  We provide a calculus for the presentation of closed 3-manifolds via
nullhomotopic filling Dehn spheres and we use it to define an invariant of
closed 3-manifolds by applying the state-sum machinery. As a potential
application of this invariant, we show how to get lower bounds for the Matveev
complexity of P2-irreducible closed 3-manifolds. We also describe an easy
algorithm for constructing a nullhomotopic filling Dehn sphere of each closed
3-manifold from any of its one-vertex triangulations.
