Homotopy Theory, Loop Spaces, and Group Cohomology

9704410科恩这个项目探索拓扑学、群论和某些结构之间的联系，这些结构受到物理学中经典的杨-巴克斯特关系的启发。主要有以下三个方面：(1)人们可以将固定流形中有限个高维球体(或悬浮体)的“编织”视为经典Artin编织群的类似。这里的主要例子是满足经典物理学中关于粒子碰撞的杨-巴克斯特关系的普适李代数。依附函数空间的同调恰好是这些“泛Yang-Baxter李代数”的泛包络代数。所考虑的空间是经典构形空间的循环空间以及M.Xicotencatl所研究的“轨道构形空间”。许多这样的空间的环空间同调被证明是由Yang-Baxter或无穷小辫子关系定义的李代数的泛包络代数。其下层流形可以是齐次空间或具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.考夫曼研究的纽结不变量中。(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。连接组织是德恩扭曲和怀特黑德产品之间的纽带。这些联系被用来给出同调计算。在这里，考虑从固定曲面到球面的映射空间。这些空间是稳定分裂的，而稳定的和是由某些曲线的模空间给出的具有基的丛的Thom复形。在有标点有限亏格的情况下，证明了同调计算类似于循环同调，并进行了一些具体的计算。证明了稳定映射类群的分类空间的正结构是分裂的。一个因子给出了球面的稳定同伦群。(3)利用组合群论中的经典方法，研究了某些空间的同伦群的挠率增长问题。在某些结合代数中，使用单纯方法和W.Magnus经典方法中的技巧来嵌入群作为单位。本文研究了双悬架同调为扭转的同伦群。主要工具是有限p-群的逆极限。这个塔是恒等式的Goodwillie塔的代数模拟；每个阶段的群都是中心扩张，每个阶段的中心由模Lie(N)的mod-p约化给出，它是n个字母秩(n-1)的对称群的模！这在同伦理论中随处可见。普通的编绳是这些问题的基础。人们还可以想象编织其他几何物体，如平面、球体或鸟取物。这些辫子迫使物理学中以前出现的对称性不再考虑粒子的碰撞，这就是众所周知的杨-巴克斯特关系。这个项目的第一部分是对高维几何对象的这些对称性的研究。接下来，考虑去掉很多孔的甜甜圈的表面。在第二个方向上，考虑该表面的“平滑变形”。这里的主要目的是分析这些“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。一个自然的延续是理解在只允许连续变形的情况下，大维度的球体如何在空间中移动。这个问题一直是解决其他问题的关键，也是同伦理论中的主要问题之一，在这个问题上取得了有趣的进展。这些球体的运动方式与时针绕时钟运动的方式很相似。实际数字相当复杂，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. ***