Mathematical Sciences: The Topology of Generalized Manifolds

点击翻译按钮获取中文摘要

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.