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.
"Some remarks on PL collapsible covers of 2-dimensional polyhedra." Osaka J. Math. 57 (1) 39 - 49, January 2020.