Differential Geometric Methods in Low-dimensional Topology

This is a survey of various applications of analytical and geometric techniques to problems in manifold topology. The author has been involved in only some of these developments, but it seems more illuminating not to confine the discussion to these. We begin by recalling the notion of a manifold. Suppose we are provided with a large collection of small paper discs. Then we can construct a wide variety of complicated objects by pasting these discs together in various fashions. Mathematically, the paper discs generalise to disjoint copies Uα of the unit ball B in some fixed Euclidean space R, where α ranges over some index set. The pasting data generalises to a collection of homeomorphisms