Certain problems regarding the algebraic topology of manifolds

数学中经常出现这样一种现象，即通过考虑一个具有某种结构的数学对象保持从该对象到其自身的映射的可逆结构空间，可以获得关于该对象的大量信息。这种被称为自同构群的空间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是特别有成效的，而且是相当普遍的。从广义上讲，这门学科的主要目的是利用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且足够有用以便于计算的不变量。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微分同胚群，对于光滑流形M，我们可以定义M的微分同胚群，记为diff(M)，它是从M到它自己的所有微分同胚群。这是自同构群的一个实例。除了由所有自同构群共享的由微分同态射的合成和逆所给出的群结构外，微分同态群还具有拓扑空间的结构。这是一个更普遍的事实的结果：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具有Whitney拓扑的拓扑空间。作为C sigma(M，N)的一个子空间，diff(M)是一个拓扑空间，进一步我们可以证明它是一个拓扑群，这意味着合成和求逆的群运算是连续的。流形的微分同胚群的研究是流形嵌入空间研究的一个重要特例。当考虑有关嵌入的问题时，自然会产生一个概念，即配置空间。流形M中n个点的构形空间，记为Confn(M)，是M的所有不同点的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到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上下文，因为任何嵌入F：M-N都会诱导出笛卡儿积上的映射，而笛卡尔积上的映射又会诱导出配置空间f：Confn(M)-Confn(N)上的映射。因此，有理由期望构形空间在构造嵌入空间的不变量方面发挥重要作用。事实上，配置空间上的一种构造的变化，即配置空间积分，已经引起了与嵌入有关的空间的一些重要不变量，这在低维和几何拓扑中产生了显著的结果。我研究生学习的目标之一是更广泛地研究流形的嵌入空间，更具体地理解这些结构。尽管位形空间积分在它们被引入的各种上下文中扮演了重要的角色，但不同已知结构之间的联系还没有被很好地理解。具体地说，大多数已知的结构涉及沿某些微分形式的纤维的积分，而缺乏更一般的同伦理论的描述。这样的描述将是重要的，使我们能够比较各种结构的配置空间积分，也将帮助我们检查相关的结构在各种系数组。科伊特切夫已经朝着这个方向迈出了一步，至少在纽结的情况下是这样。此外，我们在上面已经看到，配置空间积分取决于某些成帧数据。进一步探索这些额外数据的作用，以及我们是否可以对其他切线结构进行类似的构造，将会更加有趣[5]。该项目属于EPSRC几何和拓扑学研究领域。

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.