Descrizione

The theory of shadows of 3 and 4-manifolds represents a bridge between combinatorics of polyhedra and low-dimensional topology. On one side, it allows a purely combinatorial approach to the study of smooth 4-manifolds and, on the other side, it indicates relations between old-standing problems in group theory and recent topological results on 4-dimensional manifolds. The present Ph.D. thesis is devoted to further develop these connections and to find new applications to low-dimensional topology. The results proved, for the most part, seem to strengthen the idea that topology of 3-manifolds can be used as a guide to study the 4-dimensional case and that polyhedra can be used as a ‘bridge’: in many cases the 4-dimensional results based on shadows restrict through the theory of spines to results about 3-dimensional topology and geometry. On the 3-dimensional side, a new notion of ‘shadow-complexity’ of 3-manifolds is defined. The study of this complexity clarifies how hyperbolic geometry of 3-manifolds is intimately connected with the combinatorial structure of the polyhedra used to describe the manifolds. On the 4-dimensional side, the notion of branched shadow is introduced in order to study, through a purely combinatorial approach, differentiable objects as Spinc and almost complex structures on smooth 4-manifolds. Combinatorial sufficient conditions based on these objects are proved assuring that ‘refined’ structures on 4-manifolds exist such as integrable complex structures and Stein domain structures.