An application of Poénaru’s “zipping theory”

In this note we present an application of the “zipping theory” introduced by V. Poénaru in the 80s, aimed to kill in a controlled way all the singularities of a non-degenerate simplicial map f:X→Mf:X→M, from a simplicial complex to a manifold. By means of some elementary zipping moves  , it is possible to perform a concrete algorithm which produces the ‘smallest’ equivalence relation on XX, compatible with ff, and which kills all the singularities of ff. We use this zipping process for listing both finite 3-complexes which are homotopy equivalent to the 3-sphere, and finite 3-complexes with a finite fundamental group.