The unstable tangent fibration of a Poincaré complex is defined so that it is consistent with the manifold case. It exists uniquely for each Poincaré complex X up to fibrewise homotopy equivalence and, furthermore, if a Poincaré embedding structure exists on the diagonal X→X×X, its normal fibration is the tangent fibration.

Let f:M→N be a stable map between orientable 4-manifolds, where M is closed and N is stably parallelisable. It is shown that the signature of M vanishes if and only if there exists a stable map g:M→N homotopic to f which has only fold and cusp singularities. This together with results of Ando and Èliašberg shows that, in this situation, the Thom polynomials are the only obstructions to the elimination of the singularities except for the fold singularity. Also studied are some topological properties (including those of the discriminant set) of stable maps between 4-manifolds with only Ak-type singularities.

Let F be a non-lattice distribution function which lies in the domain of attraction of a non-normal stable distribution. Exact uniform convergence rates are obtained for the convergence of the normalised partial sums of random variables with distribution F. Second order regular variation conditions are assumed related to the domain of attraction conditions.