Some Problems in Algebraic and Geometric Topology

本文讨论了代数拓扑在高维流形分类中的应用。PI和合作者最近证明了每个有限循环空间都同伦等价于一个紧凑的光滑流形，解决了Browder提出的一个40年前的问题，激发了手术理论。Borel，Novikov，Baum-Connes和Farrell-Jones的理论断言K-和L-理论中的集合映射要么是单态的，要么是同构的。所有这些集合映射都可以用控制代数来表达，这是PIU打算继续研究的一个领域，以获得更大类的群的结果。流形是一种几何对象，局部像欧几里得空间，但可以全局弯曲，如球体。这些空间出现在数学和物理学中。流形已经被拓扑学家研究了一百多年，因为它们是理解空间本质的关键。PI建议扩展vurrent代数分类方法，以实现更深层次的几何理解。特别重要的是控制代数方法，它抓住了代数和拓扑的本质

This proposal concerns the application of Algebraic Topology to the classification of high dimensional manifolds. The PI and collaborators have recently shown that every finite loop space is homotopy equivalent to a compact, smooth manifold, solving a 40 year old problem posed by Browder, motivating Surgery Theory. This still leaves open the question whether all finite H-spaces have this property.The conjectures of Borel, Novikov, Baum-Connes and Farrell-Jones assert that the assembly maps in K- and L-theory are either monomorphisms or isomorphisms. All these assembly maps can be expressed in terms of controlled algebra, an area the PIU intends to continue working on, to obtain results for larger classes of groups.A manifold is a geometrical object which locally is like euclidean space, but may be curved globally such as a sphere. These spaces arise both in mathematics and in Physics. Manifolds have been studied by topologists for over a hundred years as they hold the key to understanding the nature of space. The PI proposes to extend vurrent algebraic classification methods to achieve a deeper geometric understanding. Of particular significance is controlled algebraic methods, which capture the essence of both algebra and topology