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
