  The aim of this paper (inspired from a problem of Habegger) is to describe
the set of cubical decompositions of compact manifolds mod out by a set of
combinatorial moves analogous to the bistellar moves considered by Pachner,
which we call bubble moves. One constructs a surjection from this set onto the
the bordism group of codimension one immersions in the manifold. The connected
sums of manifolds and immersions induce multiplicative structures which are
respected by this surjection. We prove that those cubulations which map
combinatorially into the standard decomposition of ${\bf R}^n$ for large enough
$n$ (called mappable), are equivalent. Finally we classify the cubulations of
the 2-sphere.
