Homotopy Theory, Loop Spaces, and Group Cohomology

该项目探索拓扑学、群论和某些结构之间的联系，这些结构受到物理学中经典Yang-Baxter关系的启发。出现了以下三个主要领域：(1)人们可以将固定流形中有限数量的高维球体（或悬浮）的“编织”视为经典Artin编织群的类似物。这里的主要例子是满足经典物理学中涉及粒子碰撞的Yang-Baxter关系的通用李代数。所附函数空间的同调正是这些泛Yang-Baxter李代数的泛包络代数。“考虑的空间是经典构型空间的循环空间以及M. Xicotencatl研究的‘轨道构型空间’。”其中许多空间的环空间同调被证明是由Yang- Baxter或无穷小辫状关系定义的李代数的全称包络代数。底层流形可以是齐次空间或带有欧几里得因子的流形。在“轨道构型空间”的情况下，这些相同的关系也出现在L. Kauffman研究的结不变量中。(2)研究了同伦理论中某些函数空间的选择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扭曲和Whitehead产品之间的联系。这些连接用于同调计算。这里考虑了从固定曲面到球面的映射空间。这些空间稳定分裂，而稳定和是由曲线的模空间给出基束的托姆复形。在有标记点的有限格的情况下，同调计算被证明是循环同调的类似物，同时进行了一些具体的计算。稳定映射类群的分类空间的加号构造是分裂的。一个因子给出了球的稳定同伦群。(3)利用组合群论中的经典方法研究了特定空间同伦群中挠性的生长问题。利用简化方法和经典的W. Magnus方法中的技术，将群作为单位嵌入到某些结合代数中。本文研究了同调均为扭转的双悬的同伦群。主要的工具是有限p群的逆极限。这个塔是恒等式的古德威利塔的代数模拟；每个阶段的群是中心扩展，每个阶段的中心由模Lie(n)的模p约简给出，模Lie(n)是n个秩为（n-1）的对称群的一个模！这在同伦理论中无处不在。普通的绳编织是这些问题的基础。人们还可以想象编织其他几何物体，如平面、球体或torii。这些编织力的对称性以前在物理学中由于考虑粒子碰撞而产生，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物体的对称性。接下来考虑去除许多洞的甜甜圈表面。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、代数拓扑和全纯映射的许多上下文中。一个自然的延拓是理解大维度的球体如何在空间中运动，前提是只允许连续变形。这个问题一直是其他问题的关键，也是同伦理论中取得有趣进展的主要问题之一。这些球体的运动方式与时针绕时钟转动的方式大致相同。实际的数字是相当复杂的，这个项目的一部分是理解这里的统一结构。＊＊＊

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. ***