Mathematical Sciences: The Topology of Generalized Manifolds

Frank Quinn已经证明，对于n=5，连通的广义n-流形(n-gm)X有一个关联的分解障碍，它由“局部指数”I(X)给出，它是一个与1 mod 8同余的整数。当且仅当i(X)=1时，X是可分解的。这一结果与R.D.Edwards的一个定理相结合，意味着一个n-gm X，N=5是拓扑流形当且仅当X具有不交圆盘性质(DDP)且I(X)=1。Quinn的结果中没有讨论是否存在不可分解的广义流形的问题。最近，研究人员与Steve Ferry和Shmuel Weinberger一起证明了，给定任何单连通的闭n-流形X，n=6，以及任何整数m=1(Mod 8)，都有一个n-gm X，同伦等价于M，这些例子的发现引出了关于广义流形拓扑的几个问题，这些问题构成了本项目的基础。最终目的是了解高维广义流形的一般分类模式，直接目的是发现这些空间在多大程度上表现得像真正的流形。N维拓扑流形或n-流形是一个局部看起来像n维欧几里德空间的几何对象，因此，可以使用局部坐标系来表示和分析在这些空间上模拟的科学现象。几何拓扑学的中心问题之一是给出(局部)欧氏空间的“拓扑刻画”。这里我们指的是拓扑学家可以用来检查给定空间是否是流形的一系列性质。一维或二维流形的刻画早已为人所知，但高维刻画方案很难描述和验证。詹姆斯·坎农在1978年赫尔辛基国际数学家大会上的一次演讲中猜想了维度大于4的流形的一个特征。弗兰克·奎因发现，只要另外一个整数不变量为零，坎农的标准就会给出这个问题的解决方案。最近，提出者与Steve Ferry和Shmuel Weinberger一起，发现了一大类维度大于5的几何对象，它们满足Cannon标准，但具有非零Quinn不变量。有证据表明，这些新的空间应该并入流形的类别，并发展了一个共同的理论。研究人员打算研究这一理论的基本方面及其潜在的应用。作为一个例子，他们计划将这些“广义流形”与流形上的动力学研究联系起来，证明它们可能扮演空间运动的对称轴的角色。

Frank Quinn has shown that, for n = 5, a connected generalized n-manifold (n-gm) X has an associated resolution obstruction, given by a "local index" i(X), which is an integer congruent to 1 mod 8. X is resolvable if and only if i(X)=1. This result, combined with a theorem of R. D. Edwards, implies that an n-gm X, n = 5, is a topological manifold iff X has the disjoint disks property (DDP) and i(X) = 1. Left open in the results of Quinn was the question of whether nonresolvable generalized manifolds exist. Recently, the investigators, together with Steve Ferry and Shmuel Weinberger, have shown that, given any simply connected, closed n-manifold X, n = 6, and any integer m = 1(mod 8), there is an n-gm X, homotopy equivalent to M, with i(X) = m. The discovery of these examples raises several questions about the topology of generalized manifolds and these questions form the basis of this project. The ultimate purpose is to understand the general classification scheme of generalized manifolds in high dimensions with the immediate goal of discovering to what extent these spaces behave like real manifolds. A topological manifold of dimension n, or n-manifold, is a geometric object that looks locally like euclidean space of dimension n. Thus, scientific phenomena modeled on these spaces can be expressed and analyzed, using local coordinate systems. One of the central problems of geometric topology has been to prescribe a "topological characterization" of (locally) euclidean spaces. By this we mean a list of properties that a topologist could use to check whether or not a given space is a manifold. Characterizations of manifolds of dimension one or two have been known for a long time, but higher dimensional characterization schemes have proved to be elusive to formulate and verify. A characterization of manifolds of dimension greater than four was conjectured by James Cannon in an address to the 1978 International Congress of Mathematicians in Helsinki. Frank Quinn discovered that Cannon's criteria would yield a solution to the problem, provided that an additional integer invariant is zero. Recently, the proposers, along with Steve Ferry and Shmuel Weinberger, found a large class of geometric objects of dimension greater than five that satisfy Cannon's criteria, but have non-zero Quinn invariant. There is evidence that these new spaces should be incorporated in the class of manifolds and a common theory developed. The investigators intend to study the foundational aspects of the theory and its potential applications. As an example, among other things, they plan to relate these "generalized manifolds" to the study of dynamics on manifolds by showing that they may play the role of axis of symmetry of motions in space.