spaces, configuration spaces, and hyperplane arrangements, (4) an analysis of the overlap of properties of homotopy groups with other structures such as the Lie algebra attached to the descending central series for Artin's pure braid group, and the Lie algebra obtained from the higher homotopy groups of configuration spaces, and (5) a detailed analysis of certain groups of coalgebra morphisms as well as their connections to braid groups, and homotopy theory.The main goals of the projects here involve geometry of circles moving through space. These circles can be thought of as the motions of planetary objects in orbit around each other. The interplay between the geometry of these orbits occurs in mathematics as well as in mathematical physics. Precise information concerning these orbits has provided fruitful mathematical applications to several subjects. The ultimate goal of this project is to measure different quantities in the subject as well as analyzing concrete useful answers, and computations.