A complexity of compact 3-manifold via immersed surfaces

We define an invariant, which we call surface-complexity, of compact 3-manifolds by means of Dehn surfaces. The surface-complexity is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting properties: it is subadditive under connected sum and finite-to-one on $\mathbb{P}^2$-irreducible and boundary-irreducible manifolds without essential annuli and M\"obius strips. Moreover, for these manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere, the ball, the projective space and the lens space $\mathbb{L}_{4,1}$, which have surface-complexity zero. We will also give estimations of the surface-complexity by means of ideal triangulations and Matveev complexity.