PreprintRandom collapsibility and 3sphere recognition,João Paixão and Jonathan SpreerAbstractA triangulation of a 3manifold can be shown to be homeomorphic to the 3sphere by describing a discrete Morse function on it with only two critical faces, that is, a sequence of elementary collapses from the triangulation with one tetrahedron removed down to a single vertex. Unfortunately, deciding whether such a sequence exist is believed to be very difficult in general. In this article we present a method, based on uniform spanning trees, to estimate how difficult it is to collapse a given 3sphere triangulation after removing a tetrahedron. In addition we show that out of all 3sphere triangulations with eight vertices or less, exactly 22 admit a noncollapsing sequence onto a contractible noncollapsible 2complex. As a side product we classify all minimal triangulations of the dunce hat, and all contractible noncollapsible 2complexes with at most 18 triangles. This is complemented by large scale experiments on the collapsing difficulty of 9 and 10vertex spheres. Finally, we propose an easytocompute characterisation of 3sphere triangulations which experimentally exhibit a low proportion of collapsing sequences, leading to a heuristic to produce 3sphere triangulations with difficult combinatorial properties. AMS Subject Classification: Primary 57Q15; secondary 57N12, 57M15, 90C59.
This paper is available as a pdf (752kB) file. It is also on the arXiv: arxiv.org/abs/1509.07607.
