Cubulations, immersions, mappability and a problem of Habegger

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 c onnected sums of manifolds and immersions induce multiplicative structures which are respected by this surjection. We prove that those cubulations whi ch map combinatorially into the standard decomposition of R-n for large eno ugh n (called mappable), are equivalent. Finally we classify the cubulation s of the 2-sphere. (C) Elsevier, Paris.