CONTENTS 1. Introduction 2. Background and Definitions 3. The Bistellar Flip Program 4. The Ubiquitous Poincaré Homology 3-Sphere 5. A Nonsymmetric Triangulation 3 16 on 16 Vertices 6. A Series of Non-PL d-Spheres on d+13 Vertices for d 5 7. An A 5 -Invariant Triangulation of IRIP 3 with 29 Vertices Acknowledgements References Bj orner was partially supported by the G oran Gustafsson Foundation for Research in Natural Sciences and Medicine. Lutz was supported by the graduate school \Algorithmische Diskrete Mathematik" funded by the Deutsche Forschungsgemeinschaft (DFG), grant GRK 219/2-97.We present a computer program based on bistellar operations that provides a useful tool for the construction of simplicial manifolds with few vertices. As an example, we obtain a 16-vertex triangulation of the Poincaré homology 3-sphere; we construct an infinite series of non-PL d-dimensional spheres with d + 13 vertices for d 5; and we show that if a d-manifold, with d 5, admits any triangulation on n vertices, it admits a noncombinatorial triangulation on n + 12 vertices.

Abstract(1) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with few critical cells. Our measure will depend on the topology of the space, but also on how nicely the space is triangulated.(2) The scheme we propose looks for optimal discrete Morse functions with an elementary random heuristic. Despite its naïveté, this approach turns out to be very successful even in the case of huge inputs.(3) In our view the existing libraries of examples in computational topology are 'too easy' for testing algorithms based on discrete Morse theory. We propose a new library containing more complicated (and thus more meaningful) test examples.

We study torsion in homology of the random d-complex Y ∼ Y d (n, p) experimentally. Our experiments suggest that there is almost always a moment in the process where there is an enormous burst of torsion in homology H d−1 (Y ). This moment seems to coincide with the phase transition studied in [1,20,21] , where cycles in H d (Y ) first appear with high probability.Our main study is the limiting distribution on the q-part of the torsion subgroup of H d−1 (Y ) for small primes q. We find strong evidence for a limiting Cohen-Lenstra distribution, where the probability that the q-part is isomorphic to a given q-group H is inversely proportional to the order of the automorphism group |Aut(H)|.We also study the torsion in homology of the uniform random Q-acyclic 2-complex. This model is analogous to a uniform spanning tree on a complete graph, but more complicated topologically since Kalai showed that the expected order of the torsion group is exponentially large in n 2 [14]. We give experimental evidence that in this model also, the torsion is Cohen-Lenstra distributed in the limit.

In 1970, Walkup [46] completely described the set of f -vectors for the four 3manifolds S 3 , S 2 × S 1 , S 2 × S 1 , and RP 3 . We improve one of Walkup's main restricting inequalities on the set of f -vectors of 3-manifolds. As a consequence of a bound by Novik and Swartz [35], we also derive a new lower bound on the number of vertices that are needed for a combinatorial d-manifold in terms of its β 1 -coefficient, which partially settles a conjecture of Kühnel. Enumerative results and a search for small triangulations with bistellar flips allow us, in combination with the new bounds, to completely determine the set of f -vectors for twenty further 3-manifolds, that is, for the connected sums of sphere bundles (S 2 ×S 1 ) #k and twisted sphere bundles (S 2 × S 1 ) #k , where k = 2, 3,4,5,6, 7,8,10,11,14. For many more 3-manifolds of different geometric types we provide small triangulations and a partial description of their set of f -vectors. Moreover, we show that the 3-manifold RP 3 # RP 3 has (at least) two different minimal g-vectors.

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