Interactions between geometry and topology

摘要奖：DMS-0906483主要研究者：Jean-Francois R. Lafont首席研究员（PI）建议开展一系列项目，将几何和拓扑技术结合起来，以期应用于这两个领域。一个主要组成部分，这一建议侧重于利用geometricarguments阐明代数K理论（integralgroup环）的各类群体。PI以前与Ortiz的工作提供了一个完整的算法来确定双曲3-空间等距群中格的下代数K-理论。PI打算进一步发展technologesfor计算低代数K-理论的各种其他几何显着类的群体。这种显式计算的重要性是双重的。首先，它提供了一个新技术的试验场，和一个新技术的可能来源。其次，由于代数K-理论和高维拓扑之间的密切关系，我们希望我们的计算能对高维流形的拓扑结构有所启发。PI也对更好地理解各种度量属性如何约束底层空间的拓扑感兴趣。这里有很大的自由度，取决于度量性质的含义，以及我们选择研究拓扑的哪个方面。例如，可以考虑黎曼曲率如何影响空间的有界上同调。或者人们可以尝试研究度量性质（例如G-空间公理）如何影响空间的局部拓扑。或者，人们可以尝试在一个固定的拓扑空间类（例如闭光滑流形）中研究支持各种类型度量性质的拓扑空间（例如非正曲率的黎曼度量，与局部CAT（0）度量相比）。PI打算研究与这个思想圈有关的问题。几何学是研究配备了一些附加结构的空间的性质，通常具有度量性质（即允许人们测量空间中的距离）。在拓扑学领域，人们忘记了精确的潜在距离，而只保留了空间中的点是“紧密相连”的概念。因此，拓扑通常被称为“橡胶几何”：人们可以拉伸和变形空间而不改变其底层拓扑。在拓扑学中，流形是最有趣的空间。这些空间，在小尺度上，看起来像欧几里得空间，但在大尺度上，可以弯曲。一个基本的问题在于找到检测两个流形何时彼此不同的方法。这通常是通过将某些不变量与流形相关联来实现的：如果不变量不同，那么流形也不同。代数K-理论不变量是一类在高维拓扑中特别有用的不变量。一般来说，这些不变量很难计算。PI计划开发计算这些不变量的方法，在特殊情况下，流形有一些很好的基础几何。更一般地说，PI计划研究几何结构的存在如何约束各种拓扑性质（反之亦然）。

AbstractAward: DMS-0906483Principal Investigator: Jean-Francois R. LafontThe Principal Investigator (PI) proposes to work on a series ofprojects that incorporate techniques from geometry and topology,with a view to applications to both of these fields. A majorcomponent of this proposal focuses on making use of geometricarguments to shed light on the algebraic K-theory of (integralgroup rings of ) various classes of groups. The PI's previouswork with Ortiz provided a complete algorithm to determind thelower algebraic K-theory for lattices in the isometry group ofhyperbolic 3-space. The PI intends to further develop techniquesfor computing the lower algebraic K-theory of various othergeometrically significant classes of groups. The importance ofsuch explicit computations is twofold. First of all, it providesa testing ground for conjectures, and a possible source of newconjectures. Secondly, because of the close relationship betweenalgebraic K-theory and high-dimensional topology, we expect ourcomputations to shed light on the topology of high-dimensionalmanifolds. The PI is also interested in better understanding howvarious metric properties constrain the topology of theunderlying space. There is a lot of freedom here, depending onwhat one means by a metric property, and what aspect of thetopology we choose to study. For instance, one can consider howRiemannian curvature influences the bounded cohomology of aspace. Or one can try to study how metric properties (e.g. theG-space axioms) can influence the local topology of aspace. Alternatively, one can try to study, within a fixed classof topological spaces (e.g. closed smooth manifolds), thosesupporting various types of metric properties (e.g. Riemannianmetrics of non-positive curvature, as compared to locally CAT(0)metrics). The PI intends to work on questions related to thiscircle of ideas.Geometry is the study of properties of spaces equipped with someadditional structure, usually of a metric nature (i.e. allowingone to measure distances in the space). In the field of topology,one forgets about the precise underlying distance, and onlyretain the notion of points in the space being "closetogether". As such, topology is often referred to as "rubbergeometry": one can stretch and deform spaces without changingtheir underlying topology. In topology, manifolds are the spacesof most interest. These are spaces which, on the small scale,look like Euclidean space, but on the large scale, can becurved. A basic question lies in finding ways to detect when twomanifolds are different from each other. This is usually done byassociating certain invariants to manifolds: if the invariantsare different, then the manifolds are different. One family ofinvariants which is particularly useful in high-dimensionaltopology are the algebraic K-theory invariants. In general, theseinvariants are quite hard to compute. The PI plans to developmethods for computing these invariants, in the special case wherethe manifold has some nice underlying geometry. More generally,the PI plans on studying how the presence of a geometricstructure constrains various topological properties (and viceversa).