  We analyze the topology and geometry of a polyhedron of dimension 2 according
to the minimum size of a cover by PL collapsible polyhedra. We provide partial
characterizations of the polyhedra of dimension 2 that can be decomposed as the
union of two PL collapsible subpolyhedra in terms of their simple homotopy type
and certain local properties. In the process, a special class of polyhedra of
dimension 2 appears naturally. We give a combinatorial description of the
spaces in this class, which includes all closed surfaces and the complexes
associated to one-relator presentations.
