Combinatorial triangulations of homology spheres

Combinatorial 3-Manifolds with 10 Vertices

We give a complete enumeration of combinatorial 3-manifolds with 10 vertices: There are precisely 247882 triangulated 3-spheres with 10 vertices as well as 518 vertex-minimal triangulations of the sphere product S×S and 615 triangulations of the twisted sphere product S×S. All the 3-spheres with up to 10 vertices are shellable, but there are 29 vertexminimal non-shellable 3-balls with 9 vertices.

Topology Proceedings 2 (1977) pp. 621-630: PL HOMOLOGY 3-SPHERES AND TRIANGULATIONS OF MANIFOLDS

Statement of Research

I am a doctoral student working under the supervision of Dr. Frank Lutz at the Technische Universität Berlin in Germany. In my doctoral thesis I describe the construction of some triangulated manifolds we built by hand. We applied heuristic programs to compute certain topological properties of these simplicial complexes to verify that our construction is correct. In doing so we exposed flaws in...

Stacked polytopes and tight triangulations of manifolds

Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplexwise linear embedding of the triangulation into Euclidean space is “as convex as possible”. It can thus be understood as a generalization of the concept of convexity. In even dimensions, super-neighborliness is known to be a purely combinatorial condition which implies the tightness of a triangulati...

A constructive proof of Ky Fan's generalization of Tucker's lemma

We present a constructive proof of Ky Fan’s combinatorial lemma concerning labellings of triangulated spheres. Our construction works for triangulations of Sn that contain a flag of hemispheres. As a consequence, we produce a constructive proof of Tucker’s lemma that holds for a larger class of triangulations than previous constructive proofs.