Interactions between geometry and topology

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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).