The two-category with three-manifolds as objects, h-cobordisms as morphisms, and diffeomorphisms of these as two-morphisms, is extremely rich; from the point of view of classical physics it defines a nontrivial topological model for general relativity. A striking amount of work on pseudoisotopy theory [Hatcher, Waldhausen, Cohen-Carlsson-Goodwillie-Hsiang-Madsen . . . ] can be formulated as a T...

Let g, h be marked morphisms of words. A pair (g1, h1) of marked morphisms is called a successor of (g, h) if g ◦ g1(a) = h ◦ h1(a) and g ◦ g1(b) = h ◦ h1(b) and the images of g1 and h1 are shortest possible. Successors play an important role in studying the Post Correspondence Problem. Typically, they are simpler than the original morphisms, measured by the number of suffixes of their images (...

In this paper we give a method for constructing complex valued harmonic morphisms in some pseudo-Riemannian manifolds using a parametrization of isotropic subbundles of the complexified tangent bundle. As a result we construct the first known examples of complex valued harmonic morphisms in real hyperbolic spaces of even dimension not equal to 4 which do not have totally geodesic fibres.

Weitzenböck type identities for harmonic morphisms of warped product type are developed which lead to some necessary conditions for their existence. These necessary conditions are further studied to obtain many nonexistence results for harmonic morphisms of warped product type from Einstein manifolds. Mathematics Subject Classification (2000). 58E20, 53C20, 53C25.