Title: Improving bistellar simplification

Abstract: Over a century ago, a mining engineer who dabbled a bit in mathematics and physics, too, conjectured that a 3-manifold having both the homology and fundamental group of a sphere must also be homeomorphic to the 3-sphere. This so-called (3-dimensional) Poincar\'e conjecture can be generalized to topological d-manifolds and homeomorphisms between them. And rather than the TOP category, we could instead think about the PL category, where we would work with PL manifolds and PL homeomorphisms between them. While the topological Poincar\'e conjecture has been proven in all dimensions, the PL version is still open in dimension 4; it is referred to as the smooth Poincar\'e conjecture in dimension 4 (or SPC4) since PL and DIFF coincide in dimension 4.

In the PL category, we can take advantage of the combinatorial nature of the objects at hand and consider a computational problem often referred to as sphere (or manifold) recognition. In dimension 3, the computational complexity of sphere recognition is in NP and the problem is unrecognizable (ie, undecidable) in dimension 5 and up, while the complexity of this decision problem in dimension 4 remains open.

We present our construction of one family of difficult-to-recognize 4-spheres and the heuristic algorithms that we used in an attempt to recognize them to indeed be spheres. One algorithm which will be discussed has considerable room for improvement; it uses Udo Pachner's bistellar flips.