Prof. Wolfgang Kühnel


Triangulations with few vertices

Determine the minimum number of vertices for a triangulation of a given manifold, or find "good" triangulations with a reasonable number of vertices and/or with high symmetry. Find general lower bounds for the possible numbers of vertices. A basic inequality was given in

Upper and Lower bound conjectures for combinatorial manifolds

There is the following CONJECTURE: For any 2k-dimensional combinatorial manifold M with n vertices the following inequality holds: ${{n-k-2} \choose {k+1}} \geq (-1)^k {{2k+1} \choose {k+1}}(\chi(M) - 2)$ with equality if and only if the triangulation is $(k+1)$-neighborly. For a first approach toward this conjecture see For the positive answer to the conjecture see A similar conjecture for the case of centrally-symmetric triangulations were first studied in the following work by my former student

Tight polyhedral embeddings

There is the following conjecture: For any tight polyhedral embedding of a (k-1)-connected 2k-dimensional manifold M into Euclidean d-space (not in any hyperplane) the following inequality holds: ${{d-k-1} \choose {k+1}} \leq (-1)^k {{2k+1} \choose {k+1}}(\chi(M) - 2)$ with equality if and only if the image is a (k-1)-neighborly subcomplex of the d-dimensional simplex. For partial results see More recently new tight polyhedral embeddings of 4-manifolds were investigated, see For tight polyhedral submanifolds of higher-dimensional octahedra see There is a close connection with the subject "Upper and lower bound theorem", because k-Hamiltonian 2k-manifolds in the boundary complex of a cross polytope belong to both categories, at least in the centrally-symmetric case. Further progress was made in

Combinatorial manifolds satisfying complementarity, and triangulated projective planes

The problem arises from three different points of view: So far the only known examples are the 6-vertex RP2, the 9-vertex CP2 and a number of 15-vertex triangulations of an 8-manifold, presumably HP2. The case of a 27-vertex triangulation of the octonion plane is still open, see

Triangulations of the d-dimensional torus

There is the following conjecture: Any combinatorial triangulation of the d-dimensional torus must have at least 2d+3 vertices. Examples attaining this bound are available, see In the case of lattice triangulations the conjecture is proved in

Manifolds associated with abstract regular polytopes

Abstract regular polytopes are natural generalizations of the Platonic solids and the other classical regular polytopes. The requirement is that the automorphism group acts transitively on flags. So in particular one can transform any point to any other and then, while keeping the point fixed, any edge to any other and so on. For a monograph on the subject see Among the abstract regular polytopes there are 4-polytopes with toroidal facets. When transforming these abstract toroidal facets into solid tori, a 3-manifold can be associated with the polytope, see The higher-dimensional case is still work in progress.