Certain problems regarding the algebraic topology of manifolds

数学中经常出现这样的情况：通过考虑从一个对象到它自身的可逆结构的映射空间，我们可以得到关于一个具有某种结构的数学对象的大量信息。这样的空间被称为自同构群，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有效，而且非常普遍。广义地说，这门学科的主要目的是用代数不变量来研究空间。理想情况下，我们希望使用的不变量足够有趣，足以包含大量信息，并且足够有用，易于计算。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构，对于光滑流形M，我们可以定义M的微分同构群，用Diff(M)表示，它是M到它自身的所有微分同构的空间。这是一个自同构群的实例。微分同构群除了具有所有自同构群所共有的、由微分同构的复合和反转所给出的群结构外，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，从M到N的光滑函数空间用C σ （M， N）表示是一个拓扑空间，具有Whitney拓扑。作为C σ （M， N）的一个子空间，Diff(M)是一个拓扑空间，并且可以证明它是一个拓扑群，即复合和反演的群运算是连续的。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，一个自然产生的概念是配置空间。流形M中n个点的位形空间，用Confn(M)表示，是M中所有不同点的n元组的空间，或等价于n个粒子在不碰撞的情况下移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都会在笛卡尔积上产生一个映射，而笛卡尔积又会在构型空间F: Confn(M)-Confn(N)上产生一个映射。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，构型空间上的构造的变化，构型空间积分，已经产生了一些与嵌入相关的重要空间不变量，这些不变量在低维和几何拓扑中产生了显著的结果。我的研究生学习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。尽管构型空间积分在引入它们的各种环境中扮演着重要的角色，但不同已知结构之间的联系还没有得到很好的理解。特别是，大多数已知的结构涉及沿某些微分形式的纤维的积分，并且缺乏一般的，更同伦理论的描述。这样的描述对于我们比较构形空间积分的各种构造是很重要的，也有助于我们检查不同系数群上的相关构造。Koytcheff b[4]已经朝着这个方向迈出了一步，至少在结的情况下是这样。此外，我们在上面已经看到，构型空间积分依赖于某些帧数据。进一步探索这些额外数据的作用以及我们是否可以为其他切线结构b[5]建立类似的结构将是很有趣的。该项目属于EPSRC几何和拓扑研究领域。

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.