Cubulations, immersions, mappability and a problem of Habegger

Details Cubulations, immersions, mappability and a problem of Habegger Cubulations, immersions, mappability and a problem of Habegger

1998-04-08

arxiv.org/abs/math/9804046v2

Ann.Sci.Ecole Norm.Sup., 32(1999), 681-700.

Mathematics

Geometric Topology

Revised version, Ann.Sci.Ecole Norm. Sup. (to appear)

Scientific paper

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.

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