We show that groups satisfying Kazhdan’s property $(T)$ have no unbounded actions on finite dimensional $CAT\left(0\right)$ cube complexes, and deduce that there is a locally $CAT\left(-1\right)$ Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally $CAT\left(0\right)$ cube complex.

Using the ‘Riemann Problem with zeros’ method, Ward has constructed exact solutions to a $\left(2+1\right)$–dimensional integrable Chiral Model, which exhibit solitons with nontrivial scattering. We give a correspondence between what we conjecture to be all pure soliton solutions and certain holomorphic vector bundles on a compact surface.

We describe a new approach to the canonical decompositions of 3–manifolds along tori and annuli due to Jaco–Shalen and Johannson (with ideas from Waldhausen) – the so-called JSJ–decomposition theorem. This approach gives an accessible proof of the decomposition theorem; in particular it does not use the annulus–torus theorems, and the theory of Seifert fibrations does not need to be developed in advance.

We show that a homotopy equivalence between manifolds induces a correspondence between their spin${}^c$–structures, even in the presence of 2–torsion. This is proved by generalizing spin${}^c$–structures to Poincaré complexes. A procedure is given for explicitly computing the correspondence under reasonable hypotheses.

We prove a geometric refinement of Alexander duality for certain 2–complexes, the so-called gropes, embedded into 4–space. This refinement can be roughly formulated as saying that 4–dimensional Alexander duality preserves the disjoint Dwyer filtration.

In addition, we give new proofs and extended versions of two lemmas of Freedman and Lin which are of central importance in the A-B–slice problem, the main open problem in the classification theory of topological 4–manifolds. Our methods are group theoretical, rather than using Massey products and Milnor $\mu$–invariants as in the original proofs.

We define a diffeomorphism invariant of smooth 4–manifolds which we can estimate for many smoothings of ${ℝ}^4$ and other smooth 4–manifolds. Using this invariant we can show that uncountably many smoothings of ${ℝ}^4$ support no Stein structure. Gompf constructed uncountably many smoothings of ${ℝ}^4$ which do support Stein structures. Other applications of this invariant are given.

The main theorem shows that if $M$ is an irreducible compact connected orientable 3–manifold with non-empty boundary, then the classifying space $BDiff\left(Mrel\partial M\right)$ of the space of diffeomorphisms of $M$ which restrict to the identity map on $\partial M$ has the homotopy type of a finite aspherical CW–complex. This answers, for this class of manifolds, a question posed by M Kontsevich. The main theorem follows from a more precise result, which asserts that for these manifolds the mapping class group $\mathcal{\mathscr{H}}\left(Mrel\partial M\right)$ is built up as a sequence of extensions of free abelian groups and subgroups of finite index in relative mapping class groups of compact connected surfaces.