A standard task in topology is to simplify a given finite presentation of
a topological space. Bistellar flips allow to search for vertex-minimal
triangulations of surfaces or higher-dimensional manifolds, and elementary
collapses are often used to reduce a simplicial complex in size and
potentially in dimension. Simple-homotopy theory, as introduced by
Whitehead in 1939, generalizes both concepts.

We take on a random approach to simple-homotopy theory and present a
heuristic algorithm to combinatorially deform non-collapsible, but
contractible complexes (such as triangulations of the dunce hat, Bing's
house or non-collapsible balls that contain short knots) to a point.

The procedure also allows to find substructures in complexes, e.g.,
surfaces in higher-dimensional manifolds or subcomplexes with torsion in
lens spaces.

(Joint work with Bruno Benedetti, Crystal Lai, and Frank Lutz.)