We show that the homotopy type of a finite 2-complex is determined by its Euler characteristic provided its fundamental group has 4-periodic cohomology and at most 2 one-dimensional quaternionic representations. We will use this to get a partial answer to the question of whether a Poincaré 3-complex has a cellular structure with a single 3-cell and to extend results on Wall’s D2 problem, which asks whether a cohomologically 2-dimensional 3-complex is necessarily homotopic to a 2-complex. We also consider the 4-manifolds which bound a thickening of such a 2-complex, and their role in the unstable classification of 4-manifolds.