Certain problems regarding the algebraic topology of manifolds

关于流形代数拓扑的若干问题

• 批准号: 2433258
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2433258
• 关键词: Certain; problems; regarding; algebraic; topology

It often occurs in mathematics that one can get a lot of information about a mathematical object with some structure by considering the space of invertible structure preserving maps from this object to itself. Such spaces, called automorphism groups, are usually equipped with rich structure that makes them interesting objects of study. This approach is particularly fruitful and quite ubiquitous in algebraic topology. The main purpose of the subject, broadly speaking, is to study spaces by means of algebraic invariants. Ideally we wish to work with invariants that are interesting enough to contain a lot of information and useful enough to be easily computable. The rich structure of automorphism groups is, in many cases, helpful in allowing us to reach this balance. For example, we can consider smooth manifolds. The invertible maps which preserve the smooth structure are the diffeomorphisms and for a smooth manifold M we can define the diffeomorphism group of M, denoted by Diff(M), which is the space of all diffeomorphisms from M to itself. This is an instance of an automorphism group. Apart from the group structure that is shared by all automorphism groups and given by composition and inversion of diffeomorphisms, the diffeomorphism group also has the structure of a topological space. This is a consequence of a more general fact: Given two smooth manifolds M, N,the space of smooth functions from M to N denoted by C sigma(M, N)is a topological space, equipped with the Whitney topology. As a subspace of C sigma(M, N), Diff(M)is a topological space, and furthermore we can show that it is a topological group, meaning that the group operations of composition and inversion are continuous. The study of diffeomorphism groups of manifolds is an important special case of the study of embedding spaces of manifolds. A notion which arises naturally when thinking about questions regarding embeddings is that of configuration spaces. The configuration space of n points in a manifold M, denoted by Confn(M), is the space of all n-tuples of distinct points of M, or equivalently the space of all possible positions of n particles moving in without colliding. These spaces provide a natural context for talking about embeddings since any embedding F: M-N induces a map on Cartesian products which in turn induces a map on configuration spaces f: Confn(M)-Confn(N). It is reasonable, therefore, to expect that configuration spaces play an important role in constructing invariants of embedding spaces. Indeed, variations of a construction on configuration spaces, the configuration space integral, has given rise to some important invariants of spaces related to embeddings that produced remarkable results in low dimensional and geometric topology. One of my goals for my graduate studies is to study embedding spaces of manifolds broadly and more specifically understand these constructions better. Despite the important role configuration space integrals have played in the various contexts they have been introduced, the connections between the different known constructions are not yet very well understood. In particular, most known constructions involve integration along fibers of certain differential forms and a general, more homotopy-theoretic, description is lacking. Such a description would be important in allowing us to compare the various constructions of configuration space integrals and would also help us examine related constructions over various coefficients groups. Steps towards this direction, at least for the case of knots, have been taken by Koytcheff [4]. Moreover, we have seen above that the configuration space integrals depend on certain framing data. It would further be interesting to explore what the role of this extra data is and whether we can have similar constructions for other tangential structures [5]. This project falls within the EPSRC Geometry and Topology research area.

在数学中经常会出现这样的情况，即通过考虑一个具有某种结构的数学对象到它自身的可逆结构保持映射空间，可以得到关于这个对象的大量信息。这种空间称为自同构群，通常具有丰富的结构，使它们成为有趣的研究对象。这种方法在代数拓扑学中特别富有成效，并且非常普遍。广义上讲，这门学科的主要目的是通过代数不变量来研究空间。理想情况下，我们希望使用足够有趣以包含大量信息并且足够有用以易于计算的不变量。在许多情况下，自同构群的丰富结构有助于我们达到这种平衡。例如，我们可以考虑光滑流形。保持光滑结构的可逆映射是自同构，对于光滑流形M，我们可以定义M的自同构群，记为Diff（M），它是从M到它自身的所有自同构的空间。这是一个自同构群的实例。除了由所有自同构群共享的群结构以及由自同构的合成和反转给出的群结构之外，自同构群还具有拓扑空间的结构。这是一个更一般的事实的结果：给定两个光滑流形M，N，由C sigma（M，N）表示的从M到N的光滑函数空间是一个拓扑空间，具有惠特尼拓扑。作为C sigma（M，N）的一个子空间，Diff（M）是一个拓扑空间，进而我们可以证明它是一个拓扑群，这意味着群的复合和反演运算是连续的。流形的自同态群的研究是流形嵌入空间研究的一个重要特例。在思考嵌入问题时，自然会产生一个概念，那就是位形空间。流形M中n个点的位形空间，记为Confn（M），是M中所有不同点的n元组的空间，或者等价地是n个粒子在没有碰撞的情况下运动的所有可能位置的空间。这些空间为讨论嵌入提供了一个自然的背景，因为任何嵌入F：M-N都导出笛卡尔积上的映射，而笛卡尔积又导出配置空间f：Confn（M）-Confn（N）上的映射。因此，我们有理由期待构形空间在构造嵌入空间的不变量中发挥重要作用。事实上，构形空间上的一个构造的变化，构形空间积分，已经产生了一些重要的与嵌入相关的空间不变量，这些不变量在低维和几何拓扑中产生了显着的结果。我的研究生学习目标之一是广泛地研究流形的嵌入空间，更具体地理解这些结构。尽管重要的作用配置空间积分发挥了各种情况下，他们已经介绍了，不同的已知结构之间的联系还没有很好地理解。特别是，大多数已知的结构涉及集成沿着纤维的某些微分形式和一般的，更同伦理论，描述是缺乏的。这样的描述将是重要的，使我们能够比较各种结构的位形空间积分，也将帮助我们检查相关的建设在各种系数组。Koytcheff [4]已经朝着这个方向迈出了一步，至少对于结的情况是这样。此外，我们在上面已经看到，位形空间积分取决于某些框架数据。进一步探索这些额外数据的作用以及我们是否可以为其他切向结构提供类似的结构将是有趣的[5]。该项目属于EPSRC几何和拓扑研究领域的福尔斯。