Homotopy Theory, Loop Spaces, and Group Cohomology

小行星9704410 这个项目探讨拓扑学，群论， 以及受到物理学中经典杨-巴克斯特关系启发的某些结构。 三个主要领域如下：（1）人们可以将固定流形中有限数量的高维球体（或悬浮体）的“编织”视为经典阿丁编织群的类似物。 这里的主要例子是满足经典物理学杨-巴克斯特关系的泛李代数，它涉及粒子的碰撞。 附加函数空间的同调正是这些“泛杨-巴克斯特李代数”的泛包络代数。所考虑的空间是经典构形空间的圈空间以及M. Xicotencatl. 许多这些空间的循环空间的同源性被证明是一个李代数，是定义在杨-巴克斯特或无穷小辫子关系的通用包络代数。 基础流形可以是齐次空间或具有欧几里得因子的流形。 在"轨道配置空间“的情况下，这些相同的关系也出现在纽结不变量中，正如L.考夫曼 (2)研究了同伦理论中函数空间的某些选择的结构与经典代数几何中曲线的模空间之间的密切联系。 连接组织是Dehn捻和Whitehead产品之间的联系。 这些连接用于进行同调计算。 在这里，考虑从一个固定的表面到一个球的映射的空间。 这些空间是稳定分裂的，而稳定的和项是以一定的曲线模空间为基的丛的Thom复形。 在有限亏格的情况下，证明了同调计算是循环同调的一种模拟，并进行了一些具体的计算。 证明了稳定映射类群分类空间的加结构是分裂的。 一个因子给出球面的稳定同伦群。 (3)组合群论中的经典方法被用来研究某些空间的同伦群中挠率的增长问题。 单纯形方法以及W. Magnus关于某些结合代数中作为单位的嵌入群。 研究了同调均为挠的双悬挂系的同伦群。 主要工具是有限p-群的逆极限。 这个塔是恒等式的古德威利塔的代数模拟;每一级的群都是中心扩张，每一级的中心由模Lie（n）的mod-p约化给出，Lie（n）是秩为n（n-1）的对称群的模！在同伦理论中无处不在 这些问题的基础是普通的弦编织。 人们也可以想象编织其他几何对象，如平面，球体或鸟居。 这些辫子迫使对称性发生在以前的物理学中，考虑粒子的碰撞，被称为杨-巴克斯特关系。 这个项目的第一部分是研究高维几何对象的对称性。 接下来考虑去除了许多孔的甜甜圈的表面。 在第二个方向上，考虑了该表面的"光滑变形“。 这里的主旨是分析这些"光滑变形“空间中的”洞“，因为它们出现在物理学、弦理论、几何学、代数拓扑学和全纯映射的许多背景中。 一个自然的延续是理解大尺寸的球体如何在空间中移动，只要允许连续变形。 这个问题一直是关键的人，是一个主要的问题同伦理论上有有趣的进展。 这些球体的运动方式与时针绕时钟的运动方式大致相同。 实际的数字是相当复杂的，这个项目的一部分是了解这里的统一结构。 ***

9704410 Cohen This project explores connections between topology, group theory, and certain constructions inspired by the classical Yang-Baxter relations in physics. Three main areas arise as follows: (1) One could consider ``braidings'' of a finite number of higher dimensional spheres (or suspensions) in a fixed manifold as an analogue of the classical Artin braid group. The main examples here arise as the universal Lie algebra that satisfies the Yang-Baxter relations of classical physics, which concerns collisions of particles. The homology of the attached function spaces is precisely the universal enveloping algebras of these ``universal Yang-Baxter Lie algebras.'' The spaces under consideration are loop spaces of classical configuration spaces as well as ``orbit configuration spaces'' as studied by M. Xicotencatl. The loop space homology of many of these spaces is shown to be the universal enveloping algebra of a Lie algebra that is defined in terms of the Yang- Baxter or infinitesimal braid relations. The underlying manifold can be a homogeneous space or a manifold with a Euclidean factor. In the case of ``orbit configuration spaces,'' these same relations also appear in knot invariants as studied by L. Kauffman. (2) The intimate link between the structure of certain choices of function spaces in homotopy theory and moduli spaces of curves within classical algebraic geometry is studied. The connecting tissue is the link between Dehn twists and Whitehead products. These connections are used to give homological calculations. Here, the space of maps from a fixed surface to a sphere is considered. These spaces stably split, while the stable summands are Thom complexes of bundles with base given by certain moduli spaces of curves. In the case of finite genus with marked points, homological calculations are shown to be an analogue of cyclic homology, while some specific calculations are carried out. The plus construction for the classifying space of th e stable mapping class group is shown to split. One factor gives the stable homotopy groups of spheres. (3) Classical methods in combinatorial group theory are used to attack problems on the growth of the torsion in the homotopy groups of certain spaces. Simplicial methods are used as well as techniques from classical methods of W. Magnus on embedding groups as units in certain associative algebras. Here, the homotopy groups of double suspensions all of whose homology is torsion are studied. The main tool is an inverse limit of finite p-groups. This tower is an algebraic analog of the Goodwillie tower of the identity; the groups at each stage are central extensions with centers at each stage given by the mod-p reductions of modules Lie(n), a module of the symmetric group on n letters of rank (n-1)! that has occurred ubiquitously in homotopy theory. Ordinary braidings of strings are at the foundation of these problems. One could also imagine braiding other geometric objects such as planes, spheres, or torii. These braidings force symmetries that have occurred previously in physics from considering collisions of particles and are known as the Yang-Baxter relations. The first part of this project is a study of these symmetries for higher dimensional geometric objects. Next consider the surface of a doughnut with many holes removed. In a second direction, the ``smooth deformations'' of this surface are considered. The main thrust here is to analyze the ``holes'' in these spaces of ``smooth deformations'' as they appear in many contexts in physics, string theory, geometry, algebraic topology, and holomorphic maps. A natural continuation is to understand how spheres of large dimension move around in space provided only continuous deformations are permitted. This problem has been the key to others and is one of the main problems in homotopy theory on which there has been interesting progress. These spheres move in much the same way that the hour hand moves around a clock. The actual numbers are quite complicated, and part of this project is to understand the uniform structure here. ***