Cubulations, immersions, mappability and a problem of habegger

Abstract

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 R n for large enough n (called mappable), are equivalent. Finally we classify the cubulations of the 2-sphere.

Surface cubications mod flips

“…On the other hand, by cutting the surface Σ along the arcs of ϕ C (N C ) we get a number of polygonal disks. An immersion having these two properties was called admissible in [10]. Further we have a converse for the construction given above.…”

“…If the cubication C is smooth then N C has a smooth structure and the immersion ϕ C can be made smooth by means of a small isotopy. This connection between cubications and immersions appeared independently in [2,10] but this was presumably known to specialists long time ago (see e.g. [30]).…”

“…Remark 2.2. Our methods are not combinatorial, as was the case of the sphere (see [30,10,4]), since one uses in an essential manner the identification of H 1 (Σ, ∂Σ; Z/2Z) ⊕ Z/2Z with N (Σ), which is of topological nature. The main interest in developing the topological proof below is that one can give an unifying treatment of all surfaces and the hope that these arguments might be generalized to higher dimensions.…”

“…The problem above was addressed in ( [10,11]), where we show that, in general, there are topological obstructions for two cubications being flip equivalent.…”

“…These moves have been called cubical or bubble moves in [10,11], and (cubical) flips in [4]. Notice that the flips did already appear in the mathematical polytope literature ( [5,37]).…”