Mathematical Sciences: The Topology of Generalized Manifolds

Frank Quinn证明了，对于n = 5，一个连通的 广义n-流形（n-gm）X有一个相关的归结 阻塞，由“局部索引”i（X）给出，i（X）是整数 等于1模8。 X是可分解的，当且仅当i（X）=1。 这个结果与R. D.爱德华兹暗示 一个n-gm X，n = 5，是一个拓扑流形当且仅当X有 不相交圆盘性质（DDP）和i（X）= 1。 左开在 Quinn的结果是一个问题， 存在广义流形。 最近，调查人员， 与史蒂夫·费里和什穆尔·温伯格一起，已经证明， 给定任意单连通闭n-流形X，n = 6，且任意 整数m = 1（mod 8），存在一个n-gm X，同伦等价于 M，其中i（X）= m。 这些例子的发现提出了几个 关于广义流形的拓扑问题， 这些问题构成了这个项目的基础。 最终目的 是了解广义的分类方案， 高维流形的直接目标是 发现这些空间在多大程度上像真实的 流形 一个n维的拓扑流形，或n-流形，是一个 局部看起来像欧几里得空间的几何对象， 维度;尺寸 因此，以这些空间为模型的科学现象 可以使用局部坐标系来表示和分析。 几何拓扑学的中心问题之一是 规定（局部）欧几里得“拓扑特征” 空间. 这里我们指的是一个拓扑学家 可以用来检查给定的空间是否是流形。 一维或二维流形的特征已经被 很久以前就知道了，但更高的维度特征 计划的制定和验证已被证明是难以捉摸的。 一 维数大于4的流形的特征是 詹姆斯·坎农（James Cannon）在1978年的一次演讲中 国际数学家大会在赫尔辛基。 弗兰克 奎因发现，坎农的标准将产生一个解决方案， 这个问题， 条件是附加的整数不变量为零。 最近， 提议者，沿着史蒂夫·费里和什穆尔·温伯格， 发现了一大类几何物体， 满足Cannon标准但Quinn值不为零的五个 不变的 有证据表明，这些新的空间应该是 纳入类流形和共同的理论 开发 研究人员打算研究 理论及其潜在应用。 作为 例如，除其他外，他们计划将这些 “广义流形”的动力学研究流形上的 这表明它们可能扮演着 空间中的运动。

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.