  This paper has two aims. The former is to give an introduction to our earlier
work on the Hodge theory of algebraic maps and more generally to some of the
main themes of the theory of perverse sheaves and to some of its geometric
applications. Particular emphasis is put on the topological properties of
algebraic maps. The latter is to prove a motivic version of the decomposition
theorem for the resolution of a threefold Y. This result allows to define a
pure motive whose Betti realization is the intersection cohomology of Y. We
provide a few explicit computations of perverse truncations and intersection
cohomology complexes which we could not find in the literature and which may be
helpful to understand the machinery. We discuss in detail the case of surfaces,
threefolds and fourfolds. In the surface case, our "intersection forms" version
of the decomposition theorem stems quite naturally from two well-known and
widely used theorems on surfaces, the Grauert contractibility criterion for
curves on a surface and the so called Zariski Lemma.
